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Reading the Auflage to the second edition is very instructive, moreso than the Vorrede to the 1862 version.
After some 33 years Hermann is able to take a mature look at what transpired to his work.

As you will know if you read the thread

Hermann was beset about by anxieties, pressures indecisions , and saved only by his young mans energy and the determination of his convictions that he had been given a precious insight. He hoped , in vain that his Acadmic peers would see this and grant him a seat to do this crucial work with vigorous endeavour. He published his imperfections his I decisions in a uniquely potent mix of prodigy and promise. He published! And then felt damned!

For ten years he endured meagre response and scant recognition of his work or his abilities. Already struggling to find time and peace ti process and edit his ideas he let this occasional pastime languish. His printed works apart from the few in the catalogues of printers and libraries, mouldered in the warehouse and were eventually destroyed.

But the work was too important to forget, and in particular his brother who was having greater social success than Hermann required this work to be in print to establish the family name, his fathers ideas and his own critique of current philosophy. By his urgings snd discussions Hermann took up the work once more with dedication in bout 1854.

This was the tie Hamilton began to extensively work on his quaternion ideas after critical acclaim had given him a meteoric rise to fame, only later to be dashed by Lord Kelvin and others in a bitter rivalry.

Hermann started nis Ausdehnungslehre again from a different ground plane, Robert insisting that he cater for the mathematical audience rather than the Philosophical one. Thus his choice of the Euclidean format was clear , and his editing choices were guided by formal conventions. With Roberts input and editing as the publisher the work pursued a thorough and pervasive course of Rigorous Euclidean presentations and formats. All areas of Mathemtics and physics and mechanics were accessed by applications of the method.

Much of what was in the 1844 version was completed and extended; new areas were opened up and explored, and the mathematical themes of functions, integrals and differentials were fully exposited and new formats suggested.

When the work was published in 1862 Hermann expressed some disbelief in the work( mostly Roberts) because it bore no relation to the previous, imperfect piecemeal inconsisten mess he had published earlier! However that was at least his own work!

At first forv5 years it looked like Roberts plan was not working. But then in 1867 Yänkel published referring to the ideas in the -Ausdehnunglehre and the floodgates opened . It seemed as if they were waiting for Riemann to die. But more astonishingly for Hermann was the growing fan club for his 1844 work! It served that the more insightful realised that the 1844 publication was the bible to refer to! As. Consequence Hermann was able to afford to get the original work reprinted with notations, eagerly snapped up by a hungry readership. There was even talk of a colour pint edition!

Hermann list several key advances in his method made by insightful others like Stregel, but Leo his own advances made by inspirational retreading of his seminal work as a young man.

There are about 4 papers he mentions including this one on Hamiltons Quaternions. Plus he mentions and insightful review of the meaning of his Ausdehnungslehre, perhaps in distinction to Roberts, no also started publishing on the topic.

By 1877 Grassmnn was making plans to further develop this work, started in a occasional and piecemeal way; published full of imperfections in hop, dashed, snd finally completely redacted in collaboration with his brother, launching him into a celebrity he could not have dreamed of! and then he died!

In keeping with his absolute fearless clarity Hermann points out where his work was fruitful, unfruitful or even inconsistent! The work clearly was a work in progress, but it's deep I'm was to supply the modern Mathematicin, physicist, engineer and mechanic with a deep understnding of the melded connectedness of all the rapid and confusing modern developments in scientific and computational and geometric thought.

Those that valued it most were those whose philosophical natures lead them to try to understand how this modern burgeoning intellectual growth was deeply connected to each and every technological and material advance in understanding the space in which we live and of which we are composed.

There is a deep melded harmony in the way we interact with space in the sciences nd engineering and Grassmann drew attention to it through what I call Spaciometry, and the effective labelling of it nd our processes in it.

Whenever I read summary biographies of theoretical physicists or mathematical engineers of the late 19th early 20th century, those who are now regarded as pioneers are always said to have created their own, idiosyncratic vector algebra. This is so of Heaviside and in this particular instance P.A.M. Dirac whose bra-ket notation is regularly inflicted on young impressionable minds.

I suppose it is easier to make such statements than it is to delve into the murky and nefarious misdeeds in academia!

Vectors in principle as I now understand are inherent in the ancient Greek notion of a segmented line, especially as described in the Stoikeia and elaborated on by Apollonius and Archimedes. Many Pre Socratic philosophers from Thales, Herakleitos , Parmenides Luycippius and Democrites, considered the plenum to be eternally dynamic, and their drawn lines just to represent instaneous form or motion,: skesis or sketches and schematics of ephemeral manifestations in the flow of Rhea or Hekate. Phusis was that grand system of opposing entities which alone maintained a sensible equilibrium or a system of equilibria with an attendant Harmonia. These two were represented as sisters and members of the 9 muses, the attendants of Apollo the great Sun God whose beauty was admired by all!

Nice myths, and memorable stories, which in the absence of television game shows occupied the minds of young impressionable people of the time. Now these myths reappear in some detail in the console games loved by do many of today's youth.

To the wise they encoded deep philosophical truths, and Newton as an example devoted considerable effort in his youth to deciphering the physical and alchemical information encoded therein. But the line segment as a symbol belongs to the Pythagorean Eudoxus, who encoded the understanding to be found in the mosaic arrangements that decorated the Pythagorean temple walls and floors. These temples to the Pythagorean Muses were called Mousaion, whence we derived Museums and also the epithet mosaic. But ib Euclid these patterns are called epipedoi, and in the literature speripedoi. They were also called Epiphaneia, because light fell upon them and cast shadows. These shadows were marked ti designate certain times of the day or year, and so crazy "abstrac" patterns were noted in early examples of mosaics.

Eudoxus took these principles to the Pythagorean conclusion that all forms can be reduced to these dynamic shadows cast onto these mosaic floor patterns. The shadows could be quickly sketched round thus giving grammai or drawn lines in the earth. The embedded pebbles, later baked tiles enabled these lines to be counted or measured, and this was the meaning of geometry.

Geometry used these drawn lines on mosaics to measure space out to the sun, and the stars, and this was known as Astrology. Thus geometry was not about surveying the earth, rather the earth was measured to survey the stars and planets in their motions and positions .

Aristotle who through circumstance never completed his Pythagorean studies under Plato, investigated this dynamic relationship between all things that moved in his great work on metaphysics. His ideas were greatly influential on Islamic scholars and notably on Newton and many others. However they were criticised by Gilbert in his work on Magnetism and on the empirical basis of human knowledge. In this Gilbert predates Bacon, who somewhat witheringly disparaged Gilbert's great insight.

Suffice it to say Aristotle was misleading on many many points and was taken with a pinch of salt by many later philosophers, who found safety and clarity in Euclids Stoikeia. Later some questioned even Euclid!

But it must be acknowledged that in Newton all that was best in Aristotles reasonings were combined into Newtonian astrological principals, ie the Principia Mathematica. Thus the Greek notions of dynamic line segments pass on to Newton and others who following Archimrdes, Apollonius and Parcletus, to name a few , gave us a system of analytical Mrchanics. However it is Newton who combines this with the analysis of Aristotle on motion to give to us the new Mevhanical philosophy that avoids many of the ancient extremes and mistakes..

From this source most of the scientific clubs and societies in the world drew their concepts of how this world order might best be described, and what mathematical principles might best be employed in doing do. Buried deep within this apprehension was the ignoble line segment and it indissoluble link to mosaic ratios of line segments and to Eudoxian proportions. Amongst  these line segments and mechanical motions of drawings, the ubiquitous circle was also preserved, and its utility and fundamental constructive power was seen as the ultimate goal of all analysis. It's perfection could hardly be described without all it's children of which the most important was the segmented line.

Notions of straight, parallel and rotation were fundamentally and mechanically defined in terms of the circle and the rigidity of space. Even though it was impossible to find such a design in nature yet it easily dropped out of rotating every rigid object no matter how convoluted! Eudoxian proportions fundamentally rely on it, as do geometric extremes and means.

Thus the great secret and harmonium Mensurarum of Sir Roger Cotes, that which Newton intuitively perceived, and De Moivre also skilfully analysed both in trigonometric form and probability measure form  is the unit sphere and it's projected shadow the unit circle.

It is the unit circle which defines every aspect of the notion of line segments and thus every spect if the notion of so called vectors. And while Newton addressed these properties in his geometrical reasonings, yet he did not make fully clear the underlying Algebra. And While Hamilton by brute force extended the underlying algebra into 3 dimensions from his noteable concepts in his  essay on the science of Pure Time, that is conjugate functions or couples., it is remarkable that he did not fully apprehend what he was pursuing so ardently.

On the other hand, both Lagrange and Euler casually cast these things about as if they were playthings, but it is not certain that they grasped the fundamental nature of what they had adduced to Newtons great works, nor those of his acolytes De Moivre and Cotes. In any case they did not seem concerned to fundamentally give insight into how these things may be best understood.

This fell into the hands of a great an innovative teacher named Justus Grassmann, who was charged with the responsibility of bringing the academic standing go the average Prussian child to the level required for Prussia to compete with the French empire in the modern industrial world. Indeed it was the duty of all educators under the imperial seal to implement the Humboldt reforms throughout the length and breadth of the Prusdian holy roman empire.

As such, the classically trained Grassmann, educated out of the best of the French Ecole tradition, which was principally disseminated by Lagrange from Berlin, who saw to it that Legendre and the work of many others became the text books in Prusdian higher education; as such Grassmann deconstructed the work of Legendre who himself had reconstructed the works of Euclid from what sources he had. This lead Grassmann into a deep logical analysis and synthesis, that resulted in him training students in a constructive and dynamic geometry, which he could show logically hung together and underpinned Aruthmetic. Rather he was keen to show that arithmetic was the fundamental of all geometry! But he came unstuck at multiplication. Under his analysis there was no logical precursor or analogue ti multiplication. It stood in its own right as a purely geometrical construction!

In fact, in my opinion, this was a misreading of the concept of logic, perpetrated by Aristotle. Aristotle based his definition of logic on the grammar of language. While tis is a very powerful consonance it is also a fundamental dissonance. Language in the end is a verbalisation of non verbal experiences. Much of what is communicated in language is non verbal, and that means it has a referent in one or many of the other sensory modalities. Thus to understand language we have to start in the nonverbal experiential modalities. This means we have to start in our experience of interacting with space.

To cover this concept I defined the notion of Spaciometry. This  covers an individuals logos, analogos, sunthemata and summetria response. These non verbal interactions with space underpin the comparison, distinction and declarations of those distinctions we call language. Thus multiplication is without doubt a non verbal primitive. In fact I claim we have mis apprehended the primitive by calling it multiplication!

In any case this intense scrutiny and synthesis lead to Hermann developing a unique constructivist approach. This Förderung was so innocuous that it literally requires the individual to go back to the wide eyed suspension of critical faculties common among children. This concept is expressed in 2 Laws of Grassmann

AB=-BA

And AB+ BC=AC

This is the fundamental product sum laws( nb! Product sum not product and sum) of what Grassmann called his Aussere product.. The inner product is based on a different kind of line segment, what I have called the trig line segments. This produces a " Normal" or usual product but the product sim is still Grassmanns product sum law.

This product sum law makes no sense as a measurement, but as a process description of spatial behaviours it is absolutely fundamental to any algebra of space.

None of this was notated as precisely as this. In fact it was not even considered as needing notation because it was sufficiently clear in the rhetoric of any geometrical " proof" or exposition. The fact that Grassmann set out an algebraic notation for it, which is really a symbolic arithmetic for it, is one of those rare moments of wacky thinking that just proved to be so right! Recall Bombelli saying that his idea of adjugate numbers was so crazy he had to shut his eyes and muddle through! Again this was described as Akgebra, by Bombelli. As you will read in the links in the thread on Bombellis operator, if they are still active, Bombelli went on to solve quintic and degree 6 equations using his notational algebra.

This of course begs the question what is algebra? Newton hated the idea of notating all his muddled thinking by means of these symbols, but Wallis believed this could help others to emulate the genius of the very best mathematical minds.

It should be noted that despite protestations to the contrary algebra represents muddled, muddied and contortional thinking in feeling ones way to a solution. Often it requires one to assume the impossible or the unexplainable, to hold ones breath or shut ones eyes and hope to god it all works out and cancels to some easy result.

It is a tortuous exercise which only a few actually enjoy. In particular, the great savants who could calculate pi in their heads to several hundred places would find the formulaic patterns intriguing and even " nice"! But to the vast majority of us these algebraic symbols and patterns are literally " all Greek" to us.

So I say that the Algebra that Grassmann and Bombelli perceived is a symbolic procedural algebra best expressed as rhetoric. Although this is much longer to read or explain it is more user friendly. The best communicators explain the symbolic arithmetic with inviting analogies, as I remember Lancelot Hogben doing in his book Mathematics for the millions, which I read as a child and which confirmed me in my decision to become a " great" mathematician!

As you see I failed in that goal round about entering into university, when my relatively sheltered mathematical training was introduced to the harsh realities of pontification! No one would explain how all these mathematical symbols and juggling actually came into existence and had any relevant meaning in everyday life. I was astute enough to realise that my mathematics should be able to describe a door! And yet I could not describe a door mathematically.

The Grassmann concept of line segments as the basic primitives of an algebra is so far reaching that you need to see how it reaches into the mind of Dirac to realise this. You need to ubpnderstand that Hamilton recognised Grassmann as a Master, that Gibbs recognised the work ofbGrassmann as fundamental to replacing the arcane Alice in wonderland world of Quatenions, that Heaviide realised that Grassmanns principles properly understood gave him physical insight, and that Bill Clifford determined to spend his life promoting Grassmann work and analysis.

Today Norman Wildberger, David Hestenes and others are busy continuing the programme and work of Grassmann. Of course Grassmann is dead, so why name your work after him! In all fairness Grassmann is referred to as the source, but few really explain how much what they discover by using his methods of Analysis and synthesis has already been written down long ago by Grassmann.

In this thread Grasdmann deals with his issue in that particular instance of Quatenions. He does not clim to have discovered quatenionic, but rather that what can be discovered by Quatenions can more rapidly and more cogently be discovered by his algebra, and indeed already had bern by him!
http://youtube.com/watch?v=myxx2uaqPLM

By the way Grassmann fully credits the works of Lagrange and Euler and others as his inspirations.

http://youtube.com/watch?v=H8xBlLWdzBE

http://youtube.com/watch?v=GMZoXXaOFeQ

These 2 videos are preceded by 2 more that I recommend watching on Galois theory.

The work of Gauss is often difficult to place in time due to his reticence in publishing his half formed ideas. Thus he later would sardonically. Liam precedence to any published or prepublished work tht he reviewed if they contained ideas he had explored in his " notebooks!" . In fact his students continued this trend after his death often climbing intellectual priority. Only the most capable mathematicians like Hamilton, would resist such fanatical domination of mathematical thought.

It is instructive however how the ideas testify to a sort of Zeitgeist and commonality of thinking. It must be remarked however that the intellectual climate, despite the difficulties in communication, relied upon frequent and lengthy scholarly missives and letters , often hand written , but later published as memorie. In addition, true " education" in the classics required travel to the great academies in Europe, in Baghdad and Southern Spain, and infrequently to the monastic centres in the Orient. Thus Libraries and manuscripts were of invaluable assistance in transmitting wisdom and intellectual philosophies around the world, wherever traders and merchants might ply their trade.

Gauss was impatiently ambitious, wishing to make his mark as one of the great Prusdian intellectuals. Up until then, the French Evole dominated the intellectual curriculum of Europe.mfiercely patriotic Gauss wanted to deliver a sense of homegrown greatness to the Prusdian Empire, which until then imported all it's intellectual capital. The royalty of Rurope in thir meetings often boasted of their national prowess by these accomplishments. In Prussia and Russia it wasa great royal embarrassment to have mainly peasants as the main economic power, whereas the advancing industrial nations boasted of mechanisms of thought and ingenuity. Engineers and mathematicians clearly as inventors had transformed the economic status even of small islands like Great Britain!

The Humboldt reforms in Prusdia were not a luxury, they were a geopolitical necessity, which all intellectuals in Prusdia acknowledged.Gauss seems only to have concentrated on the Academic quality of the higher institutes.

What I have learned from Grassmanns Vorreden and Auflagen is that their were places of rich intellectual and cultural commodity. Stettin( Sczeczin)  was a dormitory community to such an area in the Rhone valley. Thus the intellectual thought of Rurope flowed into Stettin because the printing industry had taken a major advance in this area of Prussia texts nd folios books and manuscripts were all accessible if not in the libraries then hot off the presses in Leipzig. Justus Grassmann hermanns father , and Robert Hermanns brother ho lived with his father, whereas Hermann lived ith and was fostered by his uncle, we're extremely well read.

By all accounts it seems that Hermann was a late developer, probably due to a kind of autism and fragile health condition. Thus benefitting from the education system in Stettin that his father Justus set up required Hermann to do a great deal of self or private study. In this way his Förderung was developed without harsh academic correction . But it also meant he was unaware of certain standard ideas in the public domain. For example he had only heard of Möbius Barycentric calculus, and only read it after he had formulated his point algebra as I have called it. Again Gauss tentatively published his ideas on what are now called Gausdin Integers in about 1831 after being persuaded by Wessels paper on directed magnitudes that his use of imaginary numbers, though metaphysically and philosophically unsound in his opinion was somehow fundamentally important.

Gauss used the Eulerian i notation extensively and arguably in a way that defines it as a complex number adjugate, following Bombelli. It is not clear if Euler got beyond viewing i as an ratio which was its own negative inverse which passed beyond the infinite unbounded ness of space to return on the other side! However he thought of it, he states in his institute on arcs and integration that he could see a way through the problems of understanding the solutions to certain equations only this way! He was considering polynomials we can call circular functions, and it resulted in him linking i indissolubly with the arc length through the trig functions. It is hard to express how this notion found only in Sir Roger Votes Eork some decades earlier came into his imagination.

So Grassmann states in the 1862 Vorrede that he had not known of Gausdian integers until after his publication of the 1844 Ausdehnungslehre. He took it as a sign that he was on the right track because the Gausdian form was an example of his general concept of an Ausdehnungsgröße.

From the videos you can understand that the Schwenkunlehre of Hermann Grassmann was arguable extant in the writings of Lagrange and Lplace, and certainly in the concepts of Ruler. Thus what I have called a Grassmann Twstor is not uniquely Grassmanns. In fact I clim now after studying the Vorrede of 1844 that Grassmann goes out of his way to make plin he claims no original thought beyond his Förderung!  His mindset was totally unique and insightful and it was this mindset , this Förderung he was at pains to Promote as a method of analysis and synthesis utilising " algebraic" conventions.

By 1877 the concepts of ring and zgroup theory were firmly established in Europe as the basis of modern Algebra. In that regard, Hamilton is often hailed as the father of modern Slgrbra as a separate discipline. However as Leah's, these things are not so clear cut. Along the way many Europeans contributed to what later was gathered into the new subject domain called Algebra. This was not without a considerable academic turf war in which many reputations were made and lost! Hamilton himself became a casualty of an international backlash against the new algebras.

My opinion on numbers stands. We are better off without the convoluted concepts they have become. Instead we are better off adopting the iteration of applying spaciometric Metrons to space and singing the counting ong as we fo so! The world will be a much happier place!

One note. Often mathematicians will tell you it is impossible to trisect an angle! This is not true. If you press them they will backtrack and mumble about rulers and compasses.rulers and compasses are mechanical devices , pragmatic tools for constructing the forms which geometry then idealises. Newton firmly based geometry in an interdependent loop or pair of loops with Mechanics. There are many ingenious mechanical devices that can precisely trisect any angle!

However what is perhaps more Pythagorean in constructing this trisection is the ratio of circles. Whatever pi may or may not be we can construct circles in my ratio by their diameters or radii. Thus to mark of the third of any arc is not only possible but mechanically simpler than constructing a perfect straight edge! In fact once a circle has been created as a disc it is a curious fact that the hypo trochoid of a disc inside a circular wall in the ratio 1:2 is perhaps the best way to trace out a perfectly straight line using the disc centre. All other concepts involve defining a straight edge to define a straight edge. This convolution or tautological reasoning will iterate to a perfectly straight line over time, but a circular method is faster and more accurate.

So I get deep into the Fourier transform as an example of a Grassmann tisyor( see thread on twistor) and I am close to analysing light and magneto Thermo electro complexes as twistor complexes. But the detail is distracting.

What lifts me up out of the potential quagmire is the glistening rotating sphere of unit radius. I have seen this sphere many many times, especially in the thread on the fractal foundation of mathematics, and I habpve even called it Shunya, the source of everything to do with measurement. Shunya is indeed the Sanskrit for the fullness of everything, so the sphere is merely a representation a metaphor.

My meditation on the V9 group or possibly a ring has slowly deepened. I made the mistakes of rushing in on my thread on polynomial rotations. Understanding these product tables / matrices ( Cayley) is hard, that is interpreting them. The V9 group table can be rolled up into a torus but what is the best way to do that? A square table gives one result, forming the products into a hexagonal shaped table by judicious cutting gives another.

The physical significance then takes on if fervent interpretations with regard to the flux of vortices in space!

Again the unit circle/ sphere is involved in apprehending the spatial significance of the patterns counted. We lose sight of this by replacing Arithmoi by Numerals.

Rotation is fundamental to my thinking process and my experience and crops up everywhere, so had I understood Hermann correctly, was he seeing what I see everywhere?

So many of my insights into rotation can be linked back to him one way or another. Was I right about the cyclical rotation? And the inner product, does it behave like normal integer multiplication?

The place of Hamltons Quaternions generated more questions as well as insights. Thinking about it I realised that a better working title is " How to find the spot where the Quaternions are in the Theory of stretchy thingies!" it is quite complex, and Hermann goes into detail. Complex as it is he has not yet used complex numbers! Rather he has used cyclic rotation !line segments and cyclic rotation of the indices. What effect foes that map to in space? It is not a simple rotation, but a convoluted one.

I was right to go slow. Hermann has grasped rotation only partially. That is in keeping with his times, the full extent of rotation eluded them. Rodrigues and Hamilton were the pioneers of an understanding of 3d rotation. Hermann was not. However what he was was the instigator of a method that could build models of any kind of spatial process that had the properties of three types of group or ring products and product sums.

It is instructive to see how he characterises the 1844 Asdehnungslehre as the analytical phase of his work, and the 1862 as the Synthetical phase of his work. Thus he shows now as previously that Quaternions are a combination of products in his algebra, particularly the 2 main products: the inner and outer products.

We lose sight of the spaciometric analysis as Grassmann has chosen a more formal format : that of the Eclidean or as23 call it nowadays, the Axiomatic format or presentation of Ring or Grouo theory on the n dimensional unit sphere.

However, because it is the unit phere it is all about Ritation, and cyclical Ritation is Grassmanns key tool or apprehending and finding his way about in this lineal segment space!

What is abstract thought?

I claim that it is a misnomer for Analogous thinking. My understanding of analogous thinking starts with the Babylonian cuneiform tablets. This is the oldest script that we can decipher that represents human thought clearly making analogies. However it is my suspicion that the Harsppan and Dravidian, Indus valley civilisations represent an even older civilisation capable of analogous thought.

So the sphere , the cone, pyramids a and cubes are analogous to what exactly? These often are considered as pure abstractions of thought . I am convinced that they are perfected analogues of structures found all around us, where perfect here means constructed by rigid adherence to an integrated process by a human mechanism that itself is rigid or constrained.

As a process which is found in our environment, sequential actions and behaviours are exhibited in our subjective processes, but these occur recursively at all scales. The result of this self similarity is almost self similarity at all scales! Where we " perfect" this relation is precisely where we are incapable of distinguishing any difference. Thus our standards of comparison, contrast and distinction, the do called by me Logos Analogos response is based on the principle of exhaustion! I can apprehend no smaller difference, nor any difference at all in that which I call identical or perfect in fit!

However, in reaching that determination by some recursive , that is iterative analytical process, what do I call all the nearly or almost similar forms if not Analogues?

So it is that Grassmann great contribution to human mathematical and Scientific thought is to raise the level of analogous thinking to that of a fundamental cohesive form of analysis that generates inspirational comparisons and contrasts in a consistent procedural manner. In doing so he draws on some of the most general thinking of his times which later develops into Ring, Group and Homotopy theiry.

The quintessential example of analogous thought is in my opinion expressed by Eudoxus in books 5 and 6 of Euclids Stoikeia. The thought is clearly carried over into book 7 the folio on the Arithmoi and the definition of Monas  the .concept of unit Metron.

It is of significance that every term used in the fundamentals of group or ring theory derive their meaning from Euclids Stoikia and these 2 books in particular. Analogos is defined or exemplified at the outset as is homologous and Ido Morpheus etc. thes concepts have their origin in Pythagorean philosophy of which the Stiikeia is merely an introductory or undergraduate course!

The notion if measurement by counting a Metron is established in a kind of dance in which a Metron is laid down in a contiguous pattern. This concept of a Metron is sharpened to the concept of a Monas, which is a standard Metron unit. It is then that the process of factorisation is formalised by the so called Euclidean Algorithm and the Arithmoi are identified, defined and distinguished. This is in book 7 where the notions of artios ( perfect fit) and perisos( approximate fit) are carried forward from the concepts in books 5 and 6. Into the Namespace from which the various types of Arithmoi are named.

The proto Arithmoi deserve particular mention in that each one represents a type of product, the progenitor of a multiple form that cannot be achieved by any other means.thus the name prot which usually means first . These are the first or initial forms of a whole series of multiple forms. It is the algorithm that exposes these multiple forms and which drives the naming of them.

But the concept of multiple form derives from the previous work of Eudoxus in Books 5 andv 6 which in turn is his deep meditation on the mosaics in the Pythagorean Mousaion or temples of the Muses.

So while unit is not defined in books 5 and 6 it prepares the ground for its conception. In a sense books 5 and 6 are a more general discussion and introduction to the principals of comparison, and thus a more " algebraic" introduction. However algebra is an Arabic word and Eudoxus treatment is rhetorical , conceptual and reasonably direct. The things he refers to as magnitudes are symbolised by a line segment or a form. In this sense it is symbolic as Algrbra is symbolic , that is it uses symbols to rhetorically discuss its subject matter.

Because symbols are used  we may justifiably ask to what they refer and the answer would be twofold: a quality called magnitude in its many and various forms; a quantity of a specific magnitude. But when pressed to explain quantity and how it differs from magnitude it becomes necessary to define the concept of unit so as to be specific.

In general however, we do not wish to know specifics of quantity but rather how quantity behaves vis a vis another quantity against which it is compared. It suffices to know that one is a multiple form " of" the other either directly by being of the same qualitative magnitude, or indirectly by occupying the same quantitative space as an identical form which may be of any other quality.

In fact, the behavioural dance is the same regardless of the quality of the Metron if it remains true to the form of the Metron that is appropriate to the quality of that being" measured".

Rather than labour the point suffice it to say that the Pythagoreans would use any appropriated form in their mosaics of whatever material or roughly similar size. And yet each form would be counted " as  one".  Counting as one is by no means implying that the form is a unit!  Our behaviour in counting in this way is not to measure but to perhaps remember or encode, if not for the pure pleasure of recounting to others some conception of multiple form.

The introduction of the Monas therefore signifies the introduction of the standard unit. But we will find in higher Pythagorean philosophy that this unit represents all that is!  In this regard the Monas as the Uni-verse is equivalent to Shunya as the fulfilment of all or as everything. This it is a thing to note how Shunya becomes 0 or nothing and yet it's equivalent Monas becomes 1!there is a history behind this, that as usual involves human frailties!

Yet it is also tru that the Indians used Ek to mean the initial count just as the Greeks used en, so we must distinguish the cultural iterative name from the overarching concept they are related to.

It is therefore important to note, that Grassmann as a young man and articulately as an admirer of Hegelian logic or rather dialectic process has these concerns infusing his mind at most levels. The rise of the more philosophical treatment and debate about  the ultimate rationality of human thought being expressed in the works of the Mathematikoi, as Plato owned, and the church fathers hardly contested because they could not , also fuelled the lifelong efforts of some to establish their opinion.

Roughly Kant put it tht human rationality was discoursive, that is given by divine intervention and revelation, and the sublime nature of Mathematics, principally the Principia of Newton, established his contention.

On the other hand Ficht and others contended that mathematics was the work of mans heart and mind, constructed by applying the mind in rigours of logic and deduction based on propositions expositions which relied on very few " axioms".

Such Axioms may be of divine origin but the rest was the " handiwork" of human ingenuity and genius. The implication was that the claim that rational thought derived from god and his " logos" was either over stated or entirely false! If false it left an entranceway for men of science and engineering to claim they built and added value to the world in which they lived. Further their towering intellect was equal to the task of taming nature!

That being said Kant sought in mathematics evidence of Transcendence, that which was beyond human capabilities, while Ficht  in particular sought to demonstrate that all of mathematics to that date could be constructed.

For the general person it hardly mattered, because arithmetic was not questioned, but rather used as a model of human constructed calculus. But for Many intellectuals it was troublesome as to where to locate the source of this construction. Abel for example demonstrated a deep axiomatic group structure on the ordinary counting process., which suggested a psychological base to our conting and mathematical concepts.

Justus Grassmann took this further constructing a logical or a ground in logic for the synthesis of Mathematics. It is in this family tradition that Grassmann pursued his fleeting analogy between displacements and the arithmetical sum of 2 summands. This is how he started his analogous thinking, with reference to a product and to a sum.

What I find in his writings is not the initiation of yet a new innovation, but rather the working out and working through of the idea glimpsed so simply. It took him years of effort and Analydis to check if his IDE was "right" or rather consistent. This work of reprocessing all of the then known mathematicians to see if his idea still worked meant that he read extremely widely and persistently. He deconstructed eveything and reconstructed it according to his new principles sndr ideals. As he did so he naturally absorbed the mathematical thought and conventions of his day.

What I now find in this piece is a full acceptance and use of common mathemrucal notation, which is then deconstructed and analysed and reconstructed either as is or in a new more efficient way. Atthe heart of this resynthesis is a derived notion of the unit . This unit is a Monacical system or Einheit. These quantities in a Einheit are line segments usually and instructively set to 1 or unity. This unity then is fed into unit lengths, unit values. Allied to them are coefficients and these conceptually are Tallies or counts of how many unis.

But then , along with his contemporaries he can change these coefficients into something called a number, and this can be real or imaginary. He does not criticise these " numbers" but simply takes them on board.after analysing them as Euler did, but concluding they had an analogous and consistent geometric interpretation.

Wht I see then is his careful and extensive use of verified analogy , adjusted analogy to make the notation work mnemonically, and recursive analogy. Once he has analysed any analogy and seen how it works he then constructs a detailed working model which represents spatial interaction. In the  process of doing do he often sees where a mistake or a misapprehension has occurred in some conventional theory and corrects it.

So it is with Quaternions . His method deconstructs all spatial interactions into 3 main product groups. From these he can instruct the details of any kind of product.. Finding an author who brings to the tabs the Quaternion principles and products, Grassmann is immediately abe to show that these products, far from being " new" to hom were in fact discussed at length in ome pares dealing with general products in his Ausdehnunglehre.

The upshot of this is he can deconstruct the quaternion Algrbra into its fundmental lineal algebras in an hithertoo unheard of analytical tour de force! Not only that, but he can point out misuses of the Quaternions, and show the correct or best algebra to use in those circumstances.. Not that he is boasting because he cites those who have given the better treatment of a problem using the style of Algrbra he advocates.

He shows an ability to derive the form of algebra suited to the nature of the problem , all of which he demonstrates in detail making reference to the analogous thinking method he utilises to get his result or formula..
I have yet to finish reading the last third of the article, but already he suggests that Quaternions best role is in Sperical Gometry and trigonometry. I know for a fact Hamilton applied it irst their to great effect!. Thus it is becoming clear that the mastery of his style of thinking is of great advantage to a more fundamental pprehension of how I, and geometers and astrologers in general might better interact with space when it omes to applying Metrons and counting!

https://www.youtube.com/watch?v=saWxZs2Xo24

Hermann now concentrates his mind on the spherical trigonometry of the unit sphere.

He starts generally showing how to construct a set of line segments that represent any polygon on the surface and hw to construct Polarecken. From these he is able to apply some combinatorial rules referenced in the 1844 and 1862 Ausdehnungskehre which in effect label all the possible relationships. Thus I perceived that without specifying the details he had handles for every possible configuration of the system.

The rocess then was simply to assign details to every handle and this appeared to be done in long chains of proportionalities. .

Several substitutions allowed him to generalise certain corner angles and thus deal with spheres that were not units.

Applying the fundamental or Msin products allowed him to identify groups of orthonormalisation radii which then allowed him to create a quaternion easily.

Further he goes on to derive special cases for polygons of any number of sides.

In his treatment it is quite clear, although notationlly intricate, that Quaternions are most easily understood in terms of spherical trigonometry where no complex numbers are necessary.

In every sense this is the conclusion I came to in the fractal inundation thread, that numbers are underpinned by a more dynamic magnitude, the trigonometric ratios.

This understanding really means to me that numbers as such are not entities in themselves, but rather symbols for a set of relationships between line segments in space. But the curious thing is these relationships exist within the sphere, and when yo unravelled the sphere by Eulers - Cotes formula and radians or Chords, you can construct curves of any frequency that sum to any form or surface by combination. This is the deep relation of the Fourier transform as it applies to bound forms of any quantitative size.

In passing I again make reference to the principle of exhaustion . Commensurable algorithms are done to the best of our ability, and that in an exacting standard is until we drop from exhaustion. In Latin or Italian Finito means finished or completed. Infinitum means not completed and ad infinitum means to the uncompleted end! Thus ad infinitum, usually translated as an infinite process means only an exhaustive pricess which ends incomplete.

The binomial series expansions are thus incomplete expansions. Euler's expansions are incomplete expansions. To assume otherwise removes us from Archimedian and mechanical principles into the realm of fantasy.

Cantors infinities are thus fantasies. The difference in pricess he experienced is merely the difference in magnitude we all accommodate between a formal line and a formal plane and a formal volume. To call these all different infinities adds little to the realisation that they are different magnitudes!

Our most useful conception of number is to be found within the trigonometric and related logarithmic tables, which ski ground our basic notion of function. Looking at it that way enables the spaciometric mosaics , the Arithmoi of whatever form to assert their fundamental place in our interaction with pace, and our systematic proportioning. Our calculus then can be understood as the analyss and synthesis that it is and not as some arcane symbolic device.

It becomes increasingly clear that in Hermanns time Euler, Lagrange, Aplace,Gayss,and Fourier were looking for generalPrinciples of application of the mind to engineering and philosophical issues.It so happened that in Mathematical symbols they could encode such processes and demonstrate their efficacy. So it is not out of time that Hermann hold propose his method of Anslys and synthesis.. What is striking is that his nonsensical method should in fact be the way to handle this specific approach to spatial interaction.

An aside: this video game person (http://www.terathon.com/lengyel/) has a fair number of talks/papers using the works of Clifford and Grassmann.

Thanks Roquen! And welcome to the thread.

What may seem an aside to you is in fact an on point response. I find that reading Grassmann in any of his articles or works generates such flashes of insight and connection that his name is now almost a meditative Mantra!

My labour is to clear a way for everyone to see the most useful mathematical method ever constructed, but also to make it clear that it was not constructed by some incredible genius, but rather a young man, struggling with a kind of autism, and various health problems whose father was too busy to devote the time needed to care for him. Thus he was brought up by his uncle, and away from the academic pressures within his own fathers household.

Although they were a close family Hermann was definitely not considered to be the brightest of Justus Children. Nevertheless he was given a gift by God, he believed, an extra insight to compensate for his difficulties, this led him, step by step to shape his education in a particular way that no one else had been able to do. This insight is a childlike view of spatial relationships. Clifford was so struck by how elementary or babyish it was that he remarked How much it was like Alice in wonderland or even Jesus admonition about entering the kingdom like a little child.

To be sure it is rigorous and demanding requiring a flexibility of mind and a broad mathematical education to see the full beauty of it. Even Gauss was put off by the flood of new and innovative ideas expressed in the 1844 text. But the essential idea once seen is so simple that you know it is special and almost universal.

For me Mathematics has been revealed to be a sham subject. It has a place as a subject only in a larger setting which directly is computational science a subset of computer science! Bertrand Russel attempted to embed it in Philoophy, but went about it in a way that AN Whitehead thought would resolve the issues that actually failed to. The point here is Whitehead was a Grasdmann Enthusiast, but he failed to understand Grassmann properly.

The difficulty seems to be the confusion between Platonic dialectic and the more modern Hegelian dialectic that Grassmann fell in love with. The 1844 version was an abortive attempt to present both his idea of a fundamental " algebra" of space and Hegel's fundamental philosophy of all that is including that which is contradictory. Hegel's dialectic forces the mind to work though and resolve contradiction in a new whole, rather thn to ignore or hide oneself away from it.

There is a " truth" that is within and or around all contradictions that resolves the issues at different levels. When this mindset is employed a fractal complexity emerges requiring us to iterate continuously to see or experience all that can be experienced at any level.

None of this would make as much sense if the computational sciences and the computer since and technology had not employed these ideas to wonderful effect in re presenting our sensory experiences to us through the digital media both on screens and on speakers and now in haptic environments.

Grassmanns analytical and fSynthetical model is so apt for the computational sciences that it is no wonder that they have adopted it at the highest philosophical levels of their craft.

I spent some time translating and meditating on the first stanza of the Stoikeioon book 7. This was why I highly doubted the myth or false story about the Pythagoreans being in crisis when they found was in commensurable.
Monas (http://jehovajah.wordpress.com/2011/10/19/%CE%BA%CE%B1%CE%B8%CE%B5%CE%BA%CE%B1%CF%83%CF%84%CE%BF%CE%BD-%CF%84%CF%89%CE%BD-%CE%BF%CE%BD%CF%84%CF%89%CE%BD-%CE%BA%CE%B1%CF%84%CE%B1%CE%BC%CE%B5%CF%84%CF%81%CE%B7%CF%83/)

The story goes that the Pythagoreans believed there was a fundamental unit from which everything was made. Thus like Leucippus and Democrites they believed there was an atom, that is an indivisible unit of matter from which the material world was constructed. This misrepresents their philosophy grossly.

The Pythagoreans defined the concept of Monas. This definition is as general and process oriented as it is possible to be in any language! That it is presented in Book7 of a undergraduate course in Pythagorean Philosophy written by the Euclidean Platonic Academy indicates that it is not a fundamental notion , but rather a constructed notion based on a priori primitives and practices which are more fundamental. Thus the first 6 Stoikeia are fundamental primitives to ANY notion of Monas.

Thus it is with great delight to read Grassmanns opening words to this article.

The Ausdehnungslehre makes the Monas ( properly grammatically it would be Monad in this position in a sentence) an arbitrary Assumption!

The concept of a unit is completely arbitrary! In the method of the Ausdehnungslehre!

At a deeper or more primitive level that means that the unit sphere is an arbitrary concept.

It is clear, from this standpoint, that the Arithmoi on which the concepts of number are ultimately grounded in the west , are an arbitrary concept, in line with the Monas, the unit identified as one , that is its counting name is " one" ; in line with it being an arbitrary supposition. Thus numbers, far from being real are wholly subjective , and in fact completely formally defined.

There is no necessity for one and one to be two. It is that by definition alone.

Suppose I compare one object with another. How is it that I identify any object as one? It is by a process of physically separating out and placing the object protos or before all other objects. The placing down or kata of the object is that action that initiates the counting name "one".

As I repeat or iterate this process I must iterate the name in the Namespace. It is the sequence in the name space that has to be memorised to make sense of the whole of this iterative or recursive behaviour. The memorising of a cultural iteration is thus not arbitrary, but that to which it is subsequently applied is arbitrary beyond all arbitrariness!

This arbitrariness naturally means each form or object is unique, or can be named one. But each object by virtue of being unique differs in some way from every other.

Thus a unique object has to be given a unique name. We can replace "name" by " symbol or label or handle" . Later it will become clear that by substituting in this way we derive a symbolic model that can be applied in an arbitrary non sequenced way that has to be defined at every little nuanced change. This becomes the basis for symbolic ratioing, proportioning, reasoning , aggregation and factoring. It becomes a succinct way of identifying subjrctive processes of memory and recall , and allowing substantial focused manipulation of internal and external process actions.

In this way we address our existential interaction with our internal and external experience of the space within and the space without and all that populates them. These experiences I have called the set FS. It is a finite set, but it grows continually.

The set notFS is one that "ad infinitum" applies to, one we cannot finish counting ever, one which subsumed and consumes us all, whoever or whatever, in total annihilating exhaustion. But the hope is that recursion and iteration are fundamental to all things in that set, that is all things recycle in indescribably complex ways: that the possibilities are truly infinite!

As you know I do not accept that their is any process that I fo thst is totally objective nor for that matter is there anything I do that is totally subjective.

This interplay between inner and outer experiences I describe as a continuum. Thus there are no absolute poles but rather a Yin and Yang complementarity or conjugacy. In a bald sense a dynamic circularity or sphericity exists and I oscillate between the two in different frequencies for different states.

Thus when Monas was defined by the Pythagireans , the arbitrariness plas upon this cognitive dynamic deliberately. It is not meant to be confusing, but accepting of our real internal and external processes.  Languages that have self reference or Tautology reflect this insight very well, even if modern thought has forgotten the role of self reflexive grammar or tautological grammatical construction. English, and English Grammarians thus put modern thinkers st a disadvantage.

Monas is any arbitrarily defined unit( Einheit), but as such it has no name. It has a purpose that gives it a name, Katametresee , from whence I fancy Metron is derived. Whether it is so or the other way round is not important here.,the measure or bowl or the act of doling out or placing down constitute an active process that defines reflexively a Metron and thus a Monas. A Monas is a standard unit, but it was not standard by consensus( which it now is) but standard by har-monia. This idea is a dynamic flowing of rhea into a singular (mono) form. These are parts that fit perfectly together to be a harmony a flowing together into one.

Thus the standard is what we call tessellation. A Monas must tessellate in multiple forms factorised by them..

These Monas/Metron multiple forms must be perceived by a process of factorisation to be defined as Arithmos! Thus Ruclids Algorithm in book seven is the beginning, the heart and the end of that entire book. Without it the concept of Arithmos cannot be understood.

But the book moves in a different orbit. It seeks to define and clarify not Arithmos, but Arithmoi!

Thus we find very many classifications of Arithmoi by their use, their origin and their purpose. The perception of multiple form arrived at by factorisation is the key message. Thus in every sense Division and aggregation are the 2 opposing fundamental starting points of all development of the Arithmoi.

To these Arithmoi we apply the cultural counting iteration. By this means we can sequence not just monads to form Arithmoi , but at every level we can sequence Arithmoi to form mega Arithmoi, say, and we can iterate again to sequence mega Arithmoi to form plassos mega Arithmoi, and by reiteration, plassos mega Arithmoi can be sequenced into bathos plassos mega Arithmoi and so on and so forth!

Arithmos are therefore the basis of recursion or iteration using the cultural counting iteration +1, or as Norman defines it, the successor function).

In this scheme it becomes clear that 1 as a name becomes useless. We have to say 1 whatever the form is..
However the cultural iteration itself is based on the fundamental spatial experience we call order. The words sequence and order define the same spatial experience and / or action or activity. We rec ignite this by defining different word called ordinal. The cultural iteration defines for us not only the name of an arrangement or a gathering, but also a sequence order. .

The two names ordinal and cardinal are fundamental activity distinguishes, but one is procedurally prior. We must order or sequence in order to name. Thus sequencing and ordering are fundamental processes in al we do , think or say or believe, or perceive.

The concept of protos or first therefore has a prime significance. I have described how by iteration we start with many firsts of different kinds of Arithmos. And these Arithmoi also have greater aggregations of which they may be protos or first.. Thus the Pythagoreans start with a Monas / Metron , that is they make that the first in order. Because it is first it's cardinal name can be 1.

However they can also start with any Arithmos and call that cardinally 1 . When that is done it becomes clear that there is a difference in these protoi Arithmoi. Not all of them are commensurable!. Simply put some protos Arithmos cannot be used as a factor of a larger Arithmos, because it is perisos, that is approximate in fitting that larger form!
The reason becomes clear, because even if the Arithmos and the protos Arithmos are set out in their monads they differ usually by a few monads. This few is always less than the protos Arithmos.

Book 7 deals extensively with ths system and hw it is to be used, but please do not be misled by the odd and even nomenclature or the use of multiplication instead of factorisation.

You will understand why prime is the Latin word used for these Arithmoi, because it is an ordinal notion meaning the first of any group of things.

In this case, Grassmanns concept of Einheiten models this analysis of the Protoi Arithmoi, and how they fill and encapsulate space.

The really interesting question is how foes the sphere fulfill this role, when it cannot tessellate? The answer, discussed by Apollonius reemphasises the iterative sequencing at different scales and why reality will ever exhaust and consume us!

I have been busy of late and to a certain extent disenchanted.

As a young man I had fire in my belly, dragons to slay , causes to support and the world to put to rights. But now I see from the perspective of experience: youthful vigour is spent chasing paper tigers! The world like the earth rounds in huge cycles and what is curved cannot be straightened!

Nevertheless it is better to be wise than ignorant, because the wise one can find the right spot to effect a change that centuries of effort cannot. Kairos, the opportune time is always at hand , but a wise one is not.

It is instructive to read Hermann across his works. Youthful hope and idealism, replaced by hopeful endurance of rejection, and a willingness to learn from the school of knocks, to his emergence as an establishment figure mouthing those same words that at one time greeted his pubescent efforts! How what goes round comes round!

This is a rather technical paper which I will translate, slowly. The maturity of his thought in it reflects the times in which he lived. I am struck by how much that impacts on his terminology, his notational proclivities, his choice of symbols and phrases.

I remember reading how he successfully defended his primacy against St. Vainant, without ever realising the international scope of the bodies that he appealed to! Indeed the trans continental reach of these organisations was only jus being extended as electricity enabled telegraph communications to pass from the Americas to Europe and across Europe. This curious mixture of old practices and new thrusting modern conveniences was contingent on the intellectual societies of the day. Everywhere societies and guilds were springing up to pass on the new " wisdom" and practice in one form or another. Philosophical societiesn guilds of engineers and architects, Masonic and RosaeCrucian and other medieval "secret" societies or mystery schools all formed a network of " public" and private education from which some of the next generation of pioneers, inventors, and Scientists would come.

In short ignorance was no excuse, for any willing applicant as a supplicant to any of these societies , a willing postulant could be apprenticed to a master who would oversee and support the costs of their education, providing their benefactor saw grounds for doing so. Usually this was not philanthropy, although some were philanthropic. The real motivation was the gaining of some advantage, whether financial or social or both.

These " old ot" networks still exist even today and are an essential part of many social events one way or another.

Grassmann had access to the international journals of many philosophical societies, but he did not lays have time to read, keep upto date or contribute. Later in his life this changed and much of his later work is to be found in the collections of various journals and in several languages! However the main exposition of his ideas, what he wanted to promote most ardently are found in Ausdehnungslehre 1 and 2 as he refers to them.

Hamilton in his work set out new names for old long established ideas. This gave his presentations a new feel, but cut off centuries of established ' fact" forcing the reader to learn a whole new set of terms. This made old ideas seem to be newly discovered and gave cudos to the author(s), which might attract a benefactor to sponsor further " research" . Grassmann cuts through all this spin doctoring to provide the interested reader with " the Keys to modern Thought, especially mathmatical" . To paraphrase St. Vainant.

I awoke this morning, my bed pointing along the north south magnetic Ley lines, due to a rearrangement of my sons room orchestrated by my wife, expecting some new insights and Natural Energy to spark interest.

Considering again Grassmanns assertion of 3 equation groups as constraints on multiplication as a concept , using the monadic system denoted using the labels e_s and my current understanding of the commutative biproduct or bilineal product, the cyclical interchange product  with product squares all interchangeably equated and finally all product squares summing to zero, I wanted to understand my interpretation of ordinary line segments and trig line segments..

Especially, since Grassmann derived his concept under the inspiration of the great trigonometrists of his time. It seemed that my translation of " zur Seite" might be misleading me from a simple Newtonian resolution of a line segment into orthogonal line segments! That is to say that projecting vertically or Senkrecht was the process of dropping perpendiculars onto a pair of line segments orthogonal to each other but joined through one point of the originating line segment. In physics this is called resolving a line segment into its orthogonal components! However we should perhaps more simply point out the application of Pythagoras theorem and trigonometric metrication.


However Newton took the process into the next stage by also using rhombus or rhomboid forms to compound, especially when compounding for circular orbits.

The concept of an outer product as a result of line segments stepping apart in a rotational manner and contrariwise an inner product as these same lines step closer together, relies on this vertically projected line segment I called a trig line segment.

Grassmann makes some comments about it himself in which he clearly used the cosine and sine ratios And their relation to the Eulerian -Cotes radian arc measure. He also by analogy or rather logos analogos thinking connect this to the concept of the cosh and the sinh functions.

I also note that the sum of these monads/ Metron units when squared coming to 0 is a related property to the Cotes DeMoivre theorems on roots of unity.

I am reconsidering some concepts I developed of a prior or older spherical and circular geometry in which the fundamental symmetry of the circle/ spheres underpins the fundamental notion of Similar forms, the creation of parallel planes and lines , and the fundamental theorems of Thales from which Pythagoras theorem could be derived as an obvious consequence.of empirical pragmatic practice!

Our notion of proof or demonstration skews our understanding of basic mechanical and constructional insights. What is empirically found to be the case, and thus inductively always expected to be the case is different from a formal system in which a statement is deductively demonstrated to be the case. The first is pragmatic and real, the second is ideal and formal.

As a young mind it was hard to know what the words "real" and "formal" could mean, but now after much study it is clear that one is pragmatic and mechanical while the other draws on this, idealises it and removes it from any natural variation to a fixed static form. It is the Fractal , iterative Dyanmics of applying the formal to the real that enables us to apprehend that we are alive! Cogito,Sum!

Returning again to the 1877 reprint of the 1844 Seminal Ausdehnungslehre I sense how Grassmann had come to the opinion that he had perfected the flaws of the 1844 version in the 1862 version. In particular he was confirmed in this opinion by the interest and application of his ideas by other prominent thinkers in his time. I have to be wary of jumping to tht conclusion just yet!

Grassmann tried to keep the Ausdehnungslehre relevant and fresh to the developing mathematical and scientific Philosophy of his day. The New Geometries, that is projective and non Euclidean were shown by him to be best expressed by his concept of products. Others felt his ideas and methods provided a good ground for simplifying empirical research, and Schlegel,and Clebsch wrote extensive volumes on concepts that drew heavily on Grassmanns notation and style.

The concept of groups and rings Grassmann saw as the New algebra and he wrote essays highlighting the relevance of the Ausdehnungslehre to this new thrusting war zone in mathematics. He developed ideas on the 3 dimensional space, and he analysed Hamiltons Quaternions in terms of the fundamental ideas of products in the Ausdehnungslehre. In everyway Grasmann promoted and advanced his concepts as keys to the modern philosophical developments, and their advancement. And still there were kernels of ideas in his works that required greater development!


We can see much of this development in our mathematics and physics today, but what is also true is much of that has been erroneously developed!. This is not to say Grassmann is the pope infallible in the Ausdehnunglehre, as Klein invidiously implied, but rather that academicians ruthlessly baudlerised his wok and his ideas to promote themselves.  Gibbs is a case in point.

The section I am revisiting is..Vorrede p viii.

I have retranslated it and really come to the same kinds of conclusions. At this stage Hermann is concerned only with the fundamental product notion: two adjacent  " jostling" line segments form a parallelogram. Projection here is of both , one onto the other. The arithmetic product is solely the cosine , and the product is thus multiplied by a factor of cos²ø the angle measure between them not being here specified but just identified.

The later extension of projection comes from using the work of Laplace Lagrange etc as teeth cutting exercises in the development of his method.

Ot is of interest to note that this continuous development and modification led to the " perfection" of 1862. However his Vorrede there is one of the mst pessimistic that he ever wrote, and betrays anxiety about whether he had made the right choices. He refers to the impetus of his brother as the Mathematician as if ready to blame him for any failure. Thus in 1872 when in fact he received favourable responses he shows a remarkable assured ness of the unassalability of his work.  I also note a nationalist pride being portrayed in his comments on the work of Hamilton. It is not that he dismisses Hamilton, but rather he appeals to the Prussian pride in their own works and productions.

Why he asks promote an international author when homegrown authors not only tackle the subject but tackle it better!

Klein relates how in this tumultuous time of wars and revolution , when the whole geopolitical scene was delicately balance that rioters were seen in the streets shouting Grassmanns name against those shouting Hamiltons! It was perhaps diplomacy that made him and academia marginalise these excesses. Indeed most learning was from the French Ecole , but most hated Napoleonic France vehemently. This was not a good time to agitate against a French imperial power!

At any elementary level children are indoctrinated with the alphabet and a decimal numeral system of names to call out .

Both of these are cultural namespace systems, but they are taught or impressed on a child with little derivation?

They can only be derived for each individual from the space in and around them in which and of which they consist. These derivations are functionally divided into ordering space,(Let us specify ordering as arranging space relative to itself and the arranger) and ordering and constructing a model of our response to ordering space.

Ordering is a complex process involving sequencing by position, by evaluation, mental responses etc
At each process we move, count, are moved or are counted. In moving we translate, rotate, reverse reflect and many other aspects of what I shall call dance. In counting we respond sequentially rhythmically , numerically and vocatively in many aspects of what I shall call singing.

Thus in dancing and singing we account for, enjoy and respond to our experiences of all in and around us.

To focus then on 4 specific processes: dividing of wholes into ordered sequences of parts, which is factorisation, and the principal process in our nature, in increasing unit in this world.

Taking those units we have a goal for addition: to find the sum of the parts, to restore the whole.

This fundamental cycle of division or facto rising followed by addition that is summing the parts into a whole is the fundamental argument of Book 7 in Euclids Stoikeia. From this all classes of monads/ Metron or unit standards are assimilated and Arithmoi are defined and proto Arithmoi distinguished. It is here that standard Metrons are ordered into Arithmoi combinations or sums from which the pollaplasios or multiple form is distinguished and defined.
The division process naturally supports the subtraction of parts from the whole, again ordered into multiple forms.

The classic or ancient realisation was that all things can be given an alter ego, a formal representation. All things can be recorded in a mosaic form, a multiple form

And thus all multiple forms can be reduced to combinations of other multiple forms.

We can restate this, without changing its derived meaning, without giving ground to any modern interpretation as
Any product can be represented as the sum of other products.

Thus a product is the mosaic of some form derived by division. Such a product can be conceived as a sum of constituent products.

Grassmanns representational expression , for it is not an equating of equals, and neither are so many so called mathmatical equations:
AB  +  BC = AC

Is a powerful analogous expression. In it eventually Grassmann came to recognise that any product can be written as a combination of products. The fractal scale free implications are breath taking.

The rise of the consideration of ordering in in the roots of polynomial expressions that lead to group theory helped to identify ordering basics, but one fundamental arrangement process was so often ignored in the notation. It is rotation.

Cyclical rotation of letters terms etc were inconsistently apped to their spatial implications. So Cardano is left wondering what the square root of a negative expression might Be, and even what a negative quantity may be!

That reflection and rotation both seemed to be possible spatial ordering cognates, the descent from space to number made it hard to understand. In moving from ordinal or space numbers to cardinal or symbolic numbers a vital connection with space was lost.

Now I restore ordinal numerals to their preeminent place as denotations of the Arithmoi.

And the construction of products from other constituent products as fundamental to spatial ordering.

Here we shall see how Grasmann through constraints restablishes spatial ordering through the combination of constituent products of ordinal numerals

Through his monadic systems or combinations he records multiple spatial forms and how they are relatively arranged and counted, and what other multiple form may represent them either in count or effective spatial arrangement. To do so he restores rotation to its rightful position as a fundamental ordering process.

Reflection is usually added as a third process to translation and rotation. While it is useful it is a subjective intermediary. The third process that I allow is projection. This is a fundamental objective process generating shadows as objective forms.

The use of symbols and signs, in particular the contra sign must be carefully distinguished and interpreted so that what is objective and what is subjective is realised, and what is to be counted or discounted is clear.

The emotive use of terms like annihilated in physical phenomena equally must be sceptically received, for what is subjective nd counted and then discounted has never been physically present or physically removed!

While this is a head of its place, the designation of concepts like energy apart from their observables is fraught with difficulty if allowed to be considered as objective  the use and method of directed line segments provides a safeguard against such fauxes pas!

The product , one of many Grassmann constructs from his 3 product constraint groups he calls an averaged product of the 2 main products, or spatial orders, the outer product and the inner product. The inner product involves a projection process in addition to the cyclic rotation process in both products.

The constraints define what we must do and when to carry out this combined process of ordering space!

The inner product in the Vorrede has been a problem to understand , because the first ordinary line product was expressed dynamically as 2 adjacent and or thrusting together line segments. By parallel projection one may construct an intersection point which then defines the 4 points of a parallelogram.

The inner product is defined in terms of 2 line segments projecting vertically into each other to form a trig line segment in each of the ordinary line segments.  Now I have taken the inner product to be the product of these 2 adjacent trig line segments, as per the ordinary definition. Gone ver another product may be also implied. The trig line segment producted with the ordinary line segment it is lying in. This product in general would be
bcosøaa which is bcosøa² = a²bcosøu²_a

u_a is the unit line segment in the a direction

I will wait and see if he clarifies this here or in the 1862 Ausdehnunglehre

There is an argument that one should define the projection of a line segment onto another in terms of its unit line segment. This is usually called normalisation . At this stage it is an option I have yet to find Grassmanns word on.

Combinatorics and Kombinationslehre were first introduced to me by Lancelot Hogben, in relation to ordering playing cards. I only really looked at group theory last year or so when Norman started a course in it, and I took opportunity to explore a subject I could not connect with at university.

So it is not surprising then that it has only just dawned on me that the cyclic groups are precisely the same as combinatorics!
http://www.millersville.edu/~bikenaga/abstract-algebra-1/cyclic/cyclic.html

It was as a result of looking into cyclic groups that I constructed Newtonian triples.

Now combinatorics always reminds me my beginnings with permutations. From permutations I can remember selections and from these the combinatorial choice formula, and finally the link to the binomial coefficient.

In my blog I did some thinking on sequences and spatial ordering, trying to grasp at some connection to direction . I found that ordering with repeating of options and without repetition of options from an option pool had 2 formulae to give a count, but basically these counts were only significant for limitations. The infinite case was not really fathombe, or even interesting!

I came up with a fractal " cuboid" spatial arrangement for permutations , taking it beyond the table or gridlock format, and finally found I could make a nested Circe arrangement  with spirals.
The basic tree diagram for choice was what I remembered and used and found it was precisely the same as the 2d array for sequence of lengt 2 nd the 3d array for sequences of length 3 . Thus I was stymied for sequences of length greater than 3 until I came up with the nested spherical array.

Well guess what, all meters that use dials or dial equivalents are using cyclic groups, even product cyclic groups. Our number system is based on product cyclic groups.

The connection here is simple and profound. Grassmann uses the " rotation" in cyclic groups to model rotation or gyre in space.

Thus for a parallelogram the four points are cyclically ordered ABCD . The points are rotated cycliclly BCDA etc..

However we can take any 3 points , order them cycliclly and rotate among them cyclically. We can do this also fo any 2 points.

If this is done systematically, ie as a meter dial system we can actually model sequences of rotations at any level And structure reference frames , basis line segments , rotation systems etc cvordingly.

When I realised that Grassmann was using cyclic interchange to control sign changes I had no idea he was actually using cyclic groups to define spatial rotation and  orientation.

Strictly speaking I glimpsed this from the work we did in hunting for the holy grail. It lead me back to create the Newtonian triples, but I struggled to get a clear picture , a memorable understanding of what I was doing, what Grassmann was doing and what the hell were we all doing symbolically!

The cyclic " meter concept" leads me to use the fundamental expression as follows

AB + BA = AA . This I am going to call a phase product.

Similarly

AB + BC + CA = AA

Any cyclic rotation around a set of poits at the vertices of a form will begin and end on a phase point. The thing to note is that a line segment has phase points!

For a triangle we may establish a 2 dial meter based on the triangle cycle and the line segment cycle.

Thus
Setting the meter readings horizontally , a letter ( point) from each dial we get
AA
AB
BA
BB
CA
CB
AA

We can clearly see the phase points marking the beginning of a phase and its End/beginning. Without going ino detail this is precisely the role and meaning of Shunya, the alpha and the omega!

External to the meter is the motion of principal orientations, principal directions and contra directions. These are usually encoded using the +\– Direction signs. Orientation is not encoded but usually the orientation is set by convention. However, we are not to be bound by convention, and may define orientations by 2 points of a line segment . In doing so we must distinguish principal orientation and relative orientations, and certainly understand that direction is not the same concept as orientation.

Now we have the meter and phase points we can understand that for a point a phase point is not defined, for a line segment 2 phase poits may be defined  AA and BB and for any n gon n phase poits may be defined.

However when written in meter form the meter reading shows phase points as extended alilterations, ie AAAA.. Depending on the cyclical structure being recorded. ABB for example is not a phase point for the 3 meter reading( that is 3 nested cycles) but it contains a phase point for one of the nested cycles. Similarly BBA is not a phase point but it contains a higher level cycle phase point "waiting" to complete the higher level cycle.

Putting it this way I hope makes clear the sequential ordering which is the topic of combinatorics, and how it naturally encodes spatial rotations of forms, as well as contra statuses which come from the line segMent rotations! We might call these reflections, but they are rotations through a point ( or round if you want to include 2d and 3d  degrees of freedom).

Psychologically a reflection is constructed from these rotations through or round a point. Physically they may be impossible to perform as rotations round any point but the point is definitely rotated contra as it "goes" through the point of reflection. Again this is subjective processing of information returning to the source by Reflection at the point of reflection. Light does not pass through this point.

There are some fundamental issues with my understnding of Hemanns system that I can now address.

Certainly the labelling of points is the underlying handle to correctly determine wat is going on or being represented. But Hermann also uses a line segment notation.
e₁ represents AB . If written e₁ = AB I think this is a common misuse of the = sign . This occurs in the fundamental expression and is why I refer to it as an expression.
e₂ —> BC and e₃ —> AC gives
e₁  + e₂ —> e₃

This does not resonate like the point version. It contains less information and thus requires a diagram to convey that missing information. So why use it?
It is a mathematicians conceit that if the essence of a notion can be expressed in the sparest of notation, that that in itself is elegance! However for most of us it is obscurantism. However, I will allow that if a teacher employs such symbolism in rhetoric, explaining as he discourses both verbally and non verbally then it is a piece of dramatic art, and like an orchestration enhances the teachers performance in exposition.
To write such a thing in a book is however an inanity. Just as magic written down is not magic at all, but mere pedantry!

However Herman means to show the nature of the spatial ordering of products and the invariance in the form for a product of line segments as compared to a product of points. The summation is less significant, especially as it is not a sum in the arithmetical sense.

Several adjustments or constraints on use of this lineal notation are made, designed to give maximal mnemonic impact. Thus in a parallelogram ABCD AB is // to DC and has the same length coefficient so e₁ represents both these line segment and any and all parallel to it!

This deconstruction of space into parallel lines with a single lineal symbol is often ignored. What it means is that the whole of space is regularly tiled by this set up, and that what applies in one parallelogram applies in each parallelogram cell throughout space!

While this is not necessarily true it does reflect our psychological perspective. This is how we conjugate space.

The product e₁ e₂  is the sum of products in the plane, if you replace a lie segment by a sum or combination of line segments.

The use of the term combination is more in keeping with the process that is going on.

The combination of products is misleadingly presented as = . Thinking in these terms obscures the representational nature of the expression is lost. What we are doing is representing one product by a combination of other products.

The product itself represents a projective process.depending on the spatial relationship of the line segments the parallel projection involves a projection in the principal directions or one in a principal direction while the other is in the contra direction.

Also the principal rotation in the figure affects the description of the lineal combinations of the sides( line segments)

e₁  + e₂  represents both a clockwise combination and an anticlockwise rotation around the perimeter of the parallelogram. AD + DC is the clockwise rotated combination.

All these differences make it easy to make some unphysical interpretations of the spatial behaviours encoded in he 2 levels of notation.

Commutativity of lineal combinations in a parallelogram context are in fact clockwise and counterclockwise combinations.

Distributiyity of the product process over lineal combinations also needs to take caeful notice of the principal rotations as well as the directions of line segments.

This is all tied together at the level of the combinatorics meters in the previous post

The trigonometric convention for measuring angles and computing the sine and cosine and tangent Ratios has the circle divided into 4 quadrants. The convention makes the principal direction on the horizontal orientation to the page. In 3d that page swivels as easily as any universal joint!

The principal direction is then always to the right of the circles centre , on this horizontal diameter.
However this then divides the diameter into two rays, each starting at the centre and directed contra to each other. These are termed ray segments and are as such radii of the circle. This old term reflects the fixed value of the length of the diameter and thus the radius or displacement from the circles centre.

Euclid defined, or possibly Apollonius redefined this property of a circle in terms of seemeia, or indicators. These " points" lay on the intersection of every diameter with the circle form, such that every point of intersection was the same line segment( length) from a unique point called the centre.

It is hardly ever pointed out that this is a mechanical description of a drawn form. As Newton observed, Mechanicl principles lie at the base of geometry. That is to say empirical and metrical concerns precede and define the fundamentals of Geometry prior to any reasoning or ratioing.

For example the 4 quadrants are empirically determined; the 6 sectors are empirically discovered and determined, various figures are mechanically constructed.

It is because these constructions are so basic and repeatable that they are universally held to be true. The " perfection of them is hardly noticeable. However, who has not had the slightly disconcerting experience of segmenting a circular arc into 6 by the Radius displacement only to find it does not quite fit?
That is when you make the 6th segment mark it does not land on the hole made by the compass point!

As a young geometer it is easy to believe that your fumbling attempts are the problem. More skill , practice and care should eliminate the discrepancy, but it does not. Instead what we do is perfect the result. After all it is a pragmatic thing to do. However I find it highly suspicious that the perimeter length of the circle is a little over 6 1/4 radii. The curved arc naturally should be longer than the chord on which it sits, but only 1/24 of a radius longer seems too small!

We accept it because we have no other measure by which to compare. In fact the issue is neurological. Our sensors are set to enhance certain signals for definition and contrast. So the boundary of a form is enhanced to clarify it. The area or volume bounded by a perimeter is also enhanced, but when we use a Metron the surprising differences in the counts between similar figures is just one of those empirical spatial behaviours that defies simplistic pattern making and reveals the flaw in pure reason not grounded in empirical data.

This is how it is. This is what our fabulous unconscious processing delivers up to us to reason with; to compare and contrast and distinguish by symbol, name or some internal experience of memory.

 It is the " perfecting" of our memories that blinds our inquiries. Those who have insights have not obscured the flaws or ignored our process of pragmatic perfectioning.

Our conventional system of assigning Values to the ratios of line segments in a right angled triangle has been augmented over the centuries. The mechanically derived tables were initially partial and limited to a few indexing arcs.. These arcs associated with the appropriate corners became the concept of a corner angle..

The perimeter of a circle was divided in the Sumerian times by the Akkadians, into 360 parts. Each part was a day in an annual cycle of seasonal observations of the stars, moons and planets. Thus 360 was empirically derived, by marking sighting lines on a great circle.

During the Napoleonic years the Ecole in Paris decided to make the quadrant part of the metrical system. It was divided into a thousand sectors! For each sector the sine value was calculated. Needless to say it took years of calculation to complete, but was never introduced into general use! Copies of these greate tomes are in the library in the French Acadrmy.

Thus the angle, the arc and the circle are indissolubly linked. Even if you, like Norman Wildberger consider the current angle measure flawed you cannot escape the relationship between the right angled triangle and the circle.Thales theorem only asks you to accept what is empirically demonstrable: all orthogonal angles are equal. This statement can never be deduced, it must be induced from the circle properties and empirical measurement.

Before directed line segments were introduced, in particular contra line segments were distinguished by these circle measurements. Thus a diameter was evaluated as 2 radii, but the radii were rays contra from the centre. They were said to be opposite or 180° to each other. Imposing a – sign onto this ancient angle or arc measure system of defining direction was bound to lead to confusion!

In the first quadrant all ratios were positive(+). This meant that the principle orientation had the direction drawn to the right of centre, or vertically above the centre for the relative vertical orientation.

The relative vertical orientation was introduced precisely because of the right angled triangle.. Once the principal orientation is fixed the relative vertical is also fixed.

However the quadrant is limited to a quarter arc. Pragmatically this was not a problem because you only needed a quarter arc to define the other quadrants! Similarly if we defined a sector of a 1/6th arc then we would have to apply it in each sector. This means we are pragmatically forced to repeat values in each sector , and thus confuse ourselves as to relative orientation!

To overcome this 2 quadrants are used to define the trig ratios. 2 scales were written on every protractor. It was crucial to state which scale one was using.

Eventually the mathematical board decreed that 4 quadrants be used, that contra rays be used and that the ratio signs be distributed in the quadrants by the CAST mnemonic. It also decreed that the conventional principal rotation from the principal orientation was counterclockwise!

As usual this was ignored at the elementary level. So we still refer to the interior angles of closed forms and the exterior angles of the same forms., or we defy the board by rotating the figure or paper into the " legal" orientation and then measuring!

These games we play have serious consequences. We become confused and use incorrect values and our calculations then do not match the mechanical reality! Which do we believe? Unfortunately most are taught to believe the calculation!

However is the decreed method practical? For example how do you measure an angle between 2 lines when both are not horizontal? That angle is the Difference between 2 decreed angles, and that forces us to accept that angle measures are fundamentally differences between decreed angles..

Using that notion our sine table values have to guide us to the correct ratio values  for angle differences up to 2 radians.

The overlaying of the negative(–) contra sign onto the older rotated contra orientation is still a fundamental confusion in the notation. But mathematicians seem to love these quirks!

If I start with 2 contiguous rays, some call these parallel and touching, I always want to call them collinear, but perhaps I might just use collineal; then giving one the conventional principal rotation means it's relative orientation changes anti clockwise but it's principal direction remains positive (+). By this I mean moving away from the centre the arrow I attach in the initial position is fixed in that line. By this means I can perceive how my experience of its direction gradually changes from one direction on the principal orientation line to the contra direction. To see this clearly I have to do a vertical projection from a fixed point in the rotating ray.

Now I associate a decreed rotation with a projected direction on the principal orientation. Note that the ray projects a direction onto the principal orientation not just a length.

So an angle difference strictly projects a ray difference or a direction difference onto the principal orientation.

The simple minded approach often ignores these subtleties. To be fair, they arise only due to the need to be consistent, but that need arises because confusion reigns else!

Why do we make it so confusing? Or rather allow it to be so ambiguous? This ambiguity is confused with flexibility, or rather as we all know it allows a fudge factor to adjust our confused thinking to empirical evidence?,

These issues were all considered deeply by the Grassmanns in an attempt to promote rigour in these kind of discussions. Justus in particular showed a dogged determinism not to give in to the logical inconsistencies, but it is so hard to eliminate them in confused thought. Hermann by his system of notating everything at least made them obvious, even if he then was forced to define them away!

The issue of commutativity can now be tackled from this geometrical view. A line segment in a parallelogram as it rotates into the relative orientation of its adjacent line segment may have its principal direction changed to contra, but certainly as the adjacent line segment rotates into the relative orientation of the other side it has to become the contra of that orientation. .

This cyclical rotation is clearer in the point notation, but it is possible to see how the projections onto the principal orientation confirm this..

This behaviour applies solely to the parallelogram form. If we wish to use different forms we will have to accept a more complex relationship between sign change and line segment orientation change. Hermann in keeping with his scheme refers to these line segments as factors. As yet I have not found him referring to the cyclic interchange as commutativity. This is a denotation imposed by a later more robust form of group theiry, and it obscures cyclic interchange in the geometry of the parallelogram.

It is to be noted thst this interchange does not apply to triangles, because contra directions are never achieved..

Grassmann defines the inner and outer products in these cyclical terms. The outer product is where the line segments step apart from each other. He says the value gains worth as it does so. The inner product arises in the context of vertical projection, and this value gains worth as they step closer together.

This only makes sense if the arc of a circle is split into 2 by these line segments. As they step apart the arcs behave in a contra fashion, one increasing as the other decreases.

For the product to increase in both these directions the vertical projections must involve the sine and cosine ratios.  The outer product must involve the sine ratio and the inner product the cosine ratio.

As yet I have not clearly seen that in the 1844 passage in the Vorrede, but I am still looking.

It is heartening to find that even today traditional math topics are still being taught, but it is disappointing to watch how formulaic and rote and devoid of understanding much of the teaching is.

Watch these videos and try to understand how this relates to hermanns averaged product.

https://www.youtube.com/watch?v=CJNAO6LQmAw

https://www.youtube.com/watch?v=1MVn6T1nXkM

The second video gives the ancient circle theorem that explains how arcs, arc measures and angles as chord or secant intersects are related.

I have known and used this theorem since I was a lad , but the arc measure and the radian was never explained to me . Later I learned of the use of the radian as an angle measure. Even later I realised the circle arc measure itself was a universal measure of Ritation. Using parallel lines and translations I could show without angle measure that the sum of the angle measures  in a triangle, converted to arc measures was always the semi circular arc!

What I was looking for next was how to deal with arcs that were circular but not drawn from the vertex or intersection of the angle.  These videos show the sophistication of Geometrucal ideas in ancient times and how the circle was fundamental to that sophistication.

In relation to line segments and the inner and outer products it is clear to me that these products are not only related to the vertical projection of one Ono/ ino an adjacent one but also the circular arc that passes through the 2 end points of the adjacent line segments.

A vertical projection naturally establishes a right angled triangle. But a right angled triangle is a structure within a semi circle! We have two important circles both intersected by these line segments ( extended if necessary). The behaviours of these internal arc measures and external arc measures governs the apprehension of the outer and inner products..
While outer and inner products are constructions of parallelograms, these parallelograms are intimately connected to circular rotations.and arc measures. The averaged product that Grassmann constructs to describe quaternionic Ritation, or any general rotation( ie not round a circle centre) is based on this line segment intersect angle measure. The point of intersection can also be barycentrically determined as Shan by these secant formulae..

While I have more research to do to show this conclusively, it is intuitively a sound conjecture.

This implies a full trigonometric useage, even at this stage.


The restriction or constraint of a fixed direction does not prevent one from constructing a parallelogram from  any 2 line segments. Clearly the properties of parallel lines are implied here.


This definition of a product of 2 line segments implies that they rotate relative to each other, but it is not clear that this was the mechanism of the " cyclic" interchange . It took me a while to perceive that this was how he saw it in 18444 when he wrote these words.

These directed line segments or rather line segments with an arrow symbol firmly attached indicating direction , he thought of like the hands of a clock. When they point in the same direction there is clearly no parallelogram! However when they point in opposite directions this is not so clear, but he makes no comment about that here. At the beginning of the Vorrede in 1844 he does comment on oppositely directed line segments with the arrows firmly attached, but in this case the hands of this " clock" point in the same direction when at the semicircle arc rotation .

Here Grassmann is not specifying a formula, but a well known projection process! Even the other line segment is not specified. The only specification is the projection is vertically onto. There are 2 descriptions of a vertical projection of a line segment! They both relate to the right triangle.

The construction line is called dropping a perpendicular or vertical line. The direction of the line segments is important to specify where we drop the perpendicular line.

Grassmann here makes no mention of the construction line just the arithmetic product. The vertical line is also an arithmetic projection product.  The reader seems to be conducted to the line segment that is being projected onto. Suppose the line segments radiate from a vertex then the projection would involve the cosine arithmetic product. A third line segment , the proection line has to be invoked to use the sine arithmetic product.

Grassmanns method involves considering a third line segment in the plane and in this special case all 3 Strecken form a right triangle.
However he does not discuss the right triangle here, instead he says " likewise" the line segments product as this arithmetic product in the line segment.
This is the unclear part that I struggle on.

Ok, I think that I have been misunderstanding the verb darstellen in this passage! I was reading or seeing only Stellen.

It seems clearer that Hermann was comparing his ordinary line segment product with a projection of that product into one of the line segments.

That is if ab is the product of two Strecken then abcosø is the projection of that product into one of the Strecken! Apparently the representation of the shadow product is similar to the product of the ordinary line segments.

I still do not get this sentence structure here or this comparison. If I have got it wrong I will rewrite sections on trig line segments which I derived from this section.

I went back to basics.
I recognised that I had a gullibility inbuilt by almost mythical accounts of Grassmann and Clifford algebras.mbut my research showed me that Grassmnn was a man or child of his times. He was well read and educated himself through life. His great Förderung was not a new mathematics but rather a new approach to leaning memorising and manipulating mathematical ideas..

Using his approach did not mean you could skip learning mathematics, but you could learn it quicker, memorise it more securely and apply it more efficiently. In addition extended use of his system developed a creative intuition about how we interacted most effectively with space and formed formal mathmatical models.

But it's power lay in his childhood observations, which he developed refined, modified and extensively reworked throughout his lifetime. The important words are Zusammnhängend and Bearbeitung. Hermann melded, reedited, redacted, reworked everything he learned guided by one simple set of ideas. His ideas were based on his fathers presentations of " logical" geometry constructed by human action and thought, but his observation was that of a child, he was enthral led by the way the Notation behaved!

I have described these observations elsewhere, but the fundamental concept that drove him was the way the notation behaved for the geometric products. The geometric products were the areas of geometric forms.

When I first heard of exterior and interior products I immediately reflected on interior and exterior angles and their use in area formulae. I was trained in traditional mathematics, but knew nothing but a bunch of tricks and formulae. It was a disconnected mess. Consequently I did not know that much of what I was taught at elementary level was in fact based on the ideas of the Grassmanns.

The parts I have struggled with in translation have been those elementary observations and definitions of area of form! Thus when Hermann says that the product of a parallelogram is the same as the product of a rectangle I was confused, and thought he meant something else! I mean everybody except a child knows that!

I could not accept or expect that a book on higher or deeper level Maths would be looking at areas! Well yes he was and is!

The area formulae are all arithmetic products. They are also geometric products. What is the difference? The difference is what subject area they are explained in! In other words they are the same thing, but in Gometry it is required that the formulae are justified by geometric proofs. These proofs can be a bit complicated to explain.

However using Hermanns notation and system these proofs can be given quickly, providing you apply his main idea which is that a line segment should be considered as a ray segmnt( in modern geometrical terms). In addition the sign of the product must reflect the underlying ray geometry. Thus if a ray is defined, you cannot just swivel it round without paying attention to its sign! But the sign only changes every radians of rotation.

In Addition he noted a relationship in the rays of a triangle. This he captured in a fundmental congruence. Unfortunately he used = signs to represent a congruence relation. However this " bad" geometry has so many resonances with Arithmetic that it has bern hard to alter without losing he creative interplay.

For example Möbius used some similar ideas, but by being strict he lost some of the immediacy of his method and the creative flexibility. He certainly lost the connection to the geometric products of points!

I think Grassmann in this section is saying that dropping a vertical to a base in a parallelogram gives you the arithmetic formula for area of a parallelogram. This formula is still in the form of his general concept: product 2 line segments  to construct a parallelogram.

However this statement requires the following section to be clarified. But I have retranslated this section above to be clear. As it stands it has no relation to cosø or sinø, yet . The basic formula is base x height!

Each of these line segments or ray segments can be viewed as pairing with others to construct parallelograms. Using that idea it is easy to show that parallelograms are the sum of products that are rectangles!

This does not fall out from the fundamental product sum equation, but it almost does! The mathematician/ geometer has to guide the interpretation of the symbols. Once that role or uty is accepted, Grassmanns notation makes" proofs" less onerous and quicker.
In ab is the product of two line segments and I can equate that to cd + ed then I have formulaically or algebraically shown that a parallelogram is equateable to a rectangle.

The use of the terms = or congruent are problematic, but we know what we are demonstrating: how a form can be transformed into another, preserving certain invariant properties like area!

Ok, so what I have deciphered here is simply that the two products are the product of the parllelogramme sides for one and the product of the vertical height and the base or the other.
The former is called the outer product, and the latter the inner product.

The former is called the outer product because the irections of the line segments or rays spread apart to form the vertex angles . The latter is called the inner product because those sae line segments that form the rays of the sides of the pRLlelogram have to be approaching each other rotationally to enable a perpendicular to be dropped to the base.

Both these conditions must be met for the products to have a value. That is they can not be evaluated if one of these conditions fails. In this case either the outer product is null or the inner product is null . In the case where the outer product is null, the inner product is alo null or not defined. In the case where the inner product is null or not defined but the outer product is not null the line segments will be perpendicular. One might say the inner product and outer product become idntical, and can be evaluated as the arithmetic product.

The arithmetic product is the area of the rectangle that is equateable to the parallelogram. This is only 0 when the line segments are in the same direction or collineal or parallel. We could also say it is 0 when they are contra to one another.

If I am right in interpreting this section it means that the inner product is
Base x Height= line segment x line segment x sin of angle between.

Also this does not affect the designation of types of line segments I have posited in an earlier thread.

What about the dot product? That would appear to be the quotient operator in the following section.

If the outer product and the inner product evaluate to the same arithmetic quantity why remark on the differences?
The geometric value of noting these differences gives us a flexibility of approach in describing space and its behaviours. But how Grassmann uses these product patterns or constraints is at the heart of his method of analyis and synthesis .

If I have interpreted it correctly then this paper on Quaternions will demonstrate that or correct me, because it is all about these 2 product patterns conjoined to form an average product pattern.

I have now quickly surveyed the 1844 contents, addendums and combined index. The inner product as an important part of Hermanns system does not appear. The outer product and various other products and reduction combinations are the main point of the system.

The Abschatten product( the shadow cast product or cosine projection) the eingewandtes product( one relation product) and mixed products are all described in detail as types of outer products..

It is also clear that in developing his system the vertical projection and the intimate connection to the right triangle and thus through Thales with the circular disc and plane, and ultimately to the sphere and vortex was crucial to evaluation or metrical methods.  This means that the outer product could be as " abstract" or analogous as the mind could make it, but there was always a way to calculate a " numerical" answer, or really a count. What is this count?

The importance of counting is not the naming. The importance is in the ordering of space. The ordering of space is our most complex behaviour, from which we derive and induce and deduce many things.  The count became a check sum. This idea of a calculation checksum was taken up by the computer programming field in designing the format of a programme main function. In a main function a behaviour or process is laid out in terms of aggregation operations, comparison operations memory storage and movement operations, register shifts etc. the way to check this was being performed correctly was by checking bit sums.

The system evaluates in several ways. The main way is as a geometrical pattern in an orthogonal grid. This is essentially what the inner product is: an orthogonal evaluation of a general parallelogram.  The other evaluation is the calculation or count result.a final evaluation of importance is the null evaluations! These evaluations give orientation information.

The question is; why use the inner product in describing Quaternions, when it is rarely used in the 1844 version? Does it have a more significant role in the 1862 version.

Interestingly, the addendums to the 1844 version contain additional insights Hermann had just before he died. But of interest to me is addendum III , because he summarises all the main results there and gives some exercises with answers!

There is also a guide to where the artwork or figures he added in the pullout at the back should be directed!.

Originally I started on the 1862 version, but was frustrated by Hermanns constant reference back to the 1844 version. The new imprint (1877) of the 1844 version is indeed a good collection to research from.however if you just want to get the gist and use the formulae the 1862 version would be a better start.

For me, the research has proved invaluable in pointing out several errors introduced by later developers, or if you like several modifications they have made. I use the word error advisedly particularly about Gibbs version. Gibbs admitted that he really did not have a clue what Hermann was talking about in terms of his MULTI algebras. Thus the Gibbs version that underpins modern vector analysis may contain significant flaws.

The geometrical Algebra and Geometrical Cakculus approach is more consistent with the Grassmann tradition, but I would not claim that it ihas been clearly presented or taught until very recently. I am more confident now that Hermanns ideas can be quickly distinguished from those which have been added or modified by other later developers

As Hermann himself poited out, this is a work in progress , and refinements and modifications are to be expected, but it is comforting to know that we are not having the wool pulled over our eyes by Mathmagicins, and this is a genuine system that reworks everything!

So I have given a lot of thought th what " die Beziehung  zur Addition " might refer to and why it is the multiplikative Beziehung that has to apply.
Today mathematicians talk about the rules or. axioms of a group and thy specify the " operation" that essentially defines the group in terms of how combinations or associations are defined. If a group has 2 operations defined on it then it is on the way to being classed as a ring.

The interaction of those 2 operations is their Beziehung or relationship. When I was being taught arithmetic with natural numbers I was taught an " operator" precedence or a convention of operation order .BODMAS. This Mnemonic describes the relationship between all the operations in a complex calculation. In this scheme Multiplication is done before Addition , and that before Subtraction, unless a Bracket intervenes.

This is the multiplicative relationship .

How it relates to addition requires us to bracket addition, which essentially instructs us to reverse the operator precedence. In another sense it means that the final act is multiplicative. However we also perform an operation called expanding the bracket. This is called distribution of multiplication over addition.. What that means is we multiply the parts being added first and thn add the products.


This whole set of conventions is what I think this phrase is referring to .
Grassmann has a specific model
AB + BC = AC .

This is a sum but he came to regard it as a sum of products. In this case it is a product of points.
(A +C)B = AC

This simple observation reveals an analogous  form at all levels . Thus for line segment a,b,c,d where
ac is a product and  c =b+d
So ac= ab + ad

This law is crucial to constructing any algebraic ring , and for evaluating line segment products, and essentially it allows transformations to be equated under strict constraints.

There is another distributiyity rule and that is multiplication over multiplication. Strictly speaking it is multiplication over associativity. Buried within this rule is a property called commutativity. Because of its algebraic derivation commutativity does not reflect Grassmanns geometrically derived vertauschen und umkehren concepts, which are purely cyclical. This is a point I cannot emphasise enough.

Hamilton and Boole are often cited as chief instruments in the liberation of Algebra from Arithmetic. However Grassmann took the opposing direction. He sought to establish the derivation of arithmetic from general and sound geometrical principles and relationships. Thus he starts with ordering in space , and from this ordering he can develop sequencing rules that are countable. Within that framework he can develop sequences of points, line segments, plane segments etc. then the inter relationships between these elements in each sequenc can be linked to " higher" ordered sequences. Thus a sequence of 3 points can be linked to a triangle form in a plane or, more importantly a circle form in a plane. A sequence of 4 points can be linked similarly to a spherical or rectilineal form in 3d space, and so on.

From these relationships and constraints the principles of the geometries of these forms can then be derived.

In this way Hermann constructs a mathematic that fully accords with our conscious interaction with space, and so becomes an apt model by which we might describe spatial forms analogously.

There are some interesting consequences of the point product concept, but these refer to the Barycentric Calculus. Because Möbius did not include a point product in his calculus he was overtaken by Hermanns ideas in the Lineal Algebra.

AB + BC = ( A + C)B= 2PB = AC

For this to work the triangle midpoint theorem has to be applied to determine P. PB then becomes a line segment parallel and in the direction of AC . The scalar 2 which weights the point P is applied to PB  only if coincident points are acknowledged. This means that there are 2 collineal line segments ending on B. if both were summed by projecting one by the other we would get a line segment parallel to AC with the length of AC .

Returning to the single line segment case A+C=2P means the two points sum to a point which is 2 coincident points P. this point is a heavy point! We cannot locate it unles we can show AC = 2AP

Essentially this is what Grassmann defines in his point Algebra. We can locate it by bisecting the product AC and that can be done by using a pair of compasses

I thought it best to keep this reference handy to this thread.
http://books.google.co.uk/books?id=lSoJ2tJKfIEC&pg=PA250&lpg=PA250&dq=the+moment+of+a+rectangle+newton&source=bl&ots=PxkxmARNIi&sig=OfWOgOTdKZlDsV6NP7LusAehiJ0&hl=en&sa=X&ei=FInMU8qGBun07AbX4oHYDQ&ved=0CCgQ6AEwBA#v=onepage&q=the%20moment%20of%20a%20rectangle%20newton&f=false

The concept of a product being the sum of products has far reaching applications, but the essential geometry of it as rectangles and parallelograms I doubt if but a few know. Certainly Newton and latterly Grassman grasped it . The uses to which the put geometrical invariants , identities etc are perhaps best described in the Work of. Norman Eildberger as he researches these geometrical thinkers in depth.

For my purposes here it is the rectangular product which is relevant. The argument I have set that in fact this is the " inhere" product has many ramifications, not the least being that the inner product is the metric evaluation principle for the general outer produc ( sic, if the outer product is ab then this is evaluated by the inner product ABsinø, or indeed any other formulation that gives the area of a rectangle pertinent to the parallelogram form)

The use of the rectangle by Newton shows liberality of thought in modelling terms. The proposition in book 2 ( proposition 8) on which his argument or case is based needs first rejigging for the general rectangle, which is not too difficult to derive. Having done this the diagram must needs be seen as momentarily static in a flux inorder to derive Newtons flux moment for a rectangle.

The relevance here is that it now becomes an inner product in Grassmanns scheme whereby he may evaluate the flux moment of a general parallelogram.

In any case the method as defined by Newton carries over exactly into his handling of polynomial expressions. Indeed he requires that any such expression of dynamic quantities be in that form.  You will notice that Normans treatment of differential calculus , ascribed to LaGrange follows this method precisely. This should be of little surprise as LaGrange was the chief promoter of Newtonian ideas in France and Europe.

I find myself still wondering about the inner product, but I have now determined that this is due to a subconscious interaction with the English concept( geometrical) " inner". Grassmanns explanation for the worde " innere" is Annäherung. Thus this product should be expressed as the Closing in product and the exterior product as the Spreading out product.

The Spreading out product is defined in terms of the bounding line segments for the parallelogram form. The Closing in product is defined in terms of the dropped perpendicular from one of the bounding line segments onto the other  AND that other. The constraints or rules of cyclic interchange differ because the closing in product is designed to evaluate the Soreading out product.
It is clear that sometimes the perpendicular will fall beyond the line segment. However this is catered for by Extending the line segment( ie using a multiple of the line segment)

If the bounding line segments cyclically interchange then the sign of the product " switches" also. This means that the Closing in product also has a sign change, but interchanging the perpendicular and the appropriate line segment cyclically is not allowed to change the sign. That is it is a design constraint on the system of synthesis and evaluation that the closing in product does not change the sign on cyclic interchange.

The parallelogram presents 2 "rules" in one form. Summation of line segments is defined by AB + BC = AC
And product of line segments is defined by AB.AD= ABCD on cyclic interchange AD becomes –æAB because it now points in the negative direction to the positive direction established in AB initially. This is the principal orientation and the principal direction in that orientation was set by AB similarly AB becomes šBC because the principal direction was set in BC// to AD in that relative orientation. Instead Grassmann writes AD.AB = –AB .AD to denote these rules.

The evaluation of the Spreading out product fails when the bounding line segments are parallel. In that case that product should be a null parallelogram. However in that case a line can be constructed by projections However the Closing in product is arithmetically 0, so one can say the Spreading out product takes the value of the closing in product.

At the rectangle form the spreading out product is evaluated by the line segments. The Closing in product is not defined as no line segment can drop a line segment vertically onto the other, at least it is indistinguishable from the bounding line segments. Here the Closing product should also be null, but it can take the value of the spreading out product.

There are other products created by this perpendicular projection. The Shadow product for example is 0 at the same moment as the Cloing in product becomes null or void. In fact the Shaddow product is often confused with the closing in product, of which it is a part

I'll toss in some rambling.  I've never read Grassmann and only have an overview-like understanding of Clifford so it might even be pointless rambling.  Start with complex numbers: you have a scalar plus a (2D) bivector which acts as a pseudo-scalar and requires one component.  Extend to Quaternions:  you still have a scalar plus a bivector but it's now a 3D bivector so it acts a pseudo-vector and requires 3 components.  This is the source of all historical confusion and the why the 3D complex number equivalent requires 4 values instead the expected 3.  Gibbs and Heavyside jump in an and are only interested in the bivector part of the quaternion product.  Given two bivectors A and B then their product (in quaternion-speak) is .  The minus sign of the dot product portion seemed pointless so they drop it and change the signature of that of (what's now) a vector.  The source of the minus sign is that the product effectively adds angular information and they were interested in relative (angle subtraction).  So they really wanted and not .

Since there are more than one inner/outer product pairs definable for Quaternions, which are you referring to (I probably could have deduced by context but I'm too lazy).

As I suspected , Hermann and Robert develop the closing in product in the 1862 Redaction and completion of the 1844 work, if it could ever be called complete.

Perusing the Index to necessary or needed " art expressions" and their full explanations I notice that Strecke now makes a late entrance into the system, replaced by Stüfe and Grôße as fundamental building blocks in the system..

There are a number of products and multiplication processes of which the Aussere and the Innere are the main concepts . The mixed product also is introduced.

While this redaction undoubtedly saved the work from obscurity, it's main fault being its obvious incoherence in 1844, nevertheless it represents a substantial concession by Hermann to Robert who was a classically trained mathematics teacher and a rising star in the district.

It is perhaps well to remember how Hermann came to his Förderung via Strecken rather than through Number or Zahl. Hermann links directly to book2 of the Stoikeia.

Thanks Roquen. All contributions to the thread are welcome.

As I recall Hamilton was a chief advocate for a sub field of Mathematics called Algebra or symbolic logic and symbolic arithmetic. Algebra was developing through the higher arithmetics dealing with polynomials , power series inductive logic and much more. His impetus besides Aristotle, Newton and a relentless uncle was a friend who was humiliated for exploring the imaginary logarithms as they were then called. So in 1831 he published a paper on conjugate couples or the Science of Pure Time.

This landmark paper was a vehicle for the mathesis or mathematical doctrine of the imaginaries. Using a set of abstract moments in time he developed the ordinal arithmetic  of a single moment, then a couple or a beginning and ending moment. His main sounding board against whom he measured himself was LaGrange and he cleverly develops a mathematics supposedly not based on length! However despite its brilliance that proposition fails in that he has to resort to a notion he freely calls a step which is an analogy for a time step .

So, in some quite wonderful mathematical prise he develops arithmetic up to the level of higher arithmetics and logarithms and trigonometric ratios. He does so in a single lineal progression on the assumption that time progresses one way, but our minds conduct us in either direction in time! He uses a ellipsis to go from ordinal " numbers" to cardinal numbers, and similarly he uses a complex conjugate differential relationship, in which he slips the orthogonal axes without justification to develop the rules of the imaginary functions, which after he reduces to "numbers".

At the end of this tour de Force he was able to justify his friends assertions re imaginary logarithms, and along with Boole to raise the status of Algebra as a subject. He thus managed to annoy many establishment Mathematicians who went gunning for him. When he eventually revealed his concept of the Quaternions he was riding high socially and internationally, but Kelvin and Lewis Carrol waged a campaign against his" nonsense" mathematics.
Grassmann on the other hand was struggling to find time to put his ideas and insights into a coherent whole! Eventually in 1844 he published what he had in the midst of great doubt and indecision about whether he had done a good enough job. It was not good enough and was rejected by the mathematical and philosophical establishment in Prussia. However it gained a small and influential Audience internationally and sparked off a revolutionary subculture with surprising consequences.

By1862 he had refined the ideas and presentation enough to gain growing respect in Prussia, but puzzlement in Britain and America. However Bill Clifford got it, as did Hamilton years before him. Gibbs however did not. He could not understand the multiple Algebras but made notes any way, which combined Hamilton and Grassmann ideas and notations, but avoided the imaginaries.

As a Protogé of Kelvin, his notes became a chief weapon in diminishing the influence and acceptance of Hamoltons Quatsrnion vector concept in American universities. Kelvin also forced Maxwell to abandon his acceptance and promotion of Quaternions. Thus Hamilton was marginalised by his peers for daring to go against the establishment.

This historic battle has lead to many confusions at the basis of the modern notation you are now using. It is these fundamental confusions I am investigating so carefully.

It never ceases to amaze me how naive my assumptions are initially, but that is a good thing.

When looking into fundmental issues it is best to start as a little child. So for example I recall Gauss publishing his ruminations on the imaginary quantities in 1830 in haste, after finding a forgotten 1817 paper by Wessel in which he set out his theory of Directed Numbers. At the time Cauchy, Argand and others had been working on applying these strange adjugate quantities in France, but everyone was edgy about their metaphysical credibility , and Gauss was no exception.

Soaring above these workers were the ideas of Newton, Sir Roger Cotes and DeMoivre and of course ! I will add Euler to this list of Sublimes because they fully worked out the major conceptual base of the mathesis of the imaginaries.

Historically then we have a divided corpus of philosophers and practitioners. The sublimes saw what had to obtain, but try as they might they knew their contemporaries would find it hard to understand. Then the followers on like Gauss and LaGrange attempted to grasp the " reality" of these concepts rather than the sublimity or unreality of them. That these were formal constructs of the mind was not sufficient!

Eventually Wessel showed a solid , grounded practical use of them that Gauss could understand and he then published his ruminations, and his school claimed preeminence over French concepts, and indeed another claim. They even tried to browbeat Hamilton into conceding Gauss as prior to his 4 component Algebra, but Hamilton was not bullied, and his claim upheld..

The mathematical societies were divided. In Europe there was a willingness to explore these imaginary quantities, but in. Britain and America it was considered crazy Talk! In fact the negative numbers were still an issue at elementary levels! still !!

In 1844 Grassmann barely had time to pick his nose let alone research Gauss and Hamiltons work. However Robert did have time and opportunity to research thses things, and By 1862 Gauss concepts, Rienanns concepts are referred to drectly in the text, although preeminence is ascribed dubiously.

Some historical notes (from memory so don't read too much into it.  Probably some of this is in http://www.lrcphysics.com/storage/documents/Hamilton%20Rodrigues%20and%20Quaternion%20Scandle.pdf).  The situation is pretty understandable.  Prior to Hamilton and Grassmann there were only integers, reals and complex number for algebras.  All of which behaved in a similar fashion.  Then all of a sudden there are constructs that behaved completely differently and had unexpected structure and the "why" was missing.  Hamilton's notation was pretty awful and then Gibbs and Heaviside came along with vectors which had clean notation (taken from Clifford).  As an aside I just re-read Coxeter's 1946 "Quaternions and Reflections" which someone twittered an he's still using Hamilton's notation.

Now Gibbs (physicist) and Heaviside (engineer) totally focused on practical applications of vector analysis/calculus whereas Hamilton's camp mostly focused on the pure mathematics side.  Hamilton even resisted some practical application, as an example Cayley almost immediately formulated a generalized 3D rotation (p' = qpq^*) and for years Hamilton only talked about his broken version (I'd have to look the form up).  Soon thereafter linear algebra pops up and between vectors and LA applied mathematics had tools they could work with.  Applied mathematics always gets the lion's share of attention.

The flamewar started between the two camps and Hamilton's group came off looking like crackpots.  On Lord Kelvin it should be interesting to note that he was very close to Peter Tait and they wrote "Treatise on Natural Philosophy" together.  Tait was in the hotspot of the flamewar, so personal feelings probably have alot to do with Lord Kelvin's dislike of Quaternions.

It is clear, at least to me that the difference in the coherence of the 1862  work to that of the 1844 work represents a maturation of thought, collaboration and confidence in the format for presentation. In that regard I can see the hand of both Robert and Hermann.

As a successful publisher Robert had a clear editorial view.mit is his critique of the 1844 version which lead to Hermann bndoning his stated plans for the 2 volume versions and melding the ideas together into a new single volume presentation that was coherent. It was Robert who stated the 1844 version contained a difficulty which mathematicians could not overcome, and thus the work was more for the classically prepared scholar of Philosophy who had studied the Latin versions of philosophy! Ordinary mathematicians just were not going to get it, in addition to their dread of philosophy.

Robert also had another agenda of his own. He was highly critical of the non scientific basis of Philosophy in general, and his aim was to reveal that basis as flawed and to replace it with a scientific one that his father Justus, and others and he was developing. This was to be his master work and he needed Hermanns work to be well accepted so as to promote his own similar philosophical ideas.

Since Pythagoras it has been known that there are 2 descriptions of the world of conscious experience. One is as a collection of forms in space, the other is as a huge universal mosaic of monads. Plato and Socrates playfully added another important element to the experience: is it real or is it a shadow? This is the Platonic theory of Forms/ Ideas.

These distinctions have characterised philosophical debate and exposition ever since. For example Dedekind starts with the notion of ideas in developing his concept of Number in his famous Essay explaining the Dedekind cut. Hertz in his treatise on mechanics starts with a distinction between images and what is sensed, and how we must choose those experiences whose sensory image is invariant as our notion of reality.

Thus any simplistic interpretation of the word or action of geometry is completely misleading. I intuitively recognised this and coined the conceptual word Spaciometry to free my mind. Hermann considered with Robert calling the ideas they discussed Formenlehre, the study or theory of form. However he decided against that because it seemed not to capture the broad range of processes involved in mechanics and mathematics. Certainly geometry would not do as a title since even in Hermanns day Geometry had slipped into a position of low esteem, unjustifiably!  

They decided to study the behaviour of extensive magnitudes(Ausdehnung Größen), a philosophical concept perhaps highlighted by Descartes, but debated by Leubniz and Spinoza vociferously. At the time Kant had managed to broker a kind of peace between the warring philosophical schools who saw in his defense of pure reason a solution to their issues. Reason at the time meant belief in the Divine reason from God, not our scientific rationality or Logical positivism of today, which classes that Reason as irrationality!

The extensive magnitudes are experiences, but how do we determine real from shadow? Hermanns Hegelian dialectic solution was that we keep both distinctions and work them together and see what comes out as a product! This is the Hegelian dialectic process, which is a version of the Platonic dialectic process. However Hegel's dialectic is notably different and extremely logically rigorous in a geometrical / spaciometric way. It make full use of the Euclidean philosophy of the Stoikeia to describe our experiences whether real or Formal. Formal here means having an idea/ form in the platonic sense, while real represents a decision outside of the rules of the Socratic/Platonic game.

In one sense today we acknowledge the impact of more than one sense on our experience. The Socratic game is essentially visual. Thus we can define real as the other sensory modalities of a form experience. Hermann defines it in behavioural outcome terms. Something is real if you can't step over it! Similarly something is formal if no matter how you try you can't bury the concept or make it disappear!

We can go back to Apollonius to get a clear picture of formal Spaciometry. It is in his studies that multiple geometries are revealed each dependent on the form analysed. It was also his method of studying these geometrical forms that is "geo" on the ground "metry" measuring and counting and ordering , that lead to coordinate geometry . But the genius of Descartes in promoting this idea as a paramount method is all we ever hear! In fact it was Wallis who established the coordinate frame we know and love to hate!

From this coordinate geometry / Spaciometry Leibniz glimpsed the possibility of an intuitive algebraic language that would remove the reams of coordinate pairs required to describe a firm and return us to the rhetoric of points lines planes and circles. At a time of geopolitical turmoil no geometers were interested and Legendre seemed to have closed the book on geometrical education anyway.

Despite Gauss concerns about the state of geometrical research, foolishly attempting to prove the parallel postulate, it was not surprising that elementary grade school teachers should explore the difficulties and come up with the solutions. The Grassmanns were in that unique position at that unique time when the Humboldt reforms meant that academia was in turmoil in Prussia and all levels of the educational system were imperially commanded to prepare Prussia for the modern scientific age!

Spaciometry of extensive Quantities , that is bounded Magnitudes, is what resulted from the research of Grassmann and others. That Spaciometry included the study not only of extension but also rotation. It also included a third concept, that of Projection. These 3 form the fundamental transformations of Spaciometry. Klein introduced a term called translation which applies to a form, but it is fundamentally a construct of the three fundamentals: extension, rotation and projection.

Hermanns contribution is his insight into how to notate orientation , and which form to use as a fundamental basis for addition and multiplication analogues for line segment quantities, in fact any quantities of form! The answer was in Book 2 of the Stoikeia of Euclid.

Thanks Roquen.
You mentioned Hamiltons broken version. I would like to know a bit more of that if it is at all different to the generally unrecognised action of doubling the angle if Ritation common to all algebraic methods. As far as I know Rodrugues is the only version that recognised the importance of half angles in 3d rotation in the 1840's paper that went completely ignored.

However I have not got that far yet in my perusal of Grassmanns analysis. For me the Eulerian definition of the semi circular radian measure as has this unfortunate consequence. Essentially it is a constant error and easily addressed.

Hamiltons Quaternions are a brute force version, and there are many quaternion and quaternion 8 groups which are commutative. I devised the Newtonian triples after researching the quaternion 8 groups, and they are commutative and not a quaternion group. They also model 3d rotation but with a twist. In the cubic they seem to give a spaghetti double form . I have not investigated them further as of yet.

Complex numbers were and are ever a puzzle. Complex magnitudes how ever have a vibrant Geometrucal analogue, and this is one of the clear statements of Hermann. Later Robert ascribes this insight to Gauss, which is in my mind dubious to say the least. His combinatorial form is a specific example of Hermanns more general philosophical description, which admittedly is incoherently put together, but nevertheless contains brilliantly clear insights.

Roquen thanks for the link which has given me more documented information to correct my statements of historical developments. In particular the later Hamilton who was a troubled man I knew, but his public personae is graphically drawn in the link although just a sketch.

The gloss on Gibbs and Heavyside is a different matter. Much more has to be said about how vectors became dominant through the work of Gibbs and Heaviside as they reworked Grassmanns ideas in relation to Hertz , LaGrange, Ampère and Clifford and Hamilton.

The Stoikeia of the Euclid school is an introductory Pythagorean philodophy course at undergraduate( below Mathematikos) level.  Over the years since its first draft it has been used in the Euclidean Platonic Academy by many great teachers but apparently Apollonius is the main critical redactor of the course that we have documentary evidence of. Only a part of his philosophical treatises have come down to us, but it seems clear that his work on Conics substantially improves on that of Euclids own.

Conics would be an advanced course perhaps requiring the qualification of Mathematikos, I am not sure. There are several other treatises of the Euclidean Platonic Academy that may also have been required to obtain that qualification including Optics and Data, to specify but two . It would appear that Apollonius felt that students or discoursers were ill equipped to properly understand Conics because the philosophy of the circle and sphere was so obscure in the Stoikeia. His recommendations infuse a greater emphasis on the properties of the sphere and circle into the Stoikeia and probably our extant versions contain these redactions of his.

In Book 2 of the Stoikeia we are introduced to the Gnomon. Without elaboration the gnomon is approached in terms of quadrilateral rectilinear forms which are constructed by rotating line segments.  Thus the gnomon consist of the rectilinear form and a necessary curved form that is associated to the rectilinear one. I called this the curved Gnomon. In fact it is usually called a sector, and all rectilinear quadrilaterals have an associated sector used to construct them.  As a consequence there is a parallel development of metrication( counting multiple forms) between rectangles and sectors, and many basic formulae have the same format.

Given a line segment AC cut internally at a point B the rectangle AB BC is constructed by rotating one of the internal segments by a quarter turn. As that segment turns it scrapes out a quarter disc . That shape and the segment not turned forms a curved gnomon. The rectangle half perimeter could be considered as a limit Gnomon, but in practice the gnomon consists in the thick form that results from rotating a carpenters ruler.

The rectangular gnomon defines proportional relationships between all parallrogramic forms, but the basis of this is the proportional relationships between the circles that share the diameter of an enclosing circle that is tangential to them. Kissing circles if you will.  Thus the gnomon proportions are stated in terms of the diameter of the parallelogram, the longest diagonal.

These proportions hold for the shortest diagonal, but then the circles do not enclose the rectilinear form also.

In passing I must note that even the notion of rectilinear is defined by dual ( iso) points obtained by intersecting arcs . I frequently rehearse how this is done from 2 arbitrary points in space whose displacement remains fixed by our decree or relativistic observation.

The gnomon resurfaces in the concepts of Grassmann as the parallelogram product. But it is the summation form that is the real innovation. The summation form is based on the translation by a lineal projection of a parallelogram. This  translation forms the basis of tessellation of a space. It requires parallel line translations , or extensions.. We call these parallel projections today.

Grassmann did not recognise this until he was working on the geometric products in the plane. In the one form he noticed both his concept of a product and his concept of a summation. Later it is noticeable how he mixes this notion of a construction product with the notion of arithmetic multiplication. This is not an obscuration, but a revelation. However the concept of numbers does obscure this process, as currently taught.

The curved gnomon is clearly not presented in these early Grassmann notes. In fact the curved gnomon is a development that Euler promotes as his radian measure. I could go back to Cotes, but the clearest presentation , and systematic development is given by Euler.

The consequence of the curved gnomon for Euler was the exponential functional relationship with the imaginary quantities. Both Grassmann and Cotes Realised that this was geometrically analogous to the arc. This related the arc to the swing of a line segment . The swing of a line segment carves out a sector. That sector is the curved gnomon.  However Grassmann encapsulates this in a rhomboidal form, and so brings it right back to a rectilineal parallelogram!

The curved Gnomon is within the Lineal Algebra, which in itself demonstrates that the formulae will be analogous. However, the curved gnomon is a more advanced philosophy as Apollonius demonstrated. The nearest we get to it today is in Spherical geometry and spherical trigonometry. Norman Wildbergers universal hyperbolic geometry is the best course I can recommend in this area.

It will be noted that notions of perpendicularity has to be generalised to make consistent headway. It is this important Pythagorean principal that is key to the curved gnomon, and the advanced conceptual difficulties encountered .

Despite Eulers clear exposition,the full understanding of the curved gnomon lies in the Newton, DeMoivre, Cotes work and theorems on the roots of unity, and .  Euler, Hamilton, Gauss and Grassmann in short order have to grapple with these concepts so clearly delineated by Newton Cotes and DeMoivre. Grassman however has a notational technology that cuts through the confusion of concepts to help express these basic notions of trigonometry more directly and simply than any other.

Frabjous joy! Kaloo Kalay!

Scanning through the 1862 version to find the definition of the closing in product has been interesting. There is little doubt that the outer product contains within it the products or multiplications of the roots of unity. This presentation of it may perhaps be unfamiliar to many, but no wonder, few today study the intriguing and mesmerising theorems of Cotes and DeMoivre for the roots of unity. They were at one time of great mathematical interest and tHe basis of many new number designates. I can think of the elliptical numbers as an example, besides the general complex numbers.

The research and work of Robert is evident in this treatment, which shows an advanced knowledge of higher mathematics, something Hermann was hard pressed to keep abreast of during the writing of his first version. Indeed Hermann admits that he did not even know of the Gaussian form of complex number notation in 1844, 13 years after Gauss had published on the subject!

However the reason for my joy is that the definition of the closing in product is as I have adduced from the translation. To say that it is ridiculously simple is to decry the complex use that Grassman puts it to, and the generalisation he exposes by it! However, it is ridiculously simple!

As I suspected the terminology " inner" is misleading which is why I banish it. The closing in product does one simple thing: it introduces distributive multiplication of bracketed forms into his algebras.

Given any arbitrary line segments emanating from a common point and spreading apart, there spreading apart product is the 2 line segments projected by each other so as to form a parallelogram.with the intersection of the projected parallel lines identifying a fourth point diagonally opposite to the point of emanation. The product is written simply by the labels for the bounding line segments put together in order of the principals of orientation and direction.

Thus if the principal orientation is horizontal and the principal direction is left to right, and the principal rotation is anticlockwise we must write for the line segments a and b  ab, providing a is horizontal and is drawn left to right . If we now write ba we imply b is horizontal and drawn left to right and a now has a relative orientation .

In this light it is clear that ba would not attract a negative sign without some other rule bring in place. This rule is cyclic interchange.. Cyclic interchange requires the line segments to circulate cyclically around the form. In a triangle cyclic interchange does not attract a sign change. However in a parallelogram cyclic interchange attracts a sign change that is reminiscent of so calle complex number multiplication!

Having established these rules for the spreading out product. Hermann established some simple rules for the sum of line segments.( historically this was his first set of rules). These rules are best expressed in poit form

AB + BC = AC .
In the case of the parallelogram product this must become AB = a AD = b AD projected parallel by AB = BC =b , so AB  + BC = AC  can be written as a + b = c, where AC is now labeled as c givin it relative orientation, and principal direction.

Well this has several notational difficulties, especially the sloppy use of= sign, and the visually clear but algebraically obscure identification of AD as abelled the same as BC !these difficulties cannot be obscured or finessed away .

The identification of the "equality" of parallel line segments has to be properly understood not in terms of " equality" but in terms of duality or congruence.

Duality is a fundamental Pythagorean notion( isos) and it is usually translated as equal , but in fact it does the notion a disservice to tie it to equality. In fact the phrase "liberty fraternity and equality" reveals how slippery the term equality is in everyday use!

As a consequence of this slipperiness we can determine where definitions have bern put in place to tempt to fix a meaning. By the same token we can appreciate where those definitions fail to capture our experience of " sameness" or congruence or identicallity!

In the work that Apollonius does on geometry, and the projective geometry of DesArgues and others duality becomes a defineable property. The best explanation of this concept I think is found in thr Universal hyperbolic geometry of Norman Wildberger.

So now we come to the closing in product! Up until this point the distribution of multiplication over addition has not appeared, but as soon as a perpendicular is dropped from one line segment to another the closing in product appears! It is called the closing in product because the line segments must be closing toward each other( or towards extensions of each other) for the perpendicular to be drawn. Once the perpendicular is drawn it segments the other line segment into 2 . Both these line segments can be producted with the "dropping" line segment translated to this intersection point. This gives the original product of the bounding line segments as the sum of 2 other products of the appropriate line segments. This is the Resilt of dropping a perpendicular and this result itself is the closing in product.

It soon becomes clear that there are many other results that sum to the " original" or spreading ot product, and in fact all of these results can be defined as a closing in product. The closing in product defines the distribution of multiplication over addition in group theoretic terms. The unfortunate word inner conducts the mind elsewhere and obscures this simple realisation.

The realisation of this fact frees the constraints of " perpendicularity" . Now the projection does not have to be vertically into the other, because the distribution rule does not require it. However, the useful rule that when the closing in product disappears the line segments are orthogonal also has to be modified. This rule does not in fact hold as popularly stated. It requires several conditions to be "true". The closing in product is intertwined with the spreading out product and both must be consulted to determine relative orientations. In particular, some have loosened the strict principal rules Grassmann and his peers worked by, making it relative to the observer but more confusing to the student.

Grundsetzen or ground rules are absolutely essential to any cooperative human effort, and the fundamental ground rules since Wallis have been the orthogonal ordinate coordinate system! However prior to that DesCartes introduced the fixed generalised coordinate system, often ascribed to LaGrange, but in fact originated by DesCartes. The essential rule was that all should agree which terms were fixed before the discussion or demonstration began. These principal terms and rules defined the conclusions. Time must be spent communicating with colleagues to ensure foundational consensus. This is why conventions are important.

Because many experience confusion because the conventions are misunderstood or not known; or they do not realise that the conventions are agreed rules which one is at liberty to modify, many become disenchanted with the sophistry of so called Mathematicians, and thus with mathematics. However, the fact is that much of what is taken for granted as truth is of the same nature as mathematical conventions! Thus a vigorous mathematician ought to be able to explore the system and rules that govern his determinations, and thus ought to be an ardent philosopher!

Admittedly it may at first be uncomfortable to exist in a thought world of no certainty, only definitional propositions, but if one pursues this experience one gains great freedom. On the other hand one has the constant practice of forming ties or bonds of common terminology with ones colleagues. If this aspect is not done then one may as well be as mad as the March Hare for all the good this formulating will do you!

The closing in product therefore in hermanns day and mind tied the general spreading out product to the conventional trigonometry of his day. By working in the spreading out product mode he could simplify many manipulations of spatial forms, only evaluating at the last few lines by reducing the formulations to the closing in product.
How he does this is naturally the subject of the Ausdehnungslehre!

How this process works for the description of Quaternions is the purpose of this paper. The first thing to note is that a mixed or median product process is required! Because we are told that multiplication is based on tables we cannot grasp what a mixed or median product is!

The notion of multiplication must first be removed, replaced by the notions of a geometric product process, founded upon factorisation as a division or quotienting process, and then we can begin to understand what we are doing in manipulating space! Numbers I am afraid have obscured this fundamental role of the Arithmoi!

I have noted how Grõße and Stüfe have replace Strecke as fundamental primitives in Hermanns method of Analysis and synthesis by the 1862 redaction. Größe as a fundamental experience of extension is self evidently a fundamental primitive. One must bound Größe to get quantity.

However it is Stüfe that attracts my attention. This of course means to me a rising step. So the concept is of advancing to different levels . This is appropriate to a concept of rank, so that at each step or stage we rise to a different rank. Thus we can put all quantities into ranks. For example the rank of a point, then a line, then a plane, then a solid and then ranks of crystals or more complex solids.

The system of monads or Einheiten( rather Arithmoi) used to measure or quantify and distinguish ranks naturally must be magnitudes of independent orientations in space.. This is why the line segment was such a fundamental quantity initially.

However the notion of experience of extension and experience of rank difference must philosophically be prior to specific instances of these concepts. In other words we must have a prior sense of rank to be able to identify rank and rank differences. This notion is as simple as experiencing a rising step.

While a magnitude( Größe) is experience of extension, we also have ability to experience and decree limitations. Thus this ability underpins the notion of a staged or limited rise, hence "step." or rank.

With these two philosophical notions of a priori cognisance in place we can define sums or combinations of products as products of quantities. Also we can understand these combinations as sequences of products which form or describe ranks of quantities.

Each product itself may simply be a term for a form. A sequence of forms may provide ordinal information only. It is the combination of a sequence of terms or forms , where the forms are clearly at different ranks or independent orientations or both that fully describes a rank!

Clearly when a rank goes above a solid into a crystal form, it is possible to reduce a rank to a solid one. Thus a crystal , being a 3 dimensional object may be fully described by a system of monads involving 3 independent steps. However it may be initially easier to describe it in a higher rank form and then to see if it can be reduced to the solid rank, and what precisely that entails!

In this case it is fair to describe the rank as defining the notion of dimensions in space. Such dimensions are simply convenient orientation.

 However once again popular ideas of dimensions have been so misconstrued that we fail to recognise the n imendional nature of the space in which we live!

There is one other important aspect of Hermanns method. Since it is quite general the definitions can apply to analogous quantities which at first glance might seem qualitative rather than quantitative. However any experience with an extensive quality can be quantised!  Newton discusses how this should be done, using the invariant product forms of Spaciometry to do so.

The use of product forms for measures underlies Normans concept of spread and quadrance. While at first thought it seems a funny way to be measuring, in fact it highlights how screwed up numbers have made our perception of what we are actually doing in space!

An interesting comparison in the word step shows how Hamilton used the notion to distinguish a stride length in a certain direction, while Hermann used it to determine rank!

From a quick skim (I don't have all of these paper anymore) I'm not finding the version of Hamilition's rotation.  The Altmann paper seems to refer to it however.  I should note that I read these around 20 years ago, so it's insured that my brain has reinvented some parts. I want to remember he was breaking the equation into part (somewhat like how a euler-angle formulation works) so he could use P'=RP like forms, which only work iff R and P are orthogonal.  However I did run across a paper "ON QUATERNIONS AND THE ROTATION
OF A SOLID BODY" from 1850 where he uses Cayley's form.

I've often wondered how Olinde Rodrigues's paper became ignored.  The reason here is that Cayley not only cites it in his paper...his demonstration that the quaternion equation is a general rotation is by showing that it's equivalent to Rodrigues.

Another player in this time period is Louis Poinsot.

Roquen, the link you posted briefly refers to Hamiltons " broken" first formulation, as well as the general formulation.  It is clear enough to me that quadrant rotation is broken by his formulae because he uses the the principal function labels i,j,k instead of a pair and a conjugate k for example.

However, many more constraints then become necessary to establish a division algebra, and the whole unbelievable process ( at the time) becomes unwieldy and artificial! Part of the mystique of Quaternions was their extension of ordinary higher arithmetics without disconcerting fuss and the attraction and comfort of Pythagoras like "squares sum". In addition Hamilton invented a labelling that indexed certain standard measures and made handling the expanded brackets seem easier.

However at this juncture it is Cayley who comes up with a tabulated format( matrix) that makes the expanding brackets process more accessible than the multiple summation products!

I am pretty sure that the sum notation was a welcome innovation when first introduced, but there are many rules to its use that simply get bypassed in today's education.  Moreover , when producing sums the notation though concise , and revealing at the term level gets in the way of intuitive manipulation. Cayley's table products restore an intuitive feel to this aspect of expanding brackets.

(* in the 1862 version the sum notation is developed extensively from the outset, but section or statement 42 and Statement 100 bear directly on the innere product rules. While a Cayley table makes this so much simpler to explain there are a lot of associative manipulations that are ignored. These manipulations and mental facilities make Grassmann seem very subtle to the modern reader. However it is simply that we are unfamiliar with many standard associative manipulations in the rules of distributive multiplication over summations. These are what Grassmann referred to as the multiplicative product to the addition!)

It is worth a study.


When I was introduced to Matrices I was not given the table concept. In fact I saw the relationship between a product table and a matrix only fleetingly, because I was introduced to coefficient " matrices " first.

When later the determinant, the transpose and the sign rules of a determinant were introduced, I was already lost! The ad hoc appearance and relationship to ordered pairs of coordinates, n-tuples, vectors and Tensors was not a winning presentation for me.

Later the ot product, cross product, vector space combinations in set notation( direct sum)  seemed verbiage for verbiage sake. So now I have finally grasped the dot product, due to Normans consistent presentation of it, I can see clearly how it arose and how it relates to expanding brackets, product of sums manipulations and the Closing in product, and eventually how it is underpinned by a non geometrical , non Arithmoi " multiplication" operation.

Everything that early 19th century mathematicians before Dedekind developed were based on the Arithmoi of Ruclids Stoikeia and the Logos Anlogos calculus of homogenous magnitudes. Books 2,5,7 as fundamental grounds. Suddenly this was all thrown into question as new nonEuclidean geometries were touted. Even Gauss was reeling under the massive failure of mathematical geometers to solve the parallel postulate problem, conveyed from east or Islamic scholars to the west by Legendre.

The point is, and do not be deceived, this was a centuries long mistake of the design and structure and purpose of the Stoikeia course! There is no parallel postulate! The demands of the course required students to be technically and mechanically competent. They needed to be able to draw straight lines of any length( and that is not easy!) circles of any radius or diameter , and to enure that if a line was not at the same angle to a transverse line as another,,both in the same flat plane, that those straight lines could be accurately drawn through their intersection no matter how far away that might be!

Finally it was required that the accept without question that all right angles are the same?( Thales theorem for angle corner sub tended by a diameter at the perimeter.

Parallel lines are a defined concept found in the first definitions. The fact that other lines and curves can be joined in a kind of curved triangle was well known and well explored in Conics and spherical geometry, and spherical trigonometry! Lobachewsky, Bolyai and Gauss "disvovered" no new geometry or Spaciometry, no non Euclidean geometry, because what they found was already known in spherical geometry.

It is just one of many embarrassing periods glossed over in mathematical history.

So Gauss and Riemann and others lost confidence in Legendres interpretation of the supposed foundations of geometry. The academicians had lost their way. In that regard Grassmann restored confidence in fundamental Euclidean philosophy by tarting at the very elementary stages and gradually constructing the whole of higher mathematics on a few consistent principles nd a wonderfully useful notational technology.

Becausenhebdid not shirk from hard work he went step by step through every aspect of geometrical development of products, including the use of the summation notation.

The dot product and the closing in product are Clifford's identification of essential stps in the process of expanding brackets . These stps are made graphically clearer by using Cayley product tables.

Using Cayley product tables a variety of products as us of products can be defined and identified. It is precisely these products that Grasmann uses in his manipulations of product sums , summation products and the establishment of propositions in his system.

The dot product defines how 2 rows or lists or columns are combined to form a coefficient-basis  sum which is at heart any general product sum. This process is used again and again, but each,reapplication requires the product sum to be changed into row or a list or a column first.

To be strict this has to be designated by another process symbol, or a complex symbol. Grassmannmthought it necessar to introduce the divided Bracket symbol to represent the closing in product. Within that Clifford introduced the dot product as the initial stage of a compiles product sum construction best described by a Cayley product table with summation rules.

In Quaternions the different products have a well defined Cayley product table position , but one has to recognisenthebdot product atnabdifferent rank orbstepmfirst.
Thebcoeffcientsband the monad system arrangedbinmabcayleyntable produce more products than the required product , which is formed by products along the leading diagonal summed. These other products can be used for other purposes, but they simply are ignored.

Then using this dot product turned into a row and producing it with another arbitrary dot product turned into a column, another Cayley product matrix can be formed, from which a dot product can be constructed. Clearly this second dot product is at a different level or step or stage in the process!

At this stage the other productsvare not ignored. They now are used to form the vector combination sum, the cross product vector combination sum, so that a quaternion product can be defined as a combination of distinguished products, each separated by a label marker, or a list format or a component format( row or column). The terminologymvectornis very loose here . Some may call these formats vectors or tensors. Few will draw your attention to the position in the quaternion Cayley table! These insights are buried away in Matrix theory!

While studying Grassmanns notation it is well to remember the importance of Cayley tables. They are fundamental general layout structures for developing a number of processes from reference frame synthesis to matrix and determinant algebras, to computer memory structures . But also remember Cayley tables are way of representing processes of complex construction and production going on in the computing minds of Hamilton and Grassmann,,Gauss, Cauchy and even Japanese and Chinese savants. They are as fundamental as layout on a sheet of paper!

Of course, the method of construction Grassmann was using is more general than the Quaternions. Thus the terminology of the Quaternions does obscure these more general processes. Whle we may define a dot product for any Cayley table, the vector combination and the cross product combination only appear in the quaternion level. For this reason the more fundamental Wedge product is preferred by Clifford Algebras.mthe wedge product is a recursive or iterative product . It turns out to be connected, via the closing in product with the evaluation of the spreading out product, and the adeterminant of a matrix where defined.

Norman insists on calling it the cross product for obvious reasons, but it is not to be confused with the vector cross product

While table formats are used to define the wedge product it must be notednthatnthesenare closing in product tables not spreading out product tables. The spreading out product is associative not distributive. Any distributive table associated to the spreading out product will,be a closing in product.

Any combination of spreading out products will ultimately be a closing in product of some higher ranking or higher stage spreading out product. The aspect of reducing rank or stage to a step which is fundamentally evalateable forms a big part of sprading out product manipulations.

The closing in product in 1844 was conceived in a dynmic geometrical construction. This involved dropping a perpendicular, or projecting  vertically one adjacent line segment into another. The right triangle so formed symbolises the introduction of a reference frame.

A monad system consists in iindepent collection or set or sequence of " independent" Metrons. Metrons are experiences of extensive space which are bounded, as an inherent part of the experience. Thus each Metron is a bounded unity or unit.. By these we count , and thus quantify , those extensive experiences.

A Metron as a bound experience may have a visual component we call it form, but not necessarily. Each Metron/ monad is " independent" if it cannot be counted by ( using) any other Metron. Thus all Metron/ monads in a monad system are unique or " protos" . This means the first of its kind or type, but it is also stated as " prime" or relatively prime.

We can thus construct a monad system from prime Arithmoi , "protos Arithmos" if we wish. If we try to use the number concept here we run into all kinds of confusion, but in fact we can use Modulo arithmetics, that is groups of " numbers" with a modulo rule inherent to model such monadic systems. They are also called finite groups placed in direct combination or cross combination with each other.

In the Stoikeia these monad /Metrons were any of the standard forms/ ideas drawn and studied in the plane or in pace. The modern numerical concept or Area is a complex reduction of all these flat forms into one standard form ( square) or similarly for volume the standard form is a cube. Thus in a direct geometrical analogy the concept of a closing in product is the concept of reducing all or any form to an equivalent standard monad/ Metron mosaic, or mesh .

We can usefully analogise the spreading out product as any of the many different rocky crystal shards we see in a gravel pit for example. Here is where it gets interesting. Using these crystalline shards many mosaic forms and scenes were embedded into the floors of houses and temples. Such complex mosaics helped to establish the notion of the Arithmoi as fundamental natural nd supernatural constructs. Thus the Pythagoreans studied these patterns intensively.

It is clear that certain rocks are not jagged crystalline forms but indeed smooth ounces forms or pebbles. These properties make very little difference to the Philoophy ofnthebarithmoi, but the modern concept of area runs into some difficulty when making measures of space using these different metrons. Such a problem was and is cosmetic, but it took the concept terms of Benoit Mandelbrot to shake a centuries old mistaken geometry, and re admit the notion of scale free recursive or iterative applications of a irregular form. The word fractal wasmsupposed to describe how rough a geometry was. This was to use not precise regular Metron self similar at all scales, but rather Metrons that were "almost self similar" at every scale. This last self similarity was to be measured by a curious counting method to give a ratio or numeral which was to convey how roug the fractal geometry was!

This is a burdensome notion! But mathematicians are like that. If some scale of quantitive measure of a qualitative " feel" could be agreed then no one would doubt the mathematical nature of these " geometries".  Fortunately artists took over the development of these ideas and created a graphic icon that now bears his name and shows the dynamic properties of spatial Metrons . The convoluted fractal dimension measure , still pursued by mathematicians is pragmatically replaced by aesthetic assessments.

It is perhaps an interesting historical study to track exactly how artists advanced the development and acceptance of computers more than any other Academic group!

Nevertheless a monadic system consists of combinations of these various Metrons including spherical forms.Grassmann called these combinations Größe or magnitudes. These magnitudes encapsulated partial fom and also spatial dynamics of form transformation. but they are still pretty mysterious and experimental models of our experience of form and dynamics in space! Because we have a reliable symbolic format, a terminology that appears to be trustworthy many have moved from,spatial transformation and manipulation of these magnitudes to algebraic shenanigans!

However, the painstaking research and development of this terminology and attendant rules of application and applicability which we rely on were substantially achieved through the part time efforts of Hermann Grassmann, who constructed lineal Algebra from the intense study and redaction of the woks of the geniuses of his day with an inisght he felt was only given to him! This was substantially true at the time, although Möbius came close , and Hamilton closer still.

St Vainant apparently came up with the same insights, but this was judged to be an act of plagiarism while translating into French Hermanns key ideas!

The closing in product is defined as the product of any arbitrary Metrons or monads that form part of a monadic system. The important point that has been generalised from the early vertical projection insight is that the projection introduced a reference frame into the description of an arbitrary extensive magnitude. The closing in product defines how we are to manipulate the components of that reference frame .mthebconstraint isbthatnwe must get the same result using the spreading out product and the " underlying" closing in product.

This is the concept of invariance: Keine Abweichung!

This is virtually guaranteed by the definition. The same product rule applied to the monads works through to,apply to the combination magnitudes. At least formally the notation means the same at the level of a monad product as it does at the level of a magnitude product. Thus the closing in product is scale free in this fractal sense , and this is why our methods produce these almost self similarities.

Am I saying that we somehow create these fractal self similarities? Yes. Any system which relies on our mathematical models solely will return selfnsimilarnresults at all scales.

That is why mathematical physics etc without empirical observation can not give me a " true" absolute picture of reality. It merely gives me a model, one of many possibilities. Any one who studies Grassmanns notations will recognise how it influence Paul Dirac and his presentation of Quntum Mechnics, for example . Where the application of identical notations differs is in the empirical interpretation tied to each and every jot and tittle of the notation!

The closing in product, as Norman distinguishes, introduces metrication into any general geometric or rather spaciometric description using a standard format. The SI unitsmfor example form a monadic/ metro system for metrication of our geometrical descriptions of forms dynamics and pressures and powers in our physical world. As such, how we tie everything back to these units is our closing in product method or system of methods.

Returning to the 1844 version to seek the closing in product reveals a different focus and exposition entirely. The Abschattung product and the eingewandtes  product are the main counter points to the spreading out product. In addition the spreading out product is used to represent the 4 main operations for addition division multiplication and subtraction.

The Abschattung product and the eingewandtee product I am now going to explore briefly. Despite his general outlook and clear effort to meld together his specific and general ideas, the fields from which he drew inspiration and applications for his ideas are overpoweringly evident in the layout. Newtonian kinematics, harmonic analysis, imaginary products and projective geometry are all mashed up together in this first volume !

It turns out that the eingewandtes product , a major theoretical plank of the first volume was a dud by the 1877 revision of the 1862 version. The product which I am going to call the special pleadings product or the postulated product, requires a number of constraints to be true or fulfilled in order to determine it safely from the spreading out product and the null result.

This kind of detailed analysis relies on the concept of a system of equations or expressions by which certain relationships of quantities of magnitudes are established. The image or idea is that the spreading out product successfully manages or manipulates such a complex system of equations so that you can multiply or divide them as systems . The full set of operations addition and subtraction included apply to these systems.

While this is now dealt with as Matrix algebra, it is in fact first exposited in this complete way by The Grassmanns ad Hermann in particular. Cayley get the credit for sweetening the notational form so that all those summation signs and operations become more intuitively manageable as a product form. However this form exists as labels in Grassmanns work in the spreading out product, but it is so much clearer to represent the summands by matrix notation, or rather Cayley table notation.

However, the analogy is severely constrained! It is these constraints that lead to the exploration of and naming of different products. The postulated product is such a distinction. You must remember that a postulate is a special pleading, not a proposition as some still find it hard to understand.  Thus this product of all the products demands the most attention!

The product requires so many constraints to be in place that I am not surprised and thankful that it failed to be a fruitful idea! However at the time Hermann went to great lengths to plead for its existence, demonstrating how it should work in analogy to the spreading out product.

However in a footnote made in 1877 Hermann quickly alerts the reader to the fact that it was not a fruitful development in the method but that its ideas we're absorbed back ino the general body of the method. Although I have not looked it up, the regressive product suffers the same fate by 1877, as Hermann seems to identify them as the same kind of special pleading product.

What this means is that by 1862 the Abschattung product had become the main counterpoint to the Spreading out product, and it is there that we might trace how the closing in product achieved it form as expansion of brackets, which brackets represent the magnitudes written as the sums of monadic/ Metron systems.

As I pointed out earlier the dot product plays a crucial role and the closing in product as column row Cayley table products simplifies the Process considerably.

<a•A|b•B> is the forerunner to the Dirac Bra and Ket system of notations, and the tensir notation of contra and covariant tensors( n tuples rather than 3d vectors) and the Einstein summation conventions! All of them trace right back to this seminal work in 1844 in which the spreading out and closing in products were first identified as geometric products and then instantiated as algebraic or symbolic arithmetic forms. The chief foundation for the closing in product in 1862 was the Abschattung product, but in 1862 it took on the general Einheiten product format, which always from the outset in 1844 allowed it to adopt the most general form.

It is perhaps helpful to review Normans videos on polynumbers and on matrix multiplication. However one thing to note is that the [|] or <|> notation represents a summand of the Cayley product tables, whereas matrix multiplication represents a Cayley product table of dot products! The Cayley product table does not actually have a specific product symbol but it is a fundamental combinatorial product table and deserves one I think .
[ # ] where [a#b] is the Cayley product table for list a and  b in row column format. This could then be used to describe Hermanns combinatorial monadic system or Einheiten products.

These products as distributive rules are defined for any lists, magnitudes as monadic systems , rows columns etc, but the dot product is only defined for one to one lists at present.

This charming video is advanced conceptually. I place it here because the concepts ultimately reflect Hermanns most general ideas. At this level the discussion is algebraic and general. The algebra is then given possible interpretations in a consistent or coherent manner by empirical data, observable phenomena.

I might add that the concept" time" though emotionally locked in in our culture, is in fact a circular dynamic Metron . The monads we use are periodic motions, metronomic : like a pendulum, a metronome or a regular orbit.

Google "jehovajah time" for a deeper discussion.

Crucially time,Tyme etc are simply concepts of change in position. Thus displacement is the fundamental notion of both our concepts of velocity and time. Change in displacement is inversely proportional to change in period!  We cannot subjectively distinguish these experiences without this inverse experience.

When I am busy time slows down. But when I am not busy time speeds up. Inversely my perception of time speeds up and in the second case it slows down! It slows down when we watch it , speeds up when we watch things in complex motions!
https://www.youtube.com/watch?v=oy47OQxUBvw

It always struck me as curious that I was asked to solve a set of equations by setting the equations to Shunya!

Shunya means everything so I guess you could say that means the equations can fit everything! The alternative is the equations fit nothing!

We really use the additive identity and the multiplicative identity to solve equations. Thus 0 Shunya and 1 monad the indian and the Greek notions of the alpha and the omega of everything. So when did -1 become part of the solution set? When negative directed number were introduced!

While this is not one of Normans most helpful presentations, and contains several confusions, nevertheless it illustrates how applying the Grassmann method requires careful consideration of constraints and interpretation.

The butterfly diagrams are only of interest in tht they show how geometric conservation of Arithmoi count underpins the motions of conservation for any extensive magnitude, especially those symbolised by line segments.

http://youtu.be/OAKZ7l3O92s

The clear and rigorous use of analogous thinking is the hall mark of Hermanns heuristic method . Because of this, symbols like line segments take on a powerful anchoring role, being, becoming and giving meaning to concepts and ideas that exist only in the expositors head.

Thanks for the Link to the Video: The Dark Side of Time!

Hermann

I hope it was thought provoking, Hermann.

I have had a long standing project to deconstruct the notion of time and replace it by something more appropriate. The link between motion and spatial experience is fundamental to my notion of order . The sequence in which my experiences of satial extension and cognisant focus is laid down in my memory of perception basically is my consciousness of " time" as a dance of processes. How to symbolically represent that is one thing I am gradually learning through hermanns work , redaction and philosophy in the Ausdehnungslehre.

Heuristically he builds it on the thoughts and meditations of some of the greatest geometrical, mechanical and philosophical thinkers of humanity.

Winces steady deconstruction of these concepts is interesting because he relies on symbolic processes to models aspects of our perception and constructs a notion of process which is essentially circular or fractal. These are his distinguishable elements from which he constructs 3 d time.

Normans example is interesting in this regard since spacetime graphs essentially record velocities as slopes, but mass appears as a line segment that does not move! By extrapolation in a 3 d system mass would appear as a volume that does not move, but whose volume projects sloping spaces in solace time . This extension or radiation of mass into space time is conceptually difficult as a construct, but it contains our notions of conservation of mass, energy and momentum, geometrically expressed.

How do we interpret that in our daily experience?

In the 1844 version the Abschattung product is discussed in the light of the notion of projection. But interestingly the larger discussion was about the grounds or geometry!

It is not until 1853 about 9 years after Möbius and Gauss had seen Hermanns text that Gauss instructs Riemann to make the Grounds of Geometry the subject of his Habilitation speech. The irony was not lost on the academics of the time, who later found the history of the 1844 text very compelling as a source document!

Be that as it may I am still in the process of reading this chapter to determine its contribution to the 1862 closing in product concept. In so doing I notice how several themes have been rearranged and given different emphasis between 1844 and 1862, as Robert and Hermann perfected the exposition.

jehovajah: I hope it was thought provoking

It was more then that, I have so many idears in my mind that I have difficulties to write them all down.
For example the problem of matter an antimatter in our universe. On the macro scale we see only matter.
Antimatter can only be produced in acceleraters like Cern or through radioactive decay.

An very old idear of mine is, that at the big bang the same amount of matter and antimatter is produced. The antimatter is travelling into the past and matter into the future.
This allows to produce two universes out of nothing. It is only a rough idear and it is beyond may capabilities in general relativity an mathematics to workout it out as basic for a scientific discussion.

From this point it is also possible to think of an oscilating process of matter moving into the future and antimatter into the past.
At the present we experience a stream of matter moving into the future. May be we have simultaniously a stream of antimatter moving into the past, forming the basic construction of spacetime as we experience it today.

I personaly dont like the following idears:

• The universe started as a singularity.
• Inflations of the universe.
• Dark matter
• Unsymetrie between matter and antimatter in the known univers.

Hermann
P.S This is only a rough sketch of some idears.

Hermann, I would ask of you 2 things: one is to locate gavin wicke YouTube site and absorb and enjoy it; the second is less pleasant and that is to read through the Ausdehnungslehre  1844 regarding the Abschattung and Projection product( pp114 to 128 ) §80 onwards.

The second request is really to guage whether to devote translation time to this chapter before translating the paper which is the subject of this thread.

It seems to me that this section in 1844 is of fundamental significance, in that it shapes the popular quasi mathmatical concepts of dimensions of our current generation.

I have had many epiphanies reading Grassmann principally because these ideas permeate so much of modern physics, algebra and modern geometry. I am dismayed at the paucity of my mathematical education, but encouraged by the breadth of it . Dismayed again at the lack of philosophical contextualisation in mathematical education and confounded by the befuddlement between the competing fields of mathematics.

It seems to me that the method and analysis that the Grasmanns constructed is of such revolutionary importance that the whole of mathematical education should be revised along its line from kindergarten to higher education.

I know that this is Nrmans goal but I really think that from the first time a child touches a brick or a cuisinaire rod or some Lego or kinet construction kit, or sits in the sand and plays, the fundamentals of spatial interaction should be drawn out in the Grassmann way! In addition the crucial practical importance of symbolic notation, and the representation by symbolic sums, summands, series and combinations , equations and systems of equations , matrices and matrix groups and ring algebras  OF the basic rectilineal spatial forms should form the curriculum of mathematics up to advanced level.

Then in university undergraduate courses the philosophical background should be discussed with a broader range of groups , rings and dynamics explored in particular the swinging radial theories , the circular and spherical trigonometrists and Grometresse and the universal hyperbolic Geometries.

Number should be reduced to what it is , a sequence of symbolic ordering names, and the processes of dividing and recombining, analysis and synthesis  comparing, contrasting and identifying should be liberated.

There is a greater field into which these skills and expertise may be released: that of computational science. By this means and the inherent analogous thinking process embedded in Grassmanns heuristic Analytical system greater and more practical computational models can be built which ultimately will advance the computational hardware to make use of the dynamic plasma fluids that underpin our current understandings of space,

From Astrology to geometry let us not get lost by inane terminology and lack of imaginative analogy. Let us not obscure the fractal patterns embedded within our symbols , the scle free. Almost similar identities which in applying as basic modelling clay at all scales has enabled us to synthesis a complex pragmatic model that works well enough to guide us in the dangerous but exiting utilisation of fundsmental plasma dynamics.

We can always do better, be better, redact better, modify better. Even this insight is encoded between the 2 volumes of the Ausdehnungslehre by Hemann Grassmann.

In that regard and context I cn now address the several books on the Ausdehnungslehre that were written by Robert Grassmnn and which, though not famous are viable evn today. I have purposely avoided even reading them to not spoil my apprehension of the seminal works by Hermann. They now come itno purview.

Thrn there are the irks and ideas, philosophy and researches of Justus Grasmann, the father.

Not all of this will be fruitful to know directly, but indirectly, as the social history, the academic and scientific and political background is pieced together a better appreciation of the source and inspiration of the Hermann Grassmann exposition of the Ausdehnunglehre concept can be formulated.

Returning to the Abschattung product, it is clear that a prior exposure to the spreading oUt product theory is necessary.

The concept of elementary magnitudes comes after these discussions in 1844 but is the first concept in the 1862 redaction. The reason is basically a question of style dictated by the intended audiences. In 1862 the style is that of Euclids Stoikeia in which definitions, demands( aitemai) ,propositions and demonstrations are given with a general summary or scholium discussing items of interest or note. This is very much thr style plan for the Principia of Newton.

However in 1844 the style was a bit confused and incoherent, as a style, in which a general philosophical discussion was put forward and from this a concluding statement or proposition was invoked. These propositions then became insights for further discussions leading to further concluding statements or propositions.

The difference is marked. The 1844 style encourages exploration and development. The 1862  style of Euclid Denies these impulses! The propositions appear out of nowhere, there is no motivating discussion, and the demonstrations are final in feel and intent. This is the immutable word of a god: a theo- rem!

 It was important therefore that Hermann stood outside of academia, knocking on the door. If this had not happened we probably would not have seen the 1844 version! It is the 1844 version that sparkles, but it is the 1862 version that clarifies.

Without a doubt it is a culture shock to find systems of equations of a general nature treated of as an everyday topic. I cannot remember being taught to view these great sequences and series as anything other than a specialist oddity! Thus to find series, rows, steps and stages ,systems of this generality humbles me.

It is clear that in his day the great philosophers and mathematicians regarded these as elementary structures or objects of thought. Nevertheless they had not seen the need or the opportunity to form a symbolic algebra for them!

When Hermann says that he did not think anyone else would ever get his insight ever again with regard to his concepts, he was right. Not even Leibniz who sensed some possibility, nor Euler who revelled in these long series summands, or Gauss or Cauchy who developed the algorithms for solving such systems seemed to see that a terminological Algebra could be developed.

These long series are hard to conceptualise. I guess that Academic rigour would laugh at analogous thinking if ever it was put on paper! It would have been torn to shreds by " logicians". The use of analogy was left to the field of philosophy and religion, and had no provenance in geometry or mathematics, so called!

Some of the fun things I remember looking at were finding formulae to sum series! What I was not told was the geometrical basis to these formulations, because they were too busy foisting the number concept off on us instead of leading us to the geometrical underpinnings!

The remarkable moment in this chapter or part in the discussion of the Abschattung product is when Hemnn derives an observation that allows him to form a higher or super series of Abschattung products, by the same or almost similar means! By now I recognise this relationship as a fractal generator relationship. But the consequence of this is he is able to relate previously unrelated series together in a higher or super series!

Of course you will have to read the details to become aware of the constraints, and it is these onstraints that make the generalisations practicable. Without this insight on the product level , one is left with isolated methods and solutions.

But, the dual geometrical interpretation is even more supriing. Again his developed geometrical insight allowed him to interpret these series as systems of independent line segments, the Rechtsystem or Grundsystem of coordinates!

In expressing this by analogy he uses the concept of orthogonal or perpendicular axes/ line segments. Anyone who has had petpendicular drilled into them takes it as a literal geometric truth. However it is clear that this is an analogical use of the concpt of perpendicular or orthogonal! In my previous research to coming upon Normans universal hyperbolic geometry, I had determined the probable meaning of orthogonal in this context as being " independent" .

In a 3 dimensional system how can you get a 4 th or higher independent point or line.?

In fact if orthogonal means perpendicular, means spatially at right Nigel's you can't. But if it means independent, then one can set up independent cells or regions of space. Thus the regions themselves are independent, and the reference frame applies only to that specific region!

2 examples will suffice I think. Firstly a line segment in one orientation is independent of any other line segment in a different orientation. Secondly a parallelopiped in one orientation is independent of any other in a different orientation.,

In the case of a cube the orientations are naturally assumed to be the same up to symmetry. But that is not correct. Any cube which is not relatively reoriented by /2  is independent of the normal coordinate system!

Hermann did not express this very well, but he did keep on banging on about n dimensions of space obviating the view that there are only 3! I think if you are familiar with the roots of unity you will understand how the sine function of multiple angles are independent of each other.

The independence is visible geometrically, but because of numbers the numerical difference was obscured. The values of the functions are visually and geometrically obviously different. What was hard to state was that they are" orthogonal", Hemanns language here helped an important analogy to be made: perpendicularity is not the general property we think it is. Independence is.

We now construct reference frames of a general nature understanding that the independence of the elements of those frames may transform our view or representation, and thus we cannot say that every subjective reference frame is identical. Bearing that in mind Hermann gives us a way to come to an agreement through the Rechtsystem as a standard or fundamental reference frame! This is the result of his exploration of the Abschattung or projection product. It liberated our metric / Metron system for subjective use, but gave us a way to standardise and compare.

In the past this standardised system would have been confused with an absolute god centred system clearly this system was constructed not by god but by Hetmann Grassmanns analytical method!

In the light of these conclusions, the Einheiten system is a clear derivation from the Abschattung product or the projection product, and the Einheiten themselves are interpretable as oriented line segments which have a roots of unity representation. The closing in product is thus intimately connected with the trigonometric circular functions, lineal combinations of them and the exponential form of the Cotes-DeMoivre-Euler theorems for Hyperbolic and ordinary trig functions.

When I called the second type of Strecken trig line segments, it was without this further insight within the text. With this insight and others I am confident that line segments ordinary and trig are fundamental to both curved and straight analytical methods. Thus the use of a line segment does not imply merely straight or rectilinear motions! This is crucial in understanding Newtons derivation of the centripetal centrifugal circular and tangential force and velocity systems.

I am reading in the Astrological Principles of Newton's Natural Philosophy at the moment and am struck at the rhetorical style of the presentation. Thus, without the symbols of an algebraic exposition, Newton lays bare in a few sentences astrological relationships by geometrical considerations. It is hardly intricate or rarely lengthy and once use to the style very succinct.

These are qualities that Grassmann claimed for his algebraic presentation of various results and well known theorems ,reworked in the style and method of the Ausdehnungslehre. I have always felt that like Newton Hermann was heavily influenced by the rhetoricsl style of the Stoikeia, but it is very difficult to say which version of it was extant in his time, because Aristotle and the Islamic scholars followed a very similar rhetorical style. As a consequence the appreciation of the purity of the Pythagorean heritage of the Stoikeia is lost in the Platonic academic style that spread around the Hellenistic areas and were absorbed into Islam.

In addition, Legendre prepared his own redaction of the Stoikeia as found in Italy by the engineer Bombelli and colleagues, and which apparently was a Greek version not subjected to the Islamic programme of retranslation into Arabic. Thus it was distinct and not in the tradition of Al Jibr as promoted by Al khwarzimi, yet it contained the same ideas apparently. Only close research revealed the distinctness of the tradition, and thus the internal coherence of the Platonic Academic versions as philosophical texts, as opposed to the Aristotelian style as a logic, grammar and rhetorical geometrical / mechanical treatise.

Thus Legendre felt able to rearrange the philosophical text into a more logical and mechanical format, suitable for engineers, but which destroyed the philosophical nature of the course, and led to European mathematicians denigrating the logical consistency of the whole text. Trained under Islamic scholars these scholars chose to follow Aristotles logic in analysing the Pythagorean teachings without ever realising that Aristotle never acquired the Status of Mathematikos!

Thus his disagreements with the Athens Academy and thus the Alexandrian academy under Euclid represent a rival variant of the Platonic and thus the Pythagorean traditions.

Bombelli fused the 2 streams in his little known work on Algebra in which he introduces the freedom to use symbols in place of Islamic rhetoric, based on the rhetorical traditions of the academies at the time. Though not the first to use symbols, his algebra was quietly influential and known to Descartes, though unacknowledged, and Known to Wallis who brought out his hugely influential book on Algebra. He is known to have tried to garner Newtons use of symbols in his reasonings, begging him to reveal his symbolic Algrbra. 

Newton " refused" preferring to use the clean analytical rhetoric of the Greek fathers. However, whether he based it on the Stoikeia of Euclid, or on his reading and admiration of Aristotle I cannot say right now. What is clear is that apart from the overall style of the presentation, which is clearly Euclidean, his analysis is distinctly Newtonian.

Thus Hermann had a rich choice of formats to choose from which was the major part of his problem with coherence in the 1844 version. By 1862 Robert and he had decided on the Euclidean format, but the rhetoric was largely replaced by symbolic formulae or enhanced by symbolic formulae. This style he garnered from the works of LaGrange and LaPlace, Euclid and others, but the intensity of this style in terms of formulaic exposition is a Grassmann innovation.

While the whole world seems now to think this is a mathematical exposition style to be copied, I find it is not genteel, as the wordy but less symbolic presentation of Newton. In addition, I feel Hermann hardly " saw" the symbols. These were like meditative markers, or notes on which to expand and speak rhetorically. Thus on paper little was written, but in ones head " bombs" of connections and insights should be going off; or what is unfortunately more likely ribbons of confusion tying you into knots!

It is therefore important to read the 1844 version in which the style is more rhetorical and explanatory as well as philosophical. This then schools you not to rush over the symbols like so much spaghetti and to enjoy the sauce in which they are meant to be covered. Only in this sense would I dare to venture that Hermann makes clearer the underlying heuristic algebra I am finding in Newtons and Galileos works. But in no other sense, for as it is currently presented these algebras represent a great obfuscation of simple and easy to grasp geometrical truths when presented in the rhetorical styles even of Newton!

Hallo Jehovajah,

I have now my usal problem of having not enough time to work through all the material (I know that you are looking for someone to discuss Grassmans Work). But I have read a lot of papers on Geometric Algebra and I know understand the importance of Hermann Grassmanns work.

I was always convinced that I have been thaught the wrong mathematics when studying physics. My obmissions have been first made clear when listening to the lessons of Norman Wildberger on YouTube.
I also have to go to the work of David Hestenes an his books to get a deeper understanding of physics. I think geometric algebra is the right language to describe physics.

Hermann

Thanks for the interest and contributions  to the thread.

I am toying with the idea of going back to the Einleitung of the 1844 version and translating that as part of this thread.

Clearly the Average Product is a process that involves the closing in product combined with the spreading out product. Geometrically it is a combination of an orthogonal form with a general parallelogram form. The whole form can be evaluated as a combination of trig line segments, some in the rectangular frame the rest in a general frame.

The fundamental frame construction means I need to understand the closing in product and the spreading out product and what they represent in construction or synthesis terms.

The relationship between a line segment and a closed circular loop is expressed in the concept of trig line segments.

By analogy therefore any " string" theory as a generalisation of a line segment to a general curve expressible as a Fourier transform must be associated to a general trochoid or roulette also expressible as a Fourier transform.

The consequence of this on string theories is yet to be explored, but to be sure all general rotational motions should have an expression in this kind of generality that being the case we may then characterise the descriptors magnetic, thermic, sonic and electric in terms of some Fourier transformation  of a dynamic nature.

Dirac does not acknowledge the influence of Grassmann. Probably he did not know of his work. But everything he thought was based on the work of Grassmann , Riemann, Monge and others , especially the German theoretical physicists.

http://youtu.be/jPwo1XsKKXg the royal society channel

http://www.youtube.com/watch?v=jPwo1XsKKXg

There are 2 principles I warn against: that nature or god describes the laws it obeys " mathematically"; that mathematics is the way to understand the physical world.

Spaciometry is that practice of interacting ith space so as to apprehend, comprehend and manipulate it on a human scale. In doing so aesthetic responses tke on an important role in how an individual is drawn into and appreciates that practice and behaviour. The Logos Analogos Sunthemata Sumbola response is the goal of many Neoplatonists who do not realise they are engaging in the Pythagorean programme. As such Pythagoreans did not believe that nature was mathematical. What they demonstrated was that a mosaic can depict nature and its forms. These mosaics are necessary analogies of dynamic ideas, forms and relationships, but never sufficient.

The fundamental building blocks of these mosaics, called Arithmoi, are monads. There are many styles of Arithmoi, but the standard monad was chosen eventually to be a square. Early monads were pebbles or circular discs or counters.

The power of the mosaic cannot be overstated. Right now you are reading symbols on a high resolution mosaic! As Apple knows, the aesthetic quality of that mosaic , that screen, is crucial to its credibility. In that sense Dirac pursued the Pythagorean goal

In the Ausdehnungslehre page 11 § 10  Grassmann expounds the multiplicative relationship to addition as the distributive rule/ law of multiplication over a bracketed sum or subtraction..

It has also become clearer that the use of the = sign has 2 distinct conventional uses. The equating use and the labelling use. Thus x = y is ambiguous. Context is necessary to determine if this is an equation, and further an equation as a constraint, or a label and further a formula label or a function label.

The label idea becomes crucial when referring to and handling ideas , statements and magnitudes that are expressible by formulaic or functional products.

The importance of products, as the foundation of a metrical system can be found in the so called Euclidesn Algorithm in book 7 of the Stoikeia. Here we find the concept of Monas, and the concept of Arithmos. Here we find described different Arithmoi, and the logos analogos relationship between Arithmoi. Here we find the origin of adjugate or complex number / Arithmos where triangle, square, pentagonal Arithmoi are distinguished, but also stereos nimber/ Arithmos.

Because we have collapsed the Arithmoi into some stroke of a pen, we fail to see the geometrical firm of the stereos Arithmoi, and do we do not apprehend the use of proportion in evaluating and comprehending these numbers/ Arithmoi.

Thus a:a is a square number. We could write it as a+ ia. We could count it in terms of a fundamental Metron or a monad : 1 + i1 as a x a , but that requires us to understand multiplication and what is symbolised by x. The deconstruction of x involves the distribution rules. These are the product rules Grassmann explores geometrically and distinguishes by the rotational behaviour of a pair of line segments.

It is important to adopt the view that multiplication is not defined or understood. One must scotch the idea tha multiplication tables have any meaning. Then one can apprehend Justus Grassmanns difficulty in deriving multiplication as a logical process, a natural process of the mind that is valid in and of itself.

Multiplication has no validity except by geometrical form and order by these forms in space. The concept of multiplication  is grounded in the product, the processes or acts of producing multiple forms. Allied to this process is the iterative act of naming configurations  in the act called counting.

We produce multiple forms by division. We alo produce them by copying. These 2 processes define our apprehension of multiple form and the factoring or manufacturing of multiple structures.

How did we get the sophisticated terminology or notation we use for multiplication today? The answer is by a line segmnt symbology introduced in Book 2 of the Stoikeia . This symbology takes a drawn line , segments it and rotates the segments as arms or limbs of an angle. These arms are thn imagined or considered to " contain" a rectangle or parallelogram form. The Gnomon is introduced in this context, which is an essential part of planar proportion and metric theory, as well as rotational effects of the arms of the containing" angle"

Grassmann could only state that multiplication was logically sound because it was obvious from looking at rectangles.

However Hermann adopted the view that the geometrical forms were fundmental. Multiplication was  defined by this geometrical form. Multiplication was thus factorisation of these forms or tesselations of these foms. Purely geometrical with counting.

With this point of view Hermann was able to distinguish 2 kinds of geometrical products which behaved differently. He called the first kind the "outspread arms " product. This was the product formed by relative rotation of the sub segments of a line segment.

The first kind does not change count regardless of the rotation, but clearly the parallelogram is changing surface space. This surface space appears to squeeze and stretch .  This is the result of the containing arms closing in onto each other. Thus Hermann called this the " closing in " product. It was depicted by a vertically projected line segment onto one of the containing arms.

This depiction was the well known arithmetic product or multiplication process for the area of a parallelogram! So why not call it that?

The generality of these 2 products required a freedom of mind and thought that tying them to conventional terminology would destroy. Hermann was promoting a purely geometrical analogue of the standard arithmetic of his day. In so doing he hoped to show that mathematics was founded immovably on geometrical primitives.

This was at odds with his fathers more conventional aim which was attempting to show that arithmetic was derived from pure logical primitives. However his goal was to support a constructivist view of mathematics a view adopted by Sciller in opposition to Kant who declared mathematics to be discoursive, and thus of Divine revelatory origin.

Hermann thus aimed to support the constructivist view of his father, Justus, but emphasised the geometrical foundation over and above the logical basis. In fact Justus had become stuck by his belief that such a logical primitive basis existed, and he became stuck precisely on the logical basis of multiplication.

There are many other products that Hermann distinguishes from geometrical behaviours, but they are all related to the main 2 : the "closing in  " and the " outspreading" arms / limbs products. These products are identified as multiplication interchangeably in Hermanns development of the notion of multiplication.

Ausdehnungslehre 1844
Induction

A: deriving the Label/ handle of the pure Mathematic

1. The supreme partitioning of all Expertises is the partitioning into real expertises and formal ones, from which expertises the former ones  develop a copy of the experiential continuum  in the thought process, As the" self evidently  confronting to the thought process" entity ; and have their veracity in the super concording of the thought process with that experiential continuum.

The latter expertises on the other hand,  "self established as to the content" entities have the experiential continuum Through the Thought process, and have their veracity in the super concording  of the thought processes under itself.

Thought process exists only in relating together  on top of an experiential continuum what becomes confronting to it, and through the thought process becomes a developed Copy. Thus this experiential continuum is by the real expertises a self evident , outer side of "the thought process for itself " , existing entity,
On the other hand that experiential continuum by the formal expertises, an entity through the thought process self established, what now again places Itself in confrontation to a second thought act As experiential continuum !

Even if now the Veracity importantly rests in the super concording of the thought process with the experiential continuum, so rests it especially by the formal expertises in the super concording of the second thought act with the   "through the first thought act"  established experiential continuum, thus in the super concording of both thought acts; the demonstration in the formal expertises  therehere does not go far  out beyond the thought process itself, into an other Sphere,  rather purely tarries about in the combination of the differing thought acts. Therehere also the formal expertises do not permit to arise out from  empirically grounded statements, like how the real expertises permit, rather the definitions build their  foundation .



Footnote

Even if the formal expertises for example Arithmetic, has even yet introduced fundamental empirical statements, this should be seen as a misuse. The only way such statements can be understood or treated of are in the inter communicant Geometrical field. This is a topic I will return to once again later after a long exodus through other ideas. We,it suffices, have  to deal  necessarily with the omissions in the fundamental empirical statements of the fomal expertises!

Commentary

I see here the beginning of a Hegelian dialectic. This is the induction into the way the Ausdehnungslehre woks. Two antithetical ideas are compared nd contrasted. The ground rules are set by these preliminary discussions.

I do not happen to agree with Hermnns analysis or models, especially with regard to how the thought or mind processes work, or the underlying psychology. However the remarkable innovation is that the observer is not excluded from the discussion and so from what develops. This text claims to be a new twig or branch of Mathematicl expertise. I do not know enough of the mathmatical literature of that time to judge how groundbreaking this approach was, but judging by the response to this book in 1844 it was not an appealing or recognised pproach.

In addition we see that evn before 1844 Hermann nd others were examining the foundations of formal systems.

In the note below this section Hermnn identifies Arithmetic as just uch a formal system, where as geometry presumably was not. However in 1853 Riemann under instructions fom Gauss delivers his habilitation spee h on the hypothetical foundations of geometry. Thus Gauss, who had read the early manuscript of Hermanns work prior to 1844, considered Gometry to be a formal system. It is Sid this was due to his dismay at the shambles urtounding that wrong headed attempt to justify postulate 5 in terms of the initial Eucliden axiom.

Do not go there, is my advice.  Euclids Stoikeia is a course in philosophy not axiomatic geometry , mathmatics or anything else. The axiomatic approach derives from Kant, who derived it from Newton, who styled his Astrological Principles after the Euclidean rhetoric, no where in the Ruclidean rhetoric does it say that postulate 5 is anything other than a postulate. It is not a proposition. It is a entry requirement to do the course! To do the course you must be able to draw long straight lines that cross,  without that skill one cannot hope to measure the position of stars etc in relation to the spherical centre of the universe, earth!

So a formal expertise in Grassmanns mind was defective in that it could not resolve issues of what actually obtains in physical space, only a real expertise could do that. Thus arithmetic could not provide the answer to the question; What is multiplication?

I found one translation of eingewandtes which equates it to regressive. Their is mention of a regressive product in other translations , but here Grassman decades it as an unfruitful idea, the elements of which he combines in other ways in the 1862 Ausdehnungslehre.

Ausdehnungslehre 1844 Induction.

2. The formal expertises track either the general Laws of thought process or they track the Special, set by thought process entity,  former thought process is the dialectic( or logic.) , the latter is the  pure Mathematic.

The  contrasting statement between General and Special thus makes tenable  the partitioning of the Formal expertises into dialectic and  mathematic.

The former is a philosophical expertise in that it systematically seeks the monad in all the varieties of thought process, while on the other hand the mathematic has  the confrontationally set direction in that it apprehends each thought individually as a special entity  .

This is Spooky!

I have embarked on translating the Einleitung, thinking that I would do a gash job, just to give the gist of it. After all I spent long enough on the Vorrede , right?

So I fall asleep , after putting down a muddled first draft of section 2. Then I wake up and watch Normans new post on relativistic dot products and chromo geometry. I am not happy about the dismissal of the physics, but he is a geometrist after all. I decide that the Einleitung must be translated to give me the Grassmann insight into where Norman  was going" wrong". I return to my bookmark only to find it is now on a PDF entitled Hermann Grassmann and the Foundation of Linear Algebra???

In that piece it said the Einleitung is crucial to understanding his approach.

Get on with the translation then should I?

The fabulous thing about The Grassmanns is that they struggled to understand the principles. Granted they were not remarkable in that endeavour, as they lived in a time when intellectually that was the thing to do, but they were not spoiled by Academia! Justus wanted to serve his community and his nation and his emperor . He thus wanted the best for the deprived children of Stettin(Sczeczin). This meant he had to be pragmatic and foster his son Hermann with his brother, otherwise he would not have been able to afford the time and money to achieve his goal!

Nevertheless Hermann, though loved felt keenly this separation as a kind of rejection. He thus always tried to please his father for the rest of his life. He tried to advance his Fathers goals alongside his Brother Robert. He put himself and his growing family through the great rigours of trying to make ones mark in the Social setting of his time, including the political and educational turmoil that surrounded them. They took this as an opportunity to patriotically contribute to the future solution for Prussia, and they sought to get it right!

With that kind of background I can feel the drive and dedication in his words, the deep searching for an understanding that was sustainable and solid. His reading and research, directed by his fathers goals and ambitions as well as his own interests and abilities pushed him into desperately busy times. The early Ausdehnungslehre was squeezed out of him , partially, and a bit at a time , as opportunity allowed and inspiration motivated. At times it was breath taking, at times agonisingly slow and boring. At times he could not put down the work, and at other times he could not get to pick it up.

What it was was what it was! A collection of brilliant insights, simple arguments, heavy mathematical notstion and dense geometrical relationships, treated lightly because he had the simple structure of a geometrical group algebra in his mind and thoughts, based on the pieces of the parallelogram and the Gnomon found in Book 2 of the Syoikeia and overlooked by just about everybody else.

" I do not think that anyone else will be given this insight, even if they look hard for it for the rest of their lives!" he essentially believed. Essentially he was right. The nearest authors to it were Hamilton, Möbius and St Vainant, although in his case Plagiarism is suspected.

Today I wanted to point out that the product idea is based on the sum of products!

Thus the product idea is a self reflexive tautological concept that is almost self similar. For me this is the essential property of a fractal geometrical structure, algorithm or Algebra. It means that a product cannot be defined algebraically. Thus it is defined geometrically and denoted algebraically. Because of this inter communication between geometry and algebra we become confused about what is being referred to. Thus hermanns "law"  that empirical foundational statements must not be allowed to found formal systems is crucial to seeing clearly, and corresponding accurately.

A product being a sum of products allows us to deconstruct wholes into parts which themselves are products. It also allows us to synthesise wholes by summation of products.

The notation of his time required the summation symbol. But he used it " lightly". He was not interested in every single detail, because they were uniformly the same. He was interested in the general big picture. Thus writing the detailed markes and distinguishes were secondary to his thought. He saw the agglomeration as a whole. He conglomerates whole rows and series as wholes ore a single concept! This is fractal thinking fractal algebra to the core!

In the 1850's Cayley devised a notation , based mainly on his deep analysis of Hamiltons ideas and the systems of equations Hamilton used . This was the idea of a table or now called a matrix. This he combined with the work on determinants to sketch out a matrix Algebra. His matrix algebra had summation but no multiplication, but he devised a kind of multiplication called a composition. Today the cross product is based on that composition. However it is not a Product!

If I accept that a product must be a sum of products, theb a composition is not a product. However a composition can be a sum of compositions! Thus we have an analogy between 2 algebras, the Grassmann and the Cayley algebras. The spreading out product is a product that is geometrical and a product, and it is a fascinating insight into algebraic Grometry. But it is the closing in product that is inter communicant with the Cayley matrix composition and I want to show how that helps to eliminate the Summation sign in Grassmanns work on the Closing in product.

The dot product is defined by Grassmann in his definition of the closing in product, which is derived from his earlier definition of the shadow casting product and the projection product. The regressive or rather special pleadings product we will retire as unfruitful

The dot product is named by Bill Cliffird several years after Grassmanns death.

The dot product is a product formed from the summation of other products , which products are composed from rows or sequences or ordered sets of elements!

Now I am specifying a very general idea or process. Thus I am being specific about something very general! This is dialectic process in action! Soon I will get very general about something specific , and then the dialectic moves forward into a new synthesis of these ideas.

What are these elements? In general they cn be anything at all.  Sometimes I may experience these elements as a complete collection or structure or flow. Then I can utilise them as is in the specification of the dot product. At other times I may have to engage in a process of collecting nd ordering these elements.

What is the product composed from these elements ? This is as simply as possible to pair each element in one set or sequence or row with just one element in a second row. This is an association, a mapping or a linking of two elements . This pairing may be physicl, or by mapping . The product thus may be apparent and local but it ay alo be invisible and mentally apprehended.

When each elemental product is composed then the dot product is formed by gathering them all together. This gathering may be physical, so creating a region of these products, or mental creating a list on paper or in memory, or a transformation by counting, resulting in a label or name with more or less contextual meaning.

This is the dot product process and it's result, but now I am going to specify it for one of a number of very special cases.  The case I choose is if the elements are numerals.
I have 2 sets of ordered numerals. The sets are specified to be of the same number of elements. Each element in both sets has an ordinal numeral subscript, and the elements are paired by these subscripts. The resulting pairs are a product, but I now specify these products as factor multiplications.

The factor multiplications are performed and all the results added together to give a result which will be called the dot product of the 2 ordered sets of numerals. The result is a single number coded in numerals.

Thus there are several important processes in the dot product calculation.

Now I am going to specify the dot product for a set of numerals and a set of line segments. Both sets are ordered, with a numeral subscript. The dot product proceeds giving numeral line segment pairs. These pairs are products but they are geometrical products. All these products are summed and the result is a geometrical product a numeral line segment pair.

The summation of numeral line segment pairs is the parallelogram law of 2 line segments established by Grassmann.

Let a and b be the labels for the sets of ordered numerals. Then a•b is the label for the dot product of those 2 sets of ordered numerals. That represents a_ib_i
Where a_i,b_i are the individual numerals with their subscripts.

Let A be the label for a set of line segments and similarly B. Then a•A is the label for the dot product of one of the set of numerals with one of the set of line segments in order.
That represents a_iA_i where a_iA_i is each individual,numeral line segment pair.

Similarly for b•B

Now [a#b] is the Cayley table composition from a and b. It is an ordered  table of products a_ib_j where I,j run independently through the subscripts of the numerals.

Similarly [A#B] gives the composition table of product A_iB_j

But which of Hermanns products is similar to this of the line segments? In this case it is the closing in product [A_i|B_j].

The Cayley composition table simply needs a distinguishing mark between the line segment products to become the line segment products needed for the closing in product.

There is no summation at this stage. So I propose to extend the dot product to cover this case. In this instance the one to one mapping is Onto pairs with the same subscripts.

The closing in product is thus

[a#b] • [A#B]  which represents [a•A | b•B] and this is

a_ib_j[A_i|B_j].

I could denote this as the dot product of Square matrices.

I can define the cross producer as a numerical line segmnt  product.  This means that the line segment set of ordered line segments is dotted with a set of  numerical elements ordered by subscript. The difference in this numerical set is the process by which it is established. This process involves " cross" products of more primitive numerals. But no summation of the compound result.

Although I have called this " dotted" this is clearly not dotting because there is no summation. I perhaps should just call it pairing or dotpairing to maintain the link to the dot product . Cayley pairing is also a possibility to maintain link to the Cayley composition table. Whatever these are composition processes for products prior to summation to give an overall product type. Products thus involve composition and summation of primitive products.

The cross product is thus the beginning of many other possible product processes for numerals. These resultant ordered sets are then dotted with the line segmnt set to give a numerical line segment product.

I can define Cayley table composition using the ordered sets and arrays within the Cayley tables themselves.

[a#b] gives us a matrix A [c#d] gives us a matrix C

Collect the rows of A from the ordered array in A. Label them r,s,t.. Similarly for C label the columns m,n,o...

The composition of the Cayley tables now becomes

| r•m r•n r•o....|
|s•m s•n s•o....|
|t•m t•n t•o.....|
|........             |

This composition is clearly the same as or analogous to the fundamental Cayley composition from the ordered sets of elements. By reducing the array to rows or columns the matrix becomes an ordered set of rows or an ordered set of columns. Thus the composition is the same process at a " higher" stage of complexity.

Ausdehnungslehre 1844. Induction

3. The pure mathematic therehere is the expertise of the "Special Experiential Continuum", as of one  generated through the Thought process. The special experiential continuum apprehended in this sense We denote as a thought Form/thought  Pattern, or at worst a Form/Pattern. Therehere pure mathematic is a study of Form/ thought Form, that is the study of Pattern/ thought Pattern.

The Name "Magnitude doctrine" is not central to the fully gathered Mathematic, in which the same Name on an essential  branch of the same mathematic, on  combination theory finds no application  of  it, and on the arithmetic only in an  uncentrally related sense. There against, the expression "form " again seems to be too wide and the name thought form/ thought pattern a better fit.
Alone, "form" in its purest consideration, devoid of all real content is plainly not a different entity, as a thought form, and thus with the expression is inter communicant.

Before we move over to the partitioning  of the study of thought pattetn we must cut out a branch, which one untilhere has made judgement towards it with Injustice (an injustice to which one has maintained until now), specifically Geometry. Already out of the above set out label light shines in, that Geometry, so plainly like Mechanics, onto a real experiential continuum has to be restored. Specifically this must be Space/ 3d Space. And it is clear how the label of space in no way can be created whole  from thought Process, rather the same continues confronting, as a Given entity. Whoever is wanting to make the opposing division more important, is expected by the Muse to undertake  to do the work  the necessary derivation of 3 dimensions of space out of the  purely thought established ones, an exercise whose solution itself immediately presents as impossible!


Foot note
*the logic serves a pure mathematical side, which one can as a formal logic designate and its content jointly reprocessed according to my Bother Robert and me , and from the former in his 2nd Tome the Study of Form/ Pattern Stettin 1872 is represented in a centrally related form  
**The labels of magnitude come to be introduced into  Arithmetic through the act of Tallying or counting, the language therefore differentiates very well extending more about  or extending less  about ,(what the Number relates to ) from extending magnitude about  or diminishing magnitude about (what the magnitude relates to.)

Commentary

There is a lot of perspective changing going on in this section. Having narrowed the discussion to formal expertises Hermann then divides the formal expertises into 2 antithetical subjects: dialectic process and pure mathematic process. These are all thought processes, thought acts in general and in particular. The general thought process was examined for universal or general laws of thought. . The monad of thought was sought, so thought could be formed and measured in a Pythagorean sense. This exploration was pursued by a dialectic process, and not the traditional Socratic Platonic Pythagorean dialectic. Hermann was enamoured of the Hegelian dialectic process.

Pure mathematic was a thought process or thought activity that explored special cases of thought, established regimes of thought. Pure mathematic was not mathematics, it is more broader than that, but the term mathematic is so suggestive that this is easily overlooked. Pure Mathematik here refers to the reasonings performed habitually by a qualified Pythagorean expert, an Astrological reasoning that incorporates Geometrical representation.

This comparative set up of the division of the formal expertises conditions how we proceed with the analysis.. The general qualitative assessment of thought now needed to be transformed into a quantitative assessment. This was achieved by Hermann by an anthropomorphic or rather entititic representation of the experiential continuum. General thought processes becomes  general thought of a special entity.. In particular this works better for the special cases dealt with by pure Mathematic. Pure mathematic can be viewed as a special entity  that thinks through special cases, but only comes into existence by this thought act.

This was viewing the qualitative and expansive experiential continuum by a quantitative form developed through thought. This form or pattern is a step back from the activity of thinking to view it as a whole , and thus as a form or a pattern.

This form or pattern is usually approached from the Greek classical idea of Eideon, which is a visible form or an imagined form. It is the subject of the Platonic Socratic philosophy of Idea/ Form.

In choosing form over Idea Hermann departed from the classical concepts of surreal entititees to a promotion of real palpable entities. For example Dedekind started his concoction of numbers using just Ideas. Hermann is signalling that he is on the side of the constructive nature of mathematics and reason.

So now thought patterns are the entity that Hermann wants to go on to specify. That thought pattern is precisely the thought patterns treated philosophically in the Euclidean Stoikeia. This was an undergraduate level philosophy course defining thought patterns .. It contrasts strikingly with Aristotelian Logic , based on Aristotles grammatical analysis of language.

The special entity which comes into being as the thinking by it is being done, or it is a "by thought realised " special entity; this is the entity that applies the formal expertise of pure Mathematic. It is the special experiential continuum thought of in this particular way, and when the special experiential continuum is viewed in this way then it can be termed as a thought pattern.

When one concentrates on a triangle and specifies all its parts and relations that in and of itself is a special entity, a thought pattern specific to the triangle. This is similar for all spaciometric forms. Once that is grasped one can realise that thought patterns dominate all ones thinking. This is the pure mathematic. The dialectic therefore searches for the monad of all these special entities. This monad is the Metron from which all special entities may be constructed just like a basic unit of a mosaic.

Is there a basic unit of thought? This is the general quest of Hegelian dialectic in particular.

Commentary

In the formal expertise we define space as 3 dimensional. But we can also define it as 1,2,3....,n dimensional. However in Geometry we cannot define space as 3 dimensions because we cannot logically limit the dimensions to 3! Space is space by undeniable experience. If we label space we come across undeniable experiences: we are not limited to 3 ways of labelling space. There is no way we can logically prove that assertion that space is 3 dimensional. This makes space and geometry not part of the formal expertises, where everything has to be established by definition and thought acts that concord with definitions and other defined realities or definitional systems.

Mechanics and Geometry are part of the real expertises. We apprehend them as self evident and undeniable..their veracity confronts us every second of every minute of every day of our experiential continuum.

Since we cannot define space as 3 dimensional we cannot define it as n dimensional either. Space is as we experience it. However we can establish foal systems in which these dimensions are defined. These fomal systems must be consistent with each other. The question then is are they of any use to the real expertises? And if so how?

The most fundamental geometrical pattern is the closed loop. This is discovered by rigigid and non rigid rotations, strings and fluids to be so, undeniably in our experience. Through it we  percieve other more ubiquitous patterns: spheres vortices etc. But it is the formal constrained defined circle that we begin our study with!

This pattern is formal and abstract and not found in space. Yet everything we start to know about geometrical space depends on it! Hermann is saying this is a formal model which is not geometry or mechanics, but a formal model we apply to space. It literally tells us nothing about space. It tells us everything about how we perceive space! It tells me that we have a restricted blinkered view of space!

Nevertheless, this view has proved technologically useful, mechanics refining it to fit as is. But the success of this view has blinded us to alternatives and to freedom of thought pattern .

Their is fractal regression underpinning all that we claim to know. This has been held back from view by scared philosophers. Kant for example hid it behind his transcendent schemes. Others called it a monster! As a consequence we do not know what it is we don't know! Hermann sought to clarify this at the outset. Placing geometry among the real expertises has a profound effect on the formal expertises!

The Axiomatising of Geometry was a big injustice, it was Also a big mistake. The parallel postulate shenanigans did much to denigrate the reputation of real expertise based Grometry and Mechanics.Gauss was reputedly ashamed at this fiasco in geometry and sought a new geometry, but did not know how to get there. Hermann did, by restoring Geometry to a real experiential continuum. Space.

Ausdehnunglehre  1844
Induction §3 continued

Now  someone would (Imagine if it was like  he has to deliver this thought process), the entity still , the name of Mathematic still stretch it onto  also Geometry  to  Darling it! ; so could we indeed ourselves let this happen, even if  also it will allow ourselves   to  stand on the other side to our name The study of thought form or some random equivalent; yet still we must  in advance thereof advise of  it afar, that then that Name, because it conceals the "most differing" entity  within itself, also necessarily with time must become tossed aside as superfluous. The position of Geometry toward the theory of thought form depends on  the commensurate arrangement, in which the likening/manifesting  of space stands to  pure thought process

Now, Even If, like we were saying,  it  confronts by the thought process  that likening/ manifesting as self evident Given entity, so  therewith is still not asserted  that the likening/manifesting of Space was coming to us first out of the tracking of the space like thing , rather it is a primitive likening/ manifesting, with which the opened state of our senses for the sensible world is inherently given to us, and which is plainly adhered into us from the outset, like the body to the Soul.

Onto like cognisance (a like way of thought): it proportionally arranges itself with Time, and with the kinematics founded on the likenings/ manifestings  of Time and of Space , wherefore also the pure study of kinematics ( Phorometry)  with the same right as Geometry  one   has counted also as a mathematical expertise .

Out of the likening/ manifesting of kinematics ,  out of the manifesting of the conflicting statement from originating cause and Effecting  flows mediated the labels of the kinematical Skill;  so that thus Geometry, Phorometry and Mechanics shine out onto the primitive likenings/ manifestings of the sensible world as applications of the study of thought form/ thought Pattern.

Commentary
Hermanns use and reliance on Hegelian logic shines through this difficult section.. Here the 3 propositional sections are combined to set up the dialectic progression into the next section.

The original division represented a contradicting state of affairs or Verhältniss, the second statement narrowed it down into a contradiction between pure mathematics and the Dialectic process and the third one establishes that the study of form or thought forms by a special entity is the best way forward. These special forms I have highlighted as patterns

Then he uses Geometry to advance his case, by arguing geometry has bern unjustly treated and wrongly set up as an axiomatic discipline. In his day in Prussia that statement fell on fertile ground. It was obvious to Gauss something was wrong with geometry which had singularly failed to solve the fifth postulate problem after nearly a thousand years of trying! What was wrong Gauss did not know, and he directed Riemann to put out the call to physicists to rescue mathematics and geometry

However the Grassmanns had been working on the solution starting with Justus, who corresponded with a small international group of Ring and Group theorists and crystallographers. They felt the dynamics of crystals held the key to a real geometry one in accord with nature, not human thought.

However Justus struggled with Aristotelian logic, and foundered on the multiplication problem. He could not logically justify multiplication, all he could do was point to a geometrical figure!

Hermann on the other hand studied and read Hegel's writings and his dialectic or logic. Some have seen the similarity between Hegel and Plato in their dialectics. What no one has pointed out is how this logic differs to Aristotelian logic, upon which islamic and most European Academics relied.

Actually Socratic and Platonic dialectic derives from the Pythagorean school in which koans were given to students rather than linguistic analyses. The koans were intended to exercise every aspect of the disciples being until an epiphany occurred whereby the Musai or a Muse might reveal a profound " insight"  or a truth or ability or gift.

In this way the student was assessed to have progressed until it was agreed by the senior scholars that the student had become one of them , a Mathematikos.

This " qualification" was a discipleship based recognition, not a piece of paper. You literally became a
Lifelong " monk" or inhabitant of a "Monas" tery. Such a building was also called a Mousaion or the house of the Muses. It was decorated with " Mosaics" and it was a place of culture , learning and arts and customs.

Thus the student was immersed in a collegiate of thought, discourse debate and empirical observation and Muse led insights. Aristotle disagreed with this approach. Nevertheless while he could he remained with Plato. However he was forced to leave for geopolitical reasons, the same reasons that thrust him to the very top of the burgeoning Greek empire as tutor to the Emperor Phillips sons Alexander and Phillip.

Thus for a very brief period Aristotle outshined all Platonists. And in that position he ditched the Pythagorean system and created his own, while still claiming to be a Platonist. His school quickly passed away and but for the Islamic scholars may well have been forgotten under Plato. However he was once again raised to high prominence in the Islamic empire and thus obscured the alternative Pythagorean traditions of learning to this day.

Thus when Hegel began his philosophical work, it was not Aristotle he turned to, but to Plato, and from him eventually to his own unique style of logic , without perhaps knowing it he had recovered the original Pythagorean system in whole or in part. It was this kind of dialectic Hermann used to unravel the whole of western European thought for neatly 1000 years..

The Hegelian dialectic moves in circles, it moves in levels; it is very fractal and very thorough,and it always moves forward to a new synthesis based on the thesis and antithesis in a contradictory statement.

Normans Complexions!

Note how " vectors" easily replaces the mumbo jumbo of " number". These are special entities, special thought patterns or foms. And Norman is studying these forms in a geometrical setting.

You have to have a special state of mind or thought to apply these patterns expertly! This was Hermanns Förderung, his promoted way of analysing and analogising, apprehending(interpreting) and applying.

http://youtu.be/MTddX4Qo-io
http://www.youtube.com/watch?v=MTddX4Qo-io

Anschaung  is someone making something visible to others , making something apparent to others, likening something to another individual , manifesting something to another person or persons thought process or perception.

It is a complex reflexive in that the people ultimately perceiving do so through someone else drawing attention to Some analogy which they feel is apt and hope the perceiver will draw a perception from, conveying to their minds something of what is in his mind.

Pure mathematics is an expertise of a special entity a specific state of mind a special experiential continuum. If the special experiential continuum is understood in this way then we cn call it a thought form a pattern of thought or in a poor sense a form.
The idea that pure mathematics is the study of magnitudes, that is the experience of extensivity, is misleading! There are several branches of mathematics which have nothing to do with extensivity! On the other hand form seems to be too wide a concept, which is why the terminology thought Form or even a form of thought seems more appropriate, not so bad or poor a fit.

Granted if one restricts form to just the abstract idea without any real content, just the shell of a form then the use of the term form does correspond, but in Amy case to be clear it is best to centre on the term thought form.

Because of this possibility to introduce bad thinking into the development I suggest the word pattern or thought pattern as a more general term for the idea Hermann is trying to express. This does mean that the clear connection to form in space is less clear! It requires one to see all forms as a more general pattern of extensive experiences, but it prima facia includes all patterns made principally by drawing : thus drawings, sketches, scripts , plans, schematic diagrams, flow charts etc.

The Greek word Gramme which derives from the verb which covers all drawing actions is a good correspondence. This word is usually translated " line", but that is only part of its conceptual meaning.


Pure mathematics is thus the study of the patterns of thought of an expert in a special or specific state of mind! It is thus entirel formal, not real!

However we still want to use the word mathematics, because we love it so! For that reason Hermann extends the word across the real formal divide , temporarily. Eventually , he foresees, the word mathematics will be discarded because of its internal inconsistencies! Then the study of thought patterns will come into its own!
For now (1844) he was willing to put up with the new terminology being pushed to one side, because he also loved the word mathematics like a little darling!

Having established his terminological root he wants to go on to show how that overflows into the work and terminology of the Ausdehnungs Lehre. However to do that he has to restructure the contemporary perception of pure mathematics by first removing geometry from is deadly grip, establishing geometry as a real expertise like Mechanics, based on the real experiential continuum not the special state of mind of a few experts!

The real experiential continuum on which geometry is based , derived from and verified by is Space! "Space, the final frontier" is a well used phrase that captures the extensivenesses of the real space of Hermanns conception.

In line with his scheme or set up he now draws attention to a third set of skills called Phorometry.. This is the study of pure motion in space! Today the word has been sidelined to cover the specialist work of an optician diagnosing vision problems in the eye. This is a direct result of the dialectic tussle between pure mathematics and the study of thought patterns!

Rather than recognise the difficult conceptual problem, pure mathematicians have reacted defensively. They divided their subject into pure and applied, and left the applied mathematicians to their fate! Gradually applied mathematics has bern absorbed into Mechanics, physics, chemistry , and most importantly Computational science.

Pure mathematicians are thus a dying breed, as Hermann foresaw, and their relevance has gradually slipped into the netherworlds, backstreets and the boondocks of societal perceptions. Only in Academia have they been able to maintain some kind of mythical and mystical evanescence! However computational science in my view has made it more likely that they will soon disappear as an active cadre of intellectual expertise.

Today Mathematica is the new face of mathematics. Without realising it, and under the cover of the darling name Mathematics, Hermanns "study of thought patterns" has been realised in the computational sciences. In my opinion we need to make the break and revise the concept of mathematics along similar lines to the proposals that Hermann now proceeds to work through.

Phorometry is often confused with stereometry, and even photometry. But in Hermanns experince it was a broad field of expertise delivering direct measurement procedures, tools and equipment for making displacement measurements in space. It is a fundamental application of trigonometric concepts, and thus is allied to land survey, astronomical survey, triangulation as well as optical survey and opthamolocical survey and diagnosis!

Hermann sets out geometry, Mechanics and Phorometry as the real " mathematics"  and tHese are the thought patterns he wants to study further using inter communicant formal systems!

It is worth noting that Hermann was not wedded to the exact words Formenlehre. He was equally happy with other terminology expressing the same conceptual base.

We cannot use Grassmanns analytical and synthetical method to distort the real expertise of measurements of space. But we can use pure Mathrmatics by formal set up, or math magic to do just thst!!

http://youtu.be/nXF098w48fo
http://www.youtube.com/watch?v=nXF098w48fo

You have to beware of experts and their expertise, especially if they are based on a formal or theoretical expertise
http://youtu.be/XwhtHUiAOjM
http://www.youtube.com/watch?v=XwhtHUiAOjM

Hermann distinguishes between the expertise of pure mathematics and the expertise of real "mathematics" which he denotes as the study of patterns applied to real empirical measurement skills or crafts like Geometry, Phorometry and Mechanics.

The manifestations of kinematics leads by mitigation of the contradictory statement of cause and effect or originating change and its outworking to the labels of the kinematical skill!

The Astrological Principles of Newton and the System of worlds represents a setting out of a philosophy of quantity and it's detailed application. Because Group and Ring theiry had not been identified or isolated as subject domains, newtons presentation stands out as a landmark innovative approach to Astrological mathematics.

In the same way, Hermann Ausdehnungslehre with his Putative Schwenkungslehre stands out as a landmark revision of Newtons Astrological principles in terms of group and ring theoretic terminology.

The material and work covered by both philosophies is identical, but the presentation is almost completely antithetical. Newton was secretive and dismissive of Algebraic gibberish, while Hermann was open and enamoured of Algebra. However , without the Hegelian dialectical approach he would not have been able to encompass the breadth and depth of Newtonian thought and empirical exploration.

The underpinning lineal algebra is the same, but expressed terminologically by Hermann where Newton expressed it Rhetorically.

But why stop at Newton? We may safely return to the Euclidean Stoikeia as a philosophical treatise that encapsulates all and more of what both Hermann and Newton treat of. Leaving aside Aristotles digression from the main Pythagorean principles , it is possible to see a clear path from the Pythagorean school of thought through Plato and Euclid and Apollonius et al. to Newton and then to the Grassmanns and other group and ring theorist.

Today's use of group theory and ring theory in physics is thus not a predilection intellectual snobs( only!) but a fundamental clarification of the formal system which is applied to a real expertise system.

Where Hermann stands out, by Hegelian logical process is that he fully recognises this from the outset, where most who follow his work but not his philosophy do not.

Since we are leaving the world view of Aristotelian logic and have embarked on the journey of Hegelan logic it is probably a good idea to review chadafricans videos on YouTube regarding Hegel, and also Gregory B. Sadlers.
I can recommend those 2 channels to give you a clear introduction to an unfamiliar but important philosophical and phenomenological treatment of human experience.

http://youtu.be/7Hh8OGEsmqA

http://youtu.be/6L6zndZs8Js

Many contradictions apparent to the Aristotelian world view are now acceptable in the dialectical process of Hegel. Thus many issues that bothered me in my early translation attempts of the Vorrede are actually not harbingers of doom to the Grassmann method. In fact they drive a synthesis at a new level.

Is Hegelian logic sound?

The question itself is interesting. Do the results of the dialectic ring like a bell? This test of well foundedness or good foundry process giving a perfect result is an auditory not a visual or kinaesthetic test, leaving aside the important vibration sense through the proprioceptive sensors. Thus a grammatical language analysis, the basis of  Aristotelian logic is an inappropriate vehicle for making this judgement call. The soundness is verified by empirical concordance( übereinstimmung!) and is a phenomenological assessment. Thus from the outset of this induction Hermann has promoted the soundness of the Hegelian dialectic and how it is assessed.

Lord Xenu is another recommended YouTube channel for a more traditional overview of the philosophers in their context.

I also draw your attention to Grassmanns own criticism of the Hegelian school. Thus Hermann does not slavishly conform to any academic or scholastic consensus on Hegel, neither in his own time nor in the time and contexts of those critics and admirers who followed. In any case it is hard to say whether any such putative consensus exists!

The student Karl Marx in critiquing Hegel drew his own conclusions on inadequate research portfolios. And in many instances critics have failed to grasp or grapple with the Heglian dialectical method, systematic approach or exemplars or even Hegel's own extensive analysis and synthesis of his contemporary context.

The Hegelian logic is first proposed by some as Aristotelian, because of the kindred spirit of deep unflagging analysis and total exploration of totality. However, no Aristotelian scholar seems able to finally say Hegel is a student of Aristotle, because he appears to unfailingly contradict himself in his writings where Aristotle, par excellence taxonomised every experience into a systematic taxonomy which isolates contradictions . In that difference lies the resolution that Hegel is not a slave student of Aristotle but a master of his own philosophy, which encompasses as far as is possible all philosophy and philosophers to his time.

Philosophers after Hegel have had to deal with Hegels philosophy, and most have consigned it to impenetrable obscurity as an academic trick to remove it from the preeminent position it holds. As the saying goes "they prefer the old wine because it is sweeter!". And again" new wine must go in new wine skins, lest the skin bursts and all is lost!"

Hermann stated that he was going to use the mindset of the Hegelian school to join with his own mindset to produce a remarkable independent system of analysis and synthesis . Thus he is fully ready to tackle the contradictory statements of specific analyses to resolve them , by mediation or mitigation into a new broader whole. We see this first in his trepidation, and disconcerting horror at stumbling onto a non commutative product process! Until then his experience, training, expectation and understanding led him to see multiplication in only one way. To his credit he held his nerve, did not throw out the result as nonsense, which an Aristotelian point of view actually demands, and found a higher level of distinction( or a deeper level if you so wish) that resolved the issue.

The problem for mathematics as an axiomatic system is that many demonstrations rely on contradiction to support the validity of an assertion!. Consequently Hegelian logic, if accepted, makes pure mathematic unsound wherever it relies on a contradiction to establish a truth!

What to do? Either bury your head in the sand or move on through the Hegelian dialectic process to a new resolution.! This is why Hermann predicted the " death" of pure mathematics, the obsolescence even of the name Mathematics, as time progressed and the internal contradictions became intolerable.

Yes, for now we still treat mathematics like some darling, to be indulged, to be wreathed in the some of nostalgia; but that time is fast fading away , and the time of the Fomenlehre is upon us, the time when the patterns of thought are the paramount subject of study and application.

So now we move on by a seeming contradiction to the idea of a study of thought foms. Hermann stated that the term "the study of magnitudes" was an inadequate descriptor of pure Mathematics as a formal system, even more so when Geometry is removed from its banal clutches. And yet he calls the next section the derivation of the terms or labels for extensive magnitude theory! - or the study of extensive magnitudes!

The resolution lies in the label Ausdehnung. Rather than a dead static object Ausdehnung is a dynmic, living subjective experience of changeable and changing form/ pattern. Formenlehre begins with a primitive dynamic that manifests itself in primitive, a priori manifestations. These are real not formal experiences, so to speak . From these primitive distinctions the dialectic process begins to build a greater outcome. Ultimately the result of the dialectic process is the Monad of the Neoplatonists or the Neo Pythagoreans . The logos analogos sumbola sunthemata process that restores the summetria of the Monad from the disparate pieces into which it apparently, to the mind at least , has fragmented. This is Shunya of the Sanskrit Philsophical schools of thought , before it becomes divided by the various philosophical and Yogic traditions.

Hegel goes beyond to one more level: that of absolute mind. This is the oneness of  Buddha, the Nirvana of self awareness that references only itself. It is absolute in that sense requiring no other to validate itself. Like Aristotles unmoveable mover it is self motivating , everything else it moves relative to itself. Thus the only thing that can move it is itself . Thus it is the Absolute Mover.

In this analytical result Hegel, Newton and Aristote concur, but their synthetical paths to this description differ widely. For Hegel this absolute is real, empirical and other to the individuals subjective progression of development. To Aristotle it ultimately was a categorical taxonomy. Once he had come to this conclusion he then goes on to taxonomise the inverse, and then the reverse and so on. Newton resolves his system into n all powerful God who establishes and maintains a mechanical universe.

Regardless of these ultimate drivers Hermann pursues the liberating outcome of his analytical and synthetical method as a young enthusiast encumbered by having to make his way in the world. Thus he never attains to the final conclusion of his ideas, calling on others to carry his torch onward and upward to some presumed glorious apex of the whole redesigned and more functional Mathematics layered into and manifesting in the kinematical dynamics of Space, and Nature in and of that space.

One of the conceptual difficulties that Hegel and perhaps earlier philosophers had was terminological. How could they label what they experienced or envisaged when it was fractal in nature, construction and design, scale free in aspect?

Thanks to Benoit Mandelbrot we changed from being frightened or shushed into not discussing or peering into the infinite regression of our analysis. Unlike Kant we followed Cantor. We tried to name the infinities, and it drove him mad on several occasions. But it was Benoit who served as a collection point for many others who ventured into that domain. It was Benoit who spearheaded an offensive on the citadels of Mathrmatica and tore open a breach to let the monsters in! Zombie apocalypse! Only the Zombies were the Staid and defiant professoriate, the board of mathematicians who tried to stave off the inevitable. The breach gun of course was the rising computational tools of programmable calculators and computers.

Ultimately as my self awareness develops it will mean that my experiential continuum will become indistinguishable from the universal monad. It will be a scale free fractal alignment, not a mirroring only, as previous philosophers struggled to express this profound state of Nirvana with Shunya?
Yhe universal monad is a scale free fractal of every monad totality the " I" can conceive. Each unit or monad is essentially a universe at a different scale. It is the primitive manifestation we perceive at the sensory level on which we synthesise our dialectic constructions up to the universal monad, where we and the universal monad are no longer distinguishable, and all distinctions become meaningless.

As it is unlikely that an individual consciousness will apprehend this state sufficiently to fully communicate it. It is thus a matter of logical or analytical positivism or faith that it even exists! So ultimately the founding principle is acceptance.

Ausdehnungslehre 1844 Induction

B. Deriving the labels of the study of extending magnitudes

4. Every  "by thought reified" entity( c.f. No.3) can be reified onto a two fold cognisance : either through a single act of creating, or through a two sequence act of setting and tying together. That upon the former method reified entity is  the continuos form/ pattern, or the magnitude in a minimalist sense; that upon the latter method reified entity the discrete or binding together form.

By far the  worst simple label of  Becoming someone or something / Reification as someone or something, gives the continuous thought form/pattern. That label, by the Discrete thought pattern, before the "binding together appointed entities" status, is indeed also through thought "established", but seems for the act of binding together  as Givens! And  how out of the Givens, the discrete thought pattern, the Artform comes to be  is a straight forward "gather-together!" thought process.

The label of the Continuous Becoming  is at the easiest end  to apprehend, even if one immediately tracks it according to the analogy  of the current, discrete, Rooting and Rising cognisance ; specifically according to this label, by the continual creating, the every time reeified entity becomes firmly held; and by the new Rooting and Rising label ," bang!"-like so,  in the moment of its rooting and rising up with that label, becomes "gather together!" completed thought.

So also can one  differentiate by  the analogy way,  for  the continuous form related to the label ( Continuous Becoming), according to a twofold  act of setting and tying together;

Thus! Consider this!  Both here are united to One act ; and so with an inseperable unity are driven together. Specifically from both limbs of the binding( if we this expression by way of analogy  for a brief eyeblink hold firmly )  is that one, that already reeified entity:

the other( consideration) ,contrastingly  , which in the moment of binding, that moment itself, is a new Rooting and Rising entity , thus not one before the binding already finalised.

Both acts, that is setting and binding, fit wholly onto one another so that nothing can become bound, as it is set! and nothing  is permitted to become before set, as it is bound!

Or again spoken in the continuous  arriving expression cognisance:  that what new roots and rises up, rooting and rising up  only at the already reeified entity, is thus  a moment of becoming itself, which here appears in its further course "as if to change" .

The conflicting statement of Discrete things and Continuous things is( like all sincere conflicting statements)  a fleetingly fluid one, in which the Discrete thing can become tracked also as continuous , and vice versa the continuos thing as discrete.

The Discrete thing becomes  tracked as a Continuos  thing  even if the bound thing itself is again as a reeified entity, and the act of binding is apprehended as a moment of Becoming . And the Continuous thing becomes tracked as Discrete even if individual moments of Becoming are apprehended as direct tying tigether actions , and  which so bound become  tracked for the binding as Givens  .

Weise, werden, Erzeugen, entstehen.!  These are crucial concepts in this section.

First though it is important to realise that Herman has not yet defined a concept one might call " reality". This is perhaps why some cannot get to grips with his use of Hegelian dialectic. So far everything has been in terms of thought experience, that conscious quality one perforce has to accept because to reject it is akin to madness and permanent disorientation. Some call it awareness or consciousness it I call it an experiential continuum.

Thus thought experience in General is where Herman  situated himself.

Weise is thus that tested quality that is perceptible in another's behaviour, their style, manner, speech, practice etc. It is that quality that one most readily associates with learned behaviour, and thus from that concept it forms the basis for wit, wisdom, intelligence, intellect, a cognisance that associates to all the answers to the questions: how?, who? what?, where?, when?, and yes to a pinch, just a little portion of why?. .
In oneself one is aware of Weise as an intuitive competency , hard won by life's experience, on the edge of unconscious and conscious competency and inate know how. But it demonstrates itself not just internally and subjectively but more importantly objectively and evidently.

The idea that an entity can come into being onto Weisse is unfamiliar but sensible. Weise results in evidential behaviours, thus it is a vehicle for evidencing qualities that may be otherwise invisible or not known. Like a compact disc the grooves or embossed patterns carry information in addition to the disc itself. Thus this information reeified onto the disc just as these thought reified  entities reify "onto" Weise.

Erzeugen on the other hand is simpler to grasp and present a metaphor for. It is what a magician does whenever he pulls a rabbit out of a hat! The connection to Siemens through zug is hardly deniable, but dialectic and familial language analysis would have to be done to confirm.
Pulling things out of nowhere into everywhere, or out of the invisible into the visible, or from some state into anther state ... You get the idea.

Erzeugen is therefore a straight forward concept, direct and whole.

Entstehen however is a similarly direct but richer concept. I can see the action but to convey it in English that was the problem.  Eventually I fell upon the concept to root. Then I extended that by alliteration to : to root and rise. The concept was clearly based on the behaviour and morphology of plants. That was the clue.

From seemingly nowhere a plant seed would blow in, root into the ground and then begin to raise itself up to its full glory. Thus I realised the use of the concept stehen was denoting this standing upright, while the ent was denoting digging in to the ground to support the standing up!

Many different metaphors are thus suggested by this scenario: plant and grow, dig in and fight, dig foundations and build tall, all of which clearly capture the concept of starting from nothing and becoming something that lasts.

The concept Werden captures this in its most abstract form, although it is ultimately based on wer or who. Werden then is to become a who, a somebody. It is generalised to becoming a what or a something, and the general concept of becoming is thus established not by the beginning but by the ending. Werden, like the Greek Ontes verbalises that quality one experiences as the quality of being , of experiencing life with and through others as they experience it with and through you. But that is properly Sein. Werden is Sein with the end goal of becoming someone or something, not just experiencing .

Hermann thus utilises these concepts to establish some labels or terminology for the formal expertises at the level of reifying entities which are essentially thought patterns in some person or persons heads.

An example of reify cation is the " reality" of light waves. Most of us have not seen light waves, and yet many of us accept them as real spatial objects. This was achieved by reifying the light wave concept onto the cognisance of water waves which many of us have seen. Light waves were not pulled out of a hat! Thus they were not created as a single conceptual object at a single magical event. Rather they were gradually realised by comparing the behaviours of 2phenomena to see where they coincide. Thus the one gradually becomes the analogy for the other. On the way however, many new properties and denotations are introduced to tweak the reeified concept towards the end goal of its creators. Thus they set down the elements of the concot and combine them into whatever whole suits their purpose.

The concept of a light wave is thus a so called mathematical concept with interpretations or analogies in many spheres of application. Not only are the physical applications considered as real, but also the mathematical wave is considered as the real deal, by dim. It has thus been reaified in the body of the subject physics besides the overwhelming lack of visual confirmation.

Scientists do not present light waves instead they present Weise, in the form of Beweis. This is what is used to establish light waves. It is reeified on top of Weise!

Hermanns labels thus start with this process as he analyses how it works.

Commentary
Every reified thought can be " unfolded" onto a twofold cognisance.!

This point is crucial to his argument and dialectic. There is always at leastn2 ways of considering the same experience, to ways of expressing the same experince two ways of arriving at the same conclusions.

In point of fact there is always more than one way to come to a performance of any action so here Hermann is trying to establish the necessity and sufficiency of 2 !

He needs 2 for a dialectic process anyway so that must not be overlooked, but here he is already preparing the ground or a quantitative apprehension.

In point of fact he is tracing over the Pythagorean insights. The Pythagoreans understood pragmatically tha reality could be analogised in sketch fom. A drawn picture could capture aspects of the experiential continuum in a referential way. The difference between what we experienced and what we drew of our experience is the difference between a continual form and a discrete format!

Our senses may be thought to provide us with a continuous experience, but a picture drawn by hand or photographed provides us with a discrete experience. That discrete experince is dependent and characterised by setting and binding.

This is a rather general 2 fold process not yet defined by Hemann, but self evidently it refers to the piecemeal way we have to laboriously construct representations of our experiences which by contrast seem immediate.

For example, to build a house requires the builder to set out the bricks, and the cement and then to bind them together in a house pattern.  The idea of settings and bindings is thus a general conceptual mechanism by which he will drive his dialectic.

He shows how this 2 fold pattern can be used to account for something that on the face of it is adequately accounted for by one magical act.

This process of simplifying to one cognisance permeates physics, and has lead to procedural errors in understanding. As an example the electric fluid was described in Volta's time as consisting of 2 fluids. It was well accepted as a convincing explanation of empirical results. However Franklin decided to reduce it to one fluid of consequence..

The difficulty in explaining physics in one particular cognisance is its blinding to empirical data that shows there are dual behaviours! By accepting dual cognisances as the foundation of apprehension one comprehends more naturally the supposed contradictions that must exist in a single cognisance.

A dual cognisance arising from a unity or rather a unit arising from a dual cognisance is precisely where Hegel began his synthesis and analysis of the phenomenology of Geist( spirit/ mind).

The deepest open " secret" of the Pythagoreans is the Monad/ Metron. This concept arises out of a dualist( iso meaning dual) process of judgements and judgement calls. The monad represents the foundational unit into which the ultimate unit or Monas is deconstructed by the human thought process. The Monas is by their philosophy deconstructed into Henads and each Henads further deconstructed and so on. In contrast their Monas  is constructed or synthesised by gathering together in thought foundational monads that build into the wholes that appear as the synthesis continues to its ultimate goal, the Monas!

The open analogy of this philosophy is the Mosaic. One cannot understand the Pythagoreans without appreciating the Mosaic as a foundational cognisance, not just an art form, which it also is.

Using the mosaic analogy it is clearer to restate Hermanns thinking here in terms of a "continuous picture "and the same image as a "synthesised mosaic". This is the dual cognisance or 2 fold cognisance of any reeified thought form, at least the one that concerns Hermann at this time.

Commentary

Hermann is using 2 forms of logic, apparently deliberately. The first is the language based Aristotelian logic, the second is the analogue based Ancient logic used by the Pythagoreans and the early philosophers. Based around metaphor and analogy , simile and opposites it communicates the demonstration rhetorically and homiletically.

It is not regarded as rigorous by today's " standards" but this too is a matter of opinion!

The interleaving of the two styles is confusing, whether this was deliberate or the contemporary style of effective persuasion I do not know. Certainly he mentions a current or contemporary analogical practice for this type of argument, the rooting and rising up  analogy, and this is the analogical logic he interleaved with the linguistic logic.

The section needs careful perusal and meditation to apprehend his goal in demonstrating the proposition.

Commentary
"Reeified entities can be reeified in 2 ways, either as a continuous whole or as a combined set of discrete pieces! The first is easy to label you just pull it out of a hat whole
The second you label by setting out the discrete pieces and combining them together into a whole
But you can see the continuos whole as a discrete act, and the combining of the discrete pieces as a continuous action!
So the 2 ways are kind of mixed up in there together. They are a dual fluid kind of perception!"

Ok,this is a gross simplification of Hermanns meditative homiletic! It lacks the detailed twists and turns of the subtle realisation that things are not what they seem. You do not come away feeling that there is a fundmental shift in your perception caused by wrestling with the text!

It is therefore crucial to realise that what Hermann is writing is not an explanatory article, but a persuasive and thus hypnogogic meditative manifesto! The Hegelian dialectic is not meant to be a set of facts. It is a written dialogue of one persons process of perception conveyed to another questing fellow traveller. It isnot written as an exclusive monologue that keeps the reader outside the process. It is written as a dialogue, like Galileos Dialogo that includes Every man and his wife , kids and pets in the ongoing discussion!

This is the Socratic Platonic method, which rather than answering questions raises more questions in the minds of the participants. Each person resolves these on his own advices!

However, homiletics is a sham method that uses these techniques to convince you of a dogmatic view. It's a fine balance. Hermann holds a dogmatic view, it is his Förderung. But his view is clearly new and not dominant. Thus his dogmatism is acceptable as defending his point of view, rather than imposing a dominant societal view onto an open mind incapable of critically reviewing it.

One might also view his dogmatism as arrogance. Thus Hermann is constrained to persuade by apprehensible examples and logic by these possible reactions to his presentations. He has to walk the line between being a doormat and and a stout guardian of the prize( we call them bouncers in today's clubland scene!)

There are many versions of Hermnns ideas promulgated today, so I guess he managed to pull it off. Others however like Gibbs and Hamilon seemed to go for world domination ! They clashed and Gibbs came out the winner over Hamilton using the concepts and tools Hetmann had so generously laid out for all mankind. However Gibbs did not understnd Grassmanns Algebras, a fact he admitted in writing. Hamiltons Quaternions ontthe other hand he grasped but objected to the use of imaginary quantities! He was backed by Lord Krlvin and many other reactionaries including Lewis Carrol whose Alice in Wonderland stories contain a sustained attack on the mathesis of the imaginaries!

Gibbs " vectors" in fact contain bits and pieces from everywhere, and we're a crib sheet of useful notations and formulae . This is why he did not include Hetmanns use of the imaginary quantities of Eulet! Later developers of Gibbs crib sheet introduced imaginary quantities at a later stage, in particular Pauli justified their inclusion in describing electon behaviour data using Heidelbergs matrix Algrbra Approach( Weise!)

The point I am making is both Hamilton and Hetmann inspired by LaGrange attempted to set out a spaciometric Algebra, but from different dogmatisms. Hermanns was based on the perceptions of space inhere in our neurology, which he got at through the Hegelian dialectic method and approach. Hamilton was based on his academic status as an innovator in mathmatical Astronomy and physics and his defense of the doctrine of imaginary quantities. This was related to his religious and theosophical beliefs that imaginary quantities were somehow divine revelatory quantities, at last revealed by god to man so man might truly apprehend and comprehend nature as it really is!

Both Hermann and Hamilton believed they were doing gods work in the new scientific and poetic age, one for the Holy Roman Empire of Prussia, fast breaking up and in need of deep revisionary thought and reform; the other or the new burgeoning industrial powers of the west, brimming with new scientific and technological discovery and advancing bravely forward into a new visionary and poetic future. In fact Hamilton wanted to give up his mathematical and astronomical work to become a poet of this new breath taking vision, like Blake, Yeats and Wordsworth!

The contrast could not be more spectacular! The comparison though was their common conception of an algebra for space based on geometrivl concepts . But not the old geometry of Lesbegue ! Both extracted the Pythagorean geometry from that deadly maul! Hermann by reseating it on 3d space not Axioms, Hamilton by attempting to derive quantity from Pure Time!

Hamiltons imagination blinds him to the fact that he has to use spatial metaphors to describe his conception of time. But Hermann is trenchant , undeviating from what it is one is actually experiencing when quantising and quantifying. One cannot avoid space! Fortunately Hermann had direct access to Hegelian dialectical thought, which gave him a vehicle to express his perceptions like no other. Hamilton was trained in and constrained in Aristotelian logic. And not only constrained but devoted to his jailor! No wonder he felt trapped by his seeming success.

The more I study Hetmanns writing in these short passages the more I see the radically different perspectives that led to an overlapping terminology, not at all describing the same things!

I have adopted the word label for consistency so that Begriff which I intuitively prefer as a gripper or a handle, might reveal the concept I believe Hermann perceived. Words as labels, soon and easily reduces to symbols as labels. And in this section the labels being derived so far are the words describing the ontological aspects of things hitherto only thought of asThoughts.

Commentary
I attempted to distinguish what I feel are the labels Grassmann is deriving by capitalising the initial letter. However this does not work so well in German as the nouns are routinely capitalised in this way! However it shows how natural it is to see words as labels for concepts if one adopts this approach both in German and other languages..

The important labels by the end of this section are the Continuous and the Discrete label. To this I personally would add the Contiguous label as representing the resolution of the two other labels and incorporating the setting of discrete parts as givens for a combination process that results in a Contiguous entity.

Becoming and Rooting and Rising Up as someone or something are also fundamental labels.

http://www.youtube.com/watch?v=nsZsd5qtbo4

Video one on undecideabikity on computer phill channel

http://youtu.be/nsZsd5qtbo4

Video 2 on computerphile about undecideabikity
http://youtu.be/FK3kifY-geM
http://www.youtube.com/watch?v=FK3kifY-geM

 Ausdehnungslehre 1844 induction


5. Every special / specific entity( no.3) becomes one such  through the label of the Differing Elements group wherethrough there will be coordinated into one group various specific  entities, and through which entities, entities of the likes group will be coordinated into one; wherethrough  there  will be subordinated to the same general entities  an "Other Special Entities"  group  

The thing reified out of the likes group  we can name the algebraic form, the thing reified out of the differing elements group we can name the combinatorial form

The contradictory statement of the like things list and the differing things list is equally a case of a fleeting fluid thing. The like thing is perceived as different, already as much as one and the other like thing to it are separated in some way (and without this separation it was a single thing, thus not a like situation); the differing thing is perceived as like already as much  as an entity of both is linked, through  the practice which itself is related to both, thus "both" is one combined thing.

But now , both limbs in no way are there abouts immersed in one another so that a measure stick one was having  to apply through which to appoint how much likes  to be set between  the presenting items and how much differences , but even if also  the likes always some random value already on the differences  are adhered and vice versa still every time in the moment a unit is built by the tracking during which the latter appears to be the presuppositional foundation of the former.

Under the algebraic form is not just simply the tally numbers, but also that with which in the field of the continuous things the tally numbers are inter communicant; and under the combinatorial form is  not only the combinations but also that which is understood to be with continuous things inter communicant .

.

I have been at my daughters wedding for a few days , so I translated a section on my iPhone on the plane to and from Serbia, now my second country and where my extended family lives!

4th October is a wonderful memory  now! 4 days immersed in a different experience language and culture with ones whole family and new family ina positive celebration is life affirming!

Happy Festivities to one and all!

In attempting to answer a query I googled back to some of my earlier posts in the fractal foundations thread

http://www.fractalforums.com/mathematics/foundations-of-mathematics-axioms-notions-and-the-universal-set-fs-as-a-model/330/

It shows that the concepts I am translating from Hermanns work are ones I expressed in my own way and words in 2010.

This of course only means that these concepts are accessible to a person willing to think deeply and logically about foundations of the things we take for granted or as givens in mathematics.

Do not be fooled, therefore , into thinking mathematics is not just a product of exercising careful dialectic thought patterns!  It is intuitive only because we have synthesised it to be intuitive. Like any craftsman, we are able to construct a product that resonates intuitively with how we prefer to carry out the thought act/ process.

http://en.m.wikipedia.org/wiki/Classical_Hamiltonian_quaternions

It turns out this link is very relevant to this threads topic. The approach that Hamilton used to create/ invent quaternions relied heavily on geometric,otive tots,like Hetmnns!

The debate about infinity between Norman and Jim

http://youtu.be/WabHm1QWVCA

http://www.youtube.com/watch?v=WabHm1QWVCA

The case for infinite sets is not very coherent! But basically he is using some version of the distinctions that Hermann is proposing in the Einleitung. Thus the case is barely comprehensible without the Hegelian dialectic that underpins it. Jim is advancing an Aritotelian philosophical view, but heavily eited and critical of Aristotle! Norman is claiming to be a former Platonist looking for a philosophicl basis , and evidently interested in Aspects of Jims Aristotelian approach. Like me before reading the Vorrede I had never heard of Hegel, nor yet how influential his philosophy has been on shaping the modern geopolitical dynamics!

The Hegelian Grassmann philosophical basis for mathmatics is what I am translating now, and hermanns theory of Labels or Handles or Types, by which he avoids much of the procedural difficulties of infinite processes, infinite sets, and through which he is able to ply the Hegelian dialectical method of Alysis and synthesis.

Norman claims that infinite sets are defended by the mathematicians who wanted to found Mathematicl Analysis, especially the continuum. However algebraists do not require a continuum they need discrete labels. Thus we can see a Hegelian dialectical thesis and antithesis in this debate. the synthesis is precisely what Grassmann is dealing with in the Einleitung.

One can see from Hermnns viewpoint that Norman nd Jim are not even at the deepest split of the basis of mathematics! The deepest split in Hetmanns label types is between Algebric form and Combintorial form. In this analysis Zahl, Anzahl take on a specialignificance. They are first of ll not numbers! They are Tally marks. And these tally marks are applied not just to the continuous entities or in general the continuum, but also to the discrete entities in general the different spatially separated like things or differing things. These are perceivable as a unit by a momentary thought act in which the underlying process under consideration, at the time, requires them to be seen as Givens to e tied into a new combined thing or entity.

Thus the distinctions which so divide mathematicians in particular are fleeting and fluid.

The synthesis of the Hegelian method allows unlike things to be combined. This is a fundamental structural rule in algebra. Like things are gathered together, thus forming lists of like things which lists themselves are different " things". Thus at the level of the lists the list of the lists is a list of different subsists. This data structure in which the list idea can contain sublists is crucial to the synthetic resolution of many of the problems with the taxonomic linguistic Aritotelin analysis and approach.

Hermanns Hegelian approach sees the resolution of any distinction in a " higher" level which has the same structural format or layout( for example a list) but may have different characterising definitions. Thus whether we call a structure a list, a type, a set, a collection, a class etc, is fleeting in importance as a distinction because fluidly we can now apply the same structure at a higher or lower level in a Taxonomic structure.

The simple procedural marker is the concept of type and subtype, but also type and super type!

Continuous and discrete entities thus become fluidly combined into super type entires, and also may be constructed from subtype entities. Thus some thing that at one level may be said to be continuous at a sub level ay be made from discrete givens, and vice versa. For this to be the case we have to recognise our perception of the momentary nature of becoming! This is a deep psychological and neurological fundamental to all our systems of knowledge and expertise.hermann points this out by the use of analogical comparisons . What we think we know can be characterised by the dual cognisances of how things come to be in our consciousnesses.

Commentary

Something is special, specific or particular, not of itself but because we have a category of specifics, items which vary from one anther so no 2 are the same. But we can also have a specific item as a unique element of a category of likes . Thus this one element differs from every other, but every other element or item is identical to each of the others.

Thus the concept of special requires a group of differing items and or a group of uniformly identical items in order to support the perception of special or specific.

But these groups are structured ino lists and sublists , categories and subcategories by this same process of identifying likes and unlikes which resides in the process of forming a group of differing entities. Necessarily this process means we gather like things together in order to distinguish the differing things, and thereby we construct sub groups of like things which subgroups differ from one another precisely by what defines the elements within a subgroup.

More deeply we only have like elements if those elements differ in some way! Thus if there is no difference we cannot distinguish like elements from just one element, the same element! Unless we make or allow this distinction like things would not exist as a concept: there would only be one unique item in a sea of unique items!

However differences are also perceivable as like properties, especially if some practice or job requires both elements to be employed together to perform some task. In that case the differing elements can be seen as a combined entity necessary for the performance of some Activity

The label of the algebraic form requires a Metron. This Metron is a measure of how much like and unlike elements we might need to combine to create or synthesis some combinatorial entity. The use of a Metron encourages counting discretely, but we must not forget the continuous nature of the Metron.
Thus the algebraic form encompasses discrete continuous and contiguous entities which themselves will have a combined form or format.

But having said that this contradictory statement about likes and differing groups is only apparent. The distinctions merge but not so completely that one requires some measuring tool to assign ratios of each to every presentation! Thus they can be continuous or contiguous or discrete it does not matter, as long as it is recognised that in the moment of gazing on the apparition of them they appear as a unit with the likes being the foundational presuppositions for the unlikes and vice versa.

The underpinning idea is the structural category subcategory division of apprehension. While Aristotle is one of the great Taxonomists, it is not apparent that he used his taxonomic predilection as a tool of synthesis not just a way of recording Analysis. Hegel insists on both analysis and synthesis in his philosophy. The knower can only be realised as a self referencing entity that also references other self referencing entities is a Hegelian dialectical stage. Hermann is using these ideas and approaches to induct us into a freeer more flexible mindset.

Commentary
No.5 is a crucial bridging step to the derivation of the labels for extending magnitudes. It is worth pondering for a while.

The theory of Types or Categories that underpins it is a contribution to logic which is original to Hegel. In one great schema he sets out a basis fr Cantor set theory and Gödels incompleteness theorem. Both are compounded within his categorical system of thought. Within his synthetical system one does not come to a halt or stalemate. As in the Aristotelian system, rather one is inspired to resolve the situation dialectic ally and to move forward to a new category of representation.

If you learn nothing else from this induction learn to embrace contradiction as a creative state of affairs by which one may move forward to a new synthesis in a categorical way that frees your " Geist" from imperfect limitations.

This is no trite or simple process or procedure. Rather to resolve such contradictory situations may require huge efforts in Analydis and dsynthesis to ensure that foundations for the new resolving category are secure. Thus Hermanns one man effort in deriving the Lineal Algebra will serve as a detailed example of how this can be done.

You will note that in doing do many familiar labels and names take on an altered meaning or apprehension, and logos and analogos as conceptual thought processes are given equal weight. Validity is not just the gift of a "logical" or Aristotelian argument.it is also the gift of an argument by analogy . In this way a dialectical process can be sustained on one or both types of argumentation or demonstration.

We have been encultured into thinking" logic" is merely Aristotelian logic, and only that logic is " sound". However those who study logic know this is not the case. The so called ontological argument for the existence of God is a famous example, but the categories of argument even ibAristotelianlogic run into the thousands! Most of us only know a handful of the Arisyotelian arguments, let alone anyone else's!

In addition the Platonic and Pythagorean forms of logic are often obscured by or viewed through the Aristotelian. By introducing his system of categories and subcategories Hegel liberated the subject of logic from an academic backwater into the mainstream of geopolitical real politic and religious dialectical progression. In addition he provided a strong critical analysis of where these subjects were dodging the issues, and avoiding the inevitable progression to human absolute realisation of Geist.

Hegel is difficult because he is unrelenting. Similarly Hermann is difficult because he is unrelenting . Nothing must escape the scrutiny of analysis or the practice of synthesis.

Lord Xenus YouTube site provides a great reading on the Classical philosophers by Charlton Heston. In my opinion the section on Hegel, in 2 parts is very clear and informative .

https://www.marxists.org/reference/subject/philosophy/works/ge/gadamer.htm

This link to Gadamers essay of the Science of Logic by Hegel is a bit more technical but also helpful.

The philosophical background that Kant inspired in the German / Prussian Renaisance especially impacted on the Humboldt educational reforms through Humboldts own philosophical conceptualisation of Self actualisation. Thus the content of the German philosophical idealism was directly applicable to the process of reorganising the Prussian education system to produce home grown intellectual and technological talent.

Philosophers like Fichte provided a philosophical basis for reorganisingScirnce and mathmatics on clear logical lines. It was this kind of Philodophy that heavily directed and influenced Justus Grassmann snd Robert Grassmann. They were motivated to implement its implied programmes into the primary education of the young especially in Stettin(Sczeczin).

In this regard, Hermann introduced the more up to date critique of Fichte as advanced by Hegel into the work of the Grassmann household. However this was not an uncritical or slavish acceptance of Hegelian philosophy by Hermann. It is clear that in the Grassmann household these philosophical concepts were discussed and put through the ringer to squeeze out every last practical application. However in the main Hermann seems to have bern heavily influenced by Hegel's contemporary style and lectures.

It has been mooted that Schliermacher played an important role in formulating Hermann and Roberts views, but it is clear that the philosophically rich environment and time in which they grew up lacked no champion for modern Prussian Idealistic Philodophy! They literally could take their pick.

So why Hegel?

I think because Hermann like myself , wanted a universal all encompassing apprehension, and yet still appreciable by simple rules and patterns. Hegel's patterns, despite the complexity of the content they deal with are very easily graspable and very fractally  simple. The pattern of category and sub categories  is fundamentally a pattern of 3!

AB + BC = AC

fundamentally and resonantly models, is analogous  to, is a metaphor and simile of every structural aspect of the Hegelian system . Thus Hegels categorical system of logic, and his fundmental dialectic operation or process is all captured or alluded to in that simple childish, childlike observation Hermann made when he was a young child.

Hegel and Mathrmatics a Marxist Perspective

https://www.marxists.org/reference/subject/philosophy/works/ru/kolman.htm

A resource on Hegel.
https://www.marxists.org/reference/archive/hegel/index.htm

It is important to realise that Hegel's system was immensely influential but heavily criticised. Hermann Grassmann was one of its critics! So was Marx. Nevertheless we find mentioned a corpus of Mathemtical essays and research by Hegel which may be source material for Hermanns utilisation of the Hegelian system or method.

The attractiveness of the Hegelian approach is that you bring what you have to it, start from where you are and engage in a dialectical process a la Hegel to create your own imperfect understanding that nevertheless is a new foundation for further dialectical" purification". Ultimately you arrive at a stage which is a status of your intellectual or reasoning capability at that precise time or moment which has resolved many contradictory statements or apprehensions in a positive rather than negative direction giving you greater freedom and flexibility to progress to further resolutions and solutions.

While Hegel assumed this was a process uncovering an absolute "truth" , one is never in a position to determine that absolutely. Instead one gives or is capable of giving many processes that arrive at the same result, thus establishing a basis for moving forward in the dialectical process.in a positive fashion.

The principle of Acceptance is the most fundamental principle that I have ever formulated. It is a principle that is akin to the axiom of choice, but more fundamental. It is an empirical principle by which all empirical experience is founded. Without the principle of acceptance I assert that no thought or consciousness is distinguishable as self aware and worse still is distinguishable by any thing referred to by the label "language". Thus experience and thought of experience and self awareness of experience , being and Nothing etc are empty in precisely the sense Hegel describes and even in the sense of a holographic projection.

The limit of my labelling and there referential ability, or if you like the content of a conceptual label, is the limit of my acceptance, or the limit of acceptance. In this way acceptance is how everything that may be thought or languaged is reeified .

It does not make sense to say that without it nothing exists, because "I" saying that , do so having accepted , consciously or worse yet unconsciously that which roots and rises up in that moment of acceptance, or which "manifests" out of "somewhere", both of which labels again do so in a "moment", which itself is apprehended in that same moment.  I cannot get beyond that languaged description either by language or by thought or by experience. The moment I do go beyond,  I justify that acceptance has operated and been operating at every level . Essentially I am an entity in a fractal consciousness engaging by acceptance with a fractal distribution of that consciousness.

Moment therefore is a fundamental concept , primitive idea which the dialectic process synthesises into time, a more complex and dialectically advanced notion. Indeed the notion of a process, any process whatsoever draws upon a primitive sequencing and ordering in experience which itself is perceived in a primitive moment.

The more analysis that is attempted the more attribution and generality  has to be appended to my primitive moment. Thus "a priori " is simply Italian for acknowledging this impassable foundation of consciousness.
Kant's Transcendental schema is simply a more elaborate version, or an elaboration of the same experience.

Hegel thus starts his synthesis and dialectical process of synthesis at this general, universally so, moment.Hamilon in his Theory of couples subtitles it as a science of Pure Time, because he starts in the primitive moments of an assumed progressive sequence of moments, as the reader apprehends them by recollection.

Thus Hegel does not shrink from grasping the thorny "turbulent" or fractal nature of a priori " knowing" or knowledge. The principle of acceptance includes the principle of assumption, the analogy of rooting and rising up, the process of setting down arbitrarily, anywhere and building fom there.

The principle of acceptance is therefore just stating and identifying what every philosopher and philosophy does at its outset. It includes the concept of postulates, aitema in Greek, as well as axioms ( from the Greek for axle and axis) as predicate concepts of the subjective action of assumption or accepting. This language model or analogy may help some to apprehend what is active in the distinction being made by the statement of the principle that is acceptance, but it does not describe an active act, but rather a completed or instantaneously past action. As you may feel, we are at the ultimate limit of our ability to describe this in the language model.

A meditative experience of these things is recommended, but the subjective experince of meditation is a private profundity. It cannot describe to another what has been experienced. The best individuals can do is accept that others have had similar or identical experiences and actively share time and discourse with those they are " intuitively" drawn to. For what purpose? Well dialectically let's say it is good for a few beers!  ;D

In the previous post I highlighted the limitation of languaging. Thus at a stroke logic is limited.
However what presents itself then unfettered by language is the Sensory mesh and that Hegelian unit or category which is everything to do with the senses prior to languaging. The enity or entities I refer yo as sensory mesh or meshes and their Essential quality or qualities. The idea of Essnce here is in contrast inaction to their being or entities/ ontological status. Again the principle of acceptance underpins these a priori experiences.

The signal, or state change in a sensory mesh is of fundamental importance, mainly because it is a precognitive and for the major major part unconscious interaction between the a priori entities and structures.

I shall say that a computational machine of the autonomous sort we label as a computer would be a sufficient analogy in this discussion. The various inputs have a correspondence to senses and to empowerment. The various outputs have a correspondence with consciousness, determined action and the like. For example we may usefully correspond the output screen to the visual and imagination functions within our experiential continuum.

Between the inputs and the outputs is a technological world of a priori interaction, which, in the Hegelian Analydis must include prior and posterior action and interaction! Thus feedback and feed forward interactions must be in the processes that give conscious outputs that " seem" to require a posterior I knowledge nd knowing, or to require us to " learn" before we can act wisely or safely. However in many animal cases this " instinct" is endemic in the unfolding behaviour of a new born. Thus giving credence to information communication which is non languaged!

For want of a better description we call this "symbolic coding"! The history of symbolic coding is long, but for computing Alan Turing cannot be ignored. Because of his work in encoding and breaking codes he was able to conceive of a communication mode that was symbolic and evanescent of a natural language nd yet not a recogniseble language at all.

In this regard symbols become important. What symbols refer to becomes important, and how they refer zooms into view. Reference by simple " attachment" or tying together or association is crucial. Such an attachment could be visual , kinaesthetic or auditory, wht ties them together is more often thn not position in a sequence in a sequence of moments. To make the idea more complex it ids the time position in a time sequence that links or ties symbols to their referents.

It is tempting, like the Greek seemeia or seemeioon to think that indication links 2 things together, that some indicator or indication is sufficient to identify what things are linked, but this is not the case, evn practically. For example if one points in the direction of n object evn if it is the only object in that direction it is still not clear what is being referred to. By physically touching or tying the object to the observer it is implicitly clear what the reference is.

Thus a spatial link between symbol and reference is necessary nd this is conducting the attention along a time dependnt sequence where at each moment the symbol the linking strategy and the reference are percirved as a unit or a Heglian category.

In more direct analogy I refer you to the power of symbol, or logo or symbolic parphenelia in religious or secular contexts, pictograms, or syllabaries or syntax manuals as examples of effective communication of specific referrents whether quantitative or fomal, or qualitative and experiential.

Labels and symbols provide a means of communicating about something in a generally none specific way capable of many interpretations..

How we label,and refer to extensive and extending objects snd quantities and qualities is crucial to how these labels may be organised into notations that help or hinder our communicatin in these Language restricted domains. Principally because so much language when referring to these experiences is functionally " empty" of any additional content beyond what is being and can be labeled by a symbol .

I am continuing the transition of the Induction by Hermann Grassmann in the 1844 version of the Ausdehnungslehre, but I can already grasp the purpose of the mixed product which is the subject of this thread and indeed the subject of Grassmanns referred paper.

The Grassmann use of the Hegelian method and dialectic is sufficiently distinct to focus on it as a particular form of a more general Hegelian conception.

It is necessary to analyse anything into at least 2 related but contradicting conceptions, or cognisances. Grassmann chooses a complex contradicting pair each of which is itself a contradicting pair! The fundamental characteristic of these primitive contradicting pair is that they are continuous and discrete, and further thst distinction relies on a further or deeper but more general contradiction between sameness and differing( active sense). There is Also a completed sense ( same and difference). The consideration of these tense or time differenced experiences is also considered in the full treatment.

So we already feel the complexity, and with it the dialectical pull of a dialectical process of solution or resolution of the contradictions. We might in the vernacular call this simplification but we would be misled if we then related that to easier or more fundamental! So it is called a dialectic resolution.

The closing in product and the spreading out product are two differing processes of describing the same thing.

If we restrict the discussion to the parallelogram we might conventionally say they both refer to the area of a parallelogram. Yet in doing so we pose more questions than we answer. What is area ? Why is area calculated as it is?
The dialectic system requires 2 processes in contradictory statement form. Hermann provides this by using a label called the closing in product and a label called the spreading out product to refer to the product of a parallelogram.

These products are perceptible in the geometric expertise, not in the Arithmetic one, where area is the only conceptual relative!

To sharpen the contradiction Hermann chose labels | and ^ for the closing in product and the spreading out product for constituent elements which are entities in geometry, say points, limes, planes, volumes etc.

For these elements he chose labels usually alphabetic letters. Also, to give himself more flexibility he used tally marks( numerals) to both economise on the use of alphabetic symbols/ labels but also to subtly and efficiently indicate sameness and differing, or same and different.

The binding symbols were chosen as+*–/. The subtlety of these symbols is key to his notational representation. Consequently it is vital to read his contextual exposition of the meaning of the symbols in context. It is also vitally crucial to read and understand his Vorrede and his Induction. Without these the tendency is to follow Aristotelian logic rather then Hegelian! Plus, the dialectic process is not evident in the Aristotelian process.

Induction is an Aristotelian principle , exposited by Aristotle from older Pythagorean and Platonic teachings. Intuition and induction refer to the interaction of the Musai on the sensitive soul or spirit of a human or animate consciousness. Hegel, starting with Aristotle goes beyond, particularly to synthesis of all that is attained or apprehended by analysis as set out in a book misnamed the Metsphysics.

So Grassmanns induction sets us up for the Heglian type process. The two products are contradictory in some way. One relies only on the geometric entities the other relies on a projection of the entities into each other.

Using the projection product , the closing in product, we can derive the arithmetical formula for area of a parallelogram. The arithmetical product is a correspondence to the geometrical product which is merely an entity projected into another entity using entities of construction. What results can be labelled, and the labels themselves now reveal a common Arithmetic form! Thus b|h is the same as bh or bxh. We are here identifying a product of a geometrical process of construction with a conventional arithmetic formula!

Now what about a^b? This geometrical product which is a construction using a parallel projection has no arithmetical formulary! Thus Grassmann reveals that arithmetic is a restricted model of what is geometrically possible! Resorting to arithmetic is like poking your own eye out! And being thus self blinded do you, oh mighty pedagogue, deign to teach the young who see more perfectly than you?

What we have is 2 contradicting statements about the parallelogram for example if we restrict the entity to line segments. What is the interpretation of the spreading out product in arithmetic? It has no interpretation. Thus we must resolve to a more general concept.

What is the interpretation in Geometry? The answer to both products in this case is the parallelogram.

The succeeding question is to what use can these differing products be put in Geometry?

Further analysis was required before this could be answered and the analysis revealed several other differing products that were arguably more primitive. These Grassmann organised into 3 types. Commutative, anticommutative and imaginary or roots of unity!

I say " organised" but in fact they are mutually antithetical. They contradict in the required way to have a Hegelian category of their own. These more primitive products are not arithmetical, but Geometrical. They were universally named as Higher arithmetic, but this was and is a mistake that Hermann addresses at the outset, the misfounding of Geometry on a formal axiomatic definitional basis rather than on a real, empirical experimental one. Setting this " right" allowed him to avoid the mental contusions of those trying to make geometrical objects, entities and processes into formal arithmetical ones!. It also avoided the profound confusion!

What did he discover by this result of further analysis? The products represent different projections, vertical, parallel and ROTATIONAL. Immediately a fourth projection suggests itself: none vertical or parallel! This is precisely the principle of projective geometry pioneered by DesArgues, but of course of ancient origins.

Suffice it to say that Grassmann was now free to construct mixed geometrical products! In this case the | product was defined as a vertical projection product ( although this does not have to be the case as it could be a projective geometry projection) and the ^ product was defined as a rotational product.

How can AB be both the rotational projection and the parallel projection? This lovely contradictory statement inspired further analytical research which happily was inspired by Lagrange's Analytucal Mechanics and work done on solving the Ebb and Tide problem. Grassmann was able to identify the rhombus as the parallelogram that contained the resolution. In addition it linked not only the trigonometric projections but also the hypertrigonometric proections to the rhombus parallelogram. From this he was able to easily draw on Eukers wrk with the hyperbolic functions and establish the link to the imaginary  calculus functions!

So Grassmanns Hegelian approach was doing what Hegel said it should, bringing together contradictions into a new dialectic resolution that promoted higher and more productive reasoning!

From Euler certain factors and eigen values are transated into the mixed form for the geometric product that is an archetype for the Quaternions, that is a Geometrucal archetype from which Hamiltons Quaternioms may be " derived" by analogical equation.

Because it is an analogical equation it cannot be described in arithmetical terms. However the rise of set and Group Theory with ring theory allowed analogues to be described as operators. Thus the operation in an analogous group is cross applied to an arithmetic situation to generate a result or value which frankly would not be arrived at by ordinary arithmetic means.

Now the term Algebraic came to be used to describe these general group theoretic sets , classes, categories etc. why this is so is perhaps what Grassmann is about to explain in the next few sections of the Einleitung.

http://books.google.co.uk/books?id=lpAmDtAh-nQC&pg=PA4&dq=justus+gunther+Grassmann+verbindungslehre&hl=en&sa=X&ei=hytCVOyhKcbd7Qa1rIGACw&ved=0CCUQuwUwAA#v=onepage&q=justus%20gunther%20Grassmann%20verbindungslehre&f=false

There is a whole deep underpinning of ideas that Justus Guther Grassmann set out for school children between 7 to 9 years old and for teachers with little real understanding of geometry. These ideas were published for his school district prior to 1827, (1817) using the title " Verbindungslehre".

Prior to this study I had taken the term to mean " group Theory" at least in an early nascent sense. But as I looked into the German useage it became clear that Connection and connecting was the paramount activity. Thus it is a much broader conception than the label " group Theory" .  And yet it is here perceived as an elementary subject!

Perceptions of what school kids should know and be capable of doing change constantly, so it is of interest to read this material at some stage in the study of Hermanns background education.

It is of interest that Justus proposed these ideas prior to Dedekind and to Riemann as part of a small international group of researchers looking for a natural foundation for the sciences, the philosophies and mathematics. The idea was clearly that of a theosophical group who looked for natural analogies of spiritual or Reason based conceptions. In that regard Hegel's work and philosophy starts with the evident assumption that the processes of the natural world are the processes of the absolute, or the free Reason to which he sees everything eventually perfecting by this process itself. He called the process Dialectic, and it is a synthesis process.

The study of how all these things are connected to each other is thus important, but Justus and his group are interested in a very specific subset of things, namely the 2 dimensional geometric entities, lines ,rays and points. The connections between these things and various combinations of these thins in connection with the tally markers is what he called the study of connections.

I also recognised that Lehre is more pedagogical than the word study, and perhaps teachings or doctrine is a better emphatic translation, but that is not as reader or student friendly as study, so I will keep an eye out for any dogmatic tones I maybe overlooking.

Ausdehnungslehre   1844
Induction

6. Out of the criss crossing of these  both contradictory statements , from which the former relates itself to the artform of creating whole and the latter relates itself  to the elements of creating, go forth   the 4  categories of the thought forms/ patterns; and out of which the  to them inter communicant branches of the study of thought forms/ Patterns go forth. And indeed the discrete thoughtForm/ pattern separates initially  there according to that  in Tally mark and Combination (  bundle).

Tally mark is the algebraically discrete thought form/ pattern , that is it is the summary grabbing together of the entity  as like "set" things ; the Combination is the combinatorially discrete thought pattern/ form that is it is the summary grabbing together of the entity  as differing "set" things. The expertise of the Discrete things is thus the study of Tally markers and the study of Combinations (the study of things  connecting to things)

That herethrough the label of the Tally mark is completely  exhausted ( prescribed/assigned) and exactly circumscribed, and plainlly so the label of the Combination, is wholly hardly needy of a further after thought( proof/evidence) . And there the contradictory statements through which these Definitions have  gone forth from here , the simplest definitions, in the label of the mathematical thought form/ pattern, are without mediation with Givens .

Thus  herethrough the above derivation, in sufficiently far reaching manner is well finalised correctly.(QED!)

I make still only a remark:

How that this  contradictory statement between both thought forms is expressed, through the different signing related to their elements, onto a very pure cognisance, in which the Tally mark  linked entity with  one and the same symbol will be signified (1); the Combination linked entity  will be signified with differing things; in the remainder completely  arbitrary  signs( the printers block!)wil be utilised.

 Now hereafter every crowd of things( specified system)How that  can be so well apprehended as Tally marker ,
how that  can become  so well apprehended as Combination:  each of those according  to the differing manner of expressing tracking, well hardly permits a passing comment!


Footnote to this section page xxvi
The label of the Tally mark and the Combination has already for  17 years in " Treatises" penned of my Father , running over the label of the pure doctrine of the Tally Marks, which in the Curriculum of the Stettin High Schools from 1827 has been  distributed in print, on a completely ancestrally related manner has developed , but without having reached the Attention of a bigger Audience.

Commentary
In section 6 Hermann claims to have finalised his goal of deriving the labels of the Tally mark and the Combination. It is doubtful if the reader even knows until that point that it was a goal of his. However, in the dialectic process the usual Goal / objective and operations and testing to achieve that goal, the TOTE method is not as simple as in the syllogistic approach. This is because Analysis is specific in orientation while synthesis is general , seemingly arbitrary , in orientation.

To be sure one can have a goal in building a house, but there is so much variety possible that tying down a specific design is more of a guideline than a specific outcome! What results is what results! It may be specific to the design, but everyone knows that to achieve that level of accord requires a particularly strict and literal, even robotic diligence, and a willingness to tear down what is constructed if it deviates in any way! Only one in 10 million would ever be that demanding, and they would certainly be classed as Autistic, or worse still a perfectionist.

Hermann may be autistic, but he is certainly not a perfectionist! His rigour varies in these descriptions, because he is human and not a robot. On many occasions he asserts the conclusion as self evident, or not requiring further proof! This is always an appeal to common sense, but it is not logically " sound" in the Aristotelian sense. It is an appeal to pragmatic sensibility, and to the principle of exhaustion. What exhausts one mind or person may not exhaust another. Thus these appeals constitute a " logical" weakness in that further analysis is possible at these junctures.

The " point" is when do we stop analysis? The Pythagoreans state through Euclid that we stop when we get the signal to stop, the seemeioon! The seemeioon is often translated as point, but it literally means indicator . The indicator has no parts! This is Euclids declaration at the beginning of his philosophical synthesis the Stoikeia. Analydis stops when there are no more parts to disintegrate! But long before that analysis stops due to exhaustion of the analyser!

Hermann thinks he has gone deep enough, or reache far enough to justify his definitions of the Tally mark and the Combination. But we may go further today, especially with the fabulous tools we have for measuring and perceiving.

Nevertheless few will undertake the task of digging deeper. In fact even Hamilton was astonished at how far reaching Hermanns analysis was and is! There may be some who claim to have gone deeper or further! And I may be one of those, but I doubt it as of now. The metaphors of deeper or wider or further are empirical, and based on extending magnitudes in a real expertise. However Hermann is not concerned with the real expertises here. His sole purpose is to construct the formal expertise of Formdnlehre, or thought pattern doctrine. Until now I had not even realised this level of distinction consciously, although intuiti Ely I recognise every intention Hermann has as an experience I had when I was classing myself as seriously depressive! This was before I reclassified myself as Autistic.

Unlike Hermann I cannot exhaust a thing! I recognise constantly that what I write, know or understand is already incorrect and out of date! Thus all my writings are thought dumps, just like core dumps in the old days of computing. I do not defend a position as Aristotle may imply is possible, because I think it is impossible in a changing environment to have anything but approximate knowledge. New information leads to new insights and the overturning of old constructs .

Where Scirnce goes wrong, but they are human after all, is to try to prevent that dialectical process! A professor would rather sit on or throw out of his class a person who threatens his cherished dreams and ideas. Science would rather denounce inventors as fallacious than lose access to thrif cash cow. Etc.

We all crave certainty, but there is none in my opinion. What saved me from insanity a la Cantor was the fractal zoom! Once I could accept infinite regression, something Kant refused to do, I could leave my interminable analysis alone! I could park it anywhere and go for a walk On the beach! I could start building and synthesing anywhere just to see what resulted. I was free at last.

Hegel has given me a philosophical mentoring in a way that does not make him the Don, but rather makes the process the dialectical one a useful pattern tool to utilise. Hermann in using this tool, however modified provides me with an example of what can be constructed , if one wishes and is dedicated enough. Unfortunately, Hermann believed or acce
Ted what hewasconstructing was " the Truth", and for this he suffered great pains and
Erdonal privations . The key is not the concept" the Truth" but the concept of total acceptance and dedication. I have accepted and dedicated myself to a handful of things inmy life, and that has built my experiential continuum as is. I recognise that once I stop accepting change I will become cut off from life, thus the fewer things I have to change the easier it is to continue..

The only constant in my universe is Change! Panta Rhei!

I place this video here because the issue is dealt with in Hermanns preamble to the derivation of the 4 categories in section 6. In particular we can see the Hegelian conceptualisation of nature and natures reason and the Geist or the Absolute of reason have a realisation in this discussion.

Why do physicists struggle? Because physicists and philosophers have not applied Hegelian logic or Philosophy. Is the situation only describable by quantum mechanics? The answer is no. Prior to quantum mechanics was Hegelian logic and his Phenomenology of Geist. Bohr was wrong in accusing all western philosophers because he did not know Hegelian Philosophy. In fact he directs us to mystical mentors Buddha and pragmatists like Lao Tse who nevertheless was based in traditional Chinese philosophy, and thus Indian Philodophy, that everything is connected in a complementarity that is contradictory until resolved, precisely the doctrine of Hegel.

Do I negate mystics by Hegel? On the contrary, Hegel embraces them all, and they Hegel. What I negate is the science that is so local that Bohr has to go to china to find a philosophy that matches his thinking!

http://youtu.be/BFvJOZ51tmc

http://www.youtube.com/watch?v=BFvJOZ51tmc

If we seek a definition of Ausdehnung and Ausdehnungslehre section 7 gives us one that will blow your socks off!

As far as I am aware extensive and intensive magnitudes were a creation of Gauss. Hermann defines the intensive magnitudes as being Created through algebraically continuous likes. Whatever way you cut that you get a continuous uniformity, the kind of entity that looks the same no matter how much it is scaled, added to or subtracted. That is to say you can look into the space and not know where you are relative to anywhere else in the Spacbut the Ausdehnung or extensive magnitude , or if you like the extending or extensive magnitude is defined as a creation through combinatorially continuous differings! We could say contiguous differings form the thought pattern of the Ausdehnung, and the Ausdehnungs Lehre is the study of those contiguous differings.

But the foundation or the prior art for apprehending these Ausdehnungen is function theory and the differential and integral Calculus! Actually it is not quite that bad. All we need is the foundational concepts of these subject areas.

Ausdehnungslehre is thus the doctrin of extensive and extending magnitudes.

Here is an accessible lecture about the Issues contemporary with the work of Grasmanns Justus and Hermann. What is not clear, until you read the Astronomical Principles in particular, Leibniz as an instance Lagrange as a modification and Euler as a root and branch refunding is the role of extensive and extending magnitudes in the calculus. These are none other than the infinite sums and series with which Newton established the dynamic nature of His Fluxions, and which Leibniz eventually arrives at as a general method for integration, and Lagrange explores in terms of coefficient multipliers( a Newtonian method overlooked by others) and Euler sets out as foundational principles of his exposition of  Calculus.

http://youtu.be/D_qujUbNXHw

http://www.youtube.com/watch?v=D_qujUbNXHw

Ausdehnungslehre 1844

induction

7. Plainly, in a  similar manner the Continuous thought  pattern/ form (or the magnitude there concording ) separates itself into  the Algebraically continuous thought  pattern/ form (or the intensive  magnitude) and into the Combinatorially continuous thought pattern/ form( or the extensive magnitude). Thus The intensive magnitude is that entity which is reeified through the creating whole of the Like entity. The extensive magnitude , or the extending magnitude, is that entity which is reeified  through the creating whole of the Differing entity . That yonder entity builds The foundation of the function doctrine , of the differential and the integral Calculus   as everyway varying magnitude, this hereby entity builds The foundation of the extending/ extensive magnitude doctrine

There, from both these  branches : the former, (the study of Tally marks), is nursed as a higher branch  (to become subordinate),  but yet the latter appears as an until now unknown branch; so it is necessary, without farther delay this difficult tracking  through the label of the continuous fluid and fleeting entity to declare closely: how in the Tally mark the singularising/ unit defining steps  hereforward , in the Combination the separating/specifying of the " together!" thought steps  hereforward. (Thus also in the intensive magnitude the singularising/unit defining  of the Elements, which   in accord with  their label indeed have still separated/specified the intensive magnitude, but which Only in their "essential nature", to be like themselves,  are  making a representation of  the intensive magnitude)

On the other hand in the extensive magnitude the separating of the elements which indeed as far as they make a representation of a single/ unit  magnitude are united , but which plainly only in their separating apart from one another are "constituting" ( specifying the constituents of) the magnitude.

Thus there exists likewise the intensive magnitude -the fluidly reeified  Tally mark; the extensive magnitude - the fluidly reeified  Combination. The latter one is essentially a stepping out of one another of the elements, and  a firm attachment of the same elements as out of one another entities.

The creating whole element,  by the latter it, appears as a self varying entity, that means through a differing system of status/ condition markers it appears as a  "from afar through travelling" entity , and the Gathered System of these differing status/ condition markers plainly makes a representation of  the field of study of the extending / extensive magnitude.

On the other hand, by the Intensive magnitude,  the creating whole of the same itself delivers a continuous Rank Array,  of self like status/ condition markers,  Quantity of which is plainly the intensive magnitude .

As an exemplar for the extensive magnitude we can at the best the bounded Line( line Segment) choose!  elements of which essentially are stepping  out from one another , and plainly therethrough the lins are constituting   as extending .

On the other hand as an exemplar  of the intensive  magnitudes some kinda  "with assigned strength" endowed  Point, in which here the elements do not themselves pour their innards outwards., rather present themselves only in the climax , thus  making a perfect stair step a representation of  a climax.


Also here shows itself  the difference that is placed onto a beautiful cognisance of the denotation: specifically in terms of the intensive magnitude, which  the doctrine of functions is making content out of, one does not distinguish the elements through specific symbols  , rather where specific symbols are stepping here forward , there is the complete everyway varying magnitude therethrough signified.
On the other hand in terms of the extending magnitude,  or by the  concrete representation of which , by the line,   the differing elements also become signified With  differing symbols( from the printers blocks ) , directly how in the combination doctrine.

Also it is clear how every real magnitude can become manifested in a dual cognisance. As intensive and extensive; specifically the line also becomes manifested as an intensive magnitude ; even if one from the art form looks away  , how their elements are out one another, and simply the quantity apprehends; and plainly so can the with power endowed Point as extensive magnitude become thought, in which one the power itself sets forth in form of a line.

Historically the Discrete thing  has before developed itself under the 4 branches of Mathematics as the Continuous  thing ( there, that to which "dissected  limb Perceptions" lies nearer than this), the Algebraical thing before as Combinatorial thing( there, the like thing more easily summarily grabbed together becomes than the differing thing). There to here is the Tally marks doctrine the earliest thing to root and rise up, doctrine of combination and differential calculus are same timeish, and the doctrine of extending magnitude, from them all in their abstract thought forms/ patterns must be the latest, while on the other side to it more concrete ( althoughl of a restricted nature) development, the doctrine of space, already relates to the most earliest time.

It urns out section 7 is an important stage in the induction. There is quite a bit of it yet to translate, but I have updated the last posted translation because it revealed to me how narrow my focus had been in my initial exploration of the ideas of Hermann.

Many of you readers, like me would not have heard of Hegel, except perhaps in a Marxist lderogatory! Certainly I would not have placed him amongst any mathematical greats, and in any case mathematics is not about Philodophy?  I could not disagree more. It was my study of the foundations of mathematics, that is reading David Hilberts book at university that made an otherwise inglorious university career fundamentally worthwhile!

While I did not grasp the richness of this enquiry, but rather stumbled on it during day's of aimless mind numbing questing for the " spark" , the djin that would lay Mathematics so called, out before me like sn open book, I now realise my quest was answered in that stumbling.  Yet Hegel was not a name I recall, nor Grassmann from the first reading of that Book.

And certainly, in the thread Fractal Foundations where I track down many of the great " Mathematicl" philosophers Hegel does not appear.

The obscure nature of the Ausdehnungslehre is due almost entirely  to Hegelian logic! Once you have an experience of Hegel's logic, categories and rhetorical style you find Hermann as remarkably clear! The density of thought in Hegel's works is palpable. He is not nor was not a light read. Similarly neither was Aristotle. I am beginning to glimpse that perhaps Aristotle has been much maligned by modern critics who do not read the Greek, but rather comment on someone's translation! No doubt I shall find out in time.


Nevertheless Hegel, unlike Newton and Hamilton both admirers of Aristotle, used Aristotle as a jumping off point , following Kant's lead toward a more contemporary redaction of Aristotles ideas. Whereas Kant sought to transcend philosophical reasoning by his analytical sagacity, Hegel takes a more immersive approach.

By immersing himself in the act of philosophising Hegel draws out the distinctions of all philosophers. But then he goes further. He synthesises all these distinctions back into a whole. It is that process of analysis and re synthesi that is the dialectic process. Ultimately it is a learning process during which many seeming contradictions are resolved and the instigator comes away with a new appreciation and apprehension.

In undergoing this experience Hegel developed a pattern that progressed his study. The pattern is often referred to as thesis, antithesis and synthesis. Hegel referred to it as the dialectic . Based on the Platonic dialectic it nevertheless is new questions are not just asked and answers pursued to yet more questions etc, rather contradictory statements are pitted against one another and resolutions sought actively. There is always a resolution. Hegel was positive about this, and he calls this approach the positive dialectic.

Of course there is a negative dialectic, and in that case no resolutions are sought, rather greater distinctions and separations occur. Hermann clearly apprehended the positive dialectic as a way forward .

The second and intriguing fact about Hegel is that he wrote mathematical papers applying his method . Apparently these papers are archived somewhere but only Marx could apprehend them. It is a moot point as to how much Hegel's actual Mathematical ideas were known or available to Hermann.

In any case Hermann in his induction demonstrates what comes about by applying the Hegelian dialectic. Many contradictions or contradictory statements now make sense in this Hegelian approach. Many curious effete statements of what he has achieved that I noticed in the Vorrede now make sense. The generation of new labels now makes sense. And as you will shortly read, the duality or rather the dual affinity of the point and the line now make perfect sense.

Norman Wildberger explores this duality between point and line extensively in his Universal Hyperbolic Geometry. It is therefore a pleasant surprise to realise that it is a feature of Hermnns induction that arises naturally out of the application of the Hegelian dialectic process.

This page link to my Bombelli thread indicates my early apprehension of Grassmann, but despite its intuitive astuteness I can safely say I did not have a clue what Grassmann had done or how he arrived at his exposition. Only in this last year since translating the Vorrede and  the Einleitung have I really developed an understanding that is transforming into a skill at applying the Hegelian-Grassmann dialectic! At least in understanding his writings!

http://www.fractalforums.com/complex-numbers/bombelli-operator/msg47192/#msg47192

A quick search into Hegel's mathematical documents reveals the following

https://www.ucl.ac.uk/sts/staff/gillies/documents/1999c_GermanPofM.pdf

Hegel commented on the foundational philosophical basis of mathmatics, but no specifically mathematical papers. Thus Frege, Schiller and others with Mathematical training cross fertilised 2 traditions. In this regard Hermann Grassmann was inspired to do the same. Of the attempts it is Hermnns that survives with influence into the modern world, with Marx providing an interesting ideological alternative.

The definition of the spreading out product and the closing in product in the Vorrede was of particular difficulty for me to translate. Auseinandertreten and Annäherung are very general terms that have many meanings even in the limited context of just 2 Strecken!

I was driven to try to grasp how they related to the hands of a clock, because I had the parallelogram product , which is the construction of a parallelogram, clear in my mind. Bad as my German is , the clock hands analogy still seemed too definitive. In any case the clock hand was my explanatory interjection, Hermann does not use it , and I wondered why, if that was what he meant.

I now perceive that he was not thinking about the point rotation of the Strecken , the jostling or pushing against each other of the line segments was identifying their Contiguity!  Auseinndertreten is agin identifying the Contiguity of 2 line segments particularly in a progression of line segments. Thus even if the line segments do not have the same orientation( nb not direction) travelling along them in the same direction is continuously " stepping one out of the other". The important properties of contiguity and being beyond the boundary of a preceding line segment is perhaps not usually dwelt on, but continuous and contiguous clearly are words that came from such sorts of meditations.

The contiguous property of any 2 objects is a concept of a common contact. It only really has validity in an ideal or formal setting when any 2 juxtaposed objects can be assigned one common interface or interconnect. In the real world 2 obJects pushed together retain there 2 distinct boundaries or they meld together and lose the distinct boundary altogether! Thus contiguity is an ideal interpose between these real states. One state we call discrete, the other we call continuous and the ideal in between state we call contiguous because we posit only 1 common boundary.

Auseinandertreten thus describes this contiguous progression of one distinguished entity to another.. In the case of a triangle AB has BC stepping out from it, but AC does not step out from either , even though it is contiguous with both at its endpoints.

In the case of a parallelogram AB , BC, CD and DA form a contiguous progression around the form. This I have identified in the cyclic interchange of labels as the root idea of Hetmanns anti commutative factorisation. AB.BC as a product clearly has the line segments stepping out of one another, whereas BC.AB  does not. The second notation is related to the first by using a "– " sign. This simply means the progression is the reverse of each other.
However what Hetmann noticed was that meaning carries through from the fundamental " point" level where AB and BA are the reverse progression. This behaviour captured in the notation revealed to him the behaviour of the thinker, or the mathematician. The notation imports the observers or the constructors progressive sequencing. This is important in any synthesis process. Any manufacturing process relies on the proper sequence being followed.

Yet in arithmetic it was common to discount this progression. Here Hetmann recognised a fundamental dialectical contradictory statement of the same process. In one field it was ignored but in the geometrical construction arena it was vital. In this instance the resolution in going from geometry to arithmetic had created profound logical difficulties. It also had obscured an important natural product process. After trembling with shock for a few months Hemann embraced the distinction and moved forward, creating the concept of an anticommutative product and a whole new arithmetic based on it.

We call these differing arithmetics Algebras, chiefly because they screw with our brains, but etymologically because they relate to the Arabic word Al Jibr, which roughly translated still means screwing around with our brains!  ;D

So now Annäherung refers to the property of 2 things coming into contact. They must finally meet or collide and when they do they become contiguous. However, the directions are not the same in the objects or things so that one is not stepping out of the other. Rather both are colliding and opposing or prevented from entering the other by the interconnect or contiguous boundary.

In the triangle AC and BC are closing in on one mother nd meet at C. The orientations of the line segments means one cannot say the direction of travel is opposite. In the formal sense the direction of travel is the same for each orientation, because we define the orientation of a line segment relative to a principal orientation by rotation, and that principal orientation also is used to define its principal direction. Thus a rotation about a point does not change he principal direction of a line segment. However rotating a line segment by a half tun and then disconnecting it and translating its other end to the centre of rotation describes the concpt of opposite or opposing( gegenüber). In this case the Kline segments go from spreading out from each other by rotation to coming together or colliding by translation rotation and direction, providing there is a meet or point or interconnect, a contiguous boundary.

The ability to disconnect and translate line segments changes all these conceptual relations haphazardly, necessitating a re description of the positional relationships after each such change. Relativity is fundmental to our apprehension of everything.

In the context of the parallelogram we have line segments that progress continuously and thus can outré cyclic interchange. Or we can notate so that line segments collide with each other, or we can have paths of progression that eventually collide. Also within the parallrogram we can notate the triangle form, and so have progressions and collisions based on that level of analysi. Introducing the triangle into the parallelogram is natural for the geometrical system.

The natural introduction of the triangle is through a line from corner to corner diagonally. In geometrical texts one will find forms broken down into their " pieces" , each piece named and any important properties listed. What one did not normally see until Hermann was a listing of orientation and direction.. These were just " understood". So diagonal is an orientation, as is parallel. In the case of parallel it became fashionable to notate it with arrowheads pointing in the same direction. Direction was understood by the positioning of the point labels in the notation.

The introduction of the vertical projection, so as to utilise Pythagoras theorem or trigonometric ratio tables is a far more " mathematical" introduction of a triangle into a parallelogram. In this case the line segments, the vertical drop or perpendularity drop  and the base line segment it is dropped onto are clearly not progressively contiguous. They are Closing in on one another and meeting at a point. This is such an important construction that it does deserve a name of its own, but the closing in product is not my first choice as a name!

The name of this construction has always been presented to me as dropping a perpendicular. However when you go back to Euclids construction it is not so neatly temed. The process is described as forming a line through a point not on a given line that meets the given line at a right angle, or similarly forming a line that bisects a given line segment. In relation to that label, "closing in product" is very good indeed :D.

However, in the parallelogram we can use the diagonal in this way, or we can relabel the points so 2 adjacent sides of the parallelogram meet in this way. Given that we have such a choice it is no wonder that a formal convention is established. The formal convention that Hermann learned and knew was established by Justus , his father, in the school district of Stettin. However, Justus was not like any other primary educator of his time. He held very progressive views and philosophical positions. Thus his scheme and implementation was progressive and innovative, and gave the students of Stettin a fabulous advantage. Jakob Steiner is a former pupil of a Pestalozzi school like the one established by Justus in the Stettin school system. He, after Newton was one of the premier geometers of modern times. He eschewed using " algebra" preferring the synthetic approach. It is possible that Justus was heavily influenced by Jakobs curriculum ideas and implementation.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Steiner.html

A nice twist is that Hermann was able to take up a gymnasium seat Jakob had just left when he was called to Berlin.
So the concept of line segments stepping out from one another is clearly ambiguous enough to incorporate the swinging out from each other around a common point, but each to its own. Hermann clearly distinguishes these swinging line segments in the Vorrede , but in the context of the closing in product, the trigonometric and hyperbolic trigonometric contexts. He retains the Auseinandertreten for the progressive contiguity of line segments, whether in a straight line or in a closed petimeter.

This is a nice translation of Hermanns frustration but indomitable hope for his Förderung, his pet project and conception. Such an expression of faith is not only religious it is Hegelian in philosophical tone.

http://www-history.mcs.st-andrews.ac.uk/Extras/Grassmann_1862.html

The frustrations of the previous 17 years at this point have resigned him to this last ditch effort promoted by his publisher brother Robert. Clearly he was not expecting the interest that soon developed in his works and his ideas, especially from those who were students in one fashion or another of Hegel.

As we know, by 1877 a much happier Hermann reprints his Doctrine of Extending Magnitude without change, but with heavy annotation and several very useful articles and updates and Addenda. As a consequence of one of those Addenda I am looking at the paper that forms the title of this thread.

It has proved rcessary AND rewarding to translate the Induction of the 1844 version, because the 1862 version contains updates of these fundamental concepts, and updates in labelling style, but fundamentally the original deep conception.

When I do translate the paper on Quaternions, I hope that it's intense application of labels and symbols can be overlooked as shorthand for the ideas and distinctions exposited in the Induction.

In any case, unless you have to write the damn thing, few even read the symbolism in detail! Grassmanns method is beautiful in that the format is always AC = AB + BC , that is no matter how long the terms are on the RHS we need only write 2 of them down, the first and the last! We just put ...+..  In the middle! :D

It comes to mind that the unfruitful " special pleadings" product, the so called regressive product , the eingewandtes product was a failed attempt at developing further the concept of a closing in product. In this case, rather than using a Senkrecht or vertical projection, or even just a common projective geometrical projection, the idea was probably to reverse one element in a contiguous  progressive extension  so that it collided with its immediate predecessor. Thus instead of AB and BC it becomes AB and CB. What this product produces geometrically is a meet or a ponit of meet/ intersection, within the otherwise progressing sequence.

Although I did not go into the chapter on the Eingewandtes product, it seems that the set up requires so much special pleading that it is virtually useless in most cases. In any case the ideas of intersection or meet are subsumed within the rest of the method in a more user friendly or applicable way, even in the most general dimension case.

It maybe in today's capability of Mathematica some semblance of the idea may be useful, so it might be worth someone programming it into Mathematica.

Further thought makes me Wonder if the eingewandtes product evaluates a parallelogram using the cosine law rather than the more drect trig and Pythagoras theorem approach?  If so it would be very tricky to use in the general case, but an alternative where all else is not possible.

In addition some references to how influential Hermann and Roberts work has really been.
http://www-history.mcs.st-andrews.ac.uk/References/Grassmann.html

There is still little reference to Justus Grassmann, but that perhaps is to be expected as his work was mainly in education and educational theory .

With regard to Hegel's influence on mathematics I find only Klein, Loria and Kroenecker as direct quotes to the name in Mactutor.

http://www-history.mcs.st-andrews.ac.uk/Search/historysearch.cgi?BIOGS=1&WORD=Hegel

Commentary
Section 7 now complete enough leaves me with the distinct feeling that I was so focused on Strecke that I missed the Punkt concept. The Begränzte Linie is counterpointed by the mit Kraft begabten Punkt. The line segment is dualed with the Potential Point. Thus the notion of a Schwerpunkt is a particular label of the more general mit Kraft begabten Punkt . The dualing of the Potential Point with the line segment means a lot of point perceptions can be formulated as schematic diagrams and many formulae which give a number as a result can be represented by a geometrical system that gives a line segment as a result.

It is also of interest how Hermann distances his ideas of continuity and discontinuity from those of Dedekind. He does not percieive Discrete as a " limb cutting" exercise of a continuum! Thus he does not subscribe to Dedekinds notion of  real number as a discrete cut in a line segment continuum. Instead he promotes the Doctrine of an extending / extensive magnitude as a more general concept of how we might deal with our experiential continuum. He answers Berkley's Jibes thus in a different way than those who base differential calculus on limits and ę  and § that is epsilon delta definitions. The Fluxions as fluid and fleeting dynamic quantities represent intensive magnitudes something like a potential point. These may thus be represented by line segments and the conclusions of Calculus may be founded on these vanishingly small geometrical entities that come into being as a whole, or root and grow in a moment.

Clearly Hermann sets out a perceptual progression which lies either side of these ideas. On the one side, the formal, the doctrines of tally marks followed by that of Combination and the differential calculus , eventually these lead to the doctrine of the Extnding/ extnsive magnitude; on the other side, the real, is the concrete development of the doctrine of space, which though restricted has been pursued from the earliest times. The implication seems to be that now both sides can be developed properly and fully because of the doctrine of Extending magnitudes covering both intensive and extensive magnitudes in a unified way.

Commentary

I am going to go back and review some words in the translation prior to posting section 8 , in particular Reihe and Zustände. The established or gesetzten entititees and some notions on constraint and conditions.

I had hoped to have a clear definition of Algebraic and Combinatorial by this section, but it is clear I need to review Hermanns development or derivation of these labels also. What is certain is he does not use these labels in the narrow senses we use today.

Here is Norman storming away again on a related subject? Lol!

http://youtu.be/rMj67iNwM4g

http://www.youtube.com/watch?v=rMj67iNwM4g

Omg :embarrass: my German is so bad! I went to a learn German online site to review the basic article declinations and adjective declinations to find I had got them mixed up ! The only saving grace is that I was looking in an advanced German lesson! ;D  So maybe I did not do too bad for a beginner? Lol!

I am revisiting the translations any way so I will ponder and correct. But please if you want to throw in a translation or even give me some advice you will be more than welcome. I post in a forum not a blog because I do not mind interaction and input. So please feel free to add your 2 Deutschmarks worth! Lol!

http://www.deutsch-lernen.com/learn-german-online/03_e_adjektivdeklination_II.htm

http://www.deutsch-lernen.com/learn-german-online/02_e_adjektivdeklination_I.htm

A note to myself: nebengeordnet and untergeordnet are 2 participle ideas which are more general than coordinate and subordinate . I want to use concept of group ordered and list ordered. The point is a group order is a spatial order of entities, but a list is a scribed order on some surface.

A group may be a row , a muddle or number of entities in close " spherical" proximity, or a column. One may perceive a row as a column by physically shifting ones point of view.

A list is a scribed order starting at the top of a piece of paper and written down toward the bottom using a new line to emphasise the organisation. The idea presupposes a prior organisation of a surface into an orientation of top to bottom and a notion go ruled lines for script to sit upon . This prior organisation is in fact more fundamental than the list concept itself . Having set it up the surface may be physically rotated, or the observer may physically shift so that the ruled lines may now be perceived as ruled columns! In addition the scribe might ignore the lines and scribble notes in ny muddled organisation on the surface!

Thus to use the group ordere and list ordered denotation is not as clear cut as it may seem, but keep in mind Hermann did not want clear and hard bounded words, he wanted words which have a general application and feel, open to many and indeed any interpretation. This was how the dialectic process resolved conflicts by finding ideas and labels capable of spanning the contradictions.. Having found such words, the labels, they become the entities used to organise processes and practices, becoming specific only where absolutely necessary.

While this is meaningless unless you have engaged in the dialectic process thst leads to or derives these words , or labels( hence this Induction), the practice of motivating a mathematical definition or a formal system of definitions arises out of these kinds of considerations.  

Always remember, mathematics is an empirical subject based on lots of experiences of " arithmetic" calculations. Algebra arises from these arithmetic motivations , and principally sets out a lot of labels as markers to definitions and processes. What gets served up to you as a mathematical main course had to be first prepared from common ingredients that anybody can under–stand!, or "everywhere" stand( verstehen).

The ideas of group ordered and list ordered, not only apply to rows and columns ranks and taxonomic categories, they relate to the Matrix entity as an organisational structure in a mathematical labelling system, and to the fundamental concepts of a series and more fundamentally a sequence.

Commentary
of Labels and Handles

I suppose the easiest place to start in this forum context is with a basic programme code list.
I was taught the syntax of command code, the terminators and connectors and the list format into lines or onto separate cards!
We were told to number each line of code. This was an ordinalse of numerals an Anzahl. The numerals had no cardinality or face valuee( head count value), they were Tally markers to organise and account for items( in this case lines of code or physical cards) in a list. Ordinal numerals or Tally markers.

Because they were not part of the code they were also performing another fundamental role of Anzahl, or Tallying markers, that is to label.

A label is an identifier. It is a seemeioon precisely in the sense of Euclids Stoikeioon number one " a seemeioon is that which has no parts". As a reification of a seemeioon a label clearly has parts. These parts make a label a powerful organising entity which has a " meta data" role. In the Stoikeia the seemeia have a powerful a rioting role that organises and permits a gramme to be " drawn " in a surface which consists of these seemeia.mthus they have a magical, mystical and mythical virtue which is forcibly discounted! In the particular use of the Stoikeia they are assigned a background , menal role, not by any means to diminish their fundamental importance, as later translators neglected to observe, but to rank them in the synthetic process that Euclid and the Pythagorean school were building from their Exhaystive Analysis.

The label has this fundamental synthetic role. The Anzahl have this ordinal synthesis role. So we easily listed and numbered our lines of code.

Later we were introduced to the infamous goto statement ! This is when we realised that the numerals were now conceived as labels or markers. The goto statement is perhaps a concise experience of using a numeral as a label for a line of code. The essential idea is incorporated in Alan Turings fundamental Universal machine.

Free to see the numeral as part of the ASCII set of codes for alphanumeric symbols abd entities, programmers simply accept the one to one correspondence between a line of punched holes in a card or a tape and an alphanumeric character/ symbol. The notion of an Anzahl was broadened and deepened until a set of punched holes could represent any character in our script or printers block. Thus nw it was possible to label a line of code by any set of punched holes, and thus by any character.

With the development of the function process in computing, the practice of setting and calling sub routines the label becomes absolutely crucial. But it's role also changes.mbecause a piece of code( a function) calls another piece of cide( a labe,
Led sub routine) the label now can be perceived as a handle! The function can now handle many sub routines in a more complex process of handling( Handlung) in which the handles ( Begriff) are fundamentally just labels!

That being everyway stood or understood, labels as punched holes can be as simple or as complex as desired! As memory constraints became eased it became possiblebtonhave labels that were relevant words in the process which they labeled.mthis is where Hermann introduces the labels Algebraically and Combinatorial. These words are drawn from the common stock of mathematical words in his time and are used as labels to be attached to specific definitions later. They are not used as defined today in ome quarters, they are not specifically defined by Hermann except as labels for general distinguishable processes to which he wants to give the shpe of a mathematical subject format, without ever specifying it at this stage or level.

Thus these labels are handles to call up other handles hich eventually will call the specific defined sub routine to be applied.

Algebraic thus calls tally marked labels while Combinatorial clls differing lay marked labels which are combined by ome process which is fundamentally extensive, tally marked processes are fundamentally intensive processes.

Commentary

I have been into a department store and realised that intergeordnet and nebengeordnet have a physical reeificayion in terms of shelves and stackings. Items are organised on shelves as neighbourhood collections, and these are thus stacked or piled above other such collections, so that the collections are ranked or layered or arranged under each other .

It was also notable that inventory was organised on separate stands or Zusstänfe.

Thus the group ordered and list ordered can be extended to rank or shelves ordered and shelf group ordererd.

Commentary
On Labels and Handles

Hermann at the end of §4 declares that he has achieved his goal, that he has set forth the label for the Concepts in mathematical terminology and without any Givens. This is telling because givens serve the role of specification in mathematical problems. Without the givens problems may not have a specific solution. In fact the progression to solving such problems must include a preamble in which the solver specifies what givens must be presumed . The presumption or assumption, presupposition or fore setting of the grounds and constraints could be considered as motivators for the solution to the problem. This is akin to interpreting the solution or solutions so they can be applied .

The use of the algebraic label for the Tally Markers was an interesting puzzle. The combination label was made clear by direct examples of bundles and collections tied together or linked together in some way.. Thus the general notion of combine.

So algebraic should have this direct connection to its general content which was Tally Markers. I gradually realised that algebra was taken to mean symbolic arithmetic.  The need for Anzahl or accounting entities thus becomes understandable. The relationship to Discrete was a counterpoint to the algebraic idea of general symbols arranged in like terms and then combined as a sum. Tally marks as algebraic is thus appropriate but so far removed from much that is called Algebra today.

Symbolic arithmetic is still the favoured introduction for children into algebra. It is a historical development spearheaded by Boole and Hamilton, against the denigrating background of Arithmeticians. To free Algrbra from Arithmetic, to become a subject of its own, and thus not tied to tally marks at all.

Bombelli when he wrote his book entitled it  "Algebra the art of Arithmetic the greater part of which is Algebra!". I did not understand that title until recently because I thought algebra was a different topic to Arithmetic. Symbolic Arithmetic places Algrbra at a specific time in it development into a modern subject. The contributions of Hamilton and Boole thus diverge from Grassmann who was pursuing a traditional generalisation of Arithmetic. The path was blazed in Detail by Laplace, Lagrange and Euler.

The generalisation of arithmetic involved using the symbols to represent general or undefined quantity. What had been ignored was general and undefined extension, or extensive magnitude. This is the better and more general concept of an undefined quantity, often called a variable. Hermann was not particularly aiming for this type of generalisation, but rather it crept up on him by the work of those around him and that of his father. These general labels were invested with qualities that early Ring and Grouo theorists like Abel were beginning to divine. However a much greater influence on Hermann, that of Hegel, was to take his conception in a direction no other mathematician of his age could ever entertain!

I found this clash of labels particularly when I was deriving polynomial Titations. I really struggled to apprehend what was happening. Then again years later when developing the Newtonisn triples based on studies of the Wuaternion 8 group. It was then that I really found the meta data role of a label and handle , and how a calculation or a combinatorial form could be perceived as a label or a handle, especially to reference points in a reference frame.

There is a 1 - 1 correspondence between such labels and to graphical displays and or points in a reference frame. The concept of 2 worlds, independent and yet linked by an arithmetic results process is hard to grasp when simply presented as say the complex plane, or the vector plane. The complex plane is like a computer graphics screen that sits atop another screen so that direct correspondence is highlighted, but not explainable. If a complex reference point is moved by a complex calculation to its solution, why would that be a rotation in the real world?

It is not. But it is such an extraordinary behaviour of the calculation presented in this format that one can set it as a model of rotation. Pretty soon ones head gets tired of begging the question and just accepts that the calculation is a rotation. Well it is not. Until the advent of computers and electronic circuits it was not possible to know if such a modelling was reliable. Now we have thousands of models based on these computations that re astonishingly accurate.

However the model is flawed. When the computations involve large numerals the motion capture becomes bitty and jerky, or if the trigonometric sub routines are called the problem at 2 has to be resolved, to avoid glitches.

The general use of labels in these contexts is itself a complex co development as computational programming, electronic computers and circuits reeified the very general concepts Hermann considers in this induction. The use of computations as labels or handles is such an innovation that no one could follow Hermanns presentation of the idea until recently.

AB + BC = AC is a profound equation which contains labels, handles, summation and products  in a direct connection to Spaciometry. And of course it is entirely fractal in structure.

Reihe as a common word is apprehended by examples not dictionary definitions. Thus when I start to apprehend I need lots of examples of use, with the situation of use.

If I see a stand of trees I am struck by their spaciometric arrangement in particular any marked row or column , file or rank seems to stand out and we denote that as Reine.

This experience is scale free, because as we change scale the same experience occurs. So when we try to be specific we actually divide this experience into mutually exclusive formats. Thus to limit Reihe to a sequence, a series , a list a row a column a rank a file, an array is to confuse, the experience has many application or interpretations , but they are not the experience. A little thought reveals the general use of Language words with the referential structure that they are utilised in. Studying language reveals the limit of language. It also reveals how a new word accretes its meanings and uses as a referential marker or distinguisher.

Hegel's dialectic process gets at these limits and boundaries of language so we can look beyond them to the nonverbal experiences that form the major part of our experiential continuum. In doing so we become aware of our other senses and how our consciousness is not at all encapsulated in words, but words are attached as labels to some of our experiences .

Once I had the experience of Reihe, it becomes clear that a priori to that I had an apprehension of form for a collection of things. For example a muddle is a label describing a form. Or an array is a label attached to that regular form to which I also attach Reihe to.

But another experience is that of a planting or setting. For a collection the individual elements have a spaciometric relation to each other. But if I had an intention to lay out a garden each plant would have a setting. At the same time the planned arrangement could also be called a setting.

The vomcrpt of a status generalises the setting to a dynamic situation where the setting keeps changing. Thus a status is a setting at a moment , maybe for that moment.

Zustände is a status or setting or condition, as presented to our eyes particularly. In connection with a Reihe I wanted to translate Zustände as a symbolic marker , thinking of a symbolic algebraic term in a written series in a mathematical format. However that does not translate across to the physical spaciometric experience. The concept condition or status Marker or Setting Marker does travel, and that is the more general concept for Zustände. The direct zu and Stande means a stand to which anything or everything is brought!

Commentary

Much of this may be familiar to programmers and code writers , but this is a clear induction into the Wolfram language that very closely matches the induction into the Grassmann Language or label derivation.

It is worth getting your head round it if you want to "everyway stand"( verstehen) the doctrine  of the Ausdehnung Größe or the extending/ extensive magnitudes.

http://youtu.be/H-rnezxOCA8

http://www.youtube.com/watch?v=H-rnezxOCA8
YouTube Wolfram site

Commentary
The Reihe or data structure is the fundamental content of Wolfram
 In the Reihe are the Zustandeb, the position markers .
The concpt of place, placeholder  is crucial to digital memory and digital processing. It is so fundamental that it is self referential. A piece of hardware , a register or integrated circuit instantiated the concpt of a position marker or placeholder. Every place must have a place for me to have cognisance of it, and then to be able to indicate it.

As soon as I perceive a place it is me labelling that experience. That label becomes a handle by which I grasp and manipulate the concpt. But there are other attachments I use: a cue sound or a texture cue or even a smell cue. The concpt of a " label" is more widely distributed among the senses, and do the perception of place is more subtle. Consequently the indicators of place or position are more subtle.

Zustände in this regard is a dynamic referent that encapsulates the interchange of place with what occupies it, and alo the change between moments when the places seem fixed, there content set. The differences between Zustände at each moment gives rise to many notions or labels including , condition, status, settings.

So in wolfram the positions hold some status or signal or condition. The pattern of these positions and therefore the status pattern can be uniquely labelled. Thus one of the labels is the header Zustände. This happens to be a block of memory positions. This itself is a memory position and do is a place holder.

The self referential nature of the process of matching the " real" object to a formal label is well illustrated. It is by this tautology that distinctions are made that move the formal dynamically relative to the real. The same real referent  can attract many different labels, each purporting to attribute or draw out some finer distinction.

To go through the process is always confusing, because it requires flexibility of standing" every way standing"  not just under-standing. The dialectical process as described by Hegel gives a formal structure to the process, that Socrates or Plato did not. This structure of Hegel allows it to be used as a Tool, and that tool helps us to build logic circuits and ultimately computers as Tuing envisaged.

The Reihe as the basic data structure thus can be any configuration of memory blocks from a single block to a list to an array to a cuboid to a heap. In addition the atomic labels or handles become the "points" of synthesis, because as Euclid opined a point is that which has no parts!

These points are symbols strings, numericals. Seemeia as I said ate general indicators, mistakenly called points, and this distinction is crucial to applying the Hegelian Grassmann language o computer code design.

There is a lot of inter communication between wolfram language and the Grassmann language as derived in 1844.

Assimilating my lessons on adjectives I notice how verstehen and unterstehen, everyway stand and under stand, differ. In coming to understand these declensions I noted how I looked and questioned the table every which way! The process of coming to know a thing or pattern involves this variety of ways of looking testing and turning it over.

 Simply holding it up to the light and thus standing under it , only reveals one aspect, whereas coming at it in and standing every which way and loose relative to it gives a broad set of experiences by which real knowing of a thing may be built.

I have been delaying posting section 8 until I got some understnding of it, or some inspiration by its Muse. As usual, some other thing triggers an appreciation of a text I am working on, and the generalities of the Wolfram language are a case in point. But it is not these intricacies that contain the simple exposition. It was something else, fleeting which now escapes my words, my bonne mots!

This section should summarise Hermanns derivation because section 9 launches into a full exposition of the AusdehnungsLehre, and yet it does not quite do that. It seems to be a post ahead of what is coming as a closure of what has just been gone through.

What have we gone through in this dialectical progression? It seems like I have been taken through certain broad notions . Ideas and experiences that I have in common with others but not specified with others by debate or consultation and consensus or agreement.

A lot of these notions are not up for discussion generally, except if you have a philosopher as a friend! Commonly they are shied away from, replaced with some platitude or some workaround.

I am not decrying this pragmatic response. It has saved my life and sanity on many occasions. Keep It Simple Stupid or KISS was a mnemonic I was drilled in during sales training. As you can appreciate, I found that a hard rule to follow! Nevertheless it was and is a very serviceable rule of thumb. Literally my autism meant I could never understand why communication had to be so dumb!

That is not the same as saying my communication is so super intelligent that the rest of humanity clearly is beneath my understanding! No rather I found that I was autistic, hiding my ignorance behind grand sounding words I did not understand, but could enunciate fluently! My articulation was a skill of my Autism, not of my understanding!

I think of Stephen Fry a well known Wordsmith. Out of the sounds of words and the scripting of words he can create a beautiful entertainment. But like a song it advances no technology or makes no poor man warm in clothes and fed with food. Yet those words may emliven his spirit , set his heart and mind to achieve his needs despite the prevailing circumstance. Words do have a power if they are just so, the right word for the right time.

So Hermann has looked at these words that attempt to apprehend our most general experiences, which are right out at the edge of our intuition and of our experience of consciousness and being alive. And he does not say " now let's define them precisely!" for that is a vain agenda.

That is a vain agenda that troubles all Aristotelians. The taxonomy of apprehension that Aristotle above all established is the most Autistic Trait of a person with Aspbergers syndrome. Aristotle was a high functioning Autistic . His taxonomy dominates his logic. In accepting his logic one is forced to forever analyse, even when drawing conclusions.

Hegel recognised this, and contrasted it with the teaching methods of Plato and Socrates, teachers or mentors of Aristotle. Aristotle found his balance with Plato. If he could have stayed he would never have left him, but geopolitics played him a hand he could not avoid or refuse. He had to leave Greece , but then was drawn back into the very centre of the Greco Macedonian Empire. His unique view shaped Alexanders view of the world as a system . Helemism flows from that view.

But Plato was a Pythagorean student. We have no record that he attained the qualification of Mathematikos, but he clearly sought that qualification. Eudoxus and other Pythagoreans like Euclid taught at his Acadrmies, and established sister academies. Both Euclid and Eudoxus were Mathematikoi. They achieved that qualification. Aristotle did not because he had differing opinions. Plato may simply have loved being a perpetual student!

The core concept of Pythagorean philosophy is synthesis. The core symbol of that is the Mosaic. And the core method of achieving this synthesis was by revelatIon from the Musai or Muses.

This seems perhaps different to the Socratic Philodophy as some teach, but in fact it is very much in accord with it. The Musai are the curators of all knowledge, skill, art and aesthetics, rhetoric, poetry , dance and Musical Harmony. Phusis, the sister of Harmony is a representation of the oppositional , contradictory nature that has to be harmonised. Thus Physics deals directly with these physical contradictions or opposites, + and – , positive and negative, pros and cons , evaporation nd condensation. ,,, etc. with a view to harmonising them.

Socrates suggested that the human had a soul that simply knew everything, but a body that frustrated that knowledge , in a sense. By guiding that person through skilled questions that person could remember what his soul already knee.

This is a pure Tautology. Pure in that it is the purest form of apprehension the human mind or consciousness can achieve. Everything is already known, either by the Soul or the Musai. Humans simply have to ask the right questions to recall this knowledge.

But then there were infinite regression! Human consciousness knows about endless loops of thought. That is a fundmental limit of our ability to know. A loop is the simplest tautology. We think and " advance" our thinking by tautology. But we simply rename the loop as we go round it again. Thus it is the sme but different!

It is this Same but different, or same and different tautological structure that Socrates sought to build, and Hegel managed to build by his logical categories.

Unlike Aristotles taxonomy and Ontology ( a taxonomy of being or existence notions), Hegel sets out all the possible categories based on the distinctions in language. Thus rather than studying nature and creating a new category for each discovery, Heglian listed all the possible categories language itself can distinguish. It is a very large number, but it is not infinite.

Given or realising these categories it is possible to divide our experience into real and Fomal. In such a way that the fomal always imperfectly mirrors or captures the real.

Realising this, Hegel also realised that we have trapped ourselves by our own processes of understnding. He then realised that the way to absolute freedom is by synthesis not analysis. The synthesising of our notions actually makes a finite set of constraints, onditions or categories infinite in application, because each category can have an infinite number of Statuses or conditions!

Synthesis means we can model the whole universe with a brick or we can build from bricks a more detailed model of the universe or the universal. The single or the complex structure is the same subject, but the detail we can go to is Fractal.

Clearly it is unsatisfactory to go straight for the single brick model! But starting with the Hegelian categories and working through the thousands of them, but still a finite and doable number , we could build an absolute model, the best we can hope to achieve. Along the way we attain to what Hegel called the absolute rationality or Reason of our conscious experience, whether we categorise it as a god or Philosophical reason our aesthetics determines our individual style.

So Hermann applied this kind of systems or category logic to Mathrmatics. The first thing that wrnt out the window into the rubbish bin was the term Mathematics! He and Robert coined the term doctrine of thought patterns. Yes they allowed Mathematics to be retained for a while( because we love it so!) but ultimately because of its contradictions the name will be consigned to the dustbin of history!

Preceding with thought patterns then how do we get at the subject content of Mathematics? We do this by specifying generalities. That is we specify general notions.

There are many general notions, love, peace , space... So which ones do we choose to specify?

The sections up to 5 identify those general notions.

Now I m using the word notion here but the german in Hegel is Begriff! One of my favourite mentalist words! It is literally the grip by which we handle, think and manipulate things real or imagined..

I have purposely consistently translated that as handle/ label.. As a mathematician and familiar with Algebra and programming that is the kind of terminology I would most appreciate.. So to hear the word use to denote a notion is very interesting and satisfying.

So Hermann identifies or labels 2 general contrasting generlities : Same and Differring, but only after initiating them in the very act of reification. Hermann needs these 2 generalities in order to specify the subject area of the doctrine of thought patterns. They bring embedded within them the labels or handles for Discrete and Continuous. They also embed a dynamic activity of reification as thought patterns or thought forms, entirely set by conscious or unconscious thought processes and commands.

Hermann establishes these general notions but shapes them in Mathematical language markers! Thus he sets up a system of leading questions! The shaping readily leads the educated mind to the direction Hermann wants to go. To anyone else it undoubtedly leads to confusion!

Now, I criticised Hamilton for using legerdemain or sleight of hand when he skipped over from ordinal numbers to Cardinal numbers. I could do so because he sought to slip that distinction into a syllogistic format. Here however Hermnn is using a dialectic format, and so almost anything goes that leads to a resolution! In addition because he wants to get at Mathematics from a more general platform it is required for him to show how he does that. He does it by shaping the labels as if they were leading questions. At this general level that shaping gives you or the reader sparks of insight and connections, which a mathematician would never see!

That kind of insight justifies tha approach . Or does it? You will have to decide that yourself.

Having specified his generlities by various labels and handles he then specifically discusses the  whole lot in 4 categories and labels or attaches handles to each and within each.

The labels I hope I have identified or picked up on, but the moste important are unmissable. Intensive and extensive, step rise, step extension, climax, extending out of, algebraic, symbolic arithmetic, tally marks, combinatorial, bundle, tied together, ordering: next to or under, row, column, rank,file, array, space, constraint condition status , setting, layout. Etc. and any others I may have not mentioned in that list, such as list itself!

So we have reduced by stages and shaping from the most general notions to a set of more specific notions which will service the general Mathematical discussions Hermann wants to pursue.

So finally as a heads up in section 8 Hermann states that there is a form that is posted ahead from this discussion of the subject in 4 . It is a kind of general array found everywhere in the doctrine of the extending and extensive magnitudes. The only proviso is that its use should not require writing out the whole of it again ineach branch or an offshoot ! In that case it would be useless. However where it applies it facilitates the common development of many branches and offshoots at one go!

To Hermann that was a prize worth reaching for. He seems to label it the capstone or final array.  In true Hegelian dialectic, this new summit of realisation now becomes the foundation  of the Doctrine of Extensive or extending magnitudes.

My earlier mistake was to assume the hewing into 4 had already been done! In fact the Ausdegnungslehre that follows takes us up to and through that hewing process. The general capstone array is despatched ahead to that upcoming event.

Ausdehnungslehre 1844
Induction
8. There can become sent on ahead , to the hewing of the doctrine of Thought forms/ patterns in 4 branches, a general part which represents the general entities– that is  for all branches alike, the applicable Binding settings; and which part we can denote as the General doctrine of thought Patterns

To send ahead this part to the Whole thing is essential , so far as therethrough saves, not simply the restating of the whole of the same Capstone  rank array in all 4 branches  and the same in the differing Offshoots of the same branch ( and thuswith the developing meditatively becomes shortened off); but also  ( because) the entity  to the Essence that accords to "related together things" appears "gathered together", and steps onstage as the foundation of the Whole!

Norman on the fundamentals

http://youtu.be/Ek0URXLCZCE

http://www.youtube.com/watch?v=Ek0URXLCZCE

We can see how Grassmann avoids these issues by using generalities in an Hegelian dialectic rather than an Arustotelian logic.

Wesen is what is on the outside of an entity as it indicates what is going on in the inside of that entity. Weise is what is on the inside of a conscious entity as it exhibits it's experienceces on the outside of that entity.

Thus conscious and non conscious entities have a Wese, but mostly animate entities have a Weise.

These general ideas encapsulate the objective and subjective processes of the human consciousness in its processing of its spatial experiences.

I have just reworked section 6 and hopefully clarified some of the structural sentencing.

It seems reasonable to consider algebraic as a synonym for symbolic arithmetic. The tally mark stands for a crowd of things , and is a symbol for that crowd used in arithmetic. However now Hermann has laid out briefly another or an extended symbol typology to clearly denote or stand for a crowd of things when viewed as a combination. The closest to this conceptually is the set theory of Cantor et al.

 As usual we must resist the desire to restate Hermann in words defined and derived by someone else. His signs distinguish between crowds of things as Tally mark and/ or as Combination. The concept Menge is the a priori of the set notion, but Hermann here leaves Menge undefined. He merely distinguishes two aspects of the experience of Menge!

Norman gives an example of a discrete combinatorial thought pattern. He calls them the Dihedrons and the general part is a Schlussreihe or a " capstone " array.

As you will see the " array" transforms into several different forms: matruces, vectors, linear combinations .

The task is then to fully outline the product relationships. Also certain other transformations are highlighted. This only makes any sense when you have watched the 4 preceding videos which motivate this treatment.
http://youtu.be/QDIcznHyWTo

http://www.youtube.com/watch?v=QDIcznHyWTo

I have completed retranslating section1 with my newly acquired adjectival Skills  :D

I must say they do help to better express Hermanns thoughts rather than my own. At the same time I am surprised how little of his ideas I mangled in translation! Lol?

It is clearer now than at the outset of my translation, that Hermann wanted to purify the Label or Name. Mathematic. Do the first 3 sections are about how he fors that.

It is also clearer now that he does that by reseating Geometry onto a real experiential continuum, and by renaming the rest Formenlehre or some such. This pure Mathematic is to consist in the doctrine of Thought patterns, these thought patterns are to be inter communicant with the thought process created by the confronting behaviour of the real expertises to the thought process itself. Indeed the thought process really only exists in dealing with these confrontations and attempting thereby to relate them onto an experiential continuum, while experiencing that same thing!

That being the case the formal expertise must never be allowed to cross that important difference! Formal expertises must restrict themselves to a process that is founded on definitions snd builds super concording structures under those deginitions( or stop , if you like). It must not be confused with the process of super concording with the confronting entities of real expertises. One must be clear that the experiential continuum of a formal expertise is Set by the thought process itself, and it uses the first set entity as the experiential continuum against which all subsequent thought actions must super concord . The first self set entity is deliberately placed over against the second thought act, so as to arrange for the second thought act to super concord with it. By this repeated process the formal expertises build from their definitional foundation into an image, idea or form.

Contrast that with the process the real expertises engender by their confrontational nature: the thought process forms a persistent, insistent existing copy in the thought process itself , as the experiential continuum on the inside of the self evident experiential continuum on the outside!

Removing Geometry to its correct expertise partition enables pure mathematic as Thought Pattern doctrine to be developed on a sound basis! And sets right a long injustice done to Geometry!

So I think I have finished correcting section 3 in terms of a more literal translation to Hemanns words..
The main clarification is the manifesting of space to the thought process in it proportional arrangement is vital to developing a proper kinematical expertise.

Thus the following video shows how the kind of likenings and or manifestations used to denote something affect our modelling of it and our apprehension and ultimately how true it is to empirical or primitive experience of the senses.

Do not be blinded by advanced mathematics. If you can't experience it  " It ain't necessarily so!"
http://youtu.be/XkAPv5s92z0
http://www.youtube.com/watch?v=XkAPv5s92z0

Commentary

Hermann used Vorausschicken which means to "impel ahead", to describe how he was going to handle the general part, the capstone rank array , of the doctrine of extensive magnitudes.

It has become somewhat academic practice to set out the general terms first , at the beginning or head of a piece of work. In that regard Hermann is different in that he sets out the derivation of the general pattern first , as a motivator to studying the applications. Consequently the derived result can be posted on ahead to the places where it will be used, especially to shorten the development of those applications.

Thus rather than setting and leaving the general form at the head, the reader being expected to refer Back, Hermann prefers the reader to have the general form current with them as they need it. How he accomplishes this we will see, but it may involve referencing each place where the general form is used so the reader can go there to see it in use.

Normans teaching style follows this practice very closely, but still refers the reader backwards rather than forwards.

With the review of §7 now in hand I will have completely revisited the translation. Why I did not note that his goal was to achieve what he declared at the end of §5 , and plainly stated in the opening title of the first section just shows how myopic one can be!

Just to mark this ground breaking event and to note the use of the word " Expertise". This is an example of real expertise, and geometry, Phorometry and Mechanics, the kinematical skills Hermann alludes to in section 3

http://youtu.be/R3WwbVY3YbY
http://www.youtube.com/watch?v=R3WwbVY3YbY

Commentary
§6 discusses the categories of the thought patterns. There are 4 categories because the adjectives are discrete and continuous, algebraic and Combination. These combine to give 4 categories: discrete algebraic, discrete combination, continuous algebraic, continuous combination. Section 6 deals with the first 2.

The labels are established somewhat sloppily because basically the labels are supposed to describe what they attach to. This is supposedly so obvious as to hardly require comment! For sake of completeness however the reader should challenge that assertion before deciding whether to accept it or not.

It is clear tht Hermann wanted to get on to the next 2 categories, he had nothing new to add to these categories of the discrete labels. However a modern student would benefit from pondering the brief presentation he gives, and particularly to subvert the motion of number that clogs up the system nowadays.

Essentially Hermann attaches formal lbrls to a collection or crowd of things. One label is the tally mark, the other label is the combination way of apprehending a crowd.

Now to make these 2 ways of apprehending a crowd or a collection clear in the labels he introduces symbols to signify tally marks and symbols to signify the elements as a combination .

This 5(1) means a crowd of 5 objects . 5e means a combination of 5 objects all of which are the same or like each other. 3a+ 2b is a  combination of 5 objects 3 of which are like and 2 of which are like but differ from theother 3.

This is familiar to those who have been introduced to algebra at secondary school, but its significance is not explained. It is usually rushed over and it's connection to apprehending and labelling the experince of space is downplayed, made subject to number.

Now the signage -1, 0, 1 comes from this scheme of cognising space. Our tally numbers are not just natural numbers (1), but also extended to Integers ( –1, 0, 1) . Into integers the products of arithmetic are evaluated. The products of higher arithmetic are also evaluated into the integers. These therefore represent the evaluation of the tally marks . The integers are extended into rationals.

This we construct an arithmetic of tally marks.

What about the combination of objects, especially those same objects that we have labelled or tallied using rational tally marks.?  The combining of these elements leads us down a different apprehension of relationships in that crowd . One apprehension is the spatial relations of the elements . But we are not taught that from the outset.we are taught to evaluate the combination in the tally mark arithmetic ! The Ausdehnungs Lehre shows us that we can evaluate combinations in geometrical pattern.

Now a knotty issue about the product as a construction is also resolved. Adding parallel vectors and producting parallel vectors give the same result. Or rather it gives a dual result that appears identical  to appreciate this we have to apprehend section 7 .

In the meantime section 5 deals with the Like and Differing cognisance in detail, showing how space and spatial arrangement encodes this like and differing aspect of entities. We typically spatially arrange in arrays! The simplest array is the 1 x 1, then the 1x n, these can be arranged in rows or columns . Eventually we build arrays by tows and columns.

This organisation is not simple, it is complex . It is the most general organisation of like and Differring entities and it is a Combination.

Norman has used these fundamental organisations in his development of polynumbers, both the row and column format. Thus nebengeordnet and untergeordnet has this far reaching application, for when used in designing the basic registers of a computational logic chip the arithmetic logic unit, and the data store we reify the notion of a variable magnitude. And then in binary notation we may set out the most basic definition of a polynomial.

More fundamental than even this is the notion of a sequence. But in point of fact or reification we cannot use any other than the idea of nebengeordnet or untergeordnet . In 3 d space these sequences have a flexibility not appreciable on the page. Thus an array may indeed look like a Lego brick, but equally it may look like a tangled shagpile!

Sequence is thus a basic tautology with row, column array and rank building tautologically upon the notion.
One further use of sequence is to denote comparison of magnitudes in a sequence. Thus a sequence may be a ratio, an ordered arrangement in space, a row or column, a vector etc etc.

Like and differing Is the fundamental categorisation attribute that describes all of these entities.

The everyway standing of §7 requires some background reading .
https://www.marxists.org/reference/archive/hegel/works/hl/hl217.htm
Hegel's approach or cognisance of the topic is later and based on his categorisation of linguistic or label distinguishers. In so doing he critiques and adapts the ideas as presented by Leibniz and  Kant.
http://dataspace.princeton.edu/jspui/bitstream/88435/dsp01ww72bb53m/1/Diehl_princeton_0181D_10142.pdf

I have yet to find an originating reference to Gauss, so if he did contribute to the concept it must be in response to Kant's and Leibniz philosophical examination of extensive and Intensive properties of space and spatial objects. Gauss was as a huge force in establishing a scientific standard or convention for fundamental weights and Measures that would be internationally acceptable and applicable. The word "dimensions" and dimensional analysis derive from these kinds of concerns and movements towards a common system of units. The first and easiest to establish were the extensive magnitudes. Attempts to establish intensive magnitudes as a system of units took a lot longer and required innovative measuring techniques and comparisons. The quantity of light falling on a retina for example is an intensive measure, but it is couched in an extensive format.
http://www.arcaneknowledge.org/science/magnitude/magnitude.html

http://books.google.co.uk/books?id=IUFTcOsMTysC&pg=PA236&lpg=PA236&dq=intensive+extensive+magnitude+gauss&source=bl&ots=0uXnhRc1Uc&sig=Ghp3vDEiEaUCuRF_AQ_xQixz0Rk&hl=en&sa=X&ei=S4FkVJ6jA4_g7QaRy4G4Bw&ved=0CB8Q6AEwAA#v=onepage&q=intensive%20extensive%20magnitude%20gauss&f=false

A critical simmary of Grassmanns work . Read from the heading Grassmann to the heading Riemann.

http://books.google.co.uk/books?id=ldzseiuZbsIC&pg=PA90&lpg=PA90&dq=Gauss+intensive+magnitude&source=bl&ots=UmNFqA7C-0&sig=pbPf_uAd4Ak48ZMMhSxa8s0IpMM&hl=en&sa=X&ei=U4ZkVL2pBIKwadyjgJgL&ved=0CCIQ6AEwAQ#v=onepage&q=Gauss%20intensive%20magnitude&f=false

This book studies the impact that Gauss's Arithmetical Disquisition had on mathmatics. In particular the downgrading of Geometry as king and the establishing of Arithmetic as superior Lord!
http://books.google.co.uk/books?id=IUFTcOsMTysC&pg=PA572&lpg=PA572&dq=Gauss+on++intensive+extensive+magnitude&source=bl&ots=0uXnhRf2W9&sig=qOXmOXH4-wILOxYg_aOLbPDuPbc&hl=en&sa=X&ei=d41kVJ7FOpPxaubugfgJ#v=onepage&q=Gauss%20on%20%20intensive%20extensive%20magnitude&f=false


In a very real sense Grassmann was part of that movement away fom Euclidean Geometry as promulgated by Legendre towards a more arithmetical approach. To see that we have to understand Algebra as Symbolic Arithmetic. Or alternatively we might note that Descartes called his book bout Algebra la Geometrie, in which many arithmetical formulae were derived as well as calculated proofs demonstrated.

However Hermann moved away not to diminish geometry but to diminish Mathematics and thus arithmetic. He sought to rebuild and rework the essential notions and concepts into a pure Mathmatic established on the right footings and in the right relationship with our experiential continuum.

His method seemed obscure, because his philosophy was Hegelian not Aristotelian. He challenged the then mathematical community to put aside its dread of philosophy in general and to embrace his arguments as in a court of law in which he defended his newly built expertise. Unfortunately for him the mathematical community could not be bothered to rise to the challenge . Those that did were rewarded, but many baudlerised his work and created chimera that exist even to this day in confusion.

Intensive magnitudes have exemplars in the Point potential concepts that abound in physics, while extensive magnitudes have exemplars in the line segment combinations that again abound in physics and Mechanics. That line segments can dual for Both is a little understood conception of what Hetmann was attempting to construct. In addition the inclusion of rotation was also misunderstood.

Later philosophers of nature rediscovered the relationships and concepts Hermann has laid out systematically in his 1844 version, but still only partially. You have to know a lot to get a lot out of what Hermann was attempting to do, all on his own. Thus we must recognise how extremely widely read this man was and how well self educated in many things. What he lacked was the charisma to promote his ideas. Fortunately what he lacked his brother Robert supplied, just enough to bring this extraordinary work to the wider attention it deserved.

In section 7 we enter the realm of continuous and contiguous. As these are based on geometrical exemplars the problem of an abstract definition of continuous does not arise!  Instead the labels themselves attach to the entity with its properties, so the labels are never empty!

So to the construction of a parallelogram from a and b. Whatever the orientation and direction the construction is the same : project parallel to the other line segment through the tip of the line segment. Where these projection lines intersect is the 4 th point of the parallelogram.

However if the line segments are already parallel this process produces 2 collinear lines which extend to the same point in that same direction and with that same orientation. There is no visible Parallelogram. What there is is a line which demonstrates

 a+ b = b+ a.

The result is a segmented line that shows this summation fact as commutative. It also shows that a product is fundamentally a sum of parts. How those parts are oriented determines the outcome, just as much as the process we apply to form the product!

Because the product is a dual line , hidden behind each other we can actually write 2( a + b)(1) as the sum quantity/ length, and the tally mark sign (1) identifies what we are doing is tallying the units in a and in b.

Because they are arbitrary line segments we can in fact define their tally mark sum to be 1 providing we choose the right quantity/ "magnitudes" for the line segments. However to imply that 1 comes from the magnitude of the line segments regardless , when they are in this relation is to mislead. The summed magnitude must reflect its referent, so an area requires a figure with flat space internally, while a length requires a notional line.

The transformation from a flat figure to a line is accepted as the limit or limitation of this type of transformation, not as an identity. So for example a curved line can never transform to a straight line even in the limit of a division of that curved line. That I see it as a straight line extensively does not deny its curved intensive potential. Similarly a parallelogram that is transformed to collinear line segment pairs may appear to be a line but its intensive Parallegram nature is still present.

In a dynamic situation even one of equilibrium the notion of transforming to 0 , apparently, must be distinguished by whether it is an extensive measure, an intensive measure or both. The concept of annihilation has no place in transforming magnitudes especially from extensive to intensive forms.

In a real sense , this is the distinction Newton was grappling with in constructing the Fluxions. Where his nascent or evanescent quantities came from without actually being 0? Was difficult to describe. The ancient Greeks pragmatically applied a principle of exhaustion? Kant and others developed therefore the language to discuss these things out of the Greek concerns.

I missed this lecture in this serie on the continuum concept.

http://youtu.be/Y3-wqjV6z5E

http://www.youtube.com/watch?v=Y3-wqjV6z5E

It is relevant here because the eventual structure for extending the natural / rational numbers is based on a process of modulo or clock arithmetics based on polynomials/ polynumbers. Polynumbers are a fundamental capstone rank array in the Grassmann general sense. In this process we can label the so called complex arithmetics, the finite arithmetics, the arithmetics that require or etc.

Underpinning these processes is the general method and analysis that is Hermanns extending or extensive magnitudes.

I have already mentioned that Hermann does not subscribe to the Dedekind cut or to infinitesimals. His concept is based on extensive and intensive qualities of magnitudes. §7 which I am retranslating touches upon this label or handle for the doctrine of the extending / extensive magnitude.

It should be noted that Newton did not subscribe to infinitesimals despite what you may have heard. If you actually read the Astrological principles as foreworded by Cotes you will read his distancing his method from that of those on the continent, that of Leibniz and Cauchy and others. Newton always used the principle of exhaustion.nhis Fluxions are minuscule assigned values . Thus they always remain Archimedian and pragmatic.

This video series about laser fundamentals is a good pragmatic exemplar of a Potential Point or a " bestimmter Kraft begabte Punkt"!

http://youtu.be/rgivGZqFcfY

http://www.youtube.com/watch?v=rgivGZqFcfY

Commentary
On §7

I have at last completed a reworking of this section. If anyone else would care to translate this section and post it I would appreciate it.

This section is perhaps the real heart of Grassmanns Method of analysis and synthesis. Everything leads up to this section which is the foundation of the Ausdehnungslehre.

This is a dance of ideas and notions tracked by labels and label definitions explicit or Implicit.. The reader is by now induced into a trance of new ideas and new relationships , excitedly waiting in anticipation for the great event, or totally confused and alienated!

The great event is the splitting, the hewing and hacking into 4 parts of the thought patterns that underpin the doctrine of the thought patterns. From this one is anticipating a revolution in productivity and thought process in setting describing and solving problems in 3 d space.

Commentary on the §§1-8

Ok guys, I think I get it . It is all about the correct mindset.

When I cme upon the word Förderung I raved about it being some kind of mental state of mind, even though the translation recommends promotion!

The thing is I am an etymologist. Not a professional one but a convicted one. Thus I like to trace or track the arrival and derivation of words as patterns of sound and as scriptal patterns that is as phonetic and graphic entities. So was Hetmann. In fact Hetmann was a professor of linguistics and etymological studies. He is principally known and respected for his work on the PIE the study of the Indo European roots to languages

One of my autistic traits( I am sorry guys!) is polylogia or pollulogia. I say too much, speak or write in long sentences, advance an argument in many many words, and do not really want to stop short of the fullest expression or description!

So I have had to learn to be short, pithy and terse, cryptic and succinct!

The rhetorical style is named aphorism.

Many Indian sages and philosophers utilised this aphoristic style in writing and recording. It is said that the Sanskrit epitomises this rhetorical style.

Aphorisms are not only a wonderful economy of words but also pure poetry . Within an aphorism fact fiction myth and magic may be aesthetically combined to say something deeply profound  or nothing at all ut musical evocation!

Certain Sanskrit texts appear as nonsense rhymes, Ganitas and Ghunas that make little sense, until you know that they encode arithmetical and numerical relationships. I remember watching a traditional Indian dance on YouTube in which it was explained that the drama and movement we're all an aesthetic representation of in dance! Or another expressed the right angle triangle theorem. And yet mant hymnals in Sanskrit set out the binomial series expansion in the patterning of the rhythms nd verses and tonal changes.

In the west we hardly give credit to these reported claims of Indian Yogic Sages or Tibetan monks who preserved their wisdoms or expertises. And yet we are astonished at their sciences and skills.

So how can a mindset be imparted to another human being given the exigencies of words and understanding?

The traditional way has been through apprenticeships and Koans. This is a fundamental of Indian or Sanskrit science. A student is set seemingly impossible or contradictory tasks. The student has no real guide except the expectation of his master. Through degrees the student becomes entranced and induced by his master untill his/ her mind state is receptive to whatever suggestions the master places before it. The student is in one sense hypnotised. The language used is crucial to inducing and maintaining this altered state.

In this state fantastic unbelievable communications take place which are o ly recordable as aphorisms!

The student will forever remain a student if he does not learn how this is done. Thus the qualification of master was given to any student who not only was induced by his master but could equally induce his master! Rhetorical studies were and are crucial to achieving this qualification.

Rhetoric as you may know was the highest branch of all studies of an advanced nature in the past, especially in Indo European traditions, including the Egyptian and Babylonian mystery school systems. However as time has gone on and public education has bern advanced as it has, this has bern " dumbed down" . Not only that, but the masters of the old ways have become infiltrated by those who have not been properly trained and who have dumbed down the oldest teachings to again popularise it.

The way of a monk is a lifelong dedication not to a master but to the mastery of the ancient learnings nd mindsets. Technological advance has blown appreciation of this fine tradition out of the water. The industrial military complex wants weaponise able ideas not harmonising philosophy which develops these ideas and solutions!

Everything I have studied as so called mathematics is vhingly incomplete nd bruised, gashed and crudely hacked together to make a Frankenstein monster. The parts have been torn out of their philosophical or spiritual contexts and supporting belief systems and mashed together in a big dissonant mash up! That is what has bern hitherto called mathematics nd Science and Technology. It is also called the Standard model and it works, but it does not hang together.

Thus a brilliant insight is mixed with a dumb ass rhetoric . We go from the sublime to the ridiculous to make ome engineering masterpiece. A laser for example is described in terms of pumping up the gas until it farts out both ends!  Or you could describe it in terms of electrons jumping up and down breathing in as the go up and blowing out as they come down . Or we could explain it in terms or arcane mathematical quantum field theory in which one set of variables have a defined sequence of states, and those states either cohere or concord or they do not . Depending on the level of coherence is how much power comes out both ends of a laser.

None of these descriptions is fundamentally different to the other. They just give us different rhetorical styles for expressing or modelling the same thing.

Where the real expertise is is in the juxtaposition of real phenomena and the observation of the inter relationships of that phenomena. In that experience an engineer may develop whatever formal modelling rhetoric he likes, but the most useful advances his ability with that set up to alter and predict the outcome.

Certain models are very generally useful for particular kinds of set ups. Thus finding those set ups enables that particular model to be applied. Does that model then deserve to be given the human endowed title of " the eternal Truth"!? Hardly, and especially if it is known to be a special instance of a more general but less technologically applicable system .

All these ideas run through Hermanns mind in many contrasting ways. The problem was how to make sense of them. The solution was the great Hegelian philosophical project of his day!

While not concurring with the Hegelian school he nevertheless revered the fundamental categorisation tool and Methid Hegel had created nd utilised in his great work the Phenomenology of Geist. Later Hegel laid out his Categories and categorical method, and taught the same as a Dialectic.

Hermann mentioned 2 Förderungs, his and this categorisation method of Hegels. Using both he set out the Ausdehnungslehre. In his first attempt in 1844 he brashly challenged the establishment in mathematics and a physics to drop their horror of Philodophy and to embrace his philosophical argument for a while. Of course they did not.

In addition Hermann really really struggled to find time to give his argument the best presentation! So while on the one hand he writes passages of soaring bravura towards the end of the Vorrede he is pleading to be taken seriously with a big dollop of indulgence! One minute he is announcing the solution to all things, the next minute he is admitting he could have got it wrong in several places, or expressed it better in others etc.

Nevertheless as contradictory as this may appear it reflects the thorough relevance of the Hegelian analysis and synthesis of the phenomenology of Geist, for Hermann exhibits all the main characterisations of that work!

Hermanns resolution then was to standby his imperfect piece, to launch it onto a very much unsuspecting world in the full Hegelian expectation that dialectic process would result in its eventual perfection!

Well he had to wait nearly 17 agonising years for the pricess to give any visible signs of working!

Eventually, and this is the point, the Hegelian identified process has kicked in as Hegel taught that it does. It just in practice is not predictable how long a particular notion will take to be engaged observably in the process. And also it is not clear how it may be "promoted" in the process. It is only clear that to survive observably a notion has to be promoted.

Today we have a whole global advertising industry whose role is to study and harness this promotional aspect of the Hegelian dialectical method. The newest observed dialectical promotion process has been " dubbed" or labeled / given the handle " viral".

Grassmanns faith in his Work and in the Hegelian analysis is very touching. Only Hegel remarks that the true dialectical process is the concern not only of his new philosophical approach but Also that of Religion. Hegel provides the truly combinatorial philosophical Methid that synthesise all human expression of the Geist.

I started with the ancients,and that at least by documentary evidence includes the Sumerian and Harappan  civilisations alongside the Egyptian . I have thus contextualised Hegel in a continuous stream of human thought process of a very ancient lineage. The hope is that the astute reader will not commit the mistake of thinking there is something new under the sun! Even in these days of landing mythological ly named robots on mythological ly significant " comets"!

The first 8 sections are a meditative induction to creativity. If you want to be super creative in thought in the sciences or mathematics I would advise meditating constantly on these 8 sections . Everything else will follow from that self induced mindset., that dual  Förderung of Grassmann and Hegel.

Commentary
The Quantum and the bestimmter Kraft begabten Punkt or the exemplar of intensive magnitude.

Clearly intensive magnitude concepts go back to antiquity, but it seems Leubniz and Kant and Hegel are the western proponents of such a concept.

However rather than wade through their weighty tomes it is enough to realise that in section 7 Hermann gives his own , Hegelian inspired take on the subject of intensive and extensive magnitude. The essential point is that in a very few admittedly tersely worded sentences ( especially in the German) he evokes the metaphor of a step rise, or stair step.

Contrast that with the metaphor of an extending line segment and yo get a picture of quantum as the stair step and continuous extension as the line segment.

Quantum as the intensive magnitude can in fact have Any value. In that regard it is continuous and expressible by a line segment. But while a line segment extends beyond its bounds, that being the nature of extension, a stair step is bounded and cannot get beyond a climax.

The climax however is bestimmter which I take to mean assignable, or appointed. Thus Hermann attempts to convey a physical representation or a pragmatic notion of an intensive magnitude. Fix the boundary but vary the density within that boundary. This is a measure of intensity or density that is quantised, yet continuous!

 Ŵhen we consider the Planck constants and the Planck lengths and even the Dedekind cut we have to acknowledge as Hermann says that these concepts are cut continuous ones, like the cutting of of limbs he suggests in the German.

The Ausdehnungsgröße concept allows this quantum concept to be explored in the context of a contiguous combinatorial environment and to be represented spaciometrically by line segmented figures. Thus intensive magnitude can be explored geometrically for new insights. Quanta no longer have to be represented by discrete blocky diagrams but can be explored by Ausdehnungsgröße as combinatorial objects.

This was new and to Hermann a huge advantage of his method .

Mach, Planch, Dirac and others all were heavily influenced by this approach to seemingly discrete or intensive phenomena . Whether they understood and applied Hermann properly is a question for research. I doubt it though , because of course intensive magnitude is not a unique idea to Hermann. His method of dealing with it by Ausdehnungs Größe rather than continuous or real numbers sliced at some assigned value seems to be his though.

I must add quickly, that the ancient Greek word for quantum is Metron, which is closely associated with Monad as expressing the same or very closely related idea.

Thus monad/ Metron/ quantum express the discretisation of  what can also be experienced as continuous. Hermanns 2 or dual cognisance of reification of experiential continuum

It is to be noted that reificayion accompanies the assignment of a label or handle to the phenomenon or observation.

Consequently Arithmos and Arithmoi are specifically quantised apprehensions of our experiential object, and a mosaic is a specifically quantised creative art form, as is this touch sensitive screen upon which I write!

Norman again on the Dedekind cut. Compare with Hermanns concept in §7
http://youtu.be/4DNlEq0ZrTo

http://www.youtube.com/watch?v=4DNlEq0ZrTo

The notion of intensive nd extensive magnitude has given me pause.

So Newton clearly used the notions of intensive magnitude in his Astrological principles, but I cannot recollect any deep discussion or scholium on the notion.  Indeed he delighted in the infinite series nd the infinitesimal  quantities that often comprised these series, but they were always assignable except when they were deliberately truncated due to exhaustion, else they were not counted. Everything hadbtombe countable , be they ever so small, and his Philoophy was that being do small they may be not counted with little pragmatic effect. However some on the continent wished these infinitesimal quantities to be some occult breed, and this Newton and Cotes resolutely condemned.

Yet Berkley saw fit to attack Newton without justification, for it was clear he had not read Newtons words. The substance of his argument was : have these quantities ceased to be extensive, nd now bern givn an intensive quality which in essence is the same as any religious faith might give to clearly imperceptible matters of the spirit! In that regard how thn do scientists differ from the religious cleric whom some now hld in disdain? " first take out the beam in your eye so you can see clearly to remove the speck in the clerics eye!"

These well rounded words of rebuke shook the Mathematicl establishment to the core! Their reaction, as Norman expounds was to create some tautological nonsense called limits and the axiom of choice.

Mathematicians ought to have been aware of the difference between intensive and extensive magnitude, and thus to have appreciated the subtleties of Newtons arguments and definitions. I now realise his long introduction into his Principia was in fact dealing with the difficulty of measuring or uantifying often what is not quantifiable, and yet is appreciable as degrees or levels of intensity.

Thus Newton established a philosophy of quantity that established certain measures of intensity or density as quantified entities.the subtle use of the qualitative nme to label the quantitative measure has long been a source of confusion among physicists nd philosophers alike.

Kant, Leibniz and Hegel therefore represent the better response to the issue, but few mathematicians even Gauss, gave much thought to it at a fundamental level. Hermanns treatment is perhaps the best of the succinct versions. To be fair Gauss did much to promote an international standard of measures, but shied away from the more difficult intensive magnitudes like brightness for example.

In my opinion I have only 2 extensive experiences on which to found the concept of extensive magnitude: that is space and Time. Nd even these Hermann quslifies by the contiguous nature of his idea of extension. Because of this defining characteristic I realised that any line jagged, curved or straight which is segmented continuously meets his definition . But for an intensive magnitude only an ephemeral point will do! Yet that point cannot be "nothing", that is having no parts. It has to have power and intensity Given to it where it precisely has no extension!

And this is true also for a dynamic form in an instant. Parmenides would have us assume that the arrow was stationary at each instant, but that then leads to a contradiction in time. Instead the ancients set out the principle of exhaustion by hich an infinite series may become manageable within time simply by truncating analysis where it is " reasonable" to do so , especially to get on with the process of synthesis.

Where we stop has no justification, except that it may give thr correct answer in the summation. But rather mathematicians assume a uniformity in the measure. Each measure is like a block or stair step beyond which the analysis does not go and equally the quantity never exceeds this form.

In so doing mathematicians have deliberately avoided the non uniform contiguous format, and that has bern a great difficulty for scientists and mathematicians. Indeed in laying out his Philoophy of quantity Newton provides an extensive discussion of intensive magnitudes!

Most of us have been told that velocity nd acceleration are " vectors" . They have magnitude and direction.. However what we are not told is that in fact these notions are intensive magnitudes!. We appreciate that an object has speed, but we do not  apprehend this as an intensive quality. Instead we trot to the mantra" speed is distance over time".

Speed is nothing of the sort. Time and distance are both extensive magnitudes! What we are doing, as advised by Galileo and others is labelling a ratio or Analogos, that is similarity between these extensive magnitudes in this quotient relation and the quotient relation we have in our experience of speed. Thusly we agree that such an analogy is fit for the purpose we have in mind. For it, namely to identify the uslity of peed uniquely. Thn in addition, having identified it it serves to quantify it!

However we have learned not to press our ratioing too hard, because then all is lost in a confusion of quantities, millimetres per femto second , metres per secon, kilometres per hour light years per year all may identify the exact same intensive magnitude! We typically choose a standard format, and thus avoid the reality of a fractal scale free experience!

Commentary

Rather than pander to the idea that mathematics might be truth, Hermann lags out the psychology of its construction. Thus being limited to space and time for extension, we have to recognise all else a formal works of the imagination. Thus it is important that those theoretical formalisms cohere with the definitions. It is also vital that they are inter communicant with some real confrontational experiences. Only in that process can we dare to say they may be true, but not necessarily the Truth. Especially when the vast majority of what we want to measure is of the intensive magnitudes.

The great impact of Newtons ideals were due to his  careful alignment of them. Following Glilro he starts with undeniable abolutes( although we can now refute the Jovian system description) and by a chain of relations and analogies brought the independence down to earth, to common weights and measures proceedings.

So carefully did he arrange this labelling and proportionately relate them that they survive today As a coherent set of principlesl but they are only man made , not divined from god ?

In particular the imagination to use line segments to represent essentially potential associated with a point. In fact it was something some people were unconsciously doing, but not thinking straight. Hermann points it out as a part of the normal realisation process of thinking in joined up figures.

The introduction of the notion of intensive nd extensive magnitude is a game changer!

These are notions that mathematics has to deal with going forward. In fact it should have been dealt with starting from the time Leibniz and Kant exposited on the philosophy of them and certsinly by the time Hegel expounded on the categorisation philosophy of them. The fact is that in keeping with the times no one was interested in the deep philosophy of the topic. Leibniz had just completed a worthy Tome on his conception of Monas and monads in general when the French revolution and the industrial revolution made their entrance onto the stage, bringing in a whole cast of new characters, ideas and geopolitical and social concerns. Deep meditative philosophy was pushed out of the way as new opportunities, powers and discoveries advanced science and technology and commercialism into an unbelievably powerful position.

The Förderung of the masses was now the major political concern of social philosophers and history philosophers. How settled states and empires could lose their grip on power was of great interest and concern. The social and political histories documenting the fall and the rise of The Holy Roman Empire for example were very influential. The redaction of the Christian bible , and the exposition of Prophecy in relation to the rise and fall of empires was crucial to how several key figures were understood.

Astrology and astrologers stood on the brink of a major Maelstrom in which they knew their very worth would be re evaluated, and their lives placed into great peril.

So no, no one was really that bothered about the difference between intensive and extensive magnitude. Today however the stand off between Quantum physics and Einstein relativistic physics exists because scientists and mathematicians are I'll equipped to even understand how fundamental it is to what they do!

Many still believe they are finding the secret laws of Natura, the goddess, others think they are discovering Fact, still others think the Truth is being revealed, but few are making the distinction between intensive magnitude and extensive magnitude clear in their deliberations.

Fortunately Hermann Grassmann took this upon himself to do as a life's work, and the Ausdehnungslehre is his dialectical record of how he got on and what he achieved, and where he wanted to go next.

Until you recognise that much of what you call truth is a confused modelling of our experience of intensive and extensive magnitudes you will not quite grasp what Hermann is promoting.

As a consequence of this distinction examine

AB + BC = AC

This is a combinatorial magnitude, consisting of 2 primitive magnitudes..

Or it is a script written in a certain pattern on a computer screen .

Leaving aside the identification of what it is ( simply, the reader must decide choose aor accept its semantic status) let us contextualise it in Hermanns discovery story. Then A,B,C are points in the plane ( arbitrary). They are glossed as intensive indicators of relative position, but that is not gone into. It is demanded of you to accept them as points of position. The point is given the power or attribute to indicate a spatial position. This is a bestimmte Kraft begabte Punkt!

But then we immediately set that aside , if we ever realised it, because the author hurries us on to an extensive magnitude: the line segment that connects those points. AB is thus an extensive magnitude , and we are happy with that , even if we do not understand or every way stand it!

The three extensive magnitudes are in fact quantised, and the end points set the quanta limits for a line . segment . We are directed to enumerate, evaluate or denote a quantity called the length of this extensive magnitude. In fact we are asked to do this so often that we think that that is what this extensive magnitude Is ! We rarely recognise that we are evaluating an extensive magnitude, because no one thinks it important enough to discuss. It is pushed aside as" elementary", " clear", " trivial" , "obvious" and many many other subjective directions away from what one is Actually doing.

Thus when Hermann directs us to evaluate the direction of AB and the length of AB and to hold them fast and of equal importance , it is not the result of these evaluations he is addressing, but the process of evaluation itself.

If you recall the Wolfram Alpha language you will see just how fundamental this observation and process is. Without this drive to build calculating machines, very much enhanced by the work of Leibniz , and the polynomial or polynumber concept, we would not have ended up with Turing once and for all , presumably, tackling this supposed elementary process.

I could say Hermann tackled it before him, but that would be unjust. Nevertheless Kant,Leibniz,Hegel and Grassmann laid the foundations for the work of Turing, and the work of many others. I would include Hamilton, but his conception is particular and as he acknowledged, an instance of the more general conception of the Ausdehnungslehre.

So now we have the three extensive magnitudes, each having length and direction hovering about them . How do we proceed?

Arithmetically we could add the lengths . We obtain a numbered dimension that supposedly reorients an extension. However, in space it is representing the triangle form.. As an extension this is bounded in the form. The form has a bounded extension we identify as say " area" , but that is not what the three lengths add up to. .

We choose to identify this sum as the perimeter of the form. It is a finite extension with a variation in direction, which forms a closed "loop". That description is itself, precisely an extensive magnitude, Ausdehnungsgröße. The number or numeral 10 say , even 10cm says nothing about this magnitude. In fact it obscures it behind one form: the ruler or measuring stick!

Thus the first realisation is our notational practices are defective! On top of that they are misleading. If I say the circumference of a circle is 3.142 times the radius, I imply that I can straighten out the curve and equate it to a straight line! This I assure you we cannot do physically , nor can we do it formally. This is why Hermann points out that geometry confronts our thinking process, and so cannot be a formal thought reeified system!

Clearing that up was crucial to his understanding of the Ausdehnungs Größe. But in fact he was only echoing what Eudoxus wrote 2300 years ago: there are different kinds of magnitudes and the ratio ought to be between homogenous magnitudes! Instead we seek as a ratio between non homogenous magnitudes. Is it any wonder that the ratio is transcendental? That word simply means we humans cannot fathom it. God alone can make a curved line straight, for only God can change its essential conceived nature.

Do not be fooled by a piece of string. Rather realise we assign properties and attributes to space like objects at our own convenience. The better we account for the spaces essential nature the better the attributional model we build. Thus using intensive magnitudes as tally marks where an extensive magnitude is required as a lineal combination, is going to lead to conceptual ifficulties in the end,

Now we have ABC as an extensive magnitude, which can be separated into line segment extensive magnitudes. There is a combinatorial reationship that is very very fundamental between these 3 extensive magnitudes

AB + BC = AC

This is not an arithmetic combination, or as Hermann labels it an algebraic combination. Right there he is at odds with the modern notion of Algrbra! In fact that is the issue. His label is pertinent to his time. Algebraic meant symbolic arithmetic, not the many and varied meanings we give it today based on the writings of Hamilton nd Boole.

The statement is a combinatorial expression. Hermann was blessed with the combinatorial teachings his father had penned nd introduced into the Stettin high school system. This paper really does need to be read to understand where Hermann is coming from,

The combination of these 2 extensive magnitudes can be labelled by a third extensive magnitude. However to be a combined extensive a gnitude the 2 parts must be contiguous, so that one can say the combination is indeed an extension from the smaller into the greater.

What if they are not contiguous? No problem, they are dealt with under the discrete doctrine of the thought Patterns and as such they are dealt with as discrete tallie and discrete combinations.

The realm of the continuous provided Hermann with the neglected field of study , the continuos( contiguous) combinatorial magnitudes. These he recognised or denoted as the true or obvious candidates for the label extensive magnitude. To do so he had to identify the continuous uniform magnitude as the continuos algebraic.

Because, he claims every magnitude in that pattern is uniform we can quantise it by a uniform quantum. In so king we can study the quantum, because it is indistinguishable from the entire whole. This is one of the characteristics of an intenive magnitude, it is scale free, uniform and indistinguishable. To istinguish it we have to impose formal quantum . As quanta they are bounded, and cannot extend beyond their bound no matter how contiguous they are!

A little thought will clarify why that condition is imposed by Hermann on intensive magnitude. He does attempt to closely explain it, but my translation may not be clear enough to get that across.

Extensive magnitudes on the other hand grow out of each other. Because they are not uniform in boundary or other characteristics the contiguity is important. For Hermann the concept is tht one segmented part actually extends out of a prior one. So for example. Crystal grows in this manner, and reveals it was thought a Truro aspect of nature: extensive growth of magnitude in any direction with any intensity. The everyway varying magnitude!

Due to the Hegelian logic, Hermann can hold these things together in a combinatorial resolution. Aristotelians would be going absolutely crazy trying to pull them apart!

It is not that Hermann does not pull them apart, because he cn and dies. Rather it is that he can go beyond the snalys to the synthesis and flip between the 2. This is the power of the Hrgelian toolbox.

So finally AC is an extensive magnitude, but it is now Aldo a labell Gordon a combinatory extensive magnitude which is dissectable ( zergliedern) into 2 subordinate ( untergeordnet) extensive magnitudes.

The labelling and the ordering does not stop at this simplest form. We can expand this formula into a vast endless array of such combinations. This is the Reihe that Hemann perceives, and it is fractal.

I have been able to use the expression rank array because today, after do long a time and after Caylry and others pioneering work in establishing a notational system or them, and Clifgord work in establishing the dot nd cross pouch notation from both Grassmann and Hamiltons less apprehensible notation involving the notation we have some familiarity with a rank array. In addition Gauss elimination method crystallises some of the notion of rank. Hermann clearly had these notions in mind as he wrote which is why his original work still resonates today. Back in his day such a system was perhaps only apprehensible by LaGrange LaPlace , Gauss and Euler, but it is their summand notstion tht obscures it fom moden eyes.

The counterpoint to the extensive magnitude is the intensive magnitude( continuous), but what Grassmann points out is the duality tht exists in the notation, resulting in many insights transferring back and forth between the 2 types of magnitude. The two processes are inter communicant!

Finally the notation makes clear what is intensive and what is extensive.
Thus e is an intensive or extensive bound magnitude 5 e is a continuous intensive magnitude.e₅ is the 5th Differring part of a 5 part extensive magnitude which will be
e_I as I runs from 1 to 5.

We can reduce an extensive magnitude to n intensive one by constraining the e_I to being uniform.

There is a lot more besides this, not the least being the product notions associated with extensive magnitudes, which dawned on Hermann, according to the story,when he was looking at geometrical or trigonometrically ormulae fir te area of a rectangle/ parallelogram.,

The final part is the Schwrnkunglehre part, where he finds out about a third kind of product, the roots of unity products of Cotes and De Moivre, as expounded upon by Euler.

Whatever he came across he reworked in line with his growing confidence in his Ausdehnungs Größe concept and his growin apprehension of the intenive and extensive magnitudes and their role in describing or modelling the kinematics of the real " world" .

My simple question, directed at Newton, Hamilton and physicists in general: is Time an extensive magnitude as we are led to believe?
T Hermann and Hegel the question is : is time representable by an Intensive magnitude label?

I want to apply the Ausdehnungslehre or rather the Formenlehre to some physicl phenomena, as Ampère attempted to do. But here is a taster of the standard application to circuits, in which you may experience some confusion as to what the use of vectors is and what the graphs are doing. In fact the mixed ptiduct of this thread is relevant to this description, which I hope will become apparent by and by.

http://youtu.be/LEWKHvuRsUk
http://www.youtube.com/watch?v=LEWKHvuRsUk

However if you first get clear in your mind what an extensive magnitude is and what an intensive magnitude is , then you might orient yourself to this kind of application( which by the way I think is misleading anyway! But that is beside the point)

Commentary

There is a lot more to be meditated on with regard to Intensive and extensive magnitude, but I am going to move on.
Just realise the Formenlehre, the doctrine of thought patterns is hewn into 4

 The discrete  encapsulates 2 of the four, and very roughly is tallying and combining discrete entities real or imagined.

 The second 2 are encapsulated by the continuos and the contiguous. . These are very roughly the intensive magnitude which has to have a metric assigned or applied to it, because it is continuous, and the extensive magnitude which has to have a topological structure applied to it again because it's regions are contiguous and arbitrary.

All four categories can be arranged into an array structured labelling scheme.

The Ausdehnungslehre properly treats of the continuous categories and the treatment is fundamentally the same for both intensive and extensive magnitude ( the capstone rank array) up to a point where the details go their necessary ways.

Intensive magnitude calculation?

http://youtu.be/bCOUSJrnhOE
http://www.youtube.com/watch?v=bCOUSJrnhOE

Surely the radius is extensive right? Yes but is it extending?
An extensive magnitude is a combinatorial contiguous form. Thus the extensive magnitude would be the dynamically changing one , not a balanced one. The extensive magnitudes naturally rely on calculus ideas to be expressed.

It occurs that Normans long running argument against the real numbers is another version of the discrete Continuous divide Hermann hews out in the Formenlehre. However Herman clearly identifies these formal definitions as thought patterns, not as real entities. Thus the continuum in space is part of the real experiences Hermann calls 3d Space( Raume ) and which we have an inhered apprehension of. Thus the extensive magnitudes and the intensive magnitudes Are entities we cannot represent by numbers. Our best representation is a line segment!

In this way Hermann sidesteps this issue of numeral assignment and points to the experience of constructed geometrical/ spaciometric forms directly.

The reliance on numbers produces this logical difficulty , as Justus Grassmann identified. But deeper than that the Aristotelian logic has to be enhanced to the Hegelian one for a best practice solution. Computation then gives us " approximate or bounded solutions, but computer generated graphics can represent the extensive and intensive solutions Grassmann has identified..

Norman on computation
http://youtu.be/TiryH-c44ok
http://www.youtube.com/watch?v=TiryH-c44ok

Normans presentation of interval arithmetic.

It is iortnt to note that Norman develops natural numbers from Tally Marks in a mythical way . However he skips over the difference between intensive and extensive magnitudes at this stage, because he has a logical problem with the continuum which he has recently been addressing.

Thus in going from tally marks to intervals he utilises Wallis's number line concept which to be fair Wallis called clearly a Measurement line. Thus it is only around the time of Dedekind that the line segment as a continuum model or label was abandoned for these weird logical conundrums.

Normans Intervals are line segments of Wallis's Measurement line, with the additional property of direction encoded by ordered Arabic numerals.

http://youtu.be/xReU2BJGEw4
http://www.youtube.com/watch?v=xReU2BJGEw4
And the second one
http://youtu.be/xE4DdIwPh2s

A significant result is that the distributiyity of interval int is not an equality! What this means I think is this construction process for integers is not 2 or more dimensional . Squashing it into a line segment introduces an inequality of type.

Ausdehnungslehre 1844

Induction

C. The exposition of the label of the Extending /Extensive  Magnitude doctrine

9!  The continuous  Becoming entity, in its moment laying hewn apart, appears as a continuous rooted and risen up entity with firm holding of the already reified entity.

Considered by the Extending magnitude thought pattern, the each time new rooted and risen up entity is set as a  Differing entity  : now hereby considering, we do not hold  fixedly  the eachtime reified entity , thus we reach  to the label of the Continuous Varying entity. What this Varying entity experiences we name the Created whole Element,
And the Created whole Element in a random one of the condition/ status markers, (which marker it adopts by considering its Varying entity,) we name "an element of the continuous thought pattern".

Thus hereconcording, the gathered system of all the elements is the extending magnitude thought pattern, in and  over which the Created whole Element travels by continuous Varying entity.

The label of the Continuous Varying entity  of The Element can Only by considering the Extensive / extending magnitude step hereforward.: considering the Intensive  magnitude:-by considering the on top piling of the eachtime reified entity, only the continuous alongside State description to the becoming as a "completed empty" entity would be remaining afterwards .

In the space doctrine , the point appears as the element ; the place varying, or kinematics as its continuous varying;  the Differing positions of the point in Space as its differing status/ condition markers.

Commentary

Because this is not my social language I can translate this German according to its literal etymological roots. It is a Mechanical translation with a few nice turns of phrase but not many!

The word Ansatz and Aufbung are interesting, as is Eigen . The words Ansatz and Eigen have made it into the Maths speak vocabulary. Here I have translated them as literal to the context as I can, but I can only do this because my whole translation project is as close as I can make it to Hermanns words and sentence structure. By this I hope to capture his pattern of thought as well as his patterns of expression.

A lot of these concepts or notions are so basic that you might not recognise them when they are given this Hegelian dialectical treatment. But they have this treatment because every young mathematicians wonders why so many differing words for essentially the same mark on a piece of paper!

I get x! And I get any other letter of the alphabet, but it is not so easy to grasp Algebraic symbols!

Here Hermann attempts to attribute the properties of the extending magnitude . Drawing on the set up in §1 - 8 he quickly clarifies the existential nature of the extensive magnitude both in its extending form and in its Intensive form.

Note the use of extending ( in place of extensive) because here the point to be made is that the extensive form occupies space, but the intensive form is an "empty" space entity.

Perhaps the reader might like to ponder the meaning of Nothing and Empty, in the light of my assertion that these are concepts we truly have no way of understanding, or every way standing! My contention is our signal processors in our neural networks are always " on" , and so we can only register Something.

The consequence of this is that no matter how many metaphors we give to express the concept of empty/ nothing, we are forced to explain it as the lack of some particular thing amongst a number of other particular things! Thus we truly never experience Nothing.

In this regard Hermann makes do with the creation process creating an empty entity. This is thus not an act of omission, but an act of creation. The process has not failed to actualise, it has actualised an entity called the empty entity.

Hermann does not elaborate on this. It is a profound existential reality. Instead he points to the fact that the creation of this empty entity actually leaves a side statement or an accompanying statement of what is going on. This is the Ansatz. Like Gegensatz, the notion Satz refers to a set down expression or statement, or some set thing that symbolises some thing! It is very very general and requires context to give it it's in context shape!

Here I think Hetmann uses it to refer to the Rank Array that contains all the details of the field or object of study.

He also seems to shape the extensive magnitude in terms of elements of intensive magnitude. The status/ condition markers allow this contradictory object to exist, for how else could something be made out of ultimately nothing?

The structure which encapsulates the varying extending magnitude is also the structure that sets up a representation of the intensive and ultimately non extensive or empty entities we perceive as intensity/ density/ frequency or any other aesthetically appreciated experience. This structure in the case of intensive magnitude is the Ansatz. It represents intensive magnitude and so has no extensive evaluation!

It has  an evaluation but that is intensive in nature. So if I feel the extensive structure fits well with empirical data, that same structure evaluates to an intensive" good" .

You might ask" what use is "good" to a technologist!? Well "good" is everything to an engineer , and " better"  is better! These intensive evaluations guide the application of the actual Ansatz or supporting structure through which the extensive evaluation is being processed.

What about density and mass? These are intensive magnitudes like good and better, but in these cases the Ansatz is contextualised in the comparison of strength or force. By trial and error the products of these calculating structures or algorithms are associated to good or better outcomes. The details of the structure are often irrelevant. In the past a master mason would build a superb structure by experience. Today that structure would be evaluated by some calculating algorithm. The results would be the same. A daring design does not really rely on the algorithm, it relies on the real expertises garnered from years of trial and error. The algorithm reduces this sense or engineers sense to a formula that guides the inexperienced engineer.

Trusting the algorithm may make for safe buildings but it also covers over incompetence. A competent learning engineer will build safe structures regardless of any Ansatz or supporting set of rules or formulations or guidelines , or best practice statements or Engineering for Dummies type of presentations.

Am I decrying the algorithmic approach? No rather I am contextualising it. The Ansatz is a tool we develop to guide our trial and error approach to life . As such the whole of Hermanns work can be regarded as just Ansatz, a guide on the way to continual improvement in our interaction with space.

Aufbung I have translated as "heaping on top", eigen I translate as " inward looking" in antonymony to augen which I perceive as " outward looking" and hence Augen or eyes.

Now Reihe as rank array is increasingly fertile. Firstly I recognise by it the work done by Cayley and Gauss and others in developing this notational language. Secondly I mean to imply the generalty of Hermanns conception upon which Cayley was able to construct " his" notational device.

Every concept of products in current use, derived from Haniltons works , especially on quaternions, have a deeper ocher nd more general conception in Hermanns work. Firstly Hamilton acknowledged this directly in 1853 upon reading the Ausdehnungslehre for the first time, and secondly Hermann is discussing this precisely in the essay that is the title of this thread, to which I will return after this long exodus into the related hinterlands.

Thus I preempt myself in this translation which is not true to Hermanns context, because I do not intend to explore the murky world that leads to the current Notation as exposited by Norman Wildberger, if I can help it.

To me the interesting thing about arrays is the fundamental positional structure. Thus I can start by positions in space, and thus by representative sequences. The mosaic or proportional aspect of the positioned quantities is maintained.

From this structural representation of sequence or array, Series of products are constructed by. Matrix producing.

Now tht is intriguing,

I have been following the Philae landing with interest. This blog post and particularly the response section highlight the communication problem between the experts and their audience.

http://blogs.esa.int/rosetta/2014/11/19/did-philae-drill-the-comet/#comment-215968

While some attempt to dumb down an explanation still others attempt to big themselves up by giving a scientific sounding explanation. Others just clearly want to connect what they are taught in school with the observed phenomena. Others want to use what they have some expertise in to apply it to a problem or explanation they are unfamiliar with.

In amongst all this the actual designers and engineers release only partial information . Clearly the ESA wants to put a positive spin on the whole escapade so far!

Because we are used to rockets landing we do not expect them to bounce! Yet 2 of the most successful rovers bounced there way onto mars! I mean Spirit and Opportunity.

Despite our sophistication on earth, in space we are reduced to looking like Jackasses, and that " ain't cool!". Nevertheless it is true. We are unfamiliar with the fundamentals of our solar environment. Worse still, we are unfamiliar with the fundamentals of our labelling/ handling systems for notions. Almost nobody wants to go back and address these fundamental difficulties. Instead, as you progress in a field you generally are lead into thinking that you are oing high level research. In fact the Governments only pay for "Basic" research, that is research into fundamentals.

No professor wants to admit that they got it " wrong!" . Instead they say " We may need to revise our theories ".  The difference in words reflects the difference between being hung, drawn and Quartered by the mob and retaining credibility with patrons. People, Science , scientists have to play these games to survive.

Intensive magnitudes like Mass and weight are easily confused. But a bouncing comet lander is easier to visualise! We see how the intensive magnitudes are related or in some way evaluated finally by an extensive magnitude, the displacement of positions on a cometary body.

Ausdehnungslehre 1844
Induction

10.  The differing entity should develop itself according to a rule,  even if the created whole  result is a chosen one. The rule, by considering the simple thought pattern, should be the same for all moments of the becoming entity.

Thus the simple extended magnitude thought pattern is the thought pattern which roots and rises up through a varying of the creating whole element , a compliant varying  concording to the same rule.

The gathered totality of all the createable whole elements  concording to the rule we call  a System or a Field.

The differing quality was coming to be ; there, the  "from a Given" entity  can endlessly manifoldly be differing entities, running themselves everyway completely in the Incoherent differing quality,  If it  were not to submit to a fixed rule.

 Now, however, this rule, in the pure doctrine of the thought pattern is not chosen through some "random whichever  Content", rather, the label of the extensive magnitude is  chosen through the pure abstract idea of the rule governed Entity , and through which idea of "the same rule for all moments of the varying", the label of the Simple extending magnitude  is chosen .

Now, hereconcording, the simple extensive magnitude has the Attributes of condition and property , that even if  out of an elemental extensive magnitude  of the same simple extending magnitude a an other element b of the same simple extending magnitude hereforward goes,  through an act of varying , then out of b  through the same act of varying, a third element of the same simple extensive magnitude c  hereforward goes!

In the space doctrine is  the analogous Atttibute of Direction, the singular varyings  all embracing rule,
therefore
 the line segment in the space doctrine inter communicates to the Simple  extending magnitude,

the unending direct line intercommunicates to the complete System.

Commentary on 10

The simple or basic extending magnitude is introduced here and the notion of a gathered totality of all identically created simple ones.

It is a bit tricky or slippery to catch the specific character of this simple extensive magnitude , because Hermann wants to justify choosing or imposing an abstract rule to create or reify them.

As I understand it right now, his argument is simply that we cannot mentally deal with chaos, so we are forced to impose a rule by our own inability. However it has to be the simplest rule and as content free as possible.

The interesting thing is his choice of applying the variation to every contributing element the same way at each moment of the reification. Even that sentence does not quite catch his general idea for me.
a,b,c in the example introduce a symbolic device to denote the simple extensive magnitudes I think, but the varying , according to the rule, is not yet clear to me.

The example of Direction is not as simple as he perhaps thought because direction consists of 2 notions: orientation and translation within that orientation. So what the all encompassing rule for specific variations is escapes me at the moment.


By now then he has introduced the intensive magnitude = point correspondence and the extensive magnitude = line segment correspondence.. The = sign here clearly transated entsprechen  rather than gleich. He has not used this sign yet , but I have introduced it in advance for consideration.

Further meditation is required at this juncture, because we see before us reified the fundamental space, the infinite line with a basis line segment . Direction is mentioned but not clear unless we ignore orientation . However I am loathe to ignore orientation because that is when the confusion starts regarding the notion of direction.

Most mathematicians will state the I dimensional case by drawing a horizontal orientation. They do not realise what they are doing mentally to themselves and their view of space. When I watch astronauts on the space station I recognise how limiting this innocuous seeming practice really is. Orientation is more fundamental than translation ; rotation more fundamental than any direct line segment!

Here I would introduce the circular arc segment as well as the line segment . It can be introduced by precisely the same general set up Hermann has just laid out, and it clarifies his observation of an all encompassing rule somewhat . Changing orientation of an extensive magnitude changes every contributing element to thst extensive magnitude, but whether that follows the a,b,c explanation pattern I need to determine.

Where do these segments come from? Because Hermann has separated Dpace from the thought pattern doctrine he can justifiably seat them in the space doctrine. In which case , choosing only the line segment , the direct line segment, is an oversight on his part at this stage.

However, we know Hermann comes back to include swings and pivots, and that this was a later development in his researching of his initial ideas, but as a consequence he misses the simple rotational extension as a curved or arc segment, and concentrates, as did Euler on the swinging straightline segment.

From my vantage point I think this is a fundamental element by which his ideas can be reworked. Indeed the Twistor thread is my attempt to develop this idea on my own. However I feel that by studying how Hermann did it I can learn from him and avoid the wrong turns he admits to taking on the way to producing his masterpiece the doctrine of Extending/extensive magnitudes.

This simple rule every moment of the reifying entity must obey the sane rule, underpins the Mandelbrot fractal  , whose process   z =z² + C is the same rule applied repeatedly.

I have revised the subtleties of section 10. The qualities or attributes of a behaviour may be described as a governing rule. Thus highlighting that our perception , commonly held, that nature abides by laws or rules is our psychological perception of qualities or attributes in natural phenomena.

I use attributes to emphasise that we mentally put these qualities or rules into our description of phenomena and onto descriptive statements of phenomena. It is we who cannot abide or comprehend unruly or lawless , patternless behaviours or descriptions.

Knowing this enables us or me to look deeper into a phenomena for more complex rules or patterns which I can then attribute to the phenomena! Occams razor however advises that we utilise the simplest rule wherever we can.

The description we have of phenomena in science is thus always only the simplest one of many other possible simple descriptions, let alone the complex descriptions also possible!

It is all a matter of "taste" when it comes to descriptive models, but technological performance and compliance to observable behaviour provide corroborative evidence of utility. Whether a model is true or truer is always a validation that we must carry out, by utility mostly. To go beyond this to an abstract idea " the Truth" is clearly a personal choice, not found in reality but attributed by an individual to observable phenomena.

Commentary
If , like me , Peano meditatively translated Hetmanns doctrine into Italian, it is not surprising that he should utilise the Sigma summand notation to describe systems of elemental extensive magnitudes, of the simplest kind. However it is Levi and Riccis more elaborate notating of these Summand forms that lead o the Tensor notation. The concpt has not changed, just the notation, and thus the utility and the flexibility in labelling and thinking.

So it should not be a surprise that Cayley's matrix notation is in fact a variant notation for the same conception, here laid out generally by Hermann, but also set out specifically by Hamilton for Couples and Quaternions. The utility of Matrices over Tensors is their greater transparency to the underlying basis, and greater pecificity of that basis, especially in examples with finite elements.

Once the numbers become large or infinite, there is very little notational difference between matrices and Tensors. The concept of a " vector" therefore is confused, because it is a notational device that belongs to both notational systems.

In the space of 3dimensions we do not have this confusion if we stick to the suggested line segment exemplar. In this setting Hermann identifies an all encompassing rule that governs singular variations in direction for simple extensive magnitudes, that is line segments.

While it is common to present these attributes as falling out of some general law, yet here we Knis Hermann developed the general law by observing the specifics of line segments in their behaviours. The whole of the Ausdehnungs doctrine is thus concocted from an intensive and prolonged study of the behaviours of line segments in modelling or describing attributes and or phenomena observed in space. Because of this focus it is clear to me that Hetmann missed the significance of his generalising approach for curved line segments in general, and circular arc segments specifically. Thus when he makes hs simple correspondences for the simple extensive magnitude he misses a trick. By including the circular arc segment at this juncture and the circle as the corresponding system he would have provided a truly more general method ncompassing all forms of curve and curvature.

In addition, the category of intensive magnitude would not be modelled by a "step rise" but rather by a circular disk or a spherical ball! The advantage of this would be that the radius of curvature clearly then becomes an Inverse measure of intensity or density. Point like spaces would clearly be huge intensive magnitudes while circular disk regions would represent lower regions of intensity. The perimeters of such regions would have curvatures approximating to straight lines.

We could thus model straight line and flat plane geometry as occurring on the surface of a sphere or spheres whose near common centre is very far away or at " infinity" .

Immediately we have a polarisation into local and universal behaviours or attributes. Local attributes would tend to be more curvilinear while general or universal ones would tend to be flatter.

From the outset Galileos fractal for the solar system develops this local and far difference in absolute systems. Thus the greater system of the sun does not exclude the absolute powers of the local system of Jupiter, et al. Newton in formulating his Astronomical Principles was well aware of this feature of Galileos Fractal. The inverse square law or rule was thus a rule to be applied at all scales and all moments of the systems reification, that is as an eternal dynamic.

Is the inverse swuare law gravity? Clearly it is no such thing. It is an all encompassing general rule of Newtons system which applies to the simplest exemplars of the extensive magnitudes. To apply it to a galactic mass is therefore to invite error. It is not a rule for such a complex extensive magnitude!

What such a rule would be , or how we might develop such a rule is precisely what the doctrine of Extensive magnitudes would guide us in formulating! This is assuming the doctrine is correct and robust enough.

In any case it is clearly a system suited to the fractal nature of our experiential continuum, and still intriguingly relevant!

Ausdehnungslehre 1844

Induction
11. One applies two differing rules of varying , so the gathered totality of the every way possibility "of  both rules"  creatable whole  elements develops a representation of a system of  second step(two stages).

The rules of varying, through which the elements of this System can go hereforward  out of one another, are dependant upon those  both first elements ; bringing yet a third independent rule from afar to join the other one thus  reaches  to a System of third step(three stages) ;  and so on and so forth!
M
The space doctrine here again would like to serve as an Exemplar ( a "by considering it" game). In the same doctrine comes to be created whole, out of an elemental entity, the" like a gathering" elements of a Plane, by considering two differing directions , in which entity  specifically the creating whole element whimsically many  times forward steps   according to both directions arranged one:other, and the gathered totality of the so creatable Points( elements) in one  Entity becomes together fixed.

Thus the Plane is the system of second step(two stages/ steps);  in it an endless crowd of directions is contained, which depend upon those both first directions. One takes from afar a third independent direction to join the others,  thus, mediated by it, is created whole   the whole endless Space ( As a system of third step( 3 stages/steps)); and  here one cannot come  further as one has  on way to  three independent directions ( Ruled varying directions),, while in the pure doctrine of extending magnitude the Tally Account of the same independent directions  can raise itself up even until  in the endless entity!

Commentary on §11

Ok, so now, as I should have guessed, further clarification of the ideas in §10 swim into view.

It is hard to simply express how this exposition is so well constructed. The necessary ideas are vaguely or generally described prior to when they achieve their maximal impact in specificity. So it is helpful to re-read the earlier definitions and labels, notions and concepts. The acceptability and applicability of notions presented by Hermann in your immediate awareness hang on these prior considerations that you have accepted or allowed. Like a game of chess Hegel and so Hermann locks you in to his Schema inductively.

Consequently it is important to state that these techniques and methods of persuasion constitute the Hegelian logic, an advance on the -Aristotelian logic system. The construction is hypnogogic, homiletic , designed to convict and convince.

From the outset Hermann and his brother  Robert have been working in the framework of the early 19th century view of philosophy, Religion and Mathematics, but with a view to purify and correct it from perceived and actual error. From a religious standpoint or even a high brow ethical one this is a Laudable goal  " advancing the cause of Truth" . However I do not personally subscribe to such motivations, and consequently view all such claims as skeptically as I can.

The Hegelian logic system makes this an impossible thing to do, because it includes my and any skepticism within its operating principles. I am thus reduced, as I must be, to my innate flawed opinion, and no better or worse than any other opinion of this content.

To say that Hermann concocted his generalities out of the specifics of his contemporary experience is hardly enlightening, since Hermann specifically describes in the Vorrede exactly that! And not only that in passing but in step by step detail.  So to make this remark is not to castigate Hermann, but myself for intuitively or subconsciously seeking some kind of divine Epiphany from these general statements and their specific expositions.

That is not to say that there are no epiphanies to be had, because there are. But each is personal to myself , revealing where I have made some unjustified or justified assumption; or where I have accepted some camel while straining out a gnat; or where I simply was ignorant of how what I was taught was originated.

In section 11 Hermann introduces the coordinate system of geometry (a la DesCartes), Its very general principles in a formal statement which is meaningless until exemplified in the space doctrine.  Because of this approach, it is almost automatic to ask is there another interpretation of this general formality? That truly is the " power" or absorbing interest of Hermanns method . Thus my earlier comments about using curved segments constitute an alternative .

But Hermann dissects how we arrive at the labels and notions we use in the coordinate system. In so doing he Maps out how we might apply the general formalism in an alternative setting. We see how he adjusts tetnimology or label to a specific exemplar, and learn that this is what we must perhaps do in any alternative System.

But further still, using direction as his general rule, he now can apply differing rules. Because he has not analysed direction itself I think he missed an even greater fundamental, but that in fact is a detail to which The Method he is inducing can be easily turned. It is the Method that one must pick up on as best one can.

Because of the method and earlier distinctions he points out that while it is impossible using mathematics to construct any space whatsoever, let alone 3 dimensional space, using his method the doctrine of extending magnitudes , one can rise up to greet it and indeed rise up beyond it into endless space!

The notion of climbing or enhancing to a System or level or stage is a crucial notion in his system, and in Hegels categorical logic and Dalectical system.

We see now why it was important to remove geometry from the formal expertises. It is " Geometry" that initially establishes the validity of his general statements.

But his general statements are validated in more ways than just geometry. However, remaining with Geometry one is bound to ask what is a 4 dimensional space and can geometry represent it? The answer is yes. What one has to let go of is orthogonality. Hermanns general formalism makes no mention of orthogonality, it relies solely on Independence, that is Differring at an elemental level.

Orthogonality is a unique property of the circle and sphere, it is not a general property. The belief that space is 3 dimensional is foisted upon us when we are very young with no critical apparatus to question what we are being taught. As a crystallographer one can appreciate the need for Space to be more than 3 dimensional. With this freedom the facets and crystal organisations can be more coherently and satisfactorily described.

What do we mean by independence in space? Very simply and very naturally any 2 orientations are independent! In Hermanns notion Directions are independent. Thus we have an endless crowd of independent directions to utilise, but we need the first ones to specify all the others.

Currently we are taught that the first 2,3 etc are a basis if they are independent, but all others concocted from them are not independent! This is a language confusion between dependent and dependant. The German is abhängich, and literally means "hangs off". No direction hangs off another. However we construct directions in reference to elementals which can be whimsically chosen. Any line can be referenced by an endless crowd of related pairs, etc. These pairs,etc are in the same class of multiples. These multiples are in an Equivlence relationship. It is this equivalence relationship that is confused with dependence .

Now some have sought to show that a maximal basis exists for 1,2,3 space. In fact Hermann denounces that widespread belief in announcing the 3rd stage System. Here he  goes on to show that this is an imposed constraint that hinders rather than Aids understanding spacelike objects and their behaviours relative to themselves and others. This is alluded to through an accounting Bill , the account tally. The extensive magnitude method keeps track of its elemental constructors. Thus as long as they are independent, systems of directions can be built endlessly.

However we cannot build them in the same orthogonal manner we have built the Cartesian reference frame. This is ironic, because DesCartes did not construct this commonly accepted frame, Wallis did! DesCartes frame was in fact very very general. Today this generality is wrongly ascribed to Lagrange in terms of origin. LaGrange indeed pioneered generalised coordinate systems as a development of DesCartes original prescription.

We can live intuitively in an n-dimensional space, if we recognise orthogonality as special cases not general rules.

It is worth drawing attention to that the step rise was a label associated with Intensive magnitude particularly. Now the label is being used to denote particular systems at differing levels or stage kinds or step rise types. What is going on?

Simply the intensive magnitude at each stage is of a different kind. So a point on a line is different to a point in a plane, and likewise a point in space. The intensivity of this magnitude is increasing, but extension of this magnitude is not . Hermann called it the empty or emptying magnitude as far as extensivity goes.

To ignore the intensive magnitudes is to diminish the potential of these magnitudes. In particular, when I refer to a circular or spherical intensive magnitude I include within its intensivity its Rotation. Thus a point in space is invested with many crucial but invisible dynamic potentials which physicists may draw upon.

Homogenous coordinate systems that  Norman introduces in his universal hyperbolic series, are another construction of these extending magnitude systems, and Norman introduces another for spherical and elliptical geometries in that series.

The Riemann  projection of the complex plane onto the sphere is another extensible magnitude system, and Gauss use of the unit sphere to specify curvature and tangent planes, bundles etc is another application.

In each of Normans presentations he is careful to define or label the different magnitudes. He also utilises the row or column flexibly as he requires it to specify what he wants to distinguish.

Extensive magnitudes doctrine concentrates on the continuos or contiguous extending magnitudes, but Norman has also done work in the discrete magnitudes to bring everything into line in his thinking.

Commentary

The simplest rule is identified as direction, or interpreted as direction in space. Actually the word used is Gleicheit. The thing or entity that has the like quality.

So hermanns all encompassing rule that binds together and governs all particular variations  is direction.

Actually this is hard to fathom without realising he is meditating on the line segment label and it's relationship to the endless line notion.

We usually call this a I dimensional space. We could call it an einfach or simplest space which is directly translatable as a "one type " space.

We do not even get to consider the linguistic origin of this notion, the linguistic label, because as mathematicians , early on we are indoctrinated with the idea that the straight line is the simplest line that we can "draw" or rather draw upon for insight and geometrical construction.

In fact Euclid never said that. Very much in line with Hetmanns approach he isolates the straight line as having unusual specific properties in line with the ideas of indicators or seemeia, usually translated as" point", but as we have seen that is an interpretation of an intensive magnitude experience.( according to Hetmann).

I have translated the Greek Isos as " dual" for a number of years now, because of meditating on the idea of a point for most of my life! In that regard a "good" line is made up of dual points, not just the points on the beginning and end. And a line is any scratch or dragged mark! So Euclid identified a good line as one particular sort amongst any number of sorts of drawn lines including curved , crazy and scribbled lines.

I proposed that the progenitor of these notions was in fact the circular line, and that the dual point was the intersection of 2 circular arcs. In this proposal the easiest line to draw is a circle not a straight line.mthis is because mechanically it is easier to draw a circle than to create a ruler or a piece of twine to reify the concept of a straight line.

In any case Hermann has adopted the common belief that a straight line is the simplest mathematical/ geometrical magnitude and drawn his all encompassing rule of direction from this premise.

Needless to say I do not agree with this characterisation, but the general rule he formalises is a very clever and serviceable one. It in fact allows for any all encompassing rule that is one type. Of all the possible one types, including spiral arc segments the circle is profoundly the most useful and magical ! It marks a conception that defines a plane from 2 intersecting spheres. It can be elaborated more, but I do not have time now.

Hermanns all encompassing rule therefore is : any variation whatsoever must maintain the initial direction. This is what he is indicating with a,b,c. These extensive magnitude variations whatever they may be all have the same direction by this rule.

There are issues with this assertion of a rule of direction, which I will wait to see how Hermann sorts them out.

I have corrected some typos in the course of grappling with §11, and updated the commentary.

The point is that Hermann is formally expositing an n- directional system accountable by the doctrine of extending magnitudes.
The problem of language labels looms large. It is clear and important to realise that Hermann relies upon clear language descriptors to distinguish each notion. Thus each label or handle is specified to a task or distinction. So extending magnitudes are not the same as extending an algebraic system from a binomial to a trinomial to a 4 term etc system. In fact coordinate systems as an accounting method were ill explored prior to Hermann. This mix up was due to making geometry a formal system.mthat blocked any further insight into formal systems and their development.

It is clear that Hermann identified direction as his fundamental Rule/Law governing each elemental extensive magnitude. The exemplar of such an elemental is in his system a straight line segment. Thus every variation in a line segment obeys a fundamental " law" for that specific line segment: vary only in the direction of that line segment!

As builders of systems that gives us endless possible directions to choose from, not just 3!

So why was 3 so important ? Well it was not, but orthogonality in circles is very useful and universal. However an orthogonal pair of diameters is not so easy to construct. Thus when Thales theorem, so called, was discovered in the far history of Egypt and Mesopotamia and the Indus Valkey civilisations it was a big deal.

I say it was not important only in the context of formal systems. In geometrical and spaciometric construction systems it is one of the fundamental postulates, aitema, that a student has to demonstrate to get accepted onto a Pythagorean scholarship. Thales theorem for right angles is a construction all Stoikeia students have to demonstrate along with about 4 others.

However, in space there is no direction that is the foundation of all others. We do not even have language to identify uch a concept properly. In fact arbitrary or whimsical is the only way we can describe any such protagonist!. But as soon as we choose one we become locked into an system that is constrained by orthogonal constructions. 3 dimensions is not an absolute general property of space, but it has to be realised in any and every system we may adopt. So any n dimensionl or rather n- directional system( and the distinction is important) must be able to generate an orthogonal system within its structure.

This is not the same as saying that there are only 3 directions in real space, or that space is 3 dimnsional. It is precisely saying any n directionl space can construct 3 orthogonal directions arbitrarily.

This is crucial o the topic of this thread. Rotation can be understood as transforming from one 3 directional orthogonal system to any differing 3 directional orthogonal system. But further still rotation can be understood as transforming from one n-directional  system to any other n- directional system.

We cannot apprehend rotation , formally, if we change the structure or step of the system.

Hermann uses 2nd and 3rd step or stage or level to denote the Intrnivity of the system. The intensivity is the Account Tally of the system. It directly accounts for how many stages one has to go through to identify a singular or unique element of the system called a Point or a reference Point. The concept of a point is thereby assigned n Intensity magnitude. It occupies no extensive magnitude, rather it characterises the intrinsic nature of the extensive space. It denotes how much processing we have to do to apprehend this singular " point".

In terms of intensive magnitude , this concept is a very interesting interpretation.Nto look at a colour and distinguish its intensity requires several layers of subconscious processing, never mind the unconscious processing that underpins those layers! Thus layers , levels or stages in a processing activity well model the experience of an intensive magnitude. The fact is these processing layers may occur in parall, rather than sequentially only interacting at a given " point" ( or arbitrary point) in the processes to give the final outcome or evaluation/ experience.

In addition to parallel processing add incredible " speed" of execution and you may apprehend why our computers have become excellent models of our own subliminal processing resulting in qualitative experiences.

Nevertheless to model these intensive experiences by extensive magnitudes can lead to interpretative confusion. Hermann resolves this by denoting direction as one interpretation of the rule / law of Varying!

As I pointed out earlier, one is held in a hypnogogic state where word salads come at you with hardly ant semantic referent, hardly any meaning in a specific concrete object. Thus when supplied with a concrete idea, or st least a familiar one , the mind clings on to it for dear life! In so doing it forgets the more general vague motion which it replaced by it. However, when certain boundaries arise because of the specific nature one has accepted the vague notions once again become serviceable as a means of hopping over the boundar into an analogous interpretation. Thus , using extensive magnitudes to model intensive ones does not confine us to the limitations of Spaciometry. We may have an n- directional structure, but in terms of intensivity that models an n- process level one! An n- process level is a terminology that also misleads in terms of the previous discussion, for there are  n-paralll processess allied with n-serial processes at each parallel, and do on!

The Potential Point therefore is very important to applying the Method beyond  spatial considerations. At the same time n-directional extensive magnitudes enable us to deal with spatial attributes in the best and most appropriate way for any and every situation involving spatial extension. However, because Hermann did not initially decompose direction into orientation and translation, it appears to us that rotation is not discussed in the system other than in a mysterious way called imaginary.

In fact rotation no longer needs to be imaginary or even as a result of an all encompassing rule of direction.  It can be introduced plainly and clearly as circular arc segments. Because of notation the " imaginary " label appears in most treatments of rotation even using line segments as a method. This is because early adopters of Hetmanns method Frnkly could not Fathom it! Gibbs wrote as much to a colleague, explaining that he did not understand Grasdmanns Multiple Algebra calculus, nor did he care for Hamiltons Quaternionic one. Neither did Paili.Clifford probably was best placed to expound upon it but he died young. Consequently the imaginary labs remain confusingly, where a simple arc segment would suffice. The superscript notation this invokes hides the simple extension atound the circle.

But we have seen this intimated by Hermann in the Vorrede.

The goal I have been striking for seems suddenly in reach.
http://www.fractalforums.com/complex-numbers/the-theory-of-stretchy-thingys/150/

Because arc segments are multi directional we cannot use direction as the all encompasding rule . Instead we could use a fixed quantity called the diameter. Normaly we use the radius nowadays and we use an arc segment called the radian ( Halbmesser ) . But the diameter is more practical when using a circular disc . The rule is that all varyings maintain the disc diameter. With that rule we create independent discs a plenty!

The varyings thus extend continuously round the disc perimeter!

 Jumping way ahead: how do these relate to products??

 Every geometrical product involves a rotation of the line segments of its perimeter. Thus the circle and the angle measures are excellent models for products, the use of exponents and logarithms thus encapsulates the process of producting and is well modelled by circular arc extensions.

Details like the quarter arc and the trig ratios are the fine tuning for this system and represent transformation relations between the various systems .

No product therefore has any meaning without rotation . Regarding multples, they borrow from this formal basis to make notatiional savings.

Let a be an extensive magnitude.

Define  a+ a+ a+ a to be a varying of  a if it is a line segment.

Define  a. a. a. a to be a varying of  a if it is a circular arc segment.

Then define 4 . a to be the varying in a line gathered together label.
Define  a⁴ ti be the varying in a circular arc gathered together  label.

But 4. a is now in the form of a circular arc variation.

If we write 4 as (4/ a). a  by the notation we can write that as  a ^{1+(4/ a)}

 That is , we can write it as a circular arc varying.

The quantity 4/ a would have to be defined as some rational variation of a circular arc extensive magnitude.

This not only looks like Hermanns quotient label, the exponent looks like Napiers proportion.

The  a is not a vector or a versor it is an extensive magnitude. The bold type is to highlight the notion of a nagnitude as opposed to a tally mark.  Thus 2 is a tally mark but  2 is an extensive magnitude.

When I see  2 I expect to see some object in space that is extensive at 2 of a unit extensive object .  When I see  1² I expect to see a unit arc magnitude extended around a circle at 2 of the unit extensive arc.

The tally count or mark 2 is totally subjective. The extensive magnitudes are objective.

So 4/ a is a ratio which can be treated like a fraction, but the outcome is a proportion of magnitude.

Commentary

The introduction of ruled varyings is crucial, both psychologically and developmentally . By imposing the simplest rules or identifying the simplest rules in behaviour we thinkingly connect to space as a magnitudinal entity that varies.

We can believe the varyings are random or whimsical and arbitrary, but ultimately we accept they have thinkable patterns, that is thought patterns, and this is where the doctrine of thought forms or rather thought patterns began its intriguing mesmerising progress to now.

The varyings are ruled by simple rules, complex rules, combined rules, whatever rules we impose or identify. One of the fundamental thought patterns is Everything Flows! Panta Rhei . Consequently we crave certainty and rigidity.

The separation between real and formal expertises accommodates these 2 contradictory states. Real expertises are empirical and ever changing, formal systems are inductive, adduced and then deducted, forming rigid patterns of thought that constantly need to play catch up to real experiences.

The trick is to keep these thought pattern rules as universal as possible. In that way they apply to most changing situations. The real defined goal for thought patterns is Invariance.

So how can a rule governing varying be Invariant!? The answer is what is really meant by being as simple as possible. A simple or one type rule has to apply across many complex situations and behaviours. Thus the simplicity is not in the rule itself but in its easy apprehension in a wide range of experiences..

The simple rule : matter is neither created or destroyed, has many additional terms and caveats and special cases and special pleadings , which it has acquired over the course of its enunciation, adoption and application. The rule is not simple. It is now quite complex.

It may be stated simply but then that is not accurate. As another example: water boils at 100°c is another rule of thumb. When the caveats are added it becomes a highly complex exponential rule!

It is these more complex formulations based on these do called simple rules that attract the name Invariant. What is usually meant is the algebraic or symbolic formulation remins unchanged in almost all circumstances.

How can that happen? By progressively reacting the formula down from its full statement to some " simplified" version of it that applies to the situation at hand. Hermanns method is all about applying these sorts of principles via the form of labelling he adopts. Because he tends to drop down from general rules he is able to see symmetries which are and would be missed by other Mathematicians and physicists and philosophers.

I have jumped ahead to the rules governing the circular arc segments, particularly because I had a defective understanding of the enticement Hermann puts in the Vorrede. Now the expositional rules come more clearly into view, and my mind races!

The notation or labelling I have sketched out in he previous post shows how these rules may be masked as definitions governing the notational devices and transformations. In particular certain manipulations, like going from in line to superscript notations make distinctions that appear in the geometrical magnitude they annotate.

The extensive magnitude in a circular arc behaves in an Analogous way to that in a straight line segment but the notation is different to emphasise as an aide memoir that the extending magnitudes, and there gatherings form different end results geometrically.

Thus  a. a is  a² an arc "length" twice as long as that of  a while  a+ a is a line segment length twice as long as that of  a. We must use r _a, the radius of the circle with the arc segment  a if we want to establish trigonometric transformations relating the 2 types of extensive magnitudes. As we do that the full complexity of these patterns begin to impress on our thought process, recogniseable and indelible patterns that enable physical dynmic behaviours to be modelled by these extending / extensive magnitudes and their established labelling rules.

I hope everyone knows by now that I am not a big fan of the number concept.,in that respect I am more extreme than. Norman
http://youtu.be/jlnBo3APRlU
http://www.youtube.com/watch?v=jlnBo3APRlU

The Arithmoi and extensive magnitudes, as well as discrete entities labelled by tally marks really do not require a number concept. A structured array is fundamentally all that is needed. The Abacus is a device that illustrates this principle, but it is not the only possible accounting device used or invented in ancient times. Many sophisticated accounting devices hae been found and misunderstood by western researchers including the TaiJitu, the precursor to LaoTzus YinYang astronomical calculator , and the Antikhyterra device.

Modern computers digital and or analogue follow in a long train of accounting arrays.

One of my secret desires is to read how Napier constructed his " Artificial Numbers"  later he renamed them logarithms .

http://www.17centurymaths.com/contents/napier/constructiobookone.pdf

Babbages polynomial engine relied on extensive circular arc segments

http://youtu.be/-qx7tYWZP70
http://www.youtube.com/watch?v=-qx7tYWZP70

Watch these videos with the thought pattern " a product is the sum of simpler or primitive products".

http://youtu.be/64gIN_mPOrw
http://www.youtube.com/watch?v=64gIN_mPOrw

Note also how a "parallelogram" represents a product primitive in this method.  The spatial distribution could be set out in a rectangular block , but the lattice multiplication method shows the advantage of spatially distributing it as a parallelogram.

With this method Napier did the millions of calculations he used to tabulate or Cannonise his Artificial numbers.

Here we see Napiers artificial numbers used to represent intensive magnitude, that is pitch, rather than extensive magnitude. The keys are arranged extensively because they have to be, but they are referencing the intensive magnitude rather than representing it, thus they can be equally spaced, where the intensive magnitudes themselves are not.

http://youtu.be/CekOr1pPg58
http://www.youtube.com/watch?v=CekOr1pPg58

Our number system , the decimal and binary ones as examples share this spatiali&ntensity mix or layout. What we call a number 234 for example , is a spatial sequence of intensive valuations . We call it the place value system. Thus we distribute value in space( a geometrical device) and then ignore the spatial continuum we are using. We can instruct but we cannot ever write it down because our place value rules do not allow us to!

Extensive and intensive magnitudes do not have that issue, but rather they exhibit a different set of issues , for example tabulation is problematic.

Clearly we can use and choose whatever representations best suit our needs, and flexibly interchange between them . But. Oh no Dawg, that ain't allowed!  ;D so we just sit and whine?

Archimedes just pragmatically solved the trisecting of an angle problem by neusis. But Oh no , that was against the rules! Then he solved it using a spiral , and perhaps several other solutions too. Yet until recently students were told it could not be solved! Now after a lot of back peddling mathematicians are more guarded in their expressions. Numbers clog up the brain with endless counting, magnitude is the brain friendly solution. I can see that is finite by constructing it. I can see is finite by drawing a circle with unit diameter. I can roll a unit diameter disc and mark off a line segment in a unique one to one correspondence with the circumference. By analogy I can use one to represent the other as extensive magnitudes in my thought patterning.

What is the problem? Certain autistic traits in mathematicians!

Change of basis in the " linear" algebra setting is the analogous process to change of base in the logarithmic or exponent setting. This represents or labes also the application of the combinatorial extensive magnitude to the algebraic intensive magnitude, as Grassmann labels things.

Why that is is due to Napier. His logarithm of Sines has been a fascination of mine because I used log and sine tables in my primary education. Ratios and fraction were always a puzzle without the historical background, and similarly sine tabulations and sine ratios were an uneasy mix. The link of course is the decimal expansion of the ratio analogised as a fraction.

Later in physics the period formula for a pendulum became a confusion. How could the formula define time when it involved time in the gravitational acceleration! Only later, much later while perusing Dimensional analysis did I find that all these measures were in fact ratios set to a quants defined as Monas, that is 1, the unit for that metric / dimensional quantity. Only now have I learned that dimension as a metric is used synonymously with extensive. I knew that the Latin means a cut away measure, but of course only extensive magnitudes can be cut into quanta. A Dimension in Dimensional analysis labels a quantum or Metron that is cut from a magnitude that can vary.

Well looking back into some 19th century geometry texts I was pleased to find the trigonometric line segments for sine , cosine  and of course the endless line for tangent. This was in the context of my Newtonian research into the unit disc and sphere, so called. De Moivre and Cotes were privy to Newtons discoveries regarding this geometry, which led to works of unequalled importance until Euler.

The trig line segment is therefore that open secret where particularly for sine an extensive magnitude is represented in the tabulations as a numerical sequence in the decimal system. The decimal sequence could be expanded to as many decimal places as represent able in some calculating device.

Thus the decimal expansion has always underpinned the labelling of a ratio of extensive magnitudes. They were always understood to be approximations or incommensurables. The move to make a "real" number " has thus always been flawed. Worse still it has obscured the richer extensive and intensive magnitude experiences that until then informed mathematicians about the space in which they have their very existence.

So Napier took a sine line segment and multiplied it by 10⁷. He was using his bones to perform the calculations, as well as the prostapharaesis method . He thus had 2 methods of checking his answer as correct to 14 decimal places. Also, by using the sine sine tables of his day, some had up to 30 or so decimal places, he had a third method of checking.

Many of his calculations were " new" results in that he was interpolating by the difference methods of his day for the appropriate reference digits( that is in minutes, seconds etc of arc.) thus what he sought to do was decrease the arc measure in whole units of equal valuation, that is by 1 unit at a time, while reducing the proportion by a binomial expansion of his base or initial sine line segment. Because what he was doing we now understand as creating from his initial sine line segment a geometrical progression of sine line segments , hissing line segments were not going to reduce by a unit, as his arc segment was constrained by him to do.

In attempting to keep these changes as uniform as possible he did nearly 10 million calculations on his own to create his cannon or table. I think, as he worked through, he realised the fabulous importance of the binomial expansion in indicating where he should be on the arc segment, and the impossibility of keeping the sine line segment difference uniform if he kept the arc segment differences uniform..

In any case he did something that Burgi did not, despite using the reducing sine line segment, he made the characteristic of a logarithm very clear for the same figures in a product that differed by powers of 10.

It s Brigg who, working with Napier just before he died , was able to show him the generality of his discovery. It did not need to be tied to the sine line segment base. In fact his distinction in the characteristic meant that the following part or the mantissa was invariant for a given string in the decimal system!  By this I mean the mantissa for 4  is the same for 40,; the mantissa for 797 is the same for 79700 etc.

The logs based on the sine line segment do not show this uniformity as well as the base 10 system, where it is optimal, but it was clear enough for Napier to make the characteristic an important aspect of his training avoiding the confusion that Burgi had within his tables( using a colour code). However it is Burgi who made clear the connection between an arithmetic and a geometric progression, which is how we understand exponential ion as reference tools today.
http://www.maa.org/publications/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-joost-b-rgi-introduces-logarithms

In group theory we have had to rethink the motion of duality , using Isomorphism and mapping to capture an analogous notion . In addition practices like commutativity and associativity have also had to BR rethought. Surprisingly it seems tha Hermann did not expect some of these familiar rules to change for extensive magnitudes.

Initially one can observe that these attributes of so called number are defined for number as a factorisation of number into number. However once extensive magnitude and intensive magnitude is introduced one really has to check if products are dual.

The Euclidean process of checking involved placing one product onto another to see if they fit! Thus orientation and translation were ignored, and both obckd are manipulated by neusis into a position wher we can judge whether they fit or not.

Then there are the transformed shapes. These have to be dealt with by a smaller Magnitude that fits into both exactly, and then the count shows duality of count or not. Associativity and commutativity  are properties of the tally marks in a count .

The disregard of extensive differences in order to keep associativity and commutativity reduces the information content of the labels. Some of that information is vital, and do we must redefine duality, associativity and commutativity for extensive magnitudes and extensive processes.

In trying to understand unabhängig, usually conveying independence, I expressed it this way.
2/3 is independent of 4/6, we can readily see it is different. But they are Analogous, so we can express one as a multiple of the other .  This means at one level dependence is captured by the notion of analogy. However specifically I could say (2/3)x is a family of such multiples where x symbolises a rational tally counter , and in that precise sense only 4/6 is dependent on 2/3. However it would be more accurate to say z4/6 is a dependant of the family (2/3)x.

Considering the plane : I could use x,y to represent 2 independent , arbitrary extensive magnitudes. Their independence is assured because they do not point in the identical direction( or for this discussion, their direct contra direction). Thus nx and my will be each a family of dependent extensive magnitudes in the precise sense explained above , where n,m are rational tally counters.

Now my point is that any combination of these 2 families of extensive magnitudes could be used to define or identify every extensive magnitude in their common plane. Each of those extensive magnitudes must necessarily be independent of each other, and the "referencing" pair or as we say the basis elements. Dependence requires that the dependants of a family of line segments precisely match the dependants of the other family of line segments.

Here is the point: to say that these families are dependent on the basis families is perverse. The families are merely referenced by the basis families, to which they are wholly independent. You may see this simply by choosing 2 different families as basis elements. The directions do not alter, merely how one references them alters.

I hope that is clear enough , and explanatory of why the idea of a 3 dimensional space is perverse. We live in an n- directional space which fortunately we can reference by at least 3 independent families of extensive magnitudes, that is by a 3 axis system where no 3 axis directions lie in the same plane. Any such axis system can form a basis for referencing all other independent axis directions. Here I use axis in the familiar sense of both extensive contra directions emanating from a point called "0" on a straight line, and maintaining that straight line orientation.

In this Grassmann framework, , that is in his construction of the familiar axes, we have to remember that work needs to be done to carefully transform what we have been taught as coordinate geometry to its presentation Grassmann style.

So I am thinking in particular of parallel families of lines. Norman does this by distinguishing a point from a vector, and then using a point and a vector to identify a particular family of vrctors( extending magnitudes) in the second step / stage system that we identify with the plane.

Here we see why Hermann exposited that intensive magnitudes , like potential points can be represented by extensive/ extending magnitudes. Norman simply changes brackets to indicate what is an intensive magnitude and what is an extensive magnitude.

(a,b) + m<x,y> where m is a rational tally marker and a,b,x,y are extensive magnitudes. a,b,x,y are thought of as fixed extensions, specified by a specific tally mark and the specified unit elements for the second stage/ rank/ grade system.however the m is thought to range through all the tally markers thus delineating the whole family of that extensive magnitude. The formula consists of a combination of 2 entities a point and a vector family, solely distinguished by the brackets and the expressed rules of thought for thre differentiated extensive magnitudes.

These kinds of formulae reference the underpinning pair of extensive magnitude families that reference every direction in the plane.

Your head may be hurting doing this kind of tautological self rereferencing but that is how it has been constructed. To make it less painful we leave out the underlying details, but we must not forget them. To slip into alternative explanatory statements , without cross referencing will lead to mixups and deviations from the construction. While these may be fortuitous and open up new viewpoints they may equally mislead and hide the freedom Hermann laboured so hard to reveal to us. It is not a freedom in space, but rather a freedom thought and construction of formal representations.

How do we reference curved lines( and ultimately surfaces)?

The answer Grassmann style was in fact given millennia ago by Apollonius! We call it polar coordinate geometry , but Apollonius stated that one of the initial extensive magnitudes must be a circular arc segment. The other can be anything else ,but he and we choose the direct line segment families as the second for the plane .

In the 3 directional space, the third stage we can choose a third element as a circle, a curve or a straight line. However if we want to specify cylindrical, spherical, toroidal spaces we have to specify more than 3 elements. What that means is that we are already in more than 3 dimensional space when we do. Because we do not understand the notion of the basis elements we mislead ourselves about the dimensionality of space.

Now it is possible to use trig line segments instead of an arc segment. But to declare a trig line segment as a one element family is clearly misleading ourselves. The sine ratio depends on a 2nd stage system to be defined. Thus we are de limiting a first stage system by a second stage system, making the behaviour of the whole a 3 step process reduced to the second stage system whose characteristic is a 2 step process!

By using an arc segment we restore the 2 step process to this second stage system. As we go into space the sphere maintains the characteristic process fie a third stage system while the trig line segments multiply the steps to get to the same result. Increasing the steps in this way is akin to increasing the directions required to attain a point, and thus that is an increase in dimensions required to attain a point in that basis.

What I am referring to here is otherwise called degrees of freedom. To use trig line segments we require more degrees of freedom in our thought pattern. We do not require more extensive elements in the basis, but we have to use the other degrees of freedom referenced by that basis, even to lay out the calculation on paper.

This calculative axis or axes played an important role in Hamilton realising the Quaternions as a rotational system . He could not mentally grasp how to do it until he allowed one of his degrees of freedom or axes to be purely for evaluating the calculations . This is why ijk=–1, the 4 th axis is used to evaluate the other 3 axes( as products of rotations).

It is a strange thought pattern which we must avoid, because Hermann has given us a better one. These products can be described as arc segment extensions where I,j,k are arc segment extensive magnitudes.

Returning to Apollonius, using one of the initial elements as the circular arc segment in the plane he was able to exposit the conic section curves. Norman Wildberger, inspired by both Hermann and Apollonius et al. has produced a wonderful series called Universal hyperbolic geometry which I can recommend to you.

Ausdehnungslehre 1844
Induction
12. The differing quality of the rules draws out in order, again to their exact stipulation, a "creating whole" cognisance,  closely over  which entity passes the one system in the other system,  every way feeling it. The close overpassing of the "differing systems in one another"  therehere builds a representation of a Second natural "step rise" in the field of study of the doctrine of the extending/ extensive magnitude, and with it then is the field of study of the elementary presentation of this expertise brought to an end!

It inter communicates, this intimate overtravelling of the "systems in one another", to the swinging kinematics in the doctrine of space, and with this kinematics hangs combined the corner angle magnitude, the absolute Length, the perpendicular upright, etc; what ,  to find all its laying out, comes to be in the second volume of the doctrine of extending/ extensive magnitude.

Commentary
Well I nicknamed it the Schwenkungslehre, but Hermann refers to it as the Schwenkungsbewegung!

The second stage system can also naturally model the behaviours of the circle!

Thus the notion of a system varying within a system that also varies, both free to vary differingly, gives rise to 2 " step rises" above the elementary initials. That is there are 2 stage 2 cognisances Hermann can draw attention to by his system within a system varying structure.

Be warned: this finishes the "elementary" treatment! After this second exposition it get hard!

Commentary
The second volume never materialised. Instead Robert his brother redacted this first volume with Hermanns help , shaping it for the Mathematical audience, not the philosophical one. It was republished in 1862 and gradually won acclaim. Hermann then retasked the first book of 1844 to be a annotated precursor to the 1862 version, adding many updates in addenda which was then published in 1877.

After Hermanns death Robert published a string of versions on the Ausdehnungslehre.

I will have to read in the 1862 version to find out how Hermann presented the Schwenkungsbewegung.

Commentary
The quest for precision has been buried under the spurious creation of the real number line concept.

Engineers and scientists require precise enumerations. This arises from the need to know the trigonometric ratios precisely for navigation and astronomical purposes. Later, as engineers and chemists determined other physical ratios , the need for precision grew. Precision engineering and exact Sciences became names synonymous with the most skilled application of our discernment and knowledge across a wide range of deliverable services and industries, but few realise that this was the result of a millennia long campaign to calculate the trigonometric and latterly the logarithmic cannons or tables.

The military application to ballistics and also to air flight and eventually space flight are also cases in point. Without Astronomical trigonometry none of these things would have been possible.

It is this creating cognisance, and very much also creative cognisance, that is trigonometry that Hermann introduces here in this second natural 2nd stage and 3rd stage and eventually nth- stage systems. These are all systems within systems within systems...n times or n statements.

These. Are the rank arrays that Hermann clearly imagined, but did not notate by matrix notation, because he did not invent that notation,Cayley and others did. But they only invented the notation, Hermann conceived the content that needed the notation. That is not to say he is the prima facia inventor or creator, he clearly drew together the work of his contemporaries in a way no one else did before him, and definitely before St,Vainant who either came to similar conclusions or more likely plagiarised what Hermann had written seeing that no one was responding to the ideas contained within his first volume.

It was well known and common practice for French mpnobility to plagiarise and use their wealth and status to increase their intellectual standing. But Hermann was on guard for this and successfully defended his primacy against St.Vainant, which is unusual. But Clearly St. Vainant could not carry it off as the inventor , Hermann had put too much detail in his work for anyone but him to be familiar with all it's content!

So this second aspect of the doctrine of extending magnitudes is trigonometry Grassmann style, but it combined the incredible works of Euler,Lagrange,Gauss,Laplace and Newton, De Moivre and Cotes,Fourier into a Phenomenon! Out of this Fouriers students developed the Fourier series and the Fourier transform, clearly independently of Hermann, but not embedded in such a rich Doctrine of extensive magnitude as Hermanns. In fact it is the exponential form that Hermann elucidated that is the real beauty of the Fourier Analytical method. While Fourier was defending his assertions Hermann was creating the systematic framework of systems within systems... for its exploitation.

http://en.m.wikipedia.org/wiki/Joseph_Fourier#The_Analytic_Theory_of_Heat

We do not need the set of Real numbers, we work with the set of Calculable or commensurable numbers, which I will coin the name the set of precision numbers for.

If we must have sets, then this set contains subsets of precision decimals. The set P is a countable infinite set where the count is the decimal precision of each set.

At some stage we admit that we cannot calculate beyond a certain precision set in a pragmatic way and that's the end of it!

Norman came up with this solution for the precision of

http://youtu.be/xYPw2gY_3PI
http://www.youtube.com/watch?v=xYPw2gY_3PI

This video uses intensive and extensive magnitude. Notice how the dynamic model is an example of Hermanns second natural 2nd stage system or 3rd stage system.

http://youtu.be/stzQNjtDg0g
http://www.youtube.com/watch?v=stzQNjtDg0g

Ausdehnungslehre 1844
Induction
D. The Thought Pattern of the presentation.

The centrally active Thing of the philosophical method is: that it advances in contradicting statements, and thus reaches from the general to the specific .

The mathematical method on the other hand  advances from the simplest labels to the combined set ones, and thus  wins new and more general labels through a tying together of the specific label.

Therefore, while there (placed where it is) the Overview  highlights beforehand over the Whole work, and the Development plainly consists in the any time everyway pulling apart and limbshaping of the whole , 

here, the "on one anothe"r binding of the specific entity henceforward dominates, 

and each "in itself ending" developing rank array only builds together again a representation of  a  limb for the following  everyway binding, 

and this difference of the method lies in the label entity., because the originating  entity in the philosophy is plainly the monad  of the idea , the specific system  is plainly the derived entity,. On the other hand in the mathematic the specific system is plainly the originating entity, on the other hand the idea is plainly the last entity ,the aspired to!: 

wherethrough (all of the preceding) the "in opposing established" Advancing is given shape.

Commentary

This is an elaborate sentence for an elaborate set up!

Hermann is preparing to put contemporary mathematics ( his and ours) on trial for Slavery!

Glancing ahead I noticed how topical his phraseology was, and in fact he draws on a liberation theology and credo that was contemporary to his times. Europe was undergoing liberation by France and various other European powers. The Holy roman empire of Prussia was facing its greatest challenges and Bismarck was determined to liberate it from defeat at the hands of French interests. Slave trading was rife, but a backlash of citizen emancipation was calling the whole capitalist enterprise into question.

People everywhere embraced the cry of "Liberté!" . Here Hermann puts the contemporary practices of mathematics in the dock , especially the injustice done to geometry. This spirit heavily influenced Bertrand Russel who on reading Ausdehnungslehre , along with A N Whitehead set about connecting mathematics to philosophy. He also wrote a scathing attack on the status quo in geometry thinly veiled as an attack on Euclid. Needless to say his professors and many others were to enured in their ways to realise he was criticising them roundly, not Euclid !

Describing the Products and Summation in the Ausdehnungslehre



Beware that you do not perceive the difference that placing geometry on the real expertises basis makes a fundamental difference to what operations are allowed to represent geometrical relations!.

I struggled for a great while with how ab could be analogous to–ba if they are all true extensive magnitudes, not just formal thought set ones. In fact I did not even know the difference!

To be a " true" extensive magnitude the extending must come out of the basis element, thus a extended to á must be one continuous or contiguous whole.. If the extension is governed by a different rule, that is a second direction of varying, it is crucial that it remains contiguous. This is where the rule of placing vectors tip to tail comes from.

So now I can understand "auseinander treten", and also " aussere", it means that the extensions must be continuous or contiguous and out of the basis element.

In which case how can you switch them round? Either you must rotate them cyclically, while they remain contiguous, or you must make the extension the basis element, in which case the first extension must extend in the backward direction to its original direction. . Hence in both cases we get a " negative" or contra extension to the original.

Well, you might say , if the original a must extend backwards, surely the b must also be " flowing" backwards to support this extension?  That is an important physical observation which Hermann was not thinking of at the time, but can easily accommodate. Herman was starting with the b direction as originally defined or established( a formal modelling requirement) and extending that basis element according to the original a direction( rather orientation). He is thus obliged to extend in the reverse,backward or negative direction.

It is also clear why Hermann separated Length and Direction as rules to be attributed to a line segment , stuck to them by glue!. His analytical method makes all these distinctions clear, but as he says in the Vorrede, they can be analysed further to improve the application.

There is only so much a single man can do!

Describing the Products and Summation in the Ausdehnungslehre



Now we understand I hope what " Aussere" refers to we can better understand " innere".

The fundamental difference is that the extensions meet each other, or collide into one another. Annäherung describes how they extending approach each other closest and closer until they meet, collide, bury into each other and eventually cross or intersect one another.

Because of this projective behaviour the extending direction may be opposite to the original basis element directions, or they may not. Whatever the case, switching the basis element order or cyclically rotating it does not alter the rule that they must close in on one another. Thus the rule counteracts any change in  direction due to switching the order of the basis elements in the 2nd stage/ rank system.

The innere product clearly is not affected by sign changes!

The trig line segments are also not affected because of the projection of the vertical line segment, or dropping a perpendicular. Because of this the quotient product also uses the projection extension and similarly allows commutativity of the " factors" of the 2 stage system ,

So the n-stage system will have both innere and Aussere products, and these will depend on what the process of extension chosen is. If the process is one where contiguous extensions follow one after another, each extending out of the previous one ( or parallel to such a one and in the same direction) then the product will be anti commutative. If the extensions in Anyway collide or meet even after a sequence of outward type extensions, then product will commute about that colliding pair or system pair.

It really is quite general and gives insight onto aggregation systems. Thus addition usually models objects being brought together , and subtraction objects being separated apart, but we missed the aggregation process of sequential follow through. That is an aggregation occurs only in a specific direction, rather than from every direction. Thisbisbthe process of Aussere or extending out product. If anything aggregates in sequence it can be modelled by an spreading out product. If any process aggregates from all directions it can be modelled by an inner product.

The quotient product is special. It modems Ritation around a circular arc, and so can model spirals in combination with a extending out or extending in product. A closing in product combined with a quotient product is quite "random" and do will model inner circle trochoids, or it's inverse outer circle trochoids.

This is almost where this thread topic is going, except the mixed product chosen is an closing in and a spreading out one. This product will model trochoids made of straight line segments , both inside and outside the product shape. However to do this we have to use the closing out product on its cyclical rotational interchange mode.

The mixed products are quite general, so the extensive magnitudes may be anything from points( strictly intensive magnitude) line segments, planes, spaces etc. or alternatively arc segment, spherical surfaces or curved surfaces, trochoidal spaces etc..

As you can perhaps grasp Hermann has provided a unified method of labels and synthesis and distinctions through this approach . Using it we can construct in a single system of notation every aspect of the liberated geometry we care to model.

Describing the Products and Summation in the Ausdehnungslehre

The summation sign also now cAn be given its exact meaning. It is the fundamental system process of combination or composition. The extensive magnitude rules are constructed into systems, called ranks or steps or even stages. Each stage is ruled by some simple law or set of laws. So the order in which these rules are obeyed is important.. Using the sequence of combination starting left and moving right as a process instruction, the extensions can be carried out in contiguous or continuous order.

The + ign is a natural mathematical candidate for this combination or composition. However, in this use it does not mean aggregate from every direction, it means aggregate in sequence order from left to right.. Again aggregation is the wrong concept, it actually means extend continuously in the order of the combination obeying the rules governing the initial elements.

Within this fundamental practice other rules are installed to ensure the arithmetic operations of the Tally marks  remains in the familiar mode.

Of course the natural meaning of = changes to be consistent with labelling and assignments,

The requirement of Equality or duality imposes certain conditions on the summation statements. Thus If a,b are differing initial extensive elements in a 2nd rank / step system, then the system of a is primary, the System of b is secondary. That is the extensive families are in an order. a + b as an extensive Process  is different to b +a in which the primacy of the families of extensive magnitudes are switched.

But if we identify a point by our second stage system, that is in space, it really does not matter if our system changes in any way. The point in space is the fixed thing.  So I could change the primacy order, the initial elements and so forth and should not expect the 3d point to change because of that. The geometry is now on the real expertise basis not the formal one.

So using that point in 3d space , the "real" space , I can ground my formal systems of extensive magnitudes, or the gathered( summed) totality as an n- step/ rank/ level system, (or some say " bind"  it) to my empirical situation. Now all directions, absolute lengths , continuous and contiguous extensions are defined from this real 3d point.

a has to be specified by this point or a point derived by this system in a system. This means that I have grounded my system and applied it by essentially establishing a cage or framework of parallel lines , planes and spaces , all specifiable by unique points in real space relative to this initial fixed point in real space.

This initial real point is called the origin or O. It is NOT a point in the system. It is a grounding point in real space. The confusion arises because this clarification is never made.

When your teacher starts by saying pick an arbitrary point on the blackboard as an origin, he or she should also say that this is a real point, it is now the centre of our universe by a formal decree!

Of course what they do say is that this point is completely arbitrary, but that is not the same as distinguishing between real space and the formal model.. For example , in real space the horizontal axial family is actually running in the curved surface of the earth, thus it is an extremely large arc segment with an extremely large radius. It is not a straight line segment. Consequently our model universe is already at odds with real space!

Now consider some other assumptions we make: because the reference cage is the same at every point in the formal system based on the one real point we assume that the rule established in a local frame applies everywhere. Thus we unjustifiably assume universality of our formulae. It is our job to justify that assertion any and every time we reach a point in real space that is modelled in our formal systems.

In fact Newton was not so deceived. By establishing a spherical geometrical model of space, based on a real local point, he in fact took pains to find out several real points on earth where he could ground his model. He and his collaborators chose several points after much sifting of information held within the trading empire. He also established points for astronomical observation of comets, to establish that point itself as a real point in his system.

By measuring tides, apsides of the moon, comets, pendulum swings etc he established several real points on which to found his reference framework, and thereby to carry out his calculations to see if they resulted in observable phenomena. Of course they did, and this is why his contemporaries eventually gifted his formulae with the divine Title of Laws of nature!

However, we should not be so ready to grant such a description. Indeed we are justifiably impressed by his Astrological Principles, but few have ever read even beyond a couple of pages! The principles themselves demand that we empirically justify at each point their application. Thus as we have ventured into space and back again we have learned a thing or two. They are not laws of Nature!

Now we use so called Einsteinian relativity, which is a kind of joke, because it is in fact Newtonian, and clarified over time by LaGrange, LaPlace  and Euler as Well as Cotes, and Gauss. However it is in fact Hermanns systematic method that underpins all modern physics including Einsteins. And we still have the responsibility to verify and adjust our reference frames at each real point in space!

We now have a lander called Philae on a comet. It's job is to verify our formal systems from that dynamic point in space.

If we ground our reference frame to a specific point in real space then we can identify a specifically and b specifically, and all the family of extensive magnitudes they each belong to as systems, and then we can referent a crowd of other families as systems by these initial 2 elements. The sum represents the continuous/ contiguous process required to do this, and we can start with a specific identifiable a continuously joined to a specific idrntifisbleb. However if I switch the order then the idrntified a runs into a problem. It cannot extend out of the b except backwards. So a + b does not equate to b+ a.

However, if I now make the rule that every parallel line segment will be identified by its extensive magnitude in the system and the point of interest it goes through, I can start b at the point I started a and then continuously follow by a. What I have had to do to maintain continuity of extension is to translate the start of b to the start point of a. The whole connected idea can be represented in a parallelogram.

The points of a parallelogram specify a commutative " process object"  in the system. In that regard we must clarify precisely what we mean by commutativity and associativity and how those ideas might be applied to extensive magnitude summation, and what the consequences are.

It would seem one of the consequences is tessellation of space, or fractally uniform regionalisation.

The Ausdehnungslehre 1844
Induction

14. The mathematic  there are so wholly as the philosophical expertises in the strictest sense , thus the method in both should have some identical things, what things it plainly develops toward the expertise like things . 

Now, we embed quality of expertise by considering "a Processing Cognisance",  Even if , from the point of view of one party, the reader through it  becomes driven  to the judging of every single Truth  with quality of necessity,  even if, from the point of view of the other side,  The reader becomes placed  on the Bench to watch over the direction of the further Advances   on each point of the development .


The unremitting quality of the first "out ordering" behaviour, specifically the expertise-like Rigour, comes to be received by every "out ordering" one. What thing is the second entity causes anxiety, thus, always, is this still a point, the  Rigour from the majority of Mathematicians still does not come to be attentively observed. There openly come forward demonstrations ,  by considering which demonstrated entities one initially, (even if the keynote  statement above was not keynote!), could not fully Perceive, wherein they should travel, and through which demonstrations one then, according to the entity, one from afar every step thereafter has made  a whole LongTime blind and on the " inspired sense of wholeness"  , at last, before one it itself everyway sees,  suddenly one reaches to the "due to outer experiencing" truth .

Such a demonstration can perhaps, besides Rigour, leave  not the thing to wish over , therefore expertise-like it is not, it lacks to it the second "out ordering" nature, the Overviewed-like quality. 

Who therehere  goes according to such a demonstration, arrives not at a Free "outer judgement" style of the Truth, rather remains, if he himself does not incorporate afterhere that "sudden overview" itself , in the complete-like dependent quality from the specific cognisance in which the Truth was discovered;

And this feeling of the enslavement, what enslavement in the such case roots and rises up (while least of the Received ( dogma)! ), is a highly pressing thing. for the case which has dwelt, free and independent to think, and all what thing it takes on board, self performingly and livingly itself to Centralise at its side, 

On the other hand the reader in every point of the development on the bench is set to watch, wherein it goes, thus remains he Master over the Material, he is no longer bound at the specific thought pattern of the presentation , and the centralising at his side becomes a true reproduction

Commentary
Ansatz is in this section unpacked as "Satz anstände". It is a very general flexible way of referring to any and every type of supporting material. Thus on a map the key box is an ansatze, ant abstract prior to a major work is an Ansatz, any keynote summary or overview is an Ansatz , and any conclusion as a conclusive statement is an Ansatz etc.

Here Hermann presents 2 points of view of his concept of Rigour, which he felt was Mathematics most worrying lack, in his day, and in ours. His first view is that philosophy is a rigorous discipline, requiring everything to be judged against necessity.mbut then he realised that expertise demands the same rigour! Thus his method that he is. Promoting requires everyone and everything to be ayed out in an orderly and rigorous manner.

This was just not being done! He was anxious about this general state of affairs. That was the firs promoted mind set of this section.

The second was a hammer blow to his sensibilities: there was no overview of what Mathematics was as an expertise. Instead it seemed to progress in the most disorganised way! It progressed through blind well. Intentioned enthusiasm, and sudden insights and inspirations. Whatever you might call that process Hermann felt it could not be called professional expertise!

But now Hermann had uncovered an expertise that was rigorous, philosophical in terms of rigour, and possessing a grand overview. Nhowever the received dogma would require him to dismantle it and present it in a hampered, chained and inimicable manner. He felt enslaved by the contemporary dogma which would not allow him to bring forward the discoveries and constructions of a free thinking independent and self chosen process as they were found and how they were meant to be, in his opinion.

This was frustrating and troubling. Mathematicians were slaves , or enslaved, not Masters of their own topic!

However, if the reader would be prepared to take the lordly position, and oversee every development made by Hermann , the slave, then their reward would be a true copy of the expertise he had found, arrived at by freedom of thought , meditation and action as a lord or freeman.

Hermann thus offers to emancipate the reader from slavery to the contemporary lords of mathematics, the professoriate. However this is not a freedom to licentiousness, but to rigour and overview led development as a professional expertise, so that Everyman could stand equally as free and as brothers pursuing the same goal for mathematics, or rather the doctrine of thought patterns.


Historically, this inductive appeal hit home with Bertrand Russel, who consequently made a scathing attack on the board of English Geometers, cloaking it as an attack on Euclid.

Is Hermann right?

He certainly draws on contemporary movements in his time, but also on Hegrls famous master and slave dialectic. But what is very pertinent also is the frre school education he received in his fathers school district. Justus adopted most of the Pestslozzi school principles, which helped to form the mind and character of Jakob Steiner who attended the original founders school in Switzerland, and who was appointed to the high school in Stettin by Justus fir some time before being called to Berlin University.

The Humboldt reforms provided an exciting focus for these primary education reformers who laboured long and hard to change the old system as much as possible, and for the most part had a huge influence on preparing Prussian youth for independent, self actualised, self motivated and self developed thinking. No longer would Prussia hire in foreign expertise, no matter how great, because they would be developing their own, able to take their place on the international stage!

Surprisingly, Klein resisted this movement vehemently, despite being married to Steiners daughter. He set up the Erlangen. Project when he could, but he saw no need to overthrow Academia! In fact he was worried by the geopolitical situation and international prestige. Eventually, even Hermann modified his youthful views and so did Russell.

Ah! It was ever thus! But while it lasted the Prussian Renaissance was a powerful social and international force, and the Grassmanns exploited it better than most.

Hermann is going to go on to describe the role of Analogy( analogous thinking) Ancestry, and Criticism in developing an art oskillset of wider and further developments of each development. This is in short keys to developing creativity or creative thought patterns or skills in the doctrine of thought patterns. This is also called Heuristics.

For Mathematicians I can recommend Polyas how to solve it books, as accessible introductions to the field of Heuristics( which from the Greek peripatetic teachers means hero worship, or copying your hers principles of action).

I have not kept up to date with the field since I was a student, particularly since the advanced book I bought after the Polya was off putting! It presented the topic in a set of series of state variables, and thus long abstract summands. Had I known then precisely what I know now I would have recognised this as an attempt to carry ot Hermanns rigorous method using rank arrays, almost verbatim!

The next time I glanced over such seres was in a very brief survey of Turins state logic analyses for his Turin machine, so I have no doubt that over the past 40 years some headway has been made in heuristic algorithms based on hermanns insights here.

Machine learning probably owes its development to hermanns rigorous analysis and insistence on a summary sketch overviewe of the constraints in the field of study.

As far as I know this was how Lagrange and Euler operated so successfully and innovatively for so long .

This is a nice bedtime story for the kids! Lol!

http://youtu.be/VUdFdlQNfpg
http://www.youtube.com/watch?v=VUdFdlQNfpg

This video is showing the use of LaGrangian principles in modern Physics.  As students of LaGrange, both Hermann and Hamilton incorporate his methodology into their thought patterns. As a result Hamilton went on to recast physics in a new systematic way, a kind of matrix of Status markers , or constraint makers. This came to be called the Hamiltonian. The idea was to get the Hamiltonian into a minimum state.

Hermann did not develop into that specificity. He rather delineated the most general principles and the most general way to set up a system.

Using his approach he went on to develop the whole lineal Algebra of n-stages or n-rank algebra. Hamilton was very impressed, and rushed to catch up. In his work " The Elements" he makes the throw away remark that his discussion, of course can be generalised to n-tuples! However the readers may now realise how hard that would prove to be, if it were not for Hermanns groundbreaking lineal Algebra.

I think Hamilton would have been able to achieve it, eventually, but as it was he had Hermanns work as a guide and a common expertise and quick sketch overview generated by LaGrange and LaPlace. However he died before fully completing his work, and the tide had turned against his quaternions, so few, except his students bothered to carry on his work in his style.

Instead Grassmanns style was picked up as an " orphan" idea, few even bothering to check if he was still alive! St Vainant found out that he was very much alive! Peano on the other hand translating Hermanns ideas into Italian was able to develop and implement them in his Italian school in a way that lead to Levi and Ricci developing Tensor analysis..

Gibbs took Grassmanns and Hamiltons work and presented it as his own development. It is known that he first presented it as a set of cribb notes on useful ideas and methods for physicists. As the popularity of these notes grew Gibbs was developing into an acolyte of Kelvin, and his Fourier analytical approach. This set the ground for a statistical mechanics which seemed the way forward to deal with the increasing systems complexity. Kelvin and Maxwell had pioneered this statistical and probabilistic approach of Gauss and Boltzmann.

Gibbs had an incentive to develop his notes , but he could not understand Grassmann and he secretly loathed Hamiltons quaternions, and he found that most of the established mathematicians also hated the doctrine of the imaginaries. Kelvin in particular was very scathing, so Gibbs had to include some nonsense to cover over Hermanns use of the imaginaries.

Gibbs presented a version which came to be called vector analysis. It was incomplete and underwent further development by many including Heaviside and Dirac.

However it was the usefulness of the statistical and later probability approach that returned favour to Hamilton. And his mechanics.

The mixture of Grassmann and Hamilton occurred in this fashion but underlying this was the short lived work of Bill Clifford. And Bill Clifford was a died in the wool Grassmanian

Is the Grassmann method the way forward?

http://youtu.be/hCSLqS0jKpk
http://www.youtube.com/watch?v=hCSLqS0jKpk

Commentary

The systems within systems can be developed in 2 ways: the straight line segment way and the curved arc segment way. They may seem different but they are not.

The straight line development hides the inherent rotations within the systems. It is thus a diminished descriptor of our real experience. How we came to elevate it to a divine position akin to " the way things are" beats me.

The curved arc segment development is a much richer , more complex model of our experience. We are forced to set in place constraints. This is the LaGrangian approach to describing a system or modelling behaviours. As a consequence one develops a fuller richer realisation of a mathematical model in which explicitly the constraints set are apparent and all assumptions are explicitly laid out.

The issue then becomes about defining the system strictly enough to be able to identify measurable parameters . These then become the subject of empirical data collection.

These 2 systematic ways of describing a system that is changing caught the imagination of the brightest minds, but ultimately it was dumbed down .. Here however Hermann does not allow this, and ultimately this is why quantum and Classical mechsnics differ. However they describe the same experience from different perspectives..

That statistical and probabilistic mechanics naturally attaches to the LaGrangian point of view becomes clear. Probability is a development based on the unit circle. And statistics requires a bell shaped curve to speak generally about huge quantities. Both require an apprehension of curved variation not merely summative direct variations.

Thus Bohr and Einstein created a false dichotomy as the mathematics used was really the only difference, not the empirical behaviours. But the blinkers were on and have remained on until today!

Let us begin to see clearly, as Grassmann wished and hoped for.

Commentary
The LaGrange methodology reworked Grassmann style

http://youtu.be/zhk9xLjrmi4
http://www.youtube.com/watch?v=zhk9xLjrmi4

http://youtu.be/-QVENB3aEvY

http://youtu.be/OxcCPTc_bXw

This was one of the key methodologies Hermann worked through and reworked according to his way of out ordering . Today it is still recogniseable as Lagrange's methodology, but the notationlly style has moved on through linear Algebra. It probably has a Clifford Algebra notation now, but the spark for all of this presentational style of the substance was Hermann and Hamiltons reworking of LaGranges method. Lineal Algebra rigorous and synoptically overviewed briefly , and Quaternions dissected the details presentationally so the reader and student could grasp geometrically what was being done methodically.

You still have to drop down into the detail, and calculate the elements after you have worked out how to measure them, but essentially it is all mapped out before you, and you can sure footedly progress toward a calculated solution.

Ausdehnungslehre 1844
Induction

15.  On the entity (every time (reified) points of The development) is the art form of the wider development essentially evoked through a leading idea,  which either is not the variant entity, as an every way meditated Analogy to related and already known branches of the wisdoms, or which is, and this is the best case, a  direct ancestral linking to the" first available  "for" seeking" Truth 


 The analogy is only a necessary aid/ catalyst, there it comes in from afar to play in related subject fields ; even if it plainly does not thereupon come to sit besides,  henceforward,  through this way to heave up the relationship to a related branch, and thus to pull up a forward running analogy with this branch .


The ancestry linking seems to be strange to the study field of pure expertise and to the mostly everywhere entity, the mathematical one.  It is impossible, Alone without it,  a "random any" new truth to discover;

 through blind combination one does not arrive at the achieved results thereto , rather what on has to combine and onto which manner must be evoked by the leading idea, and this idea  further around  before it  has  everyway effected/ manifested itself   through the expertise itself  , can only in the thought pattern of the ancestral relating  appear. Therehere it is some glorifying thing this ancestry linking on the  expertise-like study field. It is specifically, even if it is from  right art form, the "in one entity together showing whole developing" rank array, which the art form drives to the new truth , but with still not "out of one the other" embedded moments of development, and therehere also, in starting, only first as dark presentiment, the "out of one the other embedding "of that moment Inheres at the same instant, the discovering of the truth and the critique of that driven everyway entity .


Footnote on page xxxii
This case steps in by considering the "here to process "expertise in active relating onto the Geometry, by the viewpoint of whose "half"  I nave mostly brought into relationship the way of analogy  before you

Commentary

Ancestral linking is a rather specific rendering. One might make it wooly and in distinct, but I feel that the direct concept clarifies the otherwise befuddling statements about a really indispensable tool for experts and mathematicians.

The use of the array concept serves a similar purpose. While it is arguable that Hermann did not think of matruces when he formulated these guidelines, it is clear that LaPlace, LaGrange and Euler did think in terms of arrays of terms organised across the page. Some were series some were tables for constructing series etc . The word used was probably systems .  It is the systematic nature of these methods that drills through, no matter how long or complex the sequence or array.

The use of Analogy is very important, but do not use the obvious well used one, it won't get you to new developments . Use ones that come out of the Farfield, or the left field. But the best way forward is if you have a leading idea that calls to ancestral relations! Such an idea admittedly is dark and blind, more of a feeling that there is a progenitor or forebear out there somewhere. Using ones expertise and this idea And the array , a snapshot of everything in its place and under its constraints, the expert can push the system back towards its origins, to at the sat suddenly discover its progenitors , new concepts of a more general applicability..

It should be clear that Hermann is very advanced in his thinking at this stage, even equal to Euler , LaGrange and LaPlace, who he was studying very closely. But he had the edge which was his Hegelian analytical and synthetical model and his notation of line segments to represent and label both intensive and extensive magnitudes. Because of this label scheme he could not only follow the leading mathematicians and engineers and Astrologers of his day, he could also out think them!

http://scienceworld.wolfram.com/biography/Laplace.html

The unified physical view is the notion that Hermann calls the Überblick, or the Übersicht. This unified view which both LaGrange and LaPace subscribed to, was expressible in terms of an array of parameters and constraints with degrees of freedom systematising what could be expected from the whole System.

The transform idea is consistent with this approach and powerfully coordinates relationships between any 2 formats, and in particular between the lineal and the curvilinear, which at the time was thought to be only approachable via the Calculus.

Hermann had a different view, as did Hamilton, and the use of quaternions and line segments considerably simplified the concepts, which otherwise remained buried in differential equations.

Helmholtz was reported to opine that differential equations were the only safe way to describe the laws of nature. Many of his time were thus convinced that mathematics was the only way to understnd the natural laws, and the higher the mathematics the more profound the truths!

My opinion is, for hat it is worth, this is patent nonsense.

In researching LaPlace I was reminded that we cannot solve the n body problem for gravitational systems, and yet we can send rockets to mars and beyond. The contradictory statement is worthy of the Hegelian dialectical treatment!

I have completed a brief survey of LaGrange and engineer, LaPlace an Astronomer and Euler a polymath, to see what contribution they made to hermanns understanding of formal expertises. It seems clear that LaGrange made the biggest impact on him while LaPlace and Euler taught and refined the differential and integral calculus .

http://youtu.be/kV3oj207pUk
http://www.youtube.com/watch?v=kV3oj207pUk

The sophistication of the calculus employed by LaPlace and Euler opened up the application of formal systems to physical empirical behaviours. In particular tools in calculus became apparent and eventually identified as grad,div, curl. The integral calculus helped to reveal the connections between recognised formulae of Newton and differential forms that seem to describe physical mechanics.

Here Euler shows that a differential equation form for a fluid point in a pressure field is still a formula of proportions captured by Newtons insight.

This was important, because LaGrange with his multivatiable approach had determined a potential differential form which he felt had to be the principle of mechanics! This was his one consistent view of mechanics and engineering, the result of his deep thinking about mechanical systems and engineering. Euler corresponded with LaGrange on this point over a number of years, and his opinion was that Newtons principles were necessary and sufficient for the whole of Mechanics  . Eventually LaGrange changed his opinion . In other words differential equations do not form the necessary and sufficient formulation to set up formal systems for mechanics! Newtons direct and simple proportional measures and ratios is the recognition that as humans we can only measure and observe directly not by differential equations or any calculus whatever. Direct experience of intensive and extensive magnitude is the only place where we can begin.

The French drive to take Newton and Leibniz calculus to its fullest extent resulted in some great work and formula building, but it did not advance basic and fundamntal research and innovation. Advances come by intuitive feelings and realisations of possible ways forward, or deeper apprehension of guiding principles, or striking analogies that break up the consensus view to turn up a new relationship between old developments and new ones.

At this level it is probably hard to think that Hermans work could be important compared to these 3, but we know how important his ideas actually were, since just about everything is now notated in vector form. In addition, much of the great 3's work would be inaccessible to the majority of those who now access it, and that is a painfully small number of people in the world.


Why have so many found Hetmanns work so powerful? It is not because it is so general, but because it uses human scale thought patterns and intuitive measures . He starts at the most mysterious and mystical and brings it right down to the pragmatic and systematic and human.

The Lagrangian principle of the minimum, or Hamiltons principle of least action.

http://youtu.be/08vJyA-XD3Q
http://www.youtube.com/watch?v=08vJyA-XD3Q

These mathematical principles appear by observer thought pattern, not really as rules or laws of nature. Hermann really finds this kind of none rigorous thinking very worrying. It is sophistry like most mathematical conservation laws. This was however what LaGrange for along time felt was a fundamental principle of Nature as did Hamilton. Euler was not convinced that Mathmagicins could deserve such a promotion to universal principle. It did not rely on any empirical behaviour, rather it relies on the mathematician fooling himself and others.

Just thought of a few new symbols for operators:-

<+,+>,<*,*>

They mean add in the direction of <,> so the result is not a heap agglomerate but rather an ordered string.

Similarly the multiplication means set out the sub products as sequences added according to <,>. So 3*>3 means 3+>3+>3.

Of course I could stack the 3's and I would use a *^ symbol  or some latex generated equivalent.

Does this help or just give us more to do in being rigorous ?

It certainly makes clear that 4*>3 is not the same as 3*>4, but it lacks the extensive magnitude data which is the really useful part of hermanns method.

Do the factors need to be distinguished in this way if they are just tally marks or counters/ scalars?

What really lies behind this notion, (ancestral linking alert!) is the orientation and translation of extensive magnitudes, and that for me is more valuable than tally mark operators.

They may serve as a bridge to the underlying extensive and intensive magnitudes.

The notion of a product in the geometry of real space is intimately connected with projection. We might call these projections constructions but I prefer to call them parallel and circular and scaling projections.

Viewed in this way the sequence of projection and other constraints become important.

The notion of a system in a system means one system is " coupled" to the other by design. Thus one projection is always followed by the other and this is ensured by the continuous/ contiguous constraint.

If the projections were not constrained to be continuous/ contiguous then in fact the projections could occur without sequence or order. Hermann carefully avoided this in §10 when he specified rule governed extensions.

Discrete extensions and the combinations do not produce geometrical products in the same way . A lot more rules and constraints are required. The potential for " chaotic" forms is greater in the discrete section.

System combination has to be individually specified and so does any potential product. Without coupling of the projection process random unpredictable forms are possible.

The notion of coupling and grounding the systems distinguishes Hermann and engineers from pure mathematicians today.

Despite the general language Hermann has one very clear goal imposed by Lagrange, the systems must conform to real engineering systems ! Thus Hermann starts with 3 points in real pace. These 3 points specify a plane in real space( really they specify a formal plane which we embed in real space) within that formal plane 3 line segments are constructible by those points , and any continuous or contiguous pair can define the rule governed elemental directions. Once chosen, they set the fundamental reference of the attributed system, and they deliver a coupled system of one element inside another element.

Contrast that with an axiomatic thought pattern. A point is specified real or formal ( already we lose track with reality!) from that point 2 orientations are whimsically chosen.( so how is that specifiable?) we draw infinite lines I each orientation through the point.( where do we fo this ? We have no plane specified to do this! Ah, of course! We draw this all on the blackboard! But I do not live in the blackboard!) then we mark of a second point on each line of orientation and define an oriented line segment thereby. Oh yes, that also defines the plane( er.. What if the paper is actually lying on a curved cylinder? Oh, we just straighten it ou? Well that's alright then!).

Now we constrain the student/ operator to only draw lines parallel to these 2 infinite lines.. Next we constrain the student to always place the tip of the line segment in one direction of the 2 possible. Then we constrain the student to only extend a line segment from its tip., in accordance with the 2 orientations.

We fo not specify a process order to these extensions. You can do them in any order you like.( whimsically). This is commutativity of extensive addition. There is no idea of a system in a system or system coupling. The coupling is imposed by the student/ operators whim, and is therefore not predictable. In any engineering system a process that can behave in this way is uncoupled and cannot be relied on, even though it never goes outside the bounds of a similar but coupled system!

Hermann called this incoherent system behaviour, there are many such systems in reality , and they are a valid field of study, but they are not conforming to the doctrine of the extending / extensive magnitude.

In such decoupled systems there is no obvious cause and effect relationship. Thus we are left with a statistical and probabilistic approach. Since Classical Mechanics is based on these coupled systems it is not hard to understand why Quantum mechanics seems untenable. The fact is, as Hermann states, we impose these rules on real dynamical systems because we can't make coherent sense otherwise.

However, what statistics and probability studies have shown us is that there is a " periodicity" to certain measurable phenomena, which very roughly approximate the conic section curves. Now Statistics was in its infancy in Hetmnns day, not even a branch of mathematics, but probability had a long history by then, nd was related to games of chance. As Einstein said, no serious philosopher would place the laws of nature as described by Newton as the handiwork of God, on the basis of a crap shoot! " God does not play dice!"

However LaPlace and LaGrange we're not of the opinion god was even a hypothetical entity in their systems ! Therefore you see the first attempts to apply statistical analysis to their observations of dynamical behaviours. In fact Gauss was rather celebrated for using statistical Anlysis of errors to correctly calculate the position of a rarely sighted comet. And Boltzman, well he was something else! His visionary thinking was far ahead of his contemporaries.

Suffice it to say that when Hetmann defines a product as an extending out one, it's constraints only make sense in a coupled system. Thus imposing ab = –ba is axiomatically constraining the system to be extensively coupled. Because the axiomatic approach has not got a clue as to the whys and wherefores these constraints are whimsical and hive no insight. Worse than that they engender a profound confusion. Our mathematician has no real geometrical intuition and cannot convey to an engineer what it is he is requiring in a real system. An engineer has to go figure that out himself!

If an engineer realised he was being taught a system originally designed by an engineer he would go to LaGrsnge or LaPlace and gain better insight. However since these 2 were highly symbolic in their presentation he would perhaps miss the lementary points on which their systems are based. This is where Hermann comes in. He exposits the elementary concepts in this induction. However, since he was attempting to get a professorship out of this insight, he starts out at a too philosophicl level for just about everyone!

So there you have it: the poor ngineers is left at the mercy of some brmused mathmatician who has his own agenda to push.

Earlier I wrote that quantum mechanics is probably based on the arc segment rather than on the direct line segment. Here I have elucidated that it is based on the statistics of uncoupled systems. The connection is the circle. As a closed system the circle naturally models uncoupled systems each circle has its own uncoupled centre. When Galileo observed the Jovian system, it was seen by him to be " absolute" which simply means independent of nothing else. The concept of uncoupled is in that precise sense absolute.

However, being absolute or uncoupled does not mean it cannot be fractal! Thus it is part of a larger system to which it's parts are coupled only through a principal centre. So what am I saying? Is an uncoupled system coupled or not? Remarkably and statistically and probabilistic ally that may depend on the scale we use to observe and measure! Thus the conundrum between quantum and classical mechanics may just be a fractal scale artefact!

The doctrine of the extending magnitude, by design, deals only with coupled systems, as Lagrange and LaPlace originally conceived mechanics, but today we have by accident and innovation combined moved on to a broader range of dynamical system.

Commentary §§14, 15

This sets straight the curved and warped opinions of Hegel!

http://youtu.be/TvFu6ak_SGk
http://www.youtube.com/watch?v=TvFu6ak_SGk

Gegensatz I have translated as "contradictory statement", so this falls into the fallacy of thesis, antithesis, and synthesis.

Gegen means against or going against. It is very general . In some cases it means "opposingly" against in others it means "at the side of" against. Thus the 3 part structure of his work does not neatly fall into the commonly stated pattern.

As I translated Hermann I noticed that the Gegensätze were sometimes alternatives or advancements or analogous ways of stating the same thing!

It seems to be important that the movement to a 3rd position is a natural occurrence of thinking about the issues! The ancestral linking description is also indicative of Hegels lecture performance. The use of contemporary analogies is also a trait of Hegel. Finally the stumbling in the dark until discovering the analogy of the right phraseology is descriptive of Hegls lecture performance

It would seem that Hermann was present at at least one of his lecture performances, even before he started to write the Ausdehnungslehre

As a working translation I would think of Gegensätze as comparing and contrasting statements. Thus Hegels method is to compare and contrast , and then draw his own conclusions or resolutions or figuring out of the issues.

The fact that he write in a structured way, a fractal pattern of 3 relates to his deep religious and mystical sensibility, and representative of the Trinity: the Father,Son, and Heilige Geist!  In this way he uses the trinity as an all pervading Analogy, and a structure of the shape or form of Gest as spirit or mind in our experiential continuum.

Ancestral linking is involved in this example of revelatory truth, the Father is intimated by the Son, and the full revelation comes through the Holy Spirit! Similarly we may have a form before us evoking in us some deeper connection. By employing Geist ( mind or spirit) that deeper connection is suddenly revealed to us!

As I reluctantly come to the end of the induction, knowing that I have next to tackle the 3 product types and the development of the mixed product rules in this paper, I feel that I m returning to the chos that is modern mathematics.

The philosophical treatment of contemporary mathmatics, hermanns contemporaneity, and yet correspondingly our time has been eye opening.  The move from the philosophy to the mathematical process so called is one filled with trepidation! The Herculean work involved in setting all processes on a sound footing is daunting. And it is not as if I have not been on this journey for a long long time, but rather as if I have found what I was looking for but now have the task of mining the riches of the gold and precious metal seams I have uncovered.


At the last my childlike vision of the underdog mathmatical hero , misunderstood by his generation has been dispelled by his own words. As a free thinker at a truly unique time in world history, Hermann Grassmann saw something buried that no one else paid ny attention to, nor valued.  His hard working spirit and youthful curiosity kept him digging at it until he was drawn into a realisation that he was onto something!

He required the technology of his day to further extract what he was uncovering. That meant that as he matured he had to learn how to use various bits of philosophical and mathmatical machinery , so he had to take courses, read books, self actuated learning, self reliant study, self directed research finding out what he needed to know and do to uncover what he had Lund and evaluate the worth of it to his society.

It gradually dawned on him that in his extensive self education nd skill learning programme as well as his socil nd cultural development, his spiritual and philosophical advancement no one had expressed or revealed what was now revealing itself to him! So he was now hooked, convinced he was receiving divine inspiration or at least Muse lead insight and gifts.

What he saw were several fundamental entities that lived and breathed at the very heart and centre of all human experience of the geometrical Space, the Raum. These entities were real and complex, but constantly there in one form or another. They were of long long provenance, salled different things over history and glimpsed many ways but not distinctly uncovered.

The more Hetmann studied the more he realised how huge his discoveries were, and that how privileged he was, because the greatest minds of his era, and arguably of any era had not uncovered it.

It turns out that what he saw and described in utter detail was the matrix array.

We call it a matrix because of Cayley and his collaborators , but prior to that it was called simply LE Tableu , the table , in French. It was studied in the form of systems of equations , mostly linear , and the determinant, which I believe Cauchy and Gauss corresponded on. The determinant of a system was clearly a concpt of the times, and Hermann knew about it from the best minds of his age. But what they had not done was penetrate to the core ideas of these systems. Neither had they worked out how to systematically do this or what tools were needed to advance this development of thinking. There was no clear thinking about these systems that appeared everywhere!

Clear thinking for Hermann arrived in the form of Gottfried Wilhelm Hegel. He was the most famous philosopher of Hermanns era, and that is saying something! For Hermann who appears to have attended at least some of his lectures, or heard about them through his brother Robert who did go to Berlin to qualify as a Mathematical Teacher, something Hetmann struggled to achieve,  the consequence was that Hegel taught in his lectures by demonstration a powerful heuristic method and approach to philosophising about just snout anything. But in particular how the historical development of ideas, and deep deep concepts of life and cultures followed a pattern of advancement toward a higher and wider and fuller capability to reason and puzzle things out.

Thus Hegel powerfully linked every aspect of any individuals experince to this grounded real experience of history. Hetmnn lived in historically tumultuous times, life and death had to be faced o some kind of Readon led or religion led thinking.

Once grasped Hermann applied this meditative praxis of Hegel to everything fearlessly. You have one life and it should be a good one , delivering something of import. Hegels method or praxis of meditation contrasts with Descartes, who spent half the morning engaged in contemplative reflection, and the rest of the day going about his other duties. Hegel on the other hand lectured profusely, prepared for lectures assiduously, and ade lecture notes copiously and haphazardly, all so that when he was in front of the students he could perform! And what a performance! Intriguing, spectacular, stuttering and full of pregnant pauses, false starts, reiteration , upon reiteration until the connection and ideas were clearly seen by him, and then presented in such a breath takinly easy flow of pure philosophical gold.

Hardworking and deep ancestor linking and metaphor making  always lead to some advancement, Hetmann learned. This is precisely what he used to develop his discovery of the connection between points and line segments as formal concepts and reference points and kinematics in 3d space.

The 1844 book was intended to be a guide to a meditative praxis which would help to work through and refine thereby, the difficulties that the matrix array was ensnared in at the then proto stage of development. Hetmann did not say in 1844 he had the solution, he just claimed he had the method that would deliver he solution!

Of course we know now that Caley and others worked hard to develop the kinds of algebraic labels Hermann envisaged and constantly discussed in 1844.

The 1862 book is a different work all together, highly because it was redacted by Robert , Hermanns brother , with Hetmanns reluctant cooperation, and aimed at Mathmaticians rather than t classics students , who would recognise the references to philosophical concepts and philosophers underpinning the 1844 version.

Because Robert, as an experienced publisher, was right about the need to redact the work for a different audience, as proved by the second versions steady and important sales, Hetmnn was not going to be ungracious and criticise it as betraying his ideas, rather he utilised the success to reprint his original with additional addendums in 1877 etc. and added copious footnotes to connect the 2.

Ausdehnunglehre 1844

Induction

16. Therehere is the expertise-like presentation to its apparent nature/ character concording to a " gripping in one another " entity of two developing rank arrays, from which arrays entity the one with consequence drives from one truth to the other , and builds a representation of the central- like contents, but the other by governing the everyway travelling itself and the thought pattern appointing
In the mathematic these two developing arrays step at the most sharpened out of one another.

There has long time already in the mathematic, and Euclid himself has given the prototype therein, been conventions, only the one developing array,  which builds a model of the contral-like content ,  to allow to step hereforward, in relating onto the other thing, but it is left up to the reader it between the lines to read here out. Also Alone how  fulfilled the side by side ordering and presentation of of that developing rank array   is to be :thus is it there yet impossible, therethrough that array ,  of which the expertise should first Learn to know, already on every   points entity of the developing the overview is waiting against to support, and to set it on the bench, independent and free further to advance.

Thereto is much more necessary, that that the reader makes possible that in that array comes to be set condition/status marker every possible way   in which marker the one embedding  cover of the truth  should himself determine   the most favourable  case .

Therefore in  that array entity, which the truth uncovers, one continuous entity finds  place to sense itself over the journey of the developing.;
  It builds a copy of itself in the array to it, a centred property-like thought array
    over the way,  which it has knocked into one,
    and over the idea, which  lies at the foundation to the whole entity

and this thought array copies a central kernel/core and spirit of its Activity, while the consequential "Dividing apart" of the truths is only the everyway bodying of that idea .

Now   wishing to emote the reader, that he,  without being guided toward such thought arrays,  who still independently should advance onto the way entity of the embedded covering,
calling him  to place himself over the one embedding the cover of the Truth , and therewith is the proportional arrangement  between him and the writer ( pen man) to switch round,
whereby then the whole " away from writing planning" of the work appears as superfluous.  

Therehere have we new Mathematicians, and particularly to begin with the French, weaving together in everyway both developing arrays.

The "magnetic" entity, what therethrough  having brought  their works on the scene, plainly remains therin, that the reader himself feels free and is not hemmed in  in  thought patterns, in which patterns , because he is not mastering them, he must follow like a minion.

Now that in the mathematic these developing arrays at the sharpened edge step  out from one other:  lying in the central property-like quality of their methods (no.13); there it specifically from the specific entity out through chain linking advances, the Monad of the idea thus is the last entity!

Therehere slowly drags along with itself the second developing array a completely "rooted opposingly  set" character , how the first entity ; and the inter penetration of both appears more difficult , how in some "random" from a different expertise.

Therefore for the Sake of this  difficulty one may not allow, how it from the German mathematicians frequently happens,  the whole everyway procedure to give up on, and to throw out in every direction !

In the laid before you published  work, I have therehere the applied indicated duty Way knocked into one,

and it shone around to me, this work by considering a new expertise,  as needful ,

as plainly at the same time the Idea of the same expertise should step initially out of Light !
(https://catholicismpure.files.wordpress.com/2014/12/shepherds.jpg?w=640)

Commentary

For some reason this has been a subtle translation with a lot going on in the ideas behind the German, so I would very much appreciate comments on how understandable this section is.

As it is the end of the induction it is necessarily difficult, because the analogy of being given birth to from the Matrix, the Womb, is subconsciously and analogically driving his description here. As you may imagine this affects the reader at many levels and I guess as translator it effects me more. Being in the womb is such a safe secure place with all your basic needs provided for, so the trauma of being birthed or ejected is real, and Hermann means to eject us into the real world which he has prepared us for!

Well I guess I should be excited to face the challenges he refers to, but right now I just want to stay in the womb just a little bit longer! :D

However this womb I can revisit as often as I like, to be rebirthed, so that is encouraging.

But what of the matrices and arrays we are now equipped to look for and develop? Well certainly Norman has been acting as a good Midwife all along, so at least we can hide under his wings for a little while longer as we grow in ability and confidence. And it is clear that if you were and are German/ Prussian Hermann wants you to do just that, to take your place in the World as a new kind of Mathematician, first equalling and then by God exceeding the French! And this Ausdehnungdlehre is the work set before you to stick you to your duties!

More than that this work shines out from the Light, it is a work of inspiration from the Holy Ghost, the divine Reason that shone upon the shepherds and set them on their journey to the infant Jesus. And this divine or Absolute Reason is what Hegel was referring to as the advancement achieved through the dialectical process. Here Hermann claims the Rank Matrix is an embodiment of that truth, that reality of Space that is at its core, and there is a second like unto it which was the more difficult to integrate, but which it was his duty and the new Mathematicians duty to achieve!

There are several key points I missed coming at Grassmann the way I did, as an enquiry about the mess I perceived as the vector product and the complex numbers.

Much has been straightened out because there was much I simply did not apprehend, even from Normn. However I started thinking that the line segment was the fundamental of Grassmanns method and idea. I have ended realising that a much more complex entity even than Tensors was what he perceived: the Matrix.

On the way I realised I had not been aware of the potential point, the counterpoint to the line segment, nor the application of extensive and intensive magnitude. Further I had not grasped the rsnk array until quite late in the sections and had not understood the structure of the systems he revealed before me.

Then he mentions the 2 types of systems covering the same step rank at the second step. Or stage system, and finally how the  n- stage system comes to be.indeed how anything comes to be is addressed! But then he goes on to an entity I barely could translate until I knew the difference the adjectival endings make: the points Entity of the developing Matrix.

What that was or is I do not yet know, but I know it is to be found in the works of LaGrange and LaPlace. The formal model of this matrix is the thought Matrix and how that differs from the LaPlace and LaGrange Matrices I do not know, or even if it is different from the second developing Matrix which posed such a difficulty at the time of his writing.

Clearly Points and Point entities suoerscede the line segments as one goes higher in the system stages but how I have no clue at this moment.

The other important message was the necessity of an overview! That and a woefully lacking rigour were 2 skill set attributes he mentioned. Given that point the next chapter or essay deals with the overview of the doctrine of the thought patterns. Whether I need it to continue with this threads aim I shall determine by and by..as at this moment I need hermanns definition of the stepping out product and some indication of his roots of unity product to be ready to translate this paper on the place of the Hamiltonian quaternions in the doctrine of extensive/ extending magnitude.

On page 64 of the 1844 Ausdehnungslehre Hermann gives an anology to convince the reader that the out stepping product is anti commutative. It is an analogy because he is wanting to establish a principle for the outstepping product . Up until then Hermann has given many pages of demonstrations of the consistency of the result from the perspective of his defined 2nd rank system.

What Hermann has done is simple and clear, once you clear your mind of other thought patterns and in particular arithmetic of numbers. He has returned to the Arithmoi and the methods for handling geometrical extensions by them. I say returned because arithmetic of numbers grew out of this spatial handling of Arithmoi. This may not be precisely how the Greeks did it but it is very close.

As Hermann advised the reader needs to read between the lines here and then read the original Stoikeia away from this work. Comparing the 2 should give the reader a self established view on the matter.

So why the analogy, which forevermore has been taken incorrectly to define the outer product? It may well define the outer product  ofab as absinø the angle between the line segment but it does not define the outstripping product of Hermanns 2nd step system.

In the second step/ rank system every line segment is referenced by 2 elemental line segments in a prescribed process. In the analogy Hermann illustrates how such a prescription of process , when rigorously applied leads to the exposited result. It is the lack of rigour , the sloppy conventional thought pattern that gives us commutativity without demonstration. In his system ab has a as the principal orientation. Thus any other orientation is defined as measured from a to b. Thus ba reverses the principal orientation. Supposing now the line segments remain the same in space( or in the plane)  because they are bound to real points then ba can only be constructed by going backwards along a. This is reflected in the way the orientation of a is defined: it is measured fromb

The reversal of the way the angle is measured is an unexpected contributor to the sign change! If we are strict we cannot avoid this result, nor should we. It tells us that we are doing way more complex things in our processe than we allow for or even recognise. The protestation that we have to simplify things else we become confused, oes not wash with Hermann. There is simple and there is sloppy!

The call for rigour is thus well demonstrated in the way his whole system of systems is set up.

Now I have used the notion of cyclical rotation to grap this point as simply as I could.mthe point about cyclical interchange is the points! As the point labels move round the or imitations and principal directions do. Not follow. Labels are formal identifiers, but orientation and principal directions are spatial/geometric realities, set as such by our own thought pattern that is granted, but set in real space nonetheless and not formal space. So I may identify real points in space to do this, and once thst is done they are set forever for all intents and purposes.

Thus on earth the north and south poles are such real set points. To be flipping real poits around is to create mental disconnect with the spatial environment we exist in, and thus to lead to confusion of mind.

So the same 2nd stage system can also record formal rotation by label interchange or real rotation by point motion in and through a fixed subsystem of real reference points. This is the important realisation: to record rotation of a point or any motion of a point we require a 4th stage system, or higher!

The way it works is by imposing rigid rules on the subsystemat as a Spaciometry and then rigid or flexible rules on a local system representing the moving body. The spaciometric and the local systems thus combine in a higher rank system to describe dynamic motion.

But there is a systematic assumption that is not rigorous in most minds and that is regarding the role of Shunya or the concept of everything!

I have written quite extensively about Shunya since I fell into its warm limpid pools during my journey in the fractal foundations thread.( google jehovajah Shunya).

Suffice it to say that Shunya means everything. It was misunderstood from the day Brahmagupta formalised the astrological concept that it is, and has created panic and dread from its inception due to Brahmagupta terming the tally marks he uncovered " misfortunes"! And they continue to psychologically cause dread and loathing today as negative numbers, and most assuredly have caused psychological confusion in the hunt for the meaning of !

However Shunya was brutally tamed by the Arabs who started the chain from Shunya meaning everything, to Sifre meaning wind which eventually became zero meaning nothing to be afraid of! Then thoroughly whipped it was assigned its duties as place holder!  However in the real world it cannot be so easily tamed. It retains its infinite power. Truly of all the mathematicians Euler is the one who explored this best in the west, driven by his intrigue of Indian mystical mathematics.

While cantor was going mad establishing his infinite set theory Euler had already used circles that intersect as diagrams to apprehend how sets come to be out of Shunya. They are mistakenly called Venn Diagrams, but before that they were called Eulers rings.

In addition Euler used his rings to picture the great Indian cycles of ages or Yugas, and he also used them to partition the collections of counting numbers. By so doing he explored the Indian Decimal system based on these cycles great and small and lead the field in modulo arithmetics and equivalence Class studies. All of this derives from his intuition that Shunya meant infinity.
http://baharna.com/karma/yuga.htm

As a consequence he came to name i for infinite magnitude but came to associate it with , through his exploration of the trig ratios of the unit circle , but also of the conic section curves. All this he brought to our attention in his inimitable clear style, revealing what Newton had kept as secrets from the world.

Thus Shunya when placed in its rightful place that is according to its astrological meaning cannot be denied as a concept of infinitude.  But Brahmagupta did not mean to say men can apprehend infinite things and infinite processes, rather he meant and does mean infinity starts and ends precisely at each individuals location, that infinity is relative and we can only grasp it as the limbs of great unfathomsble circles that start at us, go off in one direction and return from the opposite direction.

The great Indian time cycles are astrological, and if you are moving through them and increasing in fortune then what was prior to now is relatively speaking misfortune. To move backwards in these great cycles is thus to enter into misfortune. Where you are at this moment in time is the balance between fortune and misfortune.

Vice versa a man moving into misfortune in these great cycles has indeed a great deal to worry about, for how is he to reverse his fortunes, except by the Gods? The hope was that Astrologers could alleviate the situation by pointing out the Kairos, the opportune time to jump ship onto a more fortunate cycle, when and if possible.

In Indian astrology each caste was assigned its particular heavenly cycle , and since the cycles rarely changed the caste rarely changed its assigned status. Everybody was happy with that because it was in accord with the will of Brahma, and it was not a constraint on fortune or misfortune. It was a divine organisation of society concording to the heavens.

So where Shunya in the Sanskrit means swollen, pregnant and ready to deliver the ancient Indian philosophy took that to mean full, and thus time to fill a new container. Using this system of full containers they constructed the modulo 10 Arithmos as power series containers of ever increasing magnitude. The symbol for full is o. This came to be 0 in the Arabic numerals and was placed at the beginning not the end of a cycle! Geometrically it hardly matters but psychologically it reduces our understanding of the origin of all extensive magnitudes.

All extensive magnitudes come out of the origin, thus thst point must have a super potential! In fact it has an infinite potential. When we ground a formal Grassmann system on a real point we are imposing a super potential formally on that point. However, because of the way the system within systems is constructed the origin of the 1st rank system is at a different potential to the origin of the 2nd rank system, and so on through all the stages or steps of the systems. What this effectively means is that within the origin of a 2nd rank systemat a whole first rank system is contained as a potential, and indeed is mistakenly set to zero rather than to Shunya!

In the real world we cannot set the points of an nth- stage system to Absolute 0. The potential of the points drops down to the previous level, and unless there is a complete collapse to all levels or stages, remains at the level where it maintains some potential.

So the anti commutativity is a relativistic one and applies only at the level of Shunya where it is identified. At all lower levels there will be tail ends that may substantially alter the real and empirically observed behaviour of a phenomena modelled by a Grassmann system, especially if it is dynamic.

Shunya IS Everything!

In the case of the out stepping product the system process is the same For the sum. The principal system varies first then the second elemental within it varies second . Thus this coupled combination produces the diagonal labelled by the sum rule and the parallelogram fletch which is identified by the product rule. The labelling is just identifying the aspects which result from the process of stepping out.

Thus we can expect a different focus from the colliding product .as far as I can determine that process is casting a shadow, or projecting a shadow .

The out stepping product is therefore naturally physically dynamic and relates well to mechanical systems of rigid bodies. Or coupled forces.

The second colliding product which later becomes termed thus is physically about casting shadows or projection and thus dynamically relates to light and electromagnetics especially reflection, refraction and diffraction., as assumed to travel in straight lines. The product is a line segment or projected image/ shadow.

Stormin' Norman!

http://youtu.be/rCDRCGjmaO8
http://www.youtube.com/watch?v=rCDRCGjmaO8

Of course we part company at numbers!

Oh and also on the postulates( demands ) of Euclid.

After extensive scanning of the main text of the 1844 versioning have determined that I indeed do need to translate the Übersicht.

The induction , adds to the Vorrede , by design, because Hermann leads you to the importance of the induction in the Vorrede. And likewise it cannot escape notice the important role he places on the Übersicht in the Induction. I have translated the term as Overview.

However further there is the importance of the Hegelian triad , which would necessitate the Overview as bring an important part in the dialectic process.

Now Hermann bemoaned the imperfection of the 1844 version and I can concur that after scanning hundreds of pages and coming upon many surprises I was left dissatisfied at the end. The book falls short of its great potential! Some may say it still is sublimely effective, and I would concur, but it , judged by its own proclamations does not deliver that rounded wholeness expected from the Vorrede..

Indeed it was prockaimed that a second book would be necessary to complete the work, and because I know such a work was scrapped by Robert , and a different volume was masterminded by him I feel that we have missed out on an original Tour de Force.

The 1862 version is a different book, much less adherent to the philosophical and religious and mystical scheme that drove Hermann in the 1844 version . Thus hermanns copious notes and annotations and addenda are a belated attempt to unify the 2 thematically.

Hermann simply did not have time left to write the second volume as he had wished. He suffered from a kind of consumptive hay fever most of his life, he had I think 10 or so children and a wife to support and he had demanding duties as a linguistics professor working on the greatest work of linguistics ever devised, the Proto Indo European language lexicons and dictionaries. And he had a few ideas that were of importance in physics to do with the colour spectrum nd photography, as well as electromagnetism.

Had he lived longer I think he would have written his second volume as planned.

Some surprises are: that the stepping out product has an inverse, or a division to its multiplication!  As he developed the labels it became clear that the symbol was a commutative operator, unlike the stepping out product itself. This is not surprising when you drop the Maths speak and comsider the processes directly.

Extending a magnitude goes only in one Continuous direction, and summation is continuous/ contiguous.

However division is not an extensive process but a disaggregative one. Thus establishing the process by the label means that in using the label in a product one is not using a the product to multiply, but rather using it to divide, so for example the product 1/2 x 3 is a division written as a product , consequently 3 x 1/2 means precisely the same division.

How we change our view between the 2 kinds of multiplication requires a full reading, but essential the difference is in one extensive magnitude being out stepping while the other is being in stepping/ inversive. 

If the reciprocal were in fact a compression of a volumetric magnitude, we might refer to it as an intensive magnitude, but as the reduction cours as a disaggregation of the volume we can only refer to it as in stepping. Reciprocal and inversive are more mathematical terms.

How Hermann then goes on to establish the shadow casting and projection product is quite amazing, but his secret is to establish very secure algebraic labels and useage, and then to take those labels and "extending" their applicability to increasingly wider and unexpected fields.

Amongst these is the solution of systems of Equations, for which he writes down a solution in a one line Combinatorially variant set of coefficients for the system of equations. This one line answer is in fact the determinant expanded for n-equations of n unknowns, thus a nth stage system. For these equations he introduces the term Grade.

There is much more , but nothing on the circle, I thought until I understood the term Richtsystem to not refer to a Rechts angle coordinate system, but rather to a Richtung system, or in other words an orientation system, where translation is sintered out leaving just the rotationl orientation .

It was also noteable that Hermmann refers to the number count of the extensive magnitudes, not to the number per se! I related this to the Arithmoi, and indeed Hermann uses the tern Zahlengrösse which literally means tally count magnitudes and which I immediately understood as Arithmoi, that is unit magnitudes/ Metrons used to count out space. These unit magnitudes are the Euclidesn monads..

Then inexplicably Hermann starts a section called the Elementargroesse. In this he basically seems to repeat the first section called the Ausdehnungsgroesse. I was surprised, and that is why I think the overview is essential reading( as well as some sections in the Ausdehnungsgröße itself) .

Essentially think this is not to be translated elementary magnitudes, but rather elemental magnitudes of the first, second, ... n-th rank system, in other words what magnitudes make up the basis for any system, and how those magnitudes are coupled together to form an elemental magnitude for the system. Thus for lines the elementals are points for planes the elementals are lines, for space the elementals are planes, and for 4 dimensional systems the elementals are spaces.

This relates to some confusing ideas in the Clifford algebras where the outer product is used to define for example a bivector. Of course this is just a plane , but it is the elemental of a Raum or 3d space .

Now the inner product is a projected line segment , and being a line segment is the element of the plane or 2 d space. However when a line and a plane are composed in a sum , that is they are formed into a combination that represents a basis system , at least as Grassmanns colliding product composed with Grassmans outstripping product the form a basis of 3 elemental magnitudes. As such they are the elemental of a 4 dimensional space or a 4th rank system.

So why not just go straight to the 3 element basis ?

Well firstly because Hamilton, not Hermann had clearly shown the need for a 4 dimensionl basis to do rotations. Hermann therefore is responding to this fact not by claiming he had discovered it, but rather by saying, now it has been discovered let us analyse it and synthesise it by the methods of the Extending/ extensive magnitudes.

Secondly he is able to draw on the rich work he has done regarding the projective product and the cyclic interchange in the plane to explain the anti commutativity of Hamiltons quaternions.

Thirdly he can show clearly how the same extensive magnitudes characterise the rotation  by giving the point of rotation and the plane of rotation.

To understand this Normans excellent series on the quaternion shows how a composition of rotations ends up being described by just one Ritation , and in Hermans ideas each rotation has a fixed plane in which it swings , thus we need to define the plane of swing by the start line on the edge of the plane and the  finish line , by which we then can draw a parallelogram plane encapsulating the composed rotation.

Of course when you are used to rotating around axes this is difficult to conceive, but Norman uses reflection in a line or plane to compose rotation. He could equally have used just projection. Had he done so he would in fact have explained the essence of Grassmann rotation and representation of the quaternions.

(http://www.mcescher.com/wp-content/uploads/2013/10/LW303-MC-Escher-Day-and-Night-19381.jpg)


I would not want the reader to go away without appreciating the complete mundanity of Hermanns vision, how in fact these truths are everyway bodied( consubstantiated) in the objects and processes around in the world. For example I can hardly move without recognising some array , or rank array around me. Particularly when I go into a department store I see row upon row of arrays  stacked with items for sale, or enticingly or ergonomic ally designed stands offering up or setting out what the shop has to sell. The department store itself as a building is a complex array , but the important point is not to get stuck with the supposedly mathematical concept of an array.

To begin to liberate your mind I recommend looking over the work of Escher, who more than most captured the essentials of what his contemporaries were conceiving, but in particular I think gives aesthetic form to Hermans concepts of the thought array, and the developing Array , and also the points entity of the developing array.


MC Escher (https://www.google.co.uk/search?q=escher&oq=escher&aqs=chrome..69i57j0j5j0&sourceid=chrome-mobile&ie=UTF-8&hl=en-US&espv=1#hl=en-US&q=sky+and+water+i+mc+escher&stick=H4sIAAAAAAAAAGOovnz8BQMDgyYHsxCHfq6-galhbp4SF4hlZJ6WZG6pxe9YVJJZXBKSD6TL84uyL_pwxy_oOHq5fdq20JOqjyJdtpx2BgDrJYGfRgAAAA)

Consequently we are led to address the curious way we attempt to understand 3 d space. Some of us apprehend it through the great works of Sculptors and Artists and Artisans as Architects both civil and military . The Stoa in Athens is architecture that gave its name and form to a school of philosophy , the Stoics. The Peripatetics were an architectural feature frequented by Aristotle and his Students at the Lyceum and gave their name to his students and teachers who spread throughout the Hellenistic nd Arabic empires.

Art in the form of Mosaics is a ground breaking fundamental symbolic mystery or secret of the Pythagoreans, whose mystical Gematria has been misunderstood and misrepresented as mystical nonsense, whereas they are actual forms in space , standardised by the term Arithmos, made up of units not necessarily uniform tiles!

One of the great obfuscations of religious and cultural translators is to take the mundane and make it mythical or legendary. Few would wax lyrical about an abacus, and yet it is one of the fundamental calculating arrays used in the ancient Chinese and Jaoanese cultures. Many have waxed lyrical about the Indian decimal system, but they have stripped away the Indian temple and deistic mythology which helped to preserve the information in a oral tradition passed down for millennia as Ganitas or lines of instruction and construction, governed by the great Gunas identified by Indian scholars of great sophistication, and the Astrologers who devised an array of cyclical clocks known as the Yugas and other huge cycles.

These are all arrays embedded in memorable mythologies to aid the transmission by oral means of much empirical expertise.

The Kaballah cannot go unmentioned. For centuries mystified as some ancient dark magical system the Kabballah is merely the accounting and calculating systems the Arabs gathered together from all the cultures in their vast empire. But since, as all these things were, it is based on actual spatial Metrons not symbolic numerals or concepts of number divorced from space, these systems embed in a Spaciometry and are shot through with spaciometric insights. The concept of number divorces the modern mind from this and  karals it into a sterile and death dealing place. Mathematics based on that is a dead thing, soon to rot into dust and blow away in the winds of advancing history.( that is, the things to come viewed as if they had happened in the past!).

So now, we have made strenuous efforts to understand these arrays in 3 d and 2 d, that is as drawings or notation on a page. And since Regiomantus the sprawling print versions, being costly have had to be abridged somehow and it has been accepted that the printer has the final word on that . Thus we get square regimented blocks of typeface as the standard and most economical way to represent what is a living reality.

The square or rectangular arrays arose from a combination of printing convenience, cost, full utilisation of expensive materials and conciseness. However they have now constrained the thinking of many generations in a way our forebears would not recognise! A tree is an array, a cellular organism is an array , a shattered plate is an array! That is why I use the term mosaic advisedly, and the thought array may well get translated as the thought Mosaic, which is formed by apprehending and populating these arrays as formal structures with empirical truths.

Where we have been taught to fill an array with numbers as data points or measurements, Hermann conceived them as status or condition markers encapsulating the truth of a condition at a moment. The developing array allows that identified array to dynamically change and then be captured at a final moment. It is the relationship between these 2 array instances that underpins the LaGrangian method, and also develops the thought array in the reader as a dynamically developing thing.

These are or were hugely difficult arrangements to calculate, despite the notions being or becoming intuitively obvious. Between the initial and final states of the developing array all that was conceivable was that points moved relative to one another. How they moved could not ultimately be determined empirically, but certain constraints could be imposed. It therefore became important to analyse the correct constraints to see if they produced the final array by a calculation process alone.

Using this concept snd the calculus of variations idea LaGrance was able to potentially describe any dynamic system by a calculated process. To do this he of course adopted the laziest assumptions. I mean why work too hard at a complicated system? But to his amazement this lazy approach more often gave him the right answers within bounds.

Hamilton picking up on this redefined Mechanics using the Hamiltonian array as evoking the principle of least action. It is still  a standard approach today, although both Euler and LaGrange agreed that it was not a foundational principle of mechanics like the system set up by Newton.

And so we come to Einstein, who really did not have a mathematical clue how to compose his insights. His insights and instincts however are the crucial thing, that is his thought array or thought Mosaic is what has been used to apprehend rank arrays in space and to populate them with empirical measurements and then to calculate the outcomes. His type and style of thought mosaic/ array became known as special and general relativity, and was meant to be a more fundmental setting for geometry in which to apply Newtonian Mechanics and astrological principles.

Suffice it to say here that Einstein as well as being a catalyst for a brave new geometrical or spaciometric approach, was alo a political pawn in a propaganda game that existed throughout the second world war and well into the cold war period. Much of what is set out about him was myth. A lot of his ideas were taken from colleagues in Europe who were unable to publish because of the political climate, and so he published on their behalf. However it was not possible to use certain words, particularly aether, because the security services used that to identify actual enemy spies! Thus the notion of spacetime, taken from Hamiltons work was born. Spacetime has always been a mathematical or geometrical model of a rotational aether.

Einstein, later in life , when the climate was again safer gave lectures in America and Europe calling for the return of the Aether to the praxis of Physics.

As usual I originally started this post with the observation that the complex plane, the vector plane and the hyperbolic plane are just 2 dimensional slices from  a Grassmann n-th rank space or system, and therefore what we have called " imaginary" are in fact just useful parameters for thought patterns that describe the field of study.

Thus we only have translation, rotation and projection in a Grassmann system and it must be one of these that must account for the strange behaviour we first identified as . That being the case means that we can identify the other formal strange behaviours we may encounter by findng the rank of a system in which that behaviour is quite usual! The reduction or projection onto a plane will thus account for its mysterious behaviours( according to our planar point of view).

I forgot to add how the rise of video and high speed photography and filming has made much of the old calculative methods really obsolete. We do need to shift paradigms to a more fluid rendition of these developing arrays and the point entities associated with them.

Differential calculus can now be replaced or refined by high speed filming, each frame is a developing array filled with the truth. By the end of the film we can see how the point entity in the array has changed. We may not have a formula to differentiate to concur with what we can now plainly see with our own eyes frame by frame, but we can use the video data to establish the most fitting constraints on a model system  to mimic the filmed behaviour.

We now have the ability to check our physical laws and algorithms against instantaneous( almost) data sets. Instead we still assert our physical laws are correct and natural behaviours should obey them!

I cannot pass on without drawing attention to the fractal generator.

These applications are natural developments of the use of developing arrays based on poit entities . We know that interest in iterative systems of Equations, or even synthetic geometrical iterations existed before Benoit Mandelbrot. In fact he drew on a french research interest in second world war Europe in formulating his ideas. Work by Argand, Fatou and others .In America little known geometers were also researching the effects of iterating geometrical forms at different scales.

It was the arrival of very powerful iterators that made possible the formulation of Mandrlbrot's ideas. But it required the development of graphical plotters to bring fractal geometry to light and life. The discovery of the Mandelbrot set was to be the start of a whole new appreciation of the point entities and the developing arrays.

(http://i271.photobucket.com/albums/jj132/12_21_12/Mandel_zoom_07_satellite.jpg)

The use of colour coding and colour cycling added a new dimension to the imaging of these sets, and the creation or rather recreation of the geometry that first developed the arrays in LaGrange and LaPlaces mind, the thought arrays. However now it was possible , through the developing array to establish point entities and to constrain their development by simple rules over and above the elemental rules of the reference frame system itself. By colour coding certain point displacements it became clear that there was a central core/ kernel as Grassmann said , that characterised the whole system. The most famous iconic image of the Mandelbrot set in the complex plane was born.

These concepts were hard to explain to ordinary mathematicians. They had no clue or conception of what Mandrlbrot was talking about nor what Fractal Grometry even was. It required the development of new mathematicians, just as Hermann had said, engineers nd physicists who required powerful iterative techniques to solve real problems described in array format. These array systems solved linear equations for all sorts of construction and manufacturing issues, but it was the computer generated image artists who made the breakthrough Mandelbrot seemed to be describing!

Suddenly instead of numbers in endless arrays being printed, images of lines and triangles and colours were being printed. Besides the Mandrlbrot se icon that Brnoit published showing the symmetry of certain complex planar sets under simple rules. There was little else until an aeronautics engineer applied benoits methods to generate realistic mountainous terrains from iterated triangles!

Slowly but surely the geometry of the real world was revealing itself through the transformation of these point entities. According to simple scaled rules iterated a number of times to develop the arrays, and then plotted as line segments and 2 dimensional forms, then surface coloured cvordingly to ome other simple rule, or colouring array.

The history of fractal geometries until now is fascinating. While still alive Brnoit tried to spread the word about the importance of his discovery and the new field he was researching. It is fair to say that the old guard I'd not take to kindly to his brand of mathematics and self promotion, but today largely due to his efforts a new mathematical cadre is well versed in the fractal geometrical point of view. What they lack is the Hermann Grassmann method to make sound sense of it all.

Today many enthusiasts hang on to the subject by immersing themselves in a practical fractal generator way, but the complex math behind it they would find hard to understand because mathematicians do not understand it themselves! They do not understand because they often distance themselves from coding algorithms and programming. Computer scientists do understand how the programme works but they throw up their hands at the " math"!  The left hand does not know what the right hand is doing.

Fortunately, Hermann Grassmann set out a way to explain all of this in really simple terms. But it does require the reader to read the Foreword, the Induction and the Overview of the doctrine of Extending/ extensive magnitudes, and to be willing to think differently..

As I retread sections of the paper regarding Hamiltons quaternions it is clear that my thought patterns have been well inducted by the translation exercise. Not only can I see what Hermann is saying more clearly in German, but also I now have anticipations of what he should BR expounding upon, and this guides the nuanced meaning of unfamiliar words..

I will next translate the overview but must once again reiterate the harm to your imagination that the number line concept is doing.

From the outset you can feel the dynamic energy of Hermanns Spaciometry. Nothing is static, all are thrusting or jostling magnitudes! Thus counting is a necessary method of kerping track on live entities. Where the entities are and how they move relative to one another is the experience of space and time within which we dance and sing a ritualistic count , or sequence. That sequence helps us to recall the Spaciometry and the dynamics of it. . Without that indissoluble link counting and numbers die and become poisonous. With the link fully recognised then the whole system is a light and sun filled day of joyous activity!it is of interest to me that Hermann refers to Helmholtz when discussing issues about the foundations of Geometry. In all my reading do far I have only come across one mention of Gauss by Hermann and none yet of Riemann. What that may mean I can only speculate on, but certainly Hermann looked to the French Ecole as the way forward on 1844, the standard to emulate and surpass. In 1877 he  reiterates that the grounding element of geometry for the Extensive magnitude doctrine is the line segment. This is because conceptually there is a duality between line and point that is indissoluble visually. Because geometry is perceived as a visual study field this link can not be ignored, without generating harmful effects.

However Spaciometry is in fact not a visual field of study. Hermanns system allows us to discover this experience by anchoring a reference frame to real points and then exploring the role of other senses.

One by one biological and neural science has uncovered the role of other sensors, particularly the proprioceptive ones in describing our 3d experience of space. The synaesthesia of these sensors explains why we often cannot distinguish a point from a line, but on a purely auditory sense points are the only perceptible intensities/ densities. Certainly orientation , the essential relative property of a poit to a listener is distinguishable, but any line or path connecting or associated to the orientation or change in orientation is not perceived auditorially . That line is the main attributable difference in visual processing over any other sensor processing.

Because Hermann takes care to include the thinker in the experience of the subject, hermanns system is so powerful and adaptable. Others have sought to remove the thinker in order to claim some " objective" reality.  For me that stance is fraught with srniry and psychological difficulty. It dehumanises our connection to our environment and creates artefacts as Ghosts in the machine. We never truly understand space that way, and it is only now that I realise we only ever approximate to a model of dynamic space by Hermanns method, but at least we do not feel alienated from our natural environment and intuitions,

You were born into a world where animates are known to count, and to visualise magnitudes, but no other compares to the human animate in the extent and intensity of that activity. So we humans count and we calculate.

Why do we calculate? Certainly as a result it gives us no more in numeral names than counting does. And that is the mindset that focussing solely on numeral and numbers engenders.

We calculate because it saves us time and effort in and through space. For the more astute we calculate because it enables us to travel through time and space at any scale with minimal effort. Calculation is our mental process of traversing a model of our reality to end up at a particular place in our model, with a particular aspect or view of that model, at a particular time in that view.

Not only do we calculate space and time but also energy and Dynmics. Calculation helps us to " move" without moving . To travelnastrally to any plane in our model and view and experience from that plane with all the assurance our model can give, but in less time and less space than punting would take.

When we count we should dance and sing, because the alternative is to trudge and mutter as we measure through out space one unit at a time.

When Napier discovered his logarithms it was through traversing around the perimeter of the broadest circle he could imagine. Logarithms were about movement and motion, but in such a way that each calculation determined a constrained motion on the arc! Of course " serious" mathematicians pooh poohed the idea preferring Bergis more " mathematical" series explanation. The dynamics of space were " killed" off again.

Berkeley weighed into Newtons Fluxions as mere religious figments, once again refusing to accept the dynamics of the Arithmoi, and eventually confusing Dedekind into a weird set concoction so unnatural and poisonous that we are dying from so called real numbers.. Berkley initiated a movement that lead to numbers and the number concept being divorced from reality, while ironically calling themselves Real Numbers. Arithmos and magnitude based on Monas / Metron now died away, to be replaced by ... Well nothing really. These numbers were supposed to be logically defensible, but in fact they were and are the feverish macinayions of an over sophisticated sensibility.

Returning to magnitude as experiential continua , either intensive or extensive, connects us back to space , time and proprioception of these and puts us or me right back at the centre of our/ my experiential continuum. From there I can make sense of all so called higher Maths

Normans Quaternion intro (http://youtu.be/uRKZnFAR7yw).
Note the double turn or arc length .

As Norman pointed out this is a basic circle geometric theorem. The double arc length subtended a single arc length on the diametric opposite to the centre. Otherwise stated: the angle subtended at the centre is 2 times the angle at the circumference on the same side of the chord!

While angles at the centre or corner are problematic, this is only because of misdirection. We measure angle on the arc not at the centre. In addition arc have always been associated to ratios of chords to diameters , so the argument about the trigonometric functions, for me is not substantial. The real issue is the real number concept .

However the rational treatment of trigonometry is a very good approach. The traditional trigonometric equations became so slippery in the function form that most physicists resorted to trig line segments to express these relationships.

Euler's theorem for the exponential expansion of the rotation of arc can be rewritten in terms of trig line segments, and immediately becomes an Ausdehnungsgröße, so a lot of the issues Norman has are catered for by Hermann.

Now the use of reflections arises naturally in this treatment by Norman, and leads to the double angle format. What I hope to show is that rotation can be described by projections, but the projections are "Backwards" so to speak. This then should link to the chord and arc ideas of Ptolemy, and the rhomboid plane of Hermann.

Theodorus spiral  is intimately connectrd to  n- th rank space.

http://en.m.wikipedia.org/wiki/Spiral_of_Theodorus

By projecting orthogonally or vertically onto an adjacdnt line segment this spider web spiral can be built up. . Each projection is onto an independent line segment in space , and thus does not need to be constrained to the plane in which it starts.

In this regard we can see that if the cross product constraint is dropped or weakened we can develop n-th rank systems that retain an orthogonal link without insisting on mutual orthogonality..

When thinking about rohttp://youtu.be/0_XoZc-A1HUtation based on projection the spiral forms the back bone of the explanation.
Norman on Eulers composition of reflections (http://youtu.be/0_XoZc-A1HU)

Here Morman introduces an axis of rotation. And yet the majority of the development is regarding rotation about a point in the plane. This particular development uses reflection, but as indicated by Theodorus spiral we could develop the notion by projection. The constraint is backward in the sense that the projection is orthogonal or vertical to the projecting line segment . In Euclidean constructions we are taught how to drop( Senkrecht) a vertical from a point onto a given line segment or how to construct a vertical at a given poit on a line segment. Both these can be viewed as projections and therefore encoded by the colliding product for 2 line segments .

Now because hermanns fundamental system is a coupled contiguous one( continuous) then the colliding product is " framed" in this system. There is thus a line segment intersection point for any 2 independent line segments within the system so that the projection can be seen as a right angled triangle.

Here Norman deals with the issue of composing reflections by aligning one of the lines of reflection, the same composition cannot be used for a projection. To obtain a common point of rotation for a projective composition we first have to obtain a common point of intersection for all the line segments involved. This may require a translation by a parallel projection, thus making the general rotation more complex.

For the sphere however, we can use the inbuilt propert that all line segments originate from some centre which is the point of rotation. This is where Theodorus spiral may be applied. The solution is specific to the conic section curves and surfaces.

Arbitrary rotations in space are not yet discussed.

How this relates to the mixed or "averaged" product is  by process. 2 proceesses are involved: the plane of the final rotation given by the out stepping product; and the projection within that plane constituting the Ritation given by the colliding product. That there is such a resultant plane is precisely what Euler and Norman are demonstrating here. Hermanns mixed product draws on this conclusion from the beginning. Herman does not prove that there is such a plane and a single rotation in that plane in this paper, and neither does Hamilton in his quaternion papers. The proof is therefore  down to Euler.

Euler may well have used normals to great circles to establish the theorem,but normals are not required to describe a general rotation about a point in space if you have this final plane of the resultant rotation. Hermanns method concentrates on delivering that final resultant plane as a parallelogram. Thus his method and Hamiltons avoid the intermediary motions and go straight for the final solution.

Gauss in his work on surfaces relied heavily on Normals as radial extensions from a spheres centre. This made sense to him Astrologically and most engineers and mathematicians are embedded in that thought pattern, but in fact Euler demonstrated that all complex rotations in space can be summed to a single rotation. This single rotation does not describe the behaviour of the rotation, just the final position.

How we model rotational behaviour then is a different question, and requires dynamic iterative calculus, or fractal equations!

Norman calls the roles of the x,y,z axes  "distinguished" but Hermann calls them elementals. The fact that we choose a mutually orthogonal set is of little consequence for referencing all and any other line segments, but it fors give us the anomalous cross product. To extend into higher ranks we have to loosen the grip of this Anomally.

Because hermanns fundamental systems are based on elements that conform rigidly to constraints and conditions and are coupled the fundamental planar form is the parallelogram. This form enables all the desired properties of 3d space and continues on into the nth-rank spaces. It is therefore distinguishable in systems of equations , renamed nth- grade systems by Hermann.

This form however is only elemental in the 3rd rank, because for Hermann the elements of a space are of the rank below. Thus for a 0 th- rank system we would expect that to be points. In fact for Hermann the fundamental geometrical element is the line segment with its dual point-line segment nature, thus the 0th-rank is occupied by tally marks. These form a calculative axis by which all elements can be fractally dismembered or divided into Metrons/ monads.

Thus 1 is a tally mark, and only has extension when applied to a first rank system. Tally marks are thus wholly formal, and "out" of any system of geometry that Hermann can devise. We as observers therefore can apply them anywhere at will, but once attached to an extensive magnitude they must remain fixed for the duration of the exposition. By attaching them to the first rank system we obtain the first of many types of Arithmoi or Zahlengröße.

These Arithmoi are the elementals of the next rank system. Thus a parallelepiped(Spath) is a fundamental element of a 4th ranks system. Rotations and other transformations are essentially mappings by transforms from one oriented, projected or curvilinear parallelepiped to another . Transformation of basis is therefore a fundamental method of translating, rotating, reflecting and projecting spatial objects.

Merry Christmas!

http://youtu.be/6MrE_6YIkJM
http://www.youtube.com/waych?v=6MrE_6YIkJM

The important part here is the international influence of India in the 1800's . Euler was very influenced by Indian Mathematics brought to the ports where his father carried on trade as a Merchant of means.

Therefore Hegel had access to eastern and Indian knowledge.

Finally Hermann worked assiduously on the Proto Indo European languages.

The Indian influence , from Arabian imperial wisdom literature, to mathematical inclusion of Indian decimal systems , to the impact on the British Royalty of the time, the Indian influence is important and inescapable. The Upanishads and the Vedas informed Hegels Philosophy.

The general Doctrine of Thought Patterns( Ausdehnungslehre 1844)

§1. We everyway stand the rank arrays, those from the truths, under the general doctrine of thought patterns, which arrays situate themselves on top of like cognisance over all branches of the mathematic , and therehere  only the general labels of the Like quality and the Differing quality, of  the Binding and the Loosing (Separating /Sundering) are pre set out ( prior to anything else)

Therehere it must, the general doctrine of the Thought Patterns, go on ahead into all speciality branches of the mathematic, for-there there  that common branch is still not as such to hand how we require of it for our expertise
and we  still are not  permitting it to travel over, without everyway wiggling to us in unutilised endlessly running   qualities ,  so nothing remains left over  to us , as the same things here are thus far developing .

 In order to firmly set the label for the Like Quality and the Differing Quality It is here initially. There the Like entity necessarily steps here out , also therewith is already the dual quality  , and the Differing entity also as like Entity must appear, only in Differing view from afar;
 thus it appears necessary by considering over-layering tracking, to place on top( of each other) differing relationships of Like quality and Differing quality ;

Thus would  be able to  become declared ,( to the considering game),  by considering everyway equating ( likening) of two bounded lines:
 the Like quality of
the Direction
or the Length,
or the Direction And the Length,
or the direction and the laid Position ,
and so further ;
and by considering other things to everyway  equateable constrained Things   other relationships of Like quality would again come to step henceforward.

Therefore that already these relationships become other relationships , each according to the Attributed quality of the "to everyway equateable" constrained Things, delivering the demonstrable   proof for  there , that these relationships are not next of kin related to the label of the Like quality itself, rather by the contents ( lists) , onto which the same label of the Like quality becomes applied,

In Practice from 2 "equal length" line segments, (to the by considering game), we cannot say that they compared besides themselves are equal, rather only , that their length  be equal, and this length then stands plainly also in the complete relationship of the Like quality.

Thuswith  we have to the label of the Like quality its simple quality returned,

and we can by the same quality therein assign that   "the  entity” is like:-
that one from which one can continuously the same entity declare,   is "like";
or the general one, what in each common judgement themselves compared side by side  can be substituted,  is "like"

How by the same quality herein immediate nearby lies declared:
that even if 2 thought patterns are like a third, they Also themselves are like one another ,
and that the   entity which out of the Like entity  was reifying  on this same cognisance ,  once again is " like",
by the same quality lies in the daylight !

Footnote page1

•see induction No.13
•• plainly the No.5

Commentary
I will bring some of my referencing labels in line with Hermann who now uses the § sign to denote paragraphs in the Thought Patterns Doctrine Proper. So I will redact my earlier referencing as and when.

Hermann here states that the Vorrede and the Einleitung are not sufficient to wring every last drop of generalisation out of the Rank arrays before going on into the Specialties. So this general doctrine of the thought patterns( what I called the Overview previously without checking) will do precisely that as best it might.

It is to be noticed that Hermann invokes the introductory aitema or demands/ requirements to do the Course Euclid sets out, and also the Ennoia or common judgements utilised in comparisons Euclid sets out. It was and is a very misleading terminology to call these initial statements axioms. To do so requires a great befuddlement of mind, as such it was when the Arabic translators attempted to translate this document in the light of Aristotles framework.

Aristotles " Logic" is distinct from Pythagorean logic, and Hegelian logic is different again. Studying the Pythgorean philosophy to which the Stoikeia were but introductory lessons, prepared the student to enter the senior discussions is courses and researches of the Pythagoren school . Studying Hermann will take us down a different route, one propounded by Hegel.

The second in Normans series on Axiomatics

http://youtu.be/ncIgoIBLPqQ

http://www.youtube.com/watch?v=ncIgoIBLPqQ

Just beware the assumption that Euclid used the concept of Axiom! The Greek language does not sustain such a notion and the nearest Greek concept is that of axle or axis, both referring to the central rotating region of a disc..

The history of Axiomatics seems to start in Kant, and represents a powerful redaction of earlier notions, including the muddled thinking of some Islamic scholars who mistook Euclids Stoikeia as a version of Aristotelian logic, being great admirers of Aristotles works..

They seemed not to differentiate between the Pythagoreans and the Platonists, and so did not realise that Euclid was a Pythagorean Mathematikos supporting Plato's promotion of Pythagoreanism, and actually achieving the recognised status of Mathematikos among the Pythagorean scholars who were still based in monasteries( Monas - fundamental Pythgorean ideal) in southen Italy and Mediterranean Aftica.

Commentary

Here chadafrican explains the position Hermann has brought us to by the. foreword and the Induction. We reframe the development of Mathematics so called from the label or Notion of the doctrine of thought patterns. And we do so from the Pre motivated and Pre established notions or as I have translated the word Labels, which have a direct " mathematical" pedigree. Thus the a priori knowledge which we call intuition in mathematics has in fact been instilled in us by the experiences and discussions  Hermann has guided us through up to this point.

The fact that Hegel makes this " movement" clear and calls it dialectic, reveals how the mystification of skills and abilities has occurred through ignorance of this or these great circle links, these great tautologies. Those first entering upon a course of study do not know these connections, and often leave off studying before realising the full spiralling circularity. However those who are the scholars of the study often also are ignorant, because they eschew that is deny the philosophy of a subject.

Here then Hermann reframed mathematics from this grnerally aspect, and because it is circular has to touch bases with Euclid and the Pythagorean philosophers who have already established this tautological base in their philosophy and symbolised it by the Sphere. However , naively, Hermann hoped to restore and advance upon the work of these philosophers because he believed the French had brought forth a New and advanced rank array methodology . In so doing he acknowledges that the Pythagoreans had developed the first rank arrays, the Arithmoi, but now humans had advanced to the dual interlinked rank arrays, and that is where the new direction lay, requiring new Mathematicans.

Thus was required a new Stoikeia a new introductory course to the philosophy of the Hegelians . This is what he is attempting to do, one man , like Euclid setting out the foundation for a New doctrine of thought forms or thought Patterns.

http://youtu.be/u27VZWVtGuk
http://www.youtube.com/watch?v=u27VZWVtGuk

I feel I must reiterate: we have moved beyond the dual rank array of LaPlace and particularly LaGrange , although you would hardly know that from mathematicians. Computer scientists use multiple rank arrays to encapsulate reality. Quantum mechanics uses multiple  probabilistic rank arrays to portray their data set findings and models, but perhaps the most accessible example is the no humble video camera which is installed in every cheap phone! Each video frame in each short video is a rank array that encodes empirical data sets, the Codec enables those rank arrays to be pushed to and from input/ output media which we can see as a moving display.

The fact that we do not make these connections is the real shame. It allows the majority of us to be hoodwinked by the few.

The humble fractal generator is every bit as capable and tooled up for basic research as any Cray supercomputer, it just does it more slowly! Many of the artistic rendering members are achieving are of direct empirical relevance to our understanding of how our models of reality work.

Commentary

While it is not necessary to fully everyway stand Hegel to apprehend Hermann, it helps in translating hermanns dialectical presentation to have some experience of Hegels style and dialectic.

Phenomenology of Geist was the first of only 2 books he personally wrote and edited and published, all the others are constructed by his students after his death in 1831. Here Sadler reaches the concluding § in the introduction!  The rest of the book still lies yet ahead.

http://youtu.be/OvkkKnx3jos
http://www.youtube.com/watch?v=OvkkKnx3jos

The general Doctrine of the Thought Pattern (Ausdehnungslehre 1844)

§2 the second comparative statement, which we in view have pulled  to here, is of the entity of the every way knotting and sundering( binding and loosing)
Even If two magnitudes or thought patterns( which Name we prior pulled  as the general one. See Induction .3) under themselves are everyway knotted, thusly they brand them: 
limbs of the everyway knotting( knitting), 
the thought pattern, which through the knitting together of both comes to be presented, 
the out giving result of the knitting together.

Should both Limbs become distinguished, thusly we name  the one the fore limb( more forward one)  and the other the hind limb( the more afar one )

As the general sign of the knitting together we choose the sign "^"
Let now a, b be the limbs of the same, and indeed  a the fore limb and  b the hind limb, thusly we signify the out giving result of the knitting together with  (a^ b);

In which the brackets here should express: that the knitting together no more in separation of  their limbs should be manifested, rather as a monad ( unit) of labels.

The out given result of the knitting together can once again with other thought patterns be everyway  knotted, and thusly one reaches to a knitting together of more limbs  which therefore  at the immediately nearby  always appears as a knitting together of each 2 .

In order to make way for the ease of practice we enslave to ourselves  the usual "shortening off" bracket assigning, in which we specifically the together related  signs of a bracket  let go away, even if  the "of the discussed" opening sign[(]  either aside the beginning of the whole expression stands, or after an other opening  sign would  be following; to the by considering game,  in place of ((a^b)^c) we write  a^b^c

Commentary §2

The comparative statement of binding and loosing, in comparison with Like and Differing  is much stronger than I first appreciated. The idea of knotting gave way to knitting together, by typo really but then I realised how more appropriate it was.

These bonds knit the forelimb and the hind limb into a continuous unit.

Moreover this knitting is done sequentially in pairs, and consequently the fore limb is always written first before the hind limb. Changing the order of writing makes the sign change from one to the other. This sign or symbol or label is not apparent in that it is not called a sign, but it is a label and a symbol. Later a sign will be introduced which will be specifically associated with negative and positive signs. By then we may forget that every label is a symbol or sign given a designated role., and some labels change their designated role when they change their spatial position. Thus sign change is more general thn just  positive or negative, it indicates a change in role due to a change in Phil position or orientation.

As it stands we can only bind or knit 2 things together at a time right now, but this will change.

Commentary on §§1-2

Because I am reading ahead I gain insights about the purpose of the earlier paragraphs, which is useful when translating.

It is now apparent that the first paragraph on Like and Differring entities is about 4 identified ways of declaring 2 thought patterns like. Thus it is about equality, duality, congruency and identicallity , and by implication any other mode of expressing like entities. Moreover it is about when and how we declare entities to be like, and thereby the judgement we use to make such a declaration.

In a very real sense most of us have been trained to identify like entities, few have been trained to use their judgment to declare likeness or equality as a finding, especially so for mathematicins who have like entities given to them on a plate!

Comparing , contrasting and then concluding based on the results of those 2 prior processes  is essentially the 3 part structure or tool Hegel uses in his presentation. The conclusion allows a judgment to be expressed, and often recommendations for improvements . The conclusion can be as short or as long as the author wishes to expand on his or her insights. Hegel steps back to view the 3 part structure as essentially a historical or developmental process in history he called the Dialectic.

It is a clear movement to and fro between greater vision from obscurity and sometimes from grand apprehension to diminished comprehension of a new contingency, and then onwards, at each stag, in the moment,  being unaware of the necessities of the resolution that then emerges as a glimmer or a shower of sparks!

Marx and others felt that not only was this process historically inevitable, but that also acting in accord or mimicry of it was justifiable as invoking the inevitable solution/ resolution. The end thus justifying the means.

However if Hermann is taken as a careful student of Hegel we cannot justify that kind of behaviour. Rather what Hermann shows is that in analysing the current position in the dialectical process we may see more clearly where we were imperfect and hopefully synthesise a more perfect apprehension.

Thus by being rigorously analytical in the comparisons and contrasts of the concepts of like and differing we may better understand the use of the "=" label or sign, and not just in arithmetic or mathematic, but more widely and dynamically.

Certain conventions have to be set. But by setting them as simply and carefully as possible Hermann hopes to securely found the doctrine of thought patterns.

Thus he now moves onto binding and loosing . The reference here is the keys of Peter . "What is bound in heaven is bound here on earth and what is loosed here on earth is loosed also in heaven". By this he gives a certain gravitas to the rules he is "out giving" or evoking here. The evoked result or out giving result/ response, or even the responsive result are essential notions behind this knitting process..

Hermann here means to deal with fundamentally the notions of commutativity and associativity. Because we are introduced to these concepts in the arithmetical arts we do not apprehend where they derive from or what they ultimately mean, over the next few paragraphs Hermann will give concording demonstrations of these notions and their implications and in fact how knitting and unravelling , binding and loosing encode addition and subtraction at a fundamental level , but also multiplication and division at a barely less fundamental process level.

The implications for commutativity and associativity are however remarkably different for both levels of process.

After years of consideration I have concluded that it is more complete to start with a whole and then divide it. Thus division also known as subtraction at a more fundamental process level is narratively prior to multiplication and thus addition.

Thus I start with Analysis or repeated( ana) cutting( lysis) of the whole  eventually followed by Synthesis or binding( syn) creation( thesis from Zeus-made or brought about by Light/ Lightning). In this way addition and multiplication has a purpose: to reveal the whole, while division and subtraction have their purpose: to uncover the structure of the whole.

These re patently Pythgorean ideas but belong here in this kind of fundamental discussion of how we came to develop the qualification Mathematikos into the subject of. Mathematics.

Is Hermann saying this is the only synthesis possible? On the contrary, he is showing how carefully we can construct the fundamental skill sets required for an evoked resultant expertise. But it does not mean we can just choose whatever we like whenever we feel like it! The rigour and the consistency demanded for serious consideration and acceptance become the more onerous the more fanciful or complex the fundamental notions that are proposed! By always choosing the simplest notion Hermann not only lessens the burden of proof on himself, but also recommends its acceptance to us as a " trivial" matter.

Later we are surprised how sturdy an edifice he then builds on such "trivialities"!

However, some have used this freedom, and the general ignorance of the process to introduce desired trivia that in fact masquerade as trivia. As Norman points out: it is not a trivial matter to apprehend to ourselves infinite powers! The trivial nature that I speak of is not the trivia of words : that is" let us assume infinite powers", or "let us be god "; which indeed are trivia bordering on frippery, and in another age would have brought death upon those proposing it! Rather the trivial thing is that of action: can I or cannot I bind 2 limbs together? Can I bind such a bound thing to a third? Can I repeat such a process until I run out of things to bind or become exhausted or die?

Such trivia as these one can readily assent to or deny . But it is by assenting to such as these that we extend our consciousness of our ability to construct certain evoked results , and to rely on calculation as meaning something in our constructive ability.

For this reaon was at first repudiated and then by dialectical degrees bought into meaningful use after some proper adjustments in interpretation.

We shall see how Hermann sets out to do this at the beginning of his Grand enterprise.

Norman continues his development of a doctrine of thought patternS.

You can see that he has made use of Grassmanns method of analysis and synthesis, but chosen a different foundational basis: the rational numbers . I won't go into the construction of the rational numbers or rather their reification onto his cognisance of accounting, which you can find in the early part of his Maths foundations course. Suffice it to say that it is a different reification to the one Hermann lays out.

 development is of extensive or extending magnitudes, and thus his fundamental " object" is the line segment . Normans development is on discrete quanta. Later he piggy backs line segments onto this discrete foundation ( in the form of vectors )  but that is the exact reverse of Hermanns programme who in the second section of his Doctrine of Thought Patterns introduces the Elemental magnitudes.

What precisely these elemental magnitudes are I am not quite sure as yet, but they are quantitive rather than extensive , and very possibly are Intensive magnitudes. Any way Hermanns Lineal algebra is precisely based on lineal elements or line segments with extensible attributes  initially, and then extended in scope to these elemental magnitudes.

The important point for all is that the formal thought patterns are identical up to the point of application, at which point certain adjustments have to be made to accommodate the elements actually to hand.

http://youtu.be/2WH6NTciV2Q

http://www.youtube.com/watch?v=2WH6NTciV2Q

The clear reason for the reversal is the necessity for computational algorithms , analogues that machines can utilise to compute results meaningful to us. The role of Geometry has not changed . It is still the real expertise based experiential continuum that Hermann restored from the axiomatic nightmare others had placed it in.

The general Doctrine of the Thought Pattern (Ausdehnungslehre 1844)

§3 Now the special artwork of the knitting together therethrough becomes evoked, what becomes firmly held by considering the same knitting  as "advantageous result" , that brands: under  which  circumstances and in which "extensive magnitude"  the beneficial result is becoming set as remaining  like itself.

The singular every way varyings , which one , without the individually knitted together thought patterns themseves changing , can take in advance ( of specific details) is the varying of the brackets and the un-ordering of the limbs.

Let us take to mind first the knitting together thusly  that by considering 3 limbs the setting of the brackets no real distinction, that brands: no distinction of the advantageous result founds,

Therefore  that   a^(b^c) =a^b^c  exists.

Thus follows immediately nearby that one also in every multiple limbed knitting together of this artform  without its advantageous result varying the brackets can be let go away.

Because each bracket encloses immediately nearby a two limbed expression every way satisfying the there over firmly set appointed ( meaning)  , and this expression must once again as limb be evryway connected with an  other thought pattern, in short it steps an every way connecting from three thought patterns henceforward, for which we set out ahead ( of any thing else) that one could  let go away the brackets without the advantageous result of their knitting together varying.

Therefore  there, one is permitted to set in  place of each thought pattern the like entity to it , the total beneficial result, through the letting go away of each bracket also is becoming not varied.

Therefore


Or there, one is permitting in 2 expressions, which only through the setting of the brackets are distinguishing themselves continuous according to the plainly outwardly demonstrated Proposition the brackets let go away , thusly are both expressions also under  it alike, (there, they are alike to the same( brackets  losing) expression),
and one has the foregoing proposition in something of a more general thought pattern:

Commentary

When Hermann advocates rigour by golly he means it! I did not expect to be deriving propositions about the use of brackets!

You may also note that he uses one of the statements or propositions of declaring entities like to derive a subtlety of the propositions.

Because of rigour he proceeds slower than expected, so I am not used to his slow dialectical build up and often second guess him incorrectly

The un-ordering that occurs at the loosing of the brackets is the unordering of the elements that in 2's pair to form a limb. The change in sequence order is for a later developmental stage,

Commentary

Even though I have not completed transacting §4 I can now see where Hermann is going with all this bracketing not changing the advantageous result.
Firstly the word Ergebeniss itself :" out giving" is a simple enough activity but what does it signify? Eventually I have come to recognise it as the "output result". Although the action is different the similarity of giving or putting something out there wherever that may be is essentially the notion.

Thus the output is what is actually completed and knitted together.

If we consider a machine process the in put is what is placed into the process. The process then "gives out" the finished result. Thus we have the "in putting" of the ingredients and this is followed by the " out giving" of the result.

The process of inputting brackets does not change the out putted / out given result of the knitting.

Section 4 is about how the brackets change the ordering of the knitting Process but not the result of the process, the out giving/ out put result.

I glimpsed this briefly while working through Hamiltons step derivations., but could not express it clearly. The output result is unaffected by the bracketing of the elements . The bracketing of the elements are whimsical, thus for a firmly fixed sequencing( this is the Fundamental ordering) the whimsical bracketing reorients commutative and associative  arrangements of the elemental processing.

Associativity and commutativity require a fundamentally fixed ground on top of which an order of combination is formally and externally imposed without varying the order of elemental sequencing. The real elemenys remain unchanged the formal combinations of them may vary without affecting the output/ out giving result..

Now onto this we can construct a definition of Tally marks, and notation for accounting praxis and sequence of notational symbols called Tallies.

Another aspect that floats back into memory is Hamiltons development of his "Science of Pure Time" or the theory of conjugate pairs or couples. In this fantastic work, which I recommend finding and reading, Hamilton proposes to develop mathematics, including the algebra of the imaginaries on the basis of moments in time.

First he starts with single moments and plods through every relationship of difference and equality possible for single moments , and importantly how the mind is conducted by these relationships. He then moves on to the possible relationships of two moments considered by the mind as a single relational unit. This is vastly more subtle , rich and rewarding, both to read and to study.

However, for a system eschewing distance( as perhaps LaGrange had developed it) and one meant to be wholly based on time he soon begs indulgence to use distance metaphors! That being said( which in fact to my mind undermines the whole enterprise) he carries it off with such style and aplomb that one does not mind the sleight of hand he uses from time to time,( such as sliding over from ordinal to Cardinal numbers or from a lineal system to a 2 axes system hidden behind differential equations in order to rø pound on imaginary magnitudes!)

All in all it was a powerful manifesto for the imaginary doctrine he was seeking to make the heart and soul of Albebra and it was very influential at a time when mathematics was stultifying. However he ran into difficulty when it came to triples and eventually abandoned the idea after about 10 years of trying and subsumed the work in hs new discovery of Quaternions in 1843.

How ironic that in those 10 years from 1831 to about 1843 Hermann was tackling the three limb issue as part of his work 1844 Ausdehnungslehre. By starting with his insight on the 3 limb aspect of space Hermann avoided certain special forms and thought patterns that are specific to the first 2 " dimensions" and was able to quickly generalise to n- dimensions in his conception.

Both were inspired by the Work of LaGrange and the French École but both took different paths from the same or similar starting point. Hermann, being outside the normal academic system, free to think and follow his interest scientifically had the advantage. The seemingly inconsequential triangle ( not really, because triangles are a fundamental object in space, like circles) was the dynamic impulse he needed to gradually uncover the ideas of extensive nd extending magnitudes in a continuous/ contiguous setting.

The general Doctrine of the Thought Pattern (Ausdehnungslehre 1844)

§4 were( onto the other side for a "knitting together") only the everyway  toutable quality ( with a view to exchange) of the both limbs firmly set, thusly would there be drawn thereout no other following result able to come to be.

This appointed meaning yet coming  from afar to  the "in the previous § completed" meaning , thusly follows: that also by multiple limbed expressions the ordering of the limbs for the total  output result is equally valid, in which one can specifically easily show , that every  two, themselves following on top one other, are allowing  limb everyway touting ( for exchange).

In practice, according to the immediate last outwardly demonstrated  proposition (§3) two such limbs,( to which entity one  will concording demonstration provide of everyway toutable quality ) ,  can one enclose in brackets without varying the total output result, further  these limbs under themselves are everyway touting for exchange .

Without varying the  output results which out of them has been represented "knitting together" ( how we do plainly set out ahead of everything else), therefore, without the  output result of the whole "knitting together" varying, there one places every thought pattern which to its like entity  can be set

and at last now can the brackets, again so set become, how they were at the beginning.

Thuswith is the everyway toutable quality( for exchange) of two one other following limbs outwardly demonstrated. (QED)

Now therefore one may bring therethrough setting forth of this process  of every limb onto every whimsical position ; thusly is the ordering of the limbs mainly equally valid.

Therefore this Result firmly fixed together with the result of the previous paragraph

We reify such a" knitting together"  making a way by the Short route, for which the output  beside us Appointed meanings empower,
we reify naming it  a Simple entity .

Now a still further-going appointed meaning is  no more possible for the artform of the knitting, even  if one does not return onto the Nature of the knitted together thought patterns, and we therehere the achieved knitting  pace to the "ontop  loosing" , or  to the analytical process.

Commentary on §4

We are lead into §5 by the last sentence because 4 is apparently such a simple proposition. But following the demonstration is not so simple when one is unfamiliar with the referrents. However it seems we have simply shown commutativity and associativity mainly comes from insisting that output results remain the same.

This is a " foreign" implication of the previous result, in that you have to be looking for it from afar to even see that the proposition can apply to commutativity and associativity.

However an analytical or "upon- loosing" discussion is to follow to once again demonstrate the same point! http://dictionary.reference.com/browse/ana-

No change cn come if the result is fixed filmly beforehand!

The more I mull this demonstration over the more I am impressed by how inept my suggested basis for commutativity was: it was tantamount to saying lets all be sloppy thinkers!

In addition Hermann has not gone down the route of listing every possibility, as Hamilton did, so his conclusion is wide ranging and unfolding. The more you think specifically say about the powerful formula

AB + BC =AC  and all other possible like versions say AD + DC = AC the more you see that commutativity has to be based on insisting on the same result to define it.

Finally bear in mind constantly Hermann is dealing with 3 limbs not 2!

In hermanns treatment you can't have commutativity without associativity, but you can have associativity without commutativity.

One other result of this appointed meaning also comes in fom afar, and that is conservation of the output!

Thus invariance is already a theme of these 2 paragraphs and in fact develops as a powerful strategy goal for Solving omplexity dynmic systems.

The conservation laws belong to this Methodology. By that I mean they are less laws of nature and more methods of solution for dealing analytically or on top of the unravelled or loosened situation with a small part that is impossible to determine any other way.

So for example we find later Hetmann setting a system of equations to 0 because tht gve a verynlear solution path. Of purse you csn't know the details of any dynmic culture, because the method precludes that. It states tht hate er the dynmic changes they mus ll give out the same result , output must remain fixed.

We find it in the conservation of momentum. This is not a physical law, as much as it is a mathematical method of solving the momentum equation.

As young scientists we are do driven by our training and desire yo " solvr" that we do not stop to think how narrow a result is as a bit of information. The grand picture is explained by a theory or a model , the mathmatical result is often a meaningless numerical answer requiring interpretation within the theoretical explanation.

Keine Abweichen, the use of the unchanging output as. A stage in finding a minuscule part of the dynamic change occurring under that constraint.

There is one more output result from the far field.

The significance is in the fact that Hermann starts his thought patterns doctrine with the relationships between 3 points and thus 3 Strecken or line segments. From this he notices the law of 2 Strecken in the plane

AB + BC = AC

One of the fundamental "equations" of thought patterns.

This I have called the law of 3 points, the law of 2 Strecken; and the product law or the law of 3 Strecken or the parallel product I have distinguished with those labels. Of course one can reduce them to line segments connecting 3 points in space!

Notice that Last flippant remark!

Suddenly I went from the plane to 3d ( or rather n-d!) space!.

Because our knowledge of Astrology is screwed up, and by now quite paltry few will have heard of much past Pythagoras and trigonometry. But I was introduced to Herons formula as a way of finding areas of triangles. I had no clue of its significance or its history, nor indeed the significance of the Pythagorean school of thought. Without knowing it my educational background was modelled on the Aristotelian school of platonic thought, not the Euclidean! And yet Euclids name was bandied about like a religious Icon.

Wallis accused Descartes of Huge Plaguarism, taking the work of Harriot as his own! Harriot was dead and not likely to object, so why not? Descartes had to survive in Europe and that meant patronage. He was never going to work as a serf, especially with his habit of staying in bed thinking most mornings!

Thought patterns, that was his Fortè and setting up a system of algebraic or symbolic geometry was his joy.. The truth about his coordinate system, as usual is a little twisted.

I read La Geometrie briefly expecting to find the x and y axes. Instead I found reference to Ptolemaic and circle theorems based on Euclids Stoikeia apparently, but in fact drawn from the mechanical philosophical texts of Omar Khayyam and other Islamic scholars who favoured Aristotle.

Thales and Apollonius were also studied as part of the advanced curriculum in the Arabic Academies of Europe( Spain)  and Baghdad . Undoubtedly copies of the Stoikeia by the Neopythagoreans came into the possession of the Islamic scholars, but by that time they were all fully enamoured of Aristotle. Thus came centuries of wrongly interpreting the Stoikeia as a course in Geometry for " babes" who first want to learn or need to learn the elements of mechanics!

Later wallis came upon a Greek version of the Stoikeia among some papers returned by English Merchants, and Isaac Barrow returned from a Harrowing trip to the far east and Europe with papers and knowledge by studying in the great libraries and seminaries around the Arabic empire. His principal interest was in Apollonius and his Conics, of which he could only "obtain" an incomplete version.

Nevertheless for his efforts and peril he was given a chair of geometry in Cambrudgeshire ( Gresham College) to work on English translations . Somehow Wallis got hold of the documents pertaining to Euclid in Greek and began a lifelong translation of those at his seats and positions in Oxford. It is possible that Bartow sold him the material for economic reasons, and because he took on the additional burden of the seat of Mathrmatics, newly established at Cambridge university which offered a higher Stipend but more duties!

Both Wallis and directly Bartow were responsible for directing Sir Isaac Newtons studies, and Newton, having been baffled by Astrology he found in a penny pamphlet in the market fair went on to learn from barrow the fundamentals of Euclidean geometry and the fundamentals of Apollonian Conics.

Of course Wallis's books on the Stoikeia were directed reading for Newton, who surpassed both his course tutors and the authors of books he was directed to study.

So it was that Mathematics came to hold both geometrical reasonings( thought patterns) and formal Aristotelian thought patterns, Euclid versus Aristotle, in a seemingly good mix.

At least that was the case in England and the British empire. In France the mechanical philosophy was compared against documents that survived the purge of the rabid clerics who during the plague and the crusades demonised all Arabic learning, documents that were preserved in the Roman catholic libraries, and inconsistencies were found.

Consequently , since Aristotle was right, Euclid had to be wrong, and the Stoikeia was rewritten several times to make it fit this view. The most famous of these reworkings was that of LeGendre a French mechanical Engineer. His text book on the Geometry of Euclid contains so many mistaken views that it is no wonder mathmatics took a wrong and embarrassing turn!

The Fifth postulate problem arose among the Islamic scholars who in studying the Stoikeia assumed that every proposition was derived or derivable from preceding ones.. This was a very Aristotelian syllogism, based on his taxonomic predilrctions( OCD of the highest kind, probably Ausbergers Syndrome), and no one, owing from the Tekne or mechanical philosophies recognised the Stoikeia as a course in Philosophy, Pythagorean philosophy as opposed to Aristotelian Academics and Platonic style learnings.

By the time LeGendre got his hands on it the Kantian notion of Axioms was just being formulated, and the Axioms of Euclid now took on a new meaning. No longer were they just axles or axes on which the wheel of learning turned, they became self evident truths. Self evident only if you were an artisan!

Because classicl scholars could not bring themselves to sully pure, holy reason with the material mundanities of mechanics, what is self evident to a mechanic is not so to a classical scholar. What is supposedly self evident to a scholar is logic, reason, spirit. By this means mythology , opinion, convention all brcome the starting point for classical scholars and Mechanics only in so far as some mythological hero employed such artisanship!

Needless to say mathematics was in a mess by this time, and Gauss in particular was looking for a way out of it. He thought Lobochevsky might have hit upon it, completely undermining Bolyai's work in favour of his own. He certainly was not going to let Bolyai take the credit of saving Mathematics!

Later a curious manuscript came into his hands, while he was busy surveying. It was Grassmanns document under the cover of a letter by Möbius. Perusing it quickly I am sure he found it ver confusing as did Möbius, but at the same time intriguing. Because of the hypnotic quality of the writing I am sure Hermanns book set off a train of thought in Gauss mind.

Gauss fairly critiqued the style and presentation of what is an was an incomplete imperfect masterpiece. It failed on the most basi Acadrmic standard of clarity of Aristotelian logic! It failed because indeed it was not Aristotelian, it was Hegelian!

What does this have to do with Herons formula and Pythagorean philosophy? Everything. Aristotle only lead human thought down one path, while Pythagorean scholars explored the n-dimensional space in which we live. In book 2 the segmented line is introduce as a fundamental of proportion. From it the Pythagoreans set out their understanding of all proportions in all dimensions. The link was the circle or rotation of the elements of the proportions. To enter into space from the segmented line you must have at least 3 points.

If those points remain interconnected by collinearity as the line segments rotate then you have or define what is a plane! Alternatively, if a circle passes through all three points while constrained to be firmly fixed at a displace et from a point in that plane, its centre , then the circle is entirely in that plane..

These are very hard constraints to achieve mechanically in thought, but in nature rigid axial rotations and rigid cutting tools are our best approximations to all of these. The Lathe delivers all these surfaces depending on rigidity. This is " axiomatic" that based on the axial or axis , to any skilled Artisan.

Hermann thus started his method where the earlier aPythagorean rtisans said one should, on the complex combination of the circle, 3 points and 3 straight line segments, on an axially lathed surface, called a plane.

http://youtu.be/8rjxOFAzBa4

http://www.youtube.com/watch?v=8rjxOFAzBa4

What is directly relevant here is that the combinations of the 3 limbs are kept firmly fixed. Thus the everyway ordering of the limbs , according to the proposition should not change the out put result. The result for the straight line should be the same for the result for any and every triangle including Pythagoras theorem, Thales theorems etc. and this should be the same for every plane in das Raum!

Oh yes Descartes started off algebraic geometry by fixing at least 2 line segment directions in what later became called generalised coordinate style and firmly associated to LaGrange!. It was Wallis who fixed the cross axes we now so easily call Cartesian.

And Gauss backing Lobachevsky and Riemann over Grassmann and Möbius eventually proved not as fruitful as he hoped.

Some mental gymnastics are required for the next paragraph, at least in translating it that seems evident.

If you thought the induction gad prepared you for Hermanns nimbleness in logic you are only partly correct. Each established label does indeed morph and change fluidly. What yo thought was established is only fleeting. As the context changes or rather the countryside through which one is everyway journeying, so the use of labels changes with it.

In particular signs for knitting are given a more nuanced utility. It is this nuance that I call mental gymnastics. It is not that the sign or label has changed completely, rather, like a noun it becomes " decline able" the same essential notion has contextualised referrents that conduct the minds eye geometrically/ spaciometrically to various viewpoints.
The near view and the far view are but 2 of a number of spaciometric mind orienting conductions. Up on top or underneath, at the side of or moving toward , are a few others. Each viewpoint offers up a different vantage point to refine a process more nearly.

What you thought you knew becomes suddenly questionable. What you move on to becomes somehow richer by nuance.is that not the same as saying you can change the meaning of a notion whimsically? The answer is I think yes with a big BUT. The changes in notation are slight , related and advantageous. Like the latest iteration of a product they seek to retain the old while straining onto the new. The constraints in doing this are a fundamental part of the whole. The constraints not only become clearer uring this process, they inform the utiliser of an essential structure within the whole project.. They become the fundamentally unmovable ground in a dynamically shifting experince.

One thing that must be obvious, but I will state it any way: Hermann develops his method analysis and synthesis on a dynamic ground, the ground is allowed to permute. The everyway tout ability for exchange discussions are not just rhetorical, they are dealing with the permutations of sequences of " limbs" or elements as an unavoidable and fundmental dynamic.

Thus inherent within the concepts of LaGrange and so Hermann and Hamilton is the fundamental pet mutability of every " thing" one may wish to study. The act of studying therefore constrains the " thing" itself. Thus it makes clear sense to identify thre constraints as part of the process of studying,

In which case why bother? If what you seek is truth, but that truth depends on constraints, then perhaps the notion of Truth you have has to change?

Consequently invariance under every conceivable condition of permutation or interchange could be considered as the " ultimate" truth?

And what if no thing remains invariant? what then?

In and of themselves the various patio models may have a great pragmatic utility, and maybe that is sufficient reason to pursue this " style" of studying.

In any case be prepared to nimbly go where none has gone before, to the frontier of your knowledge and expertise and ability, and then Beyond!

It will become apparent, but not commented upon, that the "=" sign has made an appearance in the discussion. The "=" sign is fundamental to the labelling scheme and appears in the context of Like entities.

To say the sign is not commented upon is unfair, but pertinent. Since the Induction Hermann has been setting up the prior intuitions for Like things culminating in his 4 statements or propositions of "like" declaration. Thus Hermann has been commenting on the "=" sign all along .

It is pertinent, because the trained mathematician is too easily drawn into a narrow cootation of that sign. So narrow in fact that other signs have to be employed to denote all that Hermann means to expound as Like entities.

Therefore, as soon becomes obvious the "=" sign is not the same as the arithmetical or even mathematical counterpart. Herman uses it to mean like entities. We then have to pay attention to the rhetoric to determine which of the 4 likes he is employing and why.

In particular many " algebraic" manipulations are not the same in Hermans method . Many long strings of like things may indeed appear to be algebraic manipulations, but in fact are not. They are chains of like things governed by some simple constraint.

What then are so called algebraic manipulations?

They are a mnamonic chain of permutations of a sequence of entities. Mnemonic because they are a memory tool of a much deeper analysis, which Hermann gives, and indeed Hamilton does the same in his theory of Couples. However Hamilton derives his via a model of displacements in Time. One is amused by how clever his model is but not lead to think this is anything more thn an intellectual game.

Hermann on the other hand derives his explanations from a philosophical exposing of the functioning of the human mind. Thus, as we shall see, the effect is quite different because now we must examine ourselves and how we think and form logical and arithmetical conclusions!

Now exposed is how we cover and uncover certain thought patterns to output a conclusion or result. Now is clearly set out how we ignore certain patterns, promote parts of patterns, rely on prior propositions and insights, bend or extend notations , overlook certain details while focussing on others in order to present an output result

And to what end?

For Grassmann the grand visions of LaGrange, Euler and LaPlace speak of a higher " truth" a clearer and more able Mechanics, a powerful and more flexible dynamics that could display kinematics and Ballistics and Astrological phenomena so much more accurately.

Truly he felt what he was doing, and encouraging the Prusdian people's to excel in was nothing more than the Capstone method of all mathematics, which would reveal to us sublime truths never before grasped about the very Nature of Nature itself!

Bearing that in mind you may grasp why he was so animated to look again at the foundations of  mathematics, because he saw that he could rescue it from its profound difficulties, into which it had fallen, and liberate the minds of men as the French revolution had " liberated" the French people , starting with the Prussian intelligentsia abd the rest of the world to follow.

This was highly renaissance and Romantic thinking, and it was extant throughout a wide European demographic. It empowered a whole movement to reform the institutions of Europe, and Hermmann's works and writings reflect that background .

I have mentioned before the wide ranging impact of Hermanns seminal work, and the redaction by his Brother Robert. In context hermanns work is a masterly summary of the status of advanced mathematical thinking in Europe during his time. But even more it provided an entrance into a higher state of learning for a poorly equipped Prussian Scientific community. By it he hoped to give Prusdian science and Technology an edge to compete with the other industrialising nations.,

But , as the saying goes, A prophet is not recognised in his own countr! His work in fact inspired innovations in Italy, Britain and America sooner than it did in his own struggling country.

As far as Prusdian philosophy is concerned it is a short work, and it is incomplete, but it is very accessible to philosophers. Mathematicians on the other hand recoiled at its philosophical style, even as they recoil today at Newtons Astrological principles, and it is from these negative minds that the myth arose concerning the obscure nature of the work. It did not make it easier using the renowned Philodophy of Hegel, because the establishment in Prussia was very negative toward Hegel . But the work itself is not obscure, rather it is groundbreaking and troubling.

It forces the reader to think very rigorously and in a definite pattern!

From a line to an area, the " expressions" of like entities remain firmly fixed, when 3 points, 3 line segments 3 squares, 3 "limbs" are involved!

http://youtu.be/iMWEiPuFhBQ
http://www.youtube.com/watch?v=iMWEiPuFhBQ

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§5  The analytical everyway journeying retains therin: that one seeks the other entity  to:  the output result of the Knitting together and the one limbed entity of the same (knitting) .
Therehere there are attentive to a knitting together two analytical everyway journeying artforms, each is becoming sought according to the specifically from the knitting forelimb or hindlimb ;

And both everyway journeying artfoms  deliver then only one equal/ like output result, even if both limbs of the originating knitting are everyway toutable(for exchange) . There also  the analytical everyway journeying can become apprehended as knitting together, so we differentiate the originating  or Synthetical knitting together and the "on top " loosing or analytical knitting.

In what follows we now become immediately nearby to the synthetical knitting in the sense of the previous paragraphs as one simple knitting , in order  to set out ahead ( of everything else) and as sign for the same the sign"^ " is kept held in place,
 the far against view: for the inter communicant knitting together the around turned sign "v "  is chosen,  there here both artforms fall together from the same knitting.

and indeed thusly , that we construct here,
By considering the analytical knitting
  the output result of the synthetical knitting relative to the forelimb
-What is given( relative)to the forelimb  ?

Thusly concording to here  a v b this that thought pattern signifies, which  with b synthetically knitted gives ( the result) a, so that at all times a v b ^ b=a  therefore exists.

Herein lies "thus"- like locked in, that a v b v c denotes the indicated thought pattern  , which with c and then with c  synthetically knitted gives a as a result, that is branding therefore also according to §4 the indicated thought pattern  which with the same "Values" in the around turned succession, or with b ^c synthetically knitted gives a as a result
That brands

                     a v b v c is like (=) a v c  v b
                                                                  Is like(=) a v (b ^ c)

And there the same concluding succession for whimsically many  limbs is empowering, thusly follows, that also the ordering  arrangement of the limbs, which have analytical foresigns, like valid is, and one is permitted to close these limbs in a bracket,only if one the in bracket toward the back foresign turns around

Now Hereout follows further, that let it be

             a v (b v c) =(is like) a v b ^ c
In practice one has out of the definition of analytical knitting together

           a v b v c  ! is like(=) a v (b v c ) v c^ c;

This expression is once again everyway pleasing of the so plainly outwardly demonstrating rule
      = (is like)   a v ((b v c)^c)^c
        =(is like)   a v (b v c ^c) ^c

And this last one at last is everyway pleasing of the definition of analytical knitting together

         (is like)= a v b ^ c,

Therefore also the first expression is like to the last.

We express this result in words and we combine it with the the beforehere achieved results, thus we hold out the proposition:


This is the most general results, to which by considering the taken asside settings out ahead of everything else we can reach.

Afar against that, out of the same things does not go henceforward, that one can let go away a bracket, which an analytical sign enclosed, and a synthetical entity before itself has. Much more a new setting out ahead( of everything else) must first become constructed thereto .

Commentary §5

Hermann establishes the so called rule of signs in a much more general context! Hamilton did the same in his Theory of Pure Time, but used the limited notion of Contra in the sense of a "contra step".

Hermann establishes it using the notion of the analytical or loosing from above and the synthetical meaning building up ( from below?). The process is emphasised, which is where this distinction arises naturally and generally. However the process is too simple a description of the general setting. The everyway meandering of the analyst is what Hermann keeps to the fore. There is not one process but many interlinked processes and permutable journeys he has in mind.

I took the liberty to put in a couple of extra likes just to make clear what Hermann was doing. At times he thinks it is clearer than it is to the reader.[ However, now working on §6 I realise these will mislead the careful reader, so I have removed them and replaced them with. "!". The reader must here recall the definition of the analytical knitting to understand the clear likening he is making.]

So why do this when we can use Brahmagupta's or Bombellis rules? Because nobody understands those rules or why a misfortune of a misfortune is fortunate! Hamilton was able to show how this rule falls out naturally from the contra notion, but it is only an amusing reworking of the not understood rules. Here we consider the fundamental origin of this behaviour, how insisting on a constant result necessitates these rules, and keeps them reliable in all synthetical and analytical circumstances .

We have to analyse a thing by delving into it , and we construct that thing by placing all the anlysed parts in a synthesis. That synthesis can be in any permutation, but only ones giving the same result are chosen. These are the ones we actually understand through the analysis.

Wild synthesis is possible, but then none of these rules apply. If we apply these rules in a wild synthesis we can only achieve one result, because the rules are for that purpose.

It is clear that while matrices were not conceived clearly by Hermann, the determinant was. That is not to say Hermann invented it. Cauchy and Gauss it seems were hot on its trail through attempting to understand when systems of equations were solveable., but even they were following up on clues found by Chinese Mathematicians and Indian Astrologers,

The rank array that LaGrange and Euler and LaPlace made central to mechanics is Hermanns grand framework into which he hopes to transform all thought patterns.

Thus his presuppositional work is about dealing with n-limbed knittings or combinations, where the combinations are permutable. This is much more complex and richer than we can express, but fortunately it is fractal. What Hermann had uncovered is the Fractal power of 3 . As he frequently states if it holds for 3 then it holds for as many limbs as you like!

Thus he did not have to really work with n limbs, like a computer has to, he only had to work with 3, the rest conformed by the same rule.

http://youtu.be/67pMZbTQxP8
http://www.youtube.com/watch?v=67pMZbTQxP8

The second part

http://youtu.be/MfogGCoqnRE

http://www.youtube.com/watch?v=MfogGCoqnRE

Normans introduction to the determinant is very clear , and I particularly like the det() lable that inputs a matrux( rank array) and outputs a number!

Commentary

Hermann in this general doctrine section is deriving the BODMAS rules.

As a trained Mathematician I can safely say that no one has ever derived these rules in my education!

The tension for a trained mathematician in reading and following Hermann is down to the supercilious attitude that we know all this already ! In addition, we do not want to listen to " philosophical" rhetoric snd distinctions! Rather we seek an element with which we can count and then we count in all sorts of complicated patterns.

Thus arrogantly mathematicians will not allow a philosopher to instruct them ! In this way the reverse psychology is to call such a teacher " rtogny" or a young child attempting to teach a grandmother how to suck eggs!

Here however such an attitude is unjustified because no one has deigned it important and necessary nough to logically derive the Bracets, Ordering, Division, Multiplication Addityion Subtraction rules! In fact I do not think anyone even has a clue what The O stands for!

Here then you will benefit from carefully reviewing this section: the general doctrine of the thought patterns, which I have yet to completely translate. Much to do with sequences and series is encoded in these patterns nd propositions, and in their wide applicability. Even more so the processional nature of thought is exposited and the consequence on the counting process.

Certain procedural processes of synthesis are exposited and also of analysis , and particularly their multiple-element or multiple-facet structure including therefore their fractal nature, and their dynamic permutability. This is particularly related to their ordering in space and how they combine in space. This is the O part of the Bodmas system and explains why we failed to accept directed magnitudes and rotationl magnitudes when they appeared.

We mathematicians threw the baby out with the rhetoric when we promoted symbolic algebra over rhetoric, and numbers as arbic numerals over tally counts of moads or Arithmoi, as spaciometric structures. We lost the sense nd expectation of rotation in our tally counting, and in fact reduced it all to a single line nd thus one directionl.

Hemanns pproach thus deals with all these issues dialectically and carefully, building up all the attributes rigorously and seriously. I am sure Hamilon was deadly serious about his treatment of the structure of pure time, but because he started in a fixed line , the time progression line, he alo lost the rotation and directionl aspects of space that are encoded in the rhetoric and which eventually produced as a symbolic fom!

The Ordering of spatial elements , and the ways( directions and manners and sequence orders) in which they combine is what the O stands for and is just one of the fundsmental processes of thought we employ , along with Bracketing.

The other proceesses of Thought Synthesis and Analysis rely fundamentally on brackets and spatial ordering and permutations .

The thought processes of Division as quotients will be introduced later as will multiplication, but it is multiplication where I have an axe to grind. The word itself is fine but the process is not primary. Synthesis and anlyis are, but because synthesis is inherently Simple Aalysis has to tke the fundmental process position. Thus while it is natural to unisex that we bundle sums into larger units , in fact we do not. We divide large synthesis projects into larger units first!

We divide larger synthesis projects into larger units first

So the larger units come about as a process of division not summation. Yet it looks like summation or synthesis. Well yes it is but it is not synthesis toward the goal of synthesis, rather it is synthesis toward the goal of analysis.

Such groupings or bundles thus arise more naturally in an analytical process and in such a procees the proper identification for such a group is a Factor.

Thus we can characterise divisions as factorisation processes and therefore quotients as identifying factors within a factorisation process.

There re always 2 analytical processes involved with a simple synthesis process : one to find the forelimb and the other the hindlimb . Similarly in a factorisation process 2 processes are possible: one to find the commensurate factor for a given " factor" , which only resolves if the given factor is indeed commensurate!

The other is to find the other factor given that it is commensurate and we are given precisely how ny times.

This is equivalent to finding the forelimb and the hindlimb of a Process with 2 factors .

Without elaborating too much, the structural work on synthesis and analysis being done here cn be analogously plied to factorisation structures.

It is these factorisation structures tht have come to be called multiplications! Therefore it is erroneous to link them directly to synthesis as some do , when the situation is as complex as set out there beforehand.

Thus I would Promote BODFAS which would be Brackets Ordering Division, Factorisations,Analysis, Synthesis

Commentary

I have referred to Normans work, not only because he respects and innovates on Hermanns ideas, but because his videos relate to some aspect of the translation of Hermanns work.

Here however, the algebraic manipulation Norman refers to is purely relating to the BODFAS rules. But in the far view is the mysterious Book 2 of the Stoikeia.

It is glibly passed off as Euclidean Algebra, but what it is really is proportioning a line and then applying that proportion to quadratic or 2 dimensional figure.

Here Norman shows how simple staged or successive application of the fundmental line segments can relate shapes and areas together.
http://youtu.be/_nAgDNz6ETQ
http://www.youtube.com/watch?v=_nAgDNz6ETQ

Many lines were segmented and explored and found to relate areas like the golden ratio, or or some other pattern of areas. The Pythagoreans reveled in these affine summations, and many mythological stories contain references to divine proportions marked off on a line.

I recall a vision I had when looking at Eulers equation for polyhedrons

F +V – E–2 =0

At the time I was working in the thread Fractal Foundations of Mathrmatics and I was researching polynomials as a conceptual notion. I could see the direct connection to a spatial object in degree 3 polynomials based on Eulers equation

S + F – E + V – 2 = 0
Where S is the number of spatial objects.

I put S in because I clearly had a 1-1 polynomial equation as a cubic

0.X³ + k.X² –m.X¹ + n.X⁰ – 2 = 0

I understood that as a journey through spatial ordering starting with points, sequencing them into vertices, then connecting the vertices with points to form line segments, and at that moment both faces and the whole object was reified!

Or more mysteriously points are formed sequentially into line segments and then as each line segment extends into another a face is formed and the line segments take a turn into another plane, gradually building face by face until the whole solid is formed.

But then the proces did not stop! It continued to build a line of solod forms and then a plane of solid forms and then face by face a solid fom of solid forms!

But then it reiterated the same synthesis plan at that level over and over!

It clearly was a Fractal ! The solid form did not have to be solid throughout it could be a mesh ,mand each mesh was at a different "level" of complexity, but completely captured or encoded by a polynomial of degree n!
In which case the alternating foresigns had a meaning of direction. We had missed that simple synthetical requirement .

On the way through I realised as I had suspected that n dimensional space is where we live already, we just had to clear away the fanciful notions of needing more than three mutually orthogonal axes!

I also saw the interplay of how at each level the solid form becomes the 0 of the next or the point of the next , I could not decide because what role did the numerals play in physical space? Especially as X⁰ was already accounted for.

It has taken time but I gradually realised one thing was missing: me the observer, calculator and manipulator! The numerals represent my conscious involvement in the whole polynomial expression, in formulating the equation and in utilising the spatial orientations of the structure to unravel a commensurable Arithmos as a count tally.

I did not understand it back then but it is becoming clear now as I work through Grassmanns exposition.

Whatever we humans record and commit to myth , memory and legend, mystery teachings, curricula and public eduction systems,writings in particular and general , despite being all screwed up, messed up , chopped about etc, will still holographically reappear in someone's consciousness whole and complete!

The genes will reassemble to construct an exact replica! What cannot be guaranteed is that it will be utilised by that individual/ animate  to whom it first reappears, but eventually it will reappear to someone who can and will utilise it to human advantage.

Without elaborating the Hegelan dalectic corresponds to an evolutionary developmental schema and particularly that of a virus or junk DNA scenario. If life evanesces anywhere it will eventually attain to levels of consciousness at which we are and beyond. if the universe will allow.

This , as measured by the geometrical structuring concepts of crystals is as inevitable as the day is long. We would not accept it out of deference to God, but space is a fractal that emerges out of an infinite regress/ progress synthesis.

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§6 The new setting out ahead( of everything else) , which we are from afar to here fusing, is the setting out ahead, that the output result of the analytical knitting together  is clear (one assigned), or with other words, that,

even if the one limb of the synthetical knitting remains unvaried in everyway, the other limb therefore varying itself , then also the output result every time varies itself.

Hereout outputs itself  immediately nearby , that
                          
                              a ^ b v b =( is like) a
                                
exists, because  a ^ b v b = a denotes the thought pattern, which with  b synthetically knitted  a ^ b  gives!

Now   a is such a thought pattern ( one assigned) and is everyway pleasing of  the  unambiguously clear quality of the result, therefore the singleton thought pattern  is outwardly demonstrating the validity of the above likening as .

Hereout once again henceforward goes that

           a ^ (b v c) =( is like) a ^ b v c
exists.

In order to specifically bring the second expression around onto the first , one can replace b in it with ((b v c)^c) and hold out

a ^ b v c =( is like) a ^ ((b v c)^c) v c;

This is, according to §4

         is like(=)     a^(b v c)^c v c,
and this once again concords to the thusly plainly outwardly demonstrating Proposition
                       =(is like)a ^ (b v c)

Therefore is the first expression also like to the last : (QED)

There one can these conclusions completely soundly once again reiterate, even if multiple limbs in the bracket come to the fore, thusly one has the proposition:


What the ordering of the limbs troubles, thusly follows that

a ^ b v c =( is like) a v c^ b exists;

 because

a ^ b v c =( is like)b^ a v c = b^(a v c) =a v c^b

Thusly that :

also, therefore, we   have demonstrated as concording, the everyway toutable quality of 2 limbs, of which one limb has a synthetical foresign the other an analytical foresign,

Thusdirectly the completely clear (one assigned) quality of the analytical output result is set out before ( anything else)

And now under this setting out ahead( of everything else) the propositions of this  paragraph gain power, while the propositions of the previous also  then still gain power, even if the output result of the analytical Knitting many assigned is!

Footnotes
• By considering Game entity  of such a many assigned quality delivers nothing direct, how itself later comes to be in order to show, the extensive / extending magnitude Doctrine in abundant crowds, rather then also Arithmetic displays  itself, and it is therehere the firmly set differentiating also  important for it(the Doctrine). Specifically Addition and Multiplication as simple knittings together displaying  themselves  ; and while the subtraction always is clear ( one assigned), thusly is it only the Division, as long as the Null not as Divisor appears: the propositions of the previous § generally are of the half empowering for the division only  ,  while the propositions of this § only with the restriction empowers, that the Null does not appear as divisor. Out of the not- tracking of these circumstances must the most urgent  Objections  and everyway whirrings ( fireworks!) henceforward arise , how it also toward the division has happened!

•• a far later placed besides everyway search, upon which to base   the rules for the knitting together of multiply assigned Magnitudes  has guided me to the conviction That above all one must transform the multiply-assigned magnitudes Initially into one assigned magnitude,  importantly before one  can apply on them  a random set- knitting  . I have   of this conviction  in my  Ausdehnungslehre from 1862, in the added remarks to No,348 and to No,477 expression everyway distributed, and i have together at like moment  besides former position demonstrated, how one can transform the multiply assigned magnitudes into one assigned magnitude. Also my Arithmetic( Stettin 1860, printed and everyway laid out by R. Grassmann ) lays this conviction to ground.(1877)!

Commentary §6

The overriding result of this paragraph is to demonstrate that all the threads for the use of bracketing and ordering of any synthetical and analytical process have been woven to completion.

The implications are astonishingly far reaching!

Because of his approach Hermann sits right on the borderline between 0- dimensional systems and n- dimensional systems. He can easily pass into all of them very generally, as these sets of propositions and demonstrations show.

But he concludes by mentioning the one assigne and multiply assigned magnitudes. These are new concepts that arise out of his deep thought patterning of space. Later we will get a clearer representation of these terms, but for now just think of the process of solving a rank array matrix by Gauss's elimination method. That is we reduce a system of likes row by row until we end up with a singleton limb with a single unique valuation . From this, as we shall see we can then go back up the analytical chain and evaluate every row( or alternatively we could do it by columns).  Each row is therefore the multiply assigned magnitude, and thus the sense of multiple asignment is multiple variables/ line segments/ spaces etc, etc.

It is hard to believe that we have placed ourselves in such a powerful commanding position simply by adopting Hermanns dialectical approach!

I have one dissent and that is over Multiplication, but I have explained that already.

One of my major stumbling locks with understanding Hermanns reasoning was how he came to the conclusion AB = –BA for products.

The reason for my difficulty, and perhaps that of many mathematicians in his time: quit apart from being sloppy, and quite apart from recoiling at a bit of Philosophy was simply the ingrained weak case thought patterns!
The Arithmetical thought patterns and number bonds etc are the first we are forced to memorise without question! Consequently if ever anyone dared to ask if the thought patterns were "true" the response would range from seemingly sensible but reinforcing examples to downright vilification in front of the whole school!

One soon learned to please the teacher, but then later to be pleased by ones interaction with the material. But one never quite has the time and finally not the inclination to start over! This is where the Grassmanns were different. Each for his own reasons and predilections seemingly engendered by Justus Grassmann the father of Robert and Hermann, to start from the very beginning and build up.

Hermann was self activated by his father as was Robert, they both received a free school, Pestalozzi  like education. Robert showed early progress, Hermann seemed to be much slower. He surprised his family by winning the prestigious mechanics award for original if confusing work on the Ebb and Flow of Tides, and also because he was nearly the only entrant into the competition.. Despite this he was held back while Robert soared up to university, doctorate, and district regulator for the teaching of mathematics. Hermann had to plod on working to help pay his way and only in brief moments of liberty able to work on his pet project the Ausdehnungslehre.

Finally he printed off what he had achieved gambling on a lot of interest, but the ventUre failed to make any money whatsoever and represented a financial loss as well as a slap in the face.

We have this flawed masterpiece because Robert rescued it from oblivion. We have lost the purported second volume the Kinematics of Swinging which would have built on his paper the Ebb and flow of Tides, and instead hit a rehashed Ausdehnungslehre Robert style. Anyway, it was a modest success and Hermann eventually was happy to keep his name on it. To his joy he was now able to reprint his original 1844 one with annotations. He had moved on a long way since then but still regarded the 1844 work as the fundamental ground of his later works, articles, essays and arguments.

Without reading it one cannot appreciate how fundamentally different it is to any other philosophical treatment of Mathematical principles.

So here we see that starting with 3 elements and 2 processes, synthetic and analytical one derives results that apply to knitting together cats and dogs equally as well as any production line process and of course to addition and subtraction.

One had every reasonable expectation that these rules carry over into the product case. Dropping down from 3 elements to 2 in the context of 3 brought a surprising and troubling result! If his reasoning was correct , and it was Hegelian dialectic so it should be rigorously sound, then A general principle had been missed by all mathematicians, but especially Mechanics and Geometers: anti commutativity in any any 2 limbed combination in the context of a third limb!

Or rather we recognise it in the synthesis mode, but ignore it in the multiplication or product mode of combination,.

Hermann has yet to demonstrate this but it follows from these propositions because of their general nature and applicability to multi-limbed synthetic and analytic knittings, within a n-step/ stage system governed by rigorous rules and constraints.

Hermann in the footnotes states that we should kick up a fuss over the slack or even non tracking of these important differings and differentiation.

I can recommend Norman Wildbergers WildlinAlg series as the closest to Hermanns systems as written that I have seen as a regular university level course.  Of course some specialist have embraced Hetmanns ideas directly or indirectly through Hestenes geometric Algebra, but no one puts it like Hermann, not Robert, not Justus, not Norman not David. It is a worthwhile and educative read .

I now feel that the Ebb and Tide paper also needs to be read , but that Hermans treatment of Hamiltonian quaternions represents a maturation if those ideas even beyond swinging arms into rotating arms and arcs.

We will get thereby to the Fourier and Laplace Transforms by and by as descriptions of general kinematics in any pace including fluid Dynamics.and wave deformations in any medium including magnetic plasma currents.

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§7.  Through the analytical everyway journeying one reaches to the Indifferent and the Analytical thought pattern. The former one holds out through the analytical Knitting of two like thought patterns, therefore a v a represents the Indifferent thought Pattern, and indeed is the same independently from the Value of a .

In practice a v a =( is like ) b v b;  because b v b represents the thought pattern which with b synthetically knitted b gives as a result, such a  thought pattern is a v a, there b ^ (a v a)  = b^ a v a = b exists.

Now In the surrounding catchment in which at the like moment the output result of the analytical knitting one assigned is , therehere also  a v a becomes set like b v b. There thuswith the Indifferent thought pattern under the performed setting out ahead( of everything else) always only One  value represents, thusly the needy quality outputs itself thereout  , it  to fix( the concept) through a central sign.

We choose thereto for the blink of an eye the sign &,

and besign the thought pattern (& v a) with (v a) ,

and name (v a)  "the pure analytical" thought pattern; And indeed,  (if the synthetical Knitting the Addition was), "the negative" thought pattern!

That (a v &) and (a ^  &) are like a,
that further ^(v a) is like v a and v (v a) is like ^a directly outputs itself,
in which one only has  to substitute the plainly represented fully standing expression of this thought pattern,
  thuslike the correct quality of these likenings to overview looking all around it.

The analytical thought pattern directed toward the Addition we were naming in the special case  "the negative" thought pattern and the Indifferent, in relating onto the Addition and Subtraction, we are naming Null.

Footnotes
There is an everyway farmed out  undertaking, even if one , (to "the by considering  Game") , by considering the addition and subtraction in Arithmetic, according to which  the herehere related rules  for positive tally Markers one has the concording demonstrated, it afarhere still especially wants to establish ( as demonstrated)  for  negative tally Markers. In which one namelike the negative Tally Mark as such defines, which to a added Null gives as a result , thusly one means here with the Adding( in which the label of the same adding process is  placed upon immediately nearby only for Positive Tally Marks )
either the same Knitting together cognisances , for which the Fundamental rules empower, which invoke the general label of Addition ,

Or one other knitting together.
In the former case  the concording demonstration  is unnecessary,  the wider rules then for the negative Tally markers, are already therewith demonstrating; in the latter case it is impossiblle,  if the label of the Addition of such Tallymarks not of something still  should become concording! Thusly plainly by the "fractions" in the comparative statement leaning against  the entire Tally Markers.

Commentary on §7

The indifferent and the analytical thought pattern are two of the most general ideas regarding dynamical situations that I have ever come across.

Earlier I precluded the power of these general notions by jumping, once again, too soon to a conclusion! These ideas are not about zero or negative numbers or evaluation but about unchanging results independent of anything else and about the analytical thought pattern itself!

The analytical thought pattern is precisely the tool needed to break down a whole into its constituent parts for further examination. In arithmetic it is naturally associated with negative tally marks, but the footnote here says that for the sake of rigour that needs to be demonstrated, and suggests 2 lines of demonstration. In the first it is within the definition of summation, and so needs no proof( this is the course we have been following in these discussions) but in the second case it is a bit more tricky. We have to analyse the comparative statement comparing it against the whole of the Tallymarks concept to find those parts that give rise to the negative tally marks. It is clearly a tricky task otherwise Hermann would have demonstrated it in the 17 or so years between the 2 books. He gives no reference to where he has done this particular demonstration, but he is convinced it can be done.

In this regard Norman Wildbergers construction of the Integers is a very relevant demonstration concording to Grassmanns style..
http://youtu.be/YDBLXCFrihc
http://www.youtube.com/watch?v=YDBLXCFrihc

We shall see next where all this careful demonstrating can now take us, and it ain't to establishing Arithmetic folks! Numbers are of passing importance as you have just seen?

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§8.we have until here the label of the addition purely  formally held onto, in which (position) we appointed it through the value of certain set knittings.

This formal label remains also always of the uniquely general  knitting.  Yet this is not The Artform,  how we reach toward this label artform in the individual branches of the Mathematic  . Much more outputs itself in them out of the creation whole of the magnitudes themselves a centrally acting "knitting" cognisance, which  because therethrough , that those formal rules upon it are applicable ,  presents itself  as addition in the plainly given to us general Sense.

We track specifically two magnitudes( thought patterns), 

which through advance setting of the same created whole cognisance go henceforward , and which we  name " in like sense created whole" , thusly is clear, how one can array both so: besides one another ; 

that both a complete entity produce, in which  "of  both sides" content comes to be thought related to it, that brands: the parts which both hold within, come to be in a "together!"-Thought entity ,and this whole then with both those magnitudes  "in  like sense created whole"  comes to be thought, in a like manner.

Now it is easy to show that this knitting together is an Addition, that brands: that it is a simple one , and its analysis is a one-assigned one. Initially I can whimsically  adhere together and whimsically everyway tout ( for exchange), because the parts , which " together!"-thought become, thereby-considering   remain the same, and their succession not of other entity may (be), there they are all " like" ( as through like created whole was rooting and rising up)

Therefore it also exists  its analysis is one-assigned; because were this not the case, thusly must by considering the synthetical knitting, while the one limb entity and the output  result  the same remain, the other limb differing Values can take besides; then from these Values must be of which  one greater than the other, therefore then must to the latter yet still parts from afar to come; then therefore  also would arrive to the output result the same parts from afar, the output result also an other Output result  to become , contradicting the setting out ahead ( of everything else).  Therefore there also the inter-communicant  analytical  knitting one-assigned is, thusly is the synthetical knitting as Addition to be apprehended, the inter-communicant Analytical knitting as Subtraction, and it  to empower according to the  set down rules.related for these knittings all in §§3–7 .

It was outputting itself there in that place , that the rules of these Knittings, then also remain  unvarying in everyway to endure, even  if the limbs become negative . We everyway liken the negative magnitudes with positive ones so we can say  they be in"  running into 'against set' sense created", and thuslywell we can adhere together the magnitudes in like ( ' against set) sense created as the magnitudes in " running into' against set' sense" created  under the Name" like - artform  magnitudes" , and therefore,  the real label of the addition and subtraction for like-artform magnitudes is appointed mainly  upon this cognisance.

Commentary §8

In this paragraph Hermann demonstrates the application of his method. The previous set up has created, rooted and risen up a set of rules . These are formal rules tHey do not apply in this form to spatial magnitudes. Nor do they conform to the usual mathematical formalities of addition and subtraction, which are partly based on real magnitudes.

So now he sets out to how the formal ideas apply and then how that subjects the. Entity to the formal rules in & 3–7. However he does not apply these rules to so called numbers, but rather to distinct magnitudes as quantity magnitudes.

We start with their creation whole by some process. Well by the same process we can  adhere them together to make a combined form. This demonstrates simple synthetical knitting,

Then he demonstrates that the analytical knitting process requires that the result is one- valued. That idea is explained by example in that demonstration,

Now in demonstrating that magnitudes satisfy these 2 characteristics he relies on the real experience of magnitudes of bound quantity. He simply has not done  any formal set up to justify his assertion that there must be one magnitude bigger than another! The veracity of this assertion comes from real spatial experience of bound quantities whimsically collected.

 He then goes on to bind positive and negative magnitudes together. There is no other explanation of these negative magnitudes other than they must run into against the positive magnitudes , but be like the positive magnitudes in every other respect.  Thus yhis is a spatial identification of negative magnitudes:,they simply run in the opposing direction right into and through the positive ones.

The only way to thought bind them is to rebrand them as like artform  magnitudes, except of course we have to run in the opposing direction to the positive magnitudes.

Compare yhis construction of magnitudes to Normans formal algebraic one.. Is this less rigorous?

I do not think so because we have set out the rigorous general rules, here Hermann is simply identifying how we have to arrange real objects in real space to comply.

The rules remain enduring!

However, please note Norman has introduced an abstraction via the tally marks and the Arabic numerals, a set of labels he calls Natural numbers. He therefore has to rigorously define these labels and how they behave. Hermann has not done anything about number so far because he is focused on real spatial objects. He cannot define how real objects behave, he just has to have descriptions of what they actually do.mthus addition and subtraction are not mathematical, they are spatial interactions between spatial objects. That is where Hermann starts.

Most people will react to so called symbolic Algebra as if they were given a dry biscuit or a sour grapefruit! It is a deadly dry dire experience resulting in tasteless uninspiring forms! But it neede not be.

Firstly the forms are not the object of the algebra. Newton did not see the artistic merit of Algebra, preferring to call it analysis, tools by which he analysed the more beautiful and wonderful dynamics in the world God had created. The most beautiful expression of these analytical results was in the formal astrological and geometrical designs drawn in ancient treatises.

Wallis on the other hand felt he needed to capture the thought patterns of geniuses like Newton, Viete, Harriot, and other pioneers in symbolic algebra especially Bombelli.  But you can't see thought patterns, without training in the ways and transformations of algebraic symbols.

We can also draw hope from the role of symbols in the written arts. Not only does the typeface have a pleasing aspect , but also the referrents that the words invoke they are the wondrous  things.

So in translating hermanns writings , piece by piece one is always bordering on the valley of dry bones ! One instinctively hopes that these bones can live beyond the typeface or printers art, and read like a racy novel, or an absorbing adventure or some wondrous tale!

Ones aesthetic tastes are dashed if the baleful and unrelenting referrent is but an endless sequence of parading numbers, numerals even of the Hindu Arabic arts. Gloom descends, the will to proceed fails, rigor mortis sets in and one like mathematics feels as cold as Death!

So here we learn that analogy design, art, simple construction creativity and engaging interaction with real processes in the real world are on offer. We are no longer in the Maul of Mathematics but in the open plains and uplifting hills of the doctrine of thought patterns and cognisances of actual experirnces. The formal script can now be read as a description of dynamic life nd growth, planting, rooting and rising up, self assembling and constructing into a beautiful fractal whole.

I was pissed as hell because Hermann sidestepped the problem with the multiplication, but it was the way he leapt over it that brought me to see that I was stuck in a gestalt bind by the way I was taught to do arithmetic! No explanation, insistence on rote learning, constant urging to move along and not loiter over these "trivialities" !

So I really thought I had come upon a gold mine hen I read book seven of the Stoikeia regarding the Arithmoi. There I found the source of factorisation and multiplication as Division. But I had not read books 5 and 6 the Logos Analogos methods that underpin book 7 . I know Hemann has and what is in these books s artifice, bespoke design, construction, skilled combinings, all dynamically linked to a segmented line by parallel projection, rotational projection and perspective projection. This production of other forms by these projections are the real dynamic and organic source of Multiplication, the multiplication Hermann defines in §9

This "multiplication " or projection also encompasses ith in it " division" , again by projections, but perspective projection is the one which gives us fractions and rational numbers so called. This kind of multiplication is found naturally in the organic growth of a contiguous crystal, in the organic cellular growth and division and rearrangement found in mitosis nd meiosis.

This is the track Hemann is on, not the number/ numeral symbolic tally mark one. Even the highest accountant wants to see the figures realised in some real world object, real estate, means of production and groups of people and organisms working to produce!

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§9. We have until here only tracked one  synthetical  knitting artform, for itself and in its everyway holding out ( in space) attribute toward the inter communicant analytical knitting. It now arrives thereupon, to layout the relationship of two differing synthetical knitting artforms. At the End goal the one knitting artform must  be appointed  concording to its label  artform,  through the other knitting artform.

This label concording depends on the artform, how an expression, which both knitting cognisances holds within, without varying of the total output result, can become bespoke designed .

The  simplest artform how "both  in an expression"  knittings could come to the fore is the artform that the result of one knitting becomes "cast onto" the second knitting(, therefore if "^" and "₩" the signs of both knittings are, thusly the everyway holding out in space attribute of both depends on the bespoke designed  knittings, which are permitted with the expression a ^  b ₩ c  to become taken forward. 

Even if the second knitting should  like- measuredly relate  itself onto both limbs  of the first knitting , thusly the second knitting presents  itself as the simplest bespoke designing, that one could   "cast onto" each limb of the first knitting the  second knitting , and then these individual outputs as limbs of the first knittings cognisance one could set. 

If this bespoke designing can  come to be taken forward without varying of the total output  result , that therefore brands:  a ^  b ₩ c = (a ₩ c) ^ (b ₩ c) exists , thusly   the second knitting together, the   inter communicant  "to that  first" knittiting together,   we name "  next higher step/ stage/ rank"

In particular by considering this second knitting: both limbs are depending in like cognisance ,   on the first knitting,  
thusly, therefore, 
that concording  empowers thuslywell for the hind limb of the new everyway connecting,  how for the everyway connecting of the new forelimb, 

and  further the first knitting is a simple one, and its inter communicant analytical  knitting a one- assigned one, 

thusly name we the last one " Multiplication", 

while we for the first knitting already above  the name Addition had firmly set. 

The artform, it is this  mainly : how from forward going entities herein, that brands: even if a knitting together artform is not yet given, such a knitting artform  can come to be appointed together with the " therebesides  to affiliate itself"  higher step/ stage/ rank .

Therehere we track also the addition as the knitting of the first step/ stage/ rank,  therefore the multiplication as the knitting of the second step/ stage/ rank.

From now we choose   the usual signs appointed for this knitting together artform  at the places of   the general knitting signs, and indeed we choose for the multiplication the direct besides one another writing style.

Footnoteg
As a third step/stage/rank one would present the same principle of the Exponentiating,  but what we are over going  here is the short way.The remainder of the label-concording  for these knittings here are only a formal concording , and can come to be everyway bodied, first in the individual expertises through Real definitions lying in the Nature of the Thing.

Commentary on§9

Unterwerfen is a direct German transliteration go the Latin subjugate. Thus the design idea is that the first knitting should be su jug ate to the second knitting .  But for this to happen simply the first knitting must be within the system of the second. Hermann had already covered this requirement in the induction, so this was a simple notational reference to that system.

However, in kerping with the knitting metaphor I used a knitting term, casting on, by which knitters join together different sections of knitting making each section subject to the whole design as a part..

The German umgestalten means something like to design around a something or someone . Thus the unspecified design is closely dependent on the subject of the design. In this case the design pacification seems a bit arch, that is it has to end up looking like the distribution law diesn't it?  However the principle of the art is not to choose a specific format, but to choose The format that can be taken ever forward!

Thus perhaps for the first time , I don't know this law was identified as one of the most general and necessary laws which applies to all spaciometric knittings!. But that it is crafted and designed is not disputed here but blatant.  The Grassmanns as a family believed Mathrmatics was a constructed art.

In the previous paragraph Hemann began o fit the generalities to a more specific interpretation. Here he becomes even more specific about how these knittings are going to be distributed in space..

The everyway distributed in space brhaviour of the knittings is particularly highlighted by the analytical knitting, because it is one valued, thus a subtraction in one direction may have no effect in every other direction while decreasing the whole.. Consequently addition increases the whole , but now we know that increase may be one directional only. Analysis has revealed this possibility..

In the footnote Hermann mentions that the principle of exponents is the formall description of the design model. Consequently polynomials represent a many staged system of knittings. Laying these out in space, the everyway holding out in space of the systems gives rise to Hemanns concept of an n-stage or n- dimensional system. At every stage higher than the first the knitting process is a product, but that product by design has the structure of a sum . Thus multiplication is a sum of more primitive multiplications ad infinitum .

Why is the first knitting different? It is different by design. It is designed to be the level at which we start.  In space we will find we are free to choose that level! Today we set that level at the Planck length. However supposedly the Planck length has no preferred direction! As you can see this is crazy talk. The designer sets the system at all times, not the space!

Because of this we can never know space formally. The real experience of space is always beyond our formal designs, but we can still shape our systems around space.

One other point, the design of the product process has 3 simplest formats, parallel projection, perspective projection and circular projection. Of the circular projection we would perhaps interpret the hybrid polar coordinate system as an example ,mbut the simple true circular projection is more like a chain link, and the more circular dimensions it has the further out it can systematically reach. However it cannot reach a point by a direct route . It is a system of arc segments of a circle.

Reading ahead, there is only one more general point of significance to make, which comes about from considering division as a fraction or ratio product , that is as a multiplication product as designed, division designed in this way is not one valued, it is multi-valued, and so some supplementary rules have to be applied to the real situation, to choose the appropriate result. 

Now because of this consistent design brief of subjugating addition  and subtraction processes to multiplication ones commutativity or  exchangeability of factors is not a property of multiplication!  Hermann discusses this result in a general way and points to the uniting ability in formal arithmetic as its source. Thus for Hermann the real application of the designed system must dominate over the formal system and attribute to the foundations only what can be demonstrated at every stage in a real system according to the nature of the magnitudes.

However, at this stage Hermann considered the commutativity of the first knitting induces that the designed second knitting would be empowered to operate in both directions. What this means precisely he does not specify until later. But at this stage what is good enough for the forelimb of the new knitting process is good enough for the hindlimb.

This induction is based solely on the detailed behaviour of the first knitting, and all the possible
 Exchanges in that situation. There one can interchange the limbs in a pair binding as long as the output result is not changed.

Similarly the limbs of the new binding can be exchanged round if the output result does not change.

Later we will see this written out explicitly, but the constraint on the output is not demonstrated. Consequently Hermann realised that he could not design a process that remained unchanged at the second stage/ rank/ step. The nature of the first step  being limb like and simple constrains the output, and the everyway holding out into space attribute. Going up a stage gives extra directions , in fact manifoldly many , by which the output resultant limbs may be knitted together , and the resultant output is not invariant!

What can be carried forward then is the design of multiplication by the forelimb, and multiplication by the hindlimb . Commutativity, therefore must come from the special or specific nature of the real magnitudes.

Formal numbers are not so constrained and based upon their unreal nature , there inherent imaginary nature, commutativity is defined by the factorisation table for any given number.

The factorisation table arises by a division algorithm, thus it does not come from the multiplication process just designed by Hermann . This Euclidesn division or divisor algorithm which identifies all the whole number factors of a number , and relates indeed to the factoring of a larger object in space by a smaller object in space, became the basis of the taught methods of number bonds and factor bonds confusingly renamed multiplication.

At the same time the amalgamating of the division product with the multiplication knitting process is also confusing. Nevertheless it is consistent with the knitting process that multiplication should increase a product while decreasing it should rightly be called division .

The 2 systems are clearly of use but also should be clearly differentiated.. And commutativity, like there are only 3 dimensions of space should be let go to its special and rare place in the system of things, because it is rare that any production process is commutative.

Again the interplay of 3 limbs even if the foresign of one is completely different to all the other foresigns  reveals much about our design processes under Analysis.

Designing the second and third stage knittings represents an interesting challenge for the foundations of The doctrine of thought patterns.

We see here that Hermann elected to use simple subjugation of the first knitting to the second stage knitting. However the format he used concentrated on only 2 limbs , which is the formal knitting thought pattern admittedly, but Hermann first applied it in the context of 3 limbs in which any 2 are combined to output a resultant limb in that context .

Because the forelimb is overtly or explicitly the first knitting format, he has 3 limbs, but of differing step/ stage/ rank. Ideally one would like to see the symmetry a ₩ b ₩ c, identifying with the first development of knitting, played out according to those already set out rules.

This clearly is the associative format and it would be of interest to see if under the given constraints associativity carries forward.

In Addityion, the design Herman chose is in fact a binomial design, because the forelimb is a combination of 2 whimsically chosen limbs . Norman calls these bilinear forms but they could also be called bilimbal ones. In any case it should become apparent that the Binomial expansion will play a fundamental role in the development of the knitting processes of each stage as they trickle down to the first stage.

However again the forelimb could be chosen as a trinomial, in keeping with he initial design, and the rules for knitting together adduced from this situation.

As you can see the increase in complexity of the thought patterning soon becomes overwhelming . However I am sure today's modern symbolic systems could del ith the math, making design a lot simpler.

The impact of all this on this thread is that Hermann is designing a product rule that is mixed in order to capture the tspatial Verhältniss of the quaternions, that is the everyway holding out in space attribute of these product arrangements called quaternions. It may be that his bilimbal Analysis of products is leading to or missing a more efficient product based on these design musings.

In any case the one important concept that came to Hamilton in a flash on broom bridge was that a fourth axis was required to do the calculation evaluations with! This fourth axis was ignored simply because mathematicians were not considering the ghost in the machine, their own conscious contribution to calculation. They were assuming the bus drove itself, all the while pushing it along by their own calculation effort. From the outset they attempted to keep the human involvement out of it, foolishly because it is clear but ignored that every calculation proceeds by some agency. In there time it was the human conscious, in our time we can call upon the combination of the electronic programmed conscious, and the programmed operating system, each an electronic model of human lcalculating behaviours.

The fourth axis in quaternions represents the measuring tool or carpenters measure, by which the carpenter accounts for everything else in the proportioned way of the Pythagorean school of thought. Without it we as calculators are at a loss as to how to proceed. Ith it we are trained to run through proportioning with some alacrity, but no where as fast as a modern ompuing system.

The point is no product can ever be functional if the calculator has no clue how o proceed with it. To proceed ith a calculation one has to lay it out in space either symbolically or as actual arrangment transformations of magnitudes.

For example, a fixed plane in space hardly exists, but without the notion we would be unable to lay out calculations to determine height , distances reas etc., all of which are purely formal but also relate able to spatial objects and aspects .

When designing a product one needs to keep in mind the tessalating Fractl it produces in space .

Hermanns main design product was the retinal parallelogram , the parallelepiped , the crystall facet structure of higher dimensions / stage / rank products.

Theolar coordinate product produces Shunyasutras, that is curved annulus portions , spherical shell portions

We have seen cylindrical, conical and toroidal as well as tetrahedral product designs as fractals on the forum and in the gallery. All of these product designs are differing in complexity of expression, especially when to display the result they have to be evaluated by the standard Cartesian product , but essentially we have a general solution in the form of the exponential product.

However, the simplest design pattern Hermann chose was one in which the general product of a stage was the sum of other products from the same stage. Thus a polynomial  product

AC is the same as AB + AD , where C is B + D and is a knitting in the stage that C is at  now A can not  be at the same stage as C but it must be a differing system which subjugated C. We may be used to regarding axes as independent , that means not influenced by anything else, but in fact the whole concept of constructing systems within systems, especially as it applies to products links them together in a causal dependency .

In fact we have to realise that this is a cognisance, a way of apprehending 2 independent rules as cooperating in expressing certain forms. The cooperation or dependent - like relationship we call a function or a polynomial , a rule expressing the causl link we perceive .

Once again the cusal link is emergent in our conscious perception, not in the product. Design we are placing at the foundation of our systems. Thus it is important that the product rules we design are the simplest possible ones for the fractal division of space.


Unfortunately that design constraint only gives us a simplistic view of the attributes of space. We can design more complex systems but we need computers and motion capture and fast Fourier transforms and laPlace transforms to even approach expressing these products.

Apart from the aesthetic beauty potential in these more complex products , why go there? Astrologers and presumably Mathematicians have no real interest in such complexity, but Natural philosophers do , in order to express the behaviour of chemical and organic growth nd interaction patterns , and to model statistical and stochastic processes, especially where no causal function expression seems to exist..

Again why do we need to go there when we can video the behaviour at high speed? We seem to need to express proportional relationships in order to feel that we understand or rather everyway stand a behaviour, but real expertise is ofte based on experiences where the proportions are felt not symbolised. Nevertheless some do encode this expertise symbolically and for those these proportions make and give sense.

We need both types of expertise, clearly .

This is a challenging video so needs meditating on several times.
http://youtu.be/Tx0Mop3_LEg
http://www.youtube.com/watch?v=Tx0Mop3_LEg


The " Algebra " here is in fact based on traditional algebraic manipulations that assume commutativity. However the last 3 sections approach the topic from a Grassmann lineal algebraic/ Doctrine of thought pattern style.

All the way though however Norman reduces the products to products of limb in synthetical knittings , and indeed also in analytical knitting format.

We understand that if hermanns outputs apply to 3 then they apply to n- limbs. In this case n=4 for the first step or affine 1 point geometrical interpretation. However, much of hat Norman  demonstrates he does so without the technical thought pattern background of he doctrine explicitly referenced.

Again the commutativity of the amoral part of this presentation, obscured by the algebraic bravado, is concealed in the notions of convex, cyclic permutation , area, and displacement..

In the last 3 sections he starts to analyse the situation Grassmann style.

So let us look at the product design. It turns out that quadrance is a level or rank2 knitting , but the uadruple quadrance formula is correspondingly a stage 8 product.. Thus the everyway toutability of the limbs or line segments , and the everyway holding out of them in space attribute requires an 8 dimensional system within systems to express!

Thus Brahmaguptas formula  is a constraining of an 8 dimnsional arrangement of line segments to just 2 dimensions. The natural convex quad if cyclic is a vertical projection of an 8 dimensionl pattern in a kind of helix , Ono the plane.( Norman covers something like this in his topology series).

We are reputedly going to enter the 4 th dimension when we tackle the Quatenions , and indeed Herman produces the figure 4 in his design path or brief, but we should know that 4 axes is not equivalent to 4 dimensions . We need at least 8 symbols to express orientations and calculations and that gives a Cayley Table of 64 products for a level 2 system.  If we used Norman's quadrance we would need a level 4 system with 64^2 level 4 products.

Hermanns analytical knitting and treatment reduces these to a more manageable group size , because each product has to be rigorously applied to real space.

Norman tackling the issue of Number.

He likes to direct the reader to the real number concept, but the problem is the common concept of number itself, or in Norman speak type Nat!

Anyway the focuss here for me is the superstructure of Grassmann style ideas he uses to make headway toward the doctrine of thought patterns, or some such equivalent name.

http://youtu.be/t5gbivTuk6Q

http://youtu.be/fbPVrZG4QHc

The logical difficulty Normnan has is linguistic . The square is a constructed form. We do not need to define a segment length to construct it. However the word square in mathematics is defined as the product of 2 numbers! Quadrance is a number obtained in precisely this way, it relies on number not magnitude.

As you know Euclid and Grassmann use the concept of Magnitude, specific thought patterns impressed on the cognisance by experience( read earlier parts of the Induction). The Arithmoi exist as these magnitude experiences and the impression of "like ".

However, if you downgrade logic, as Hermann had to do with regard to the issue of multiplication, and upgrade dialectical heuristics and in particular careful design strategies, you can construct a square easily using implements. Then Norman's formulation says something about these squares.

However the square of a square Norman does not define, or rather design because he has held onto the number concept. It is indeed a difficult transition from number to geometrical magnitudes when we need squares of squares! The only solution that I know of is this design process promoted by Hermann Grassmann. And what that implies is that we have to move out of the restricted 2 and 3 dimensional thinking to higher dimensional/ step/stage/rank systems within systems.

And no we do not have to look into Star Trek warp drive space to experience this, we just need to appreciate the facet topology / Spaciometry/ geometry/ astrology of crystals.

We see tha Norman is only able to demonstrate these likes numerically if he makes use of a modern computer with very large bit sizes for integer arithmetic.: the precision of a " word" made up of a number of bytes. It is possible he could still choose a set of points that overflow the buffers! Bthe issue of approximate over Precise ,perisos over artios, is constructed within the very fabric of the measurement system. This is hidden away by the terms even and odd numbers!

The relationship/ relating of multiplication to the Addition is a phrase that jumps out at you . When first I came upon it in the Vorrede I was not sure what it meant. Later I skimmer through and found page 11, saw the symbols and jumped to the wrong conclusion that it referred to the so called distributive law.

But now I realise that it specifically refers to the designing of products that subjugate addition, and specifically where the addition is of limbs in the lower step/ stage/ rank to the step of the system created by combining2 differing systems  of the same rank in such a way as to make one one system subjugate to the other.

In that case it was a design point that the subjugating system should knit wth the limbsvofvthevsubjugate system in the simplest way, and thus make the product of the addition equal to the addition of the products of the limbs.

Just as the design raises the combined system to the next stage, so the combined system of both limbs are equally raised to  the sane step . Thus it demonstrates that addition or the first knitting is carried through every step rise in analogous fashion. The combining of the 2 higher stage " elements" may be signed the same, but the everyway toutability for exchange for these elements increases dramatically.

Whereas for a line segment bound in a line of orientation the options are swapping within the line in the plane the options are increase by the freedom to rotate. Thus not only can swapping within the lines of orientations be possible, but permutations of this, and then in Addition rotation of the forms created by the products provide additional ways of knitting together.

Constraints on these degrees of freedom are allowable so that the observer/designer can focus on a particular line of development.

So here Hermannthrough subjugate system design constraints identifies this distributive behaviour of the multiplication over/ onto the 2 limbs of the addition as desirable , and it will turn out to be feasible in many real situations. However it is not feasible without some modification in rotational and projective systems.as we shll see , and it is also not as generally thought essentially commutative.

Asyet the role of associativity so called has not been mentioned. Many of these design issues are placed before malleable minds as de facto truths rather than what they are design briefs drawn by analogy from observing production processes in both man made and Natural production cycles.

One consequence of hermanns design is that , without defining prime here, every sum of n limbs can be written as a product of prime limbs each prime limb being in a different step / stage or rank to all others.

That being the case the structure or spatial geometry that corresponds to that design is not unique , even though the limbs required to create the structures are . This again is due to the everyway toutability for exchange , the permutability of the knitting designs.

I must say that I did not feel so inspired to think further after reading Hamiltons exposition of the science of Pure Time as I do after translating Hermanns Görderung. This is what he experienced when he read Möbius work on Barycentic coordinates, the lack of product design stultified thinking beyond the exposition!

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§10. The relating of Multiplication to Addittion we have thereafar appointed, that
    (a + b)c = ac + bc
          c(a + b) = ca + cb exists,
And therethrough the label of the multiplication was firmly placed by us. Through once again wholly applying of this fundamental  rule one arrives at thuslylike to the general proposition, that one if both" Factors" divided pieces are, each piece of one with each piece of the other multiplies, and the products can be added. Hereout outputs itself for the relating of multlication to subtraction an inter-communicant rule, specifically immediately nearby, that

           (a – b)c = ac – bc  exists.

Specifically one places in order the second expression to guide back to the first, in the same place of a the to it like  (a – b) + b, so one has

(a – b)c = ((a – b) + b)c – bc,

The second expression is concording to the thusly plainly set down  rule

   = (a – b)c + bc – bc,

And this expression concording to §6

= (a – b)c,

Therefore the former expression is like the latter.

On like manner follows, even if the second "Factor" is a Difference , the inter- communicant  rule. Through once again wholly applying of the laws one arrives at the more general proposition:

Commentary on §10.

When the factors are in the form of combined limbs, the synthetic or analytic Foresign carries through to an important knitting role in the final output result. It should be noted that the synthetic and analytic signs have not changed their role in becoming foresigns. We do not have negative and positive foresigns distinguished from analytical and synthetic signs yet, although Hermann has alluded to this use of the pure analytic form. Evenso the Foresign derives its meaning from synthesis and analysis. Later foresigns will take on another significance due to their use in cyclic rotation and there too they will derive their meaning from synthesis and Analysis.

The setting of negative limbs as ones running up against and into another intimates this oriented use of the analytical sign . In this case and all systems thereafter the synthesis and Analysis within a system due to orientation and direction, adds this further attribute to the analytical sign.

Now here Hemann drags in a whole load of familiar but undefined terms from arithmetic. They are not defined within his system and that represent a drop in rigour. The issue is not the appropriateness of these terms, but the slippage into an incoherent thought pattern due to being slovenly.

As we have seen some of the differences are subtle, but the implications are profound. We know that Hermann was sorely pressed for time, and we shall see evidence of this again and again in certain ellipsis statements or finessing statements he makes. These may cover a whole can of worms if we too are not careful. The light that guides him may indeed bring him safely home, but we are prone to fall off the track if we are not similarly illuminated.

One of the rules of algebra is to gather like terms together, but here Hermann says gather like signs together irrespective of whether the terms are alike or unlike . Thus he maintains the pattern of a synthetical knitting and an analytical knitting regardless of how many limbs.

The limbs when multiplied give products " held out in space ". If you think of a Cayley table you will see the kind of array, arrangement on the page or even in real space he is envisioning.. How these products are combined has to be controlled by the originating addition and subtraction form of the factors of the product, so carrying the foresigns of the products of the limbs will help to carefully knit the correct products to each other.

A Cayley table where all the products are correctly knitted together is usually called an expansion of brackets multiplication. It turns out that by adding different products from this Cayley table some have defined the dot product and the cross product. The matrix product is a more complex product which we will see Hermann intimates in his general rank array description. But truly it was Cayley et al, that crystallised the notation and the notion of this more complex product design .

The determinant is perhaps the clearest aspect of the modern matrix Algebra that Hermann could envision, despite the laborious use of the Sigma notation.

Can now tackle ahead of time the issues Hermann has yet to tackle with his product design notion. Some of these issues he does tackle later on, but some he either does not know how to resolve, or he does not see an issue or he deliberately leaves vague to be determined by the real system in application. However for the critical reader not familiar with the new ground Hemann is breaking in clarifying the product design responsibility of the observer or manufacturer of these systems within systems , it is important to emphasise that these are not issues of incorrect or wrong thinking, but of design thought patterns toward some pragmatic outcome.

When I first came upon these ideas of interpreting arithmetic in algebra, I was actually taught by geometric forms the meaning of arithmetic. Consequently I naturally thought arithmetic arise out of Geometry . Then I found after some research that perhaps it arose out of Algebra , with Algebra arising out of Geometry. These were natural deductions from inside the mathematical glasshouse.

It was the issue with i the that forever shattered the window panes and let in a hurricane of confusion, that cleared away centuries of careful mind control to reveal that man, deluded or illuminated by whatever source had indeed constructed this system now miscalled mathematics out of thought patterns derived from experiences in a real world. It took nearly a lifetime to find an influential author of the same opinion, that being the Grassmanns and Hermann in particular.

Thus it is in my view symptomatic of the pettiness that pervades the subject boundary wars that this design process should come to be viewed in terms of right or wrong. Rather as all design processes are , it should be viewed first on utilitarian grounds, pragmatic results, functionality and aesthetics. In the course of doing this the systematic or modular nature of the design can be examined and checked for fitness for purpose . We might call this a design flaw, if it turns out to be disastrous, or quirkiness if it introduces perturbations but nothing major. These issues can be remedied at each iteration of the design process.

So in that light I had a design issue about the quirkiness of the product e ₩ e .

Firstly I had no clear idea that it was a design issue, I just thought it was wrong! Later I viewed it as an opportunity to define an output but had no clue what that " should" be , and now finally I understand hermanns perspective and the design brief he gives. His guidance is essentially keep it as simple as possible, keep it as general as possible so it can carry forward into all levels , keep it based on 3 limbs and for multiplication use the subjugation idea. Finally keep it as fractal as possible so the same Pasternak or an " almost similar" pattern runs through at every step/ stage/ rank/ level/ dimension.

Already by that slashed list one can see how many ideas and notions it has to encompass!.

Let's look at signature, so called, as a set out design solution .

Thus e ₩ e is set equal to 1, 0, –1 according to the system within systems being modelled. Notice these are numbers . In mathematical terms this may not seem strange, but in design terms it is. Norman deals with this issue from the get go be defining numbers (type Nat) and then constructing a linear Algebra from these . Thus he incorporates many issues regarding number into the basis of his system, and the reader has to choose whether he wants to go along with him or not.  However he introduces the Cartesian plane as the de facto standard, which fir many purposes it is, and this allows hom to slip the type Nat onto Wallis's orthogonal axes without causing too many raised eyebrows.

The assigning of numbers to segmented lines is counting, or tallying. In that respect the numbers have no other role than to be a Namespace, to use computer terminology. To give them an a priori existence and type is to confuse the real interaction between space and human thought patterns in my view, and I think in Hermanns view too. Nevertheless it is a design decision which is carried forward to great effect by entertaining it to Hermanns Analysis

Using this set up Norman and many others are able to define the dot product of e ₩ e and get the signature value 1. Normnan however uses the Quadrance as his definition, also giving 1 as well as many other advantages.

This is fine until you move away from orthogonal axes. Then Pythagoras theorem does not apply and the dot product has to give way to the determinant .  These are fundamental design decisions and you can only appreciate them in specificity. Norman sets out these2 issues very clearly, but of course he is still puzzling about what it all means.

By enlarge the signature one relates to the dot product which is simply the product based on Pythagoras theorem . It applies flawlessly for an orthogonal system and is used to define an orthogonal system.

In the general case, that is using line segments referred to by this orthogonal system , the product still holds good as a bilinear form and still serves a purpose of projection of one general line segment onto another.

It has taken me a while to understand this confusing use of the seemingly clear ideas expressed by Hermann , especially the dot product for which I am grateful to prof Norman Wildberger. Without his insight I do not think I would have straightened that out by myself. In this instance Hemann has no clear guidance because he did not cll it the dot product! But once you recognise it in the shadow product or the projection product you can see where Norman gains his insight from.

e ₩ e set to 0 is hermanns first design decision. This does not use a part of the Cayley table to design a product as in the dot product, it uses the full bracket expansion. In this case it would appear to be visibly true that a line segment subjugate to itself produces only an extended line segment in keeping with itself. But since by design this product should be at level 2 in the system within systems, where by design the 2 knitting processes are differing, one subjugating the other, the expectation is a flat figure, not a line.

Therefore it would appear that stage 2 elements should be flat figures , generally speaking, and not extended line segments. However, again by design, Hetmann accepts extended line segments as level 2 products.

The rejoinder is : how then is that different from lower level Addition? And the answer is : it is not , neither is it meant to be! In designing the second level product it was specifically modelled on the lower level knitting. The only reason it was called multiplication was because the name addition was already taken, but indeed it is a knitting just as addition is a knitting,mtherefore it is not odd that in the specific case e ₩ e it should become identical to the knitting in thst system.

Without the development Hermann gives it is impossible to satisfactorily resolve this niggle. But resolving it this way is very powerful, both thought pattern wise and fractally, and it seems to mimic spatial phenomena when dynamic systems Collapse into a lower dimensional state.

The difference between multiplication and addition is the orientation of the subjugation!  I can lay out a pile of bricks as a cuboid or all in a single line. The arrangement in space does not rffect the total " volume" but it certainly effects its dynamic contingencies and behaviour. A hunk of metal sinks. That same metal beaten out into a bowl may well float and carry more than its own mass!

So why choose 0? I think that we have to understand0 not as nothing but as not the emergent form, but the form just prior to emerging! Thus 0 is Shunya, full of potential, pregnant, waiting to give birth to. At level 2 a line is such an entity, it is waiting to give birth to a level 2 element. Similarly at level 3 a plane is Shunya or0 waiting to give birth to a level 3 element, a space called a parallelepiped etc. thus 0 does not mean nothing, rather it means their is a potential that needs following down a level to fully appreciate what might be occurring in a dynamic system.

Now in light of this Hermann deals with a product design where the third limb orients within the level 2 system created by the first 2 synthetic limbs. This is also a source of confusion, which Hermann and Normann following him avoid by prcifying the design. This analysis depends on the first 2 limbs being synthetically linked as the same level elements, in this case level 2. The result, due to subjugation is a parallelogram. This analysis is firmly based on the parallelogram form.

Introducing a 3rd coplanar limb now allows the analysi to proceed according to the previously laid out rules. The news issue arises precisely when we produce e ₩ e in all it's forms. The discussion above allows us to replace this by 0 , but then in certain cases that leads go an inconsistency. We get
0 = ab + ba

And for consistency sake we have to let commutativity go!

The problem is in understanding that result. By other design constraints we can show that a system that subjugates another produces the same result if the subjugation is the other way round. But once we have set a system we cannot justifiably change it in the middle of a calculation! Commutativity implies just that! The only thing we can do is cyclically move the labels around.

It takes a good while to grasp that . Moving the labels is not moving the system , but why would a calculation cyclically move labels? The answer is the calculation does not, the observer does, and we do it all the time, often confusing ourselves. Moving the labels is equivalent to us changing our viewpoint. The best example of this is the clockwise counterclockwise labelling. This is still a confusing labelling system!

So how does changing your point of view make a parallrogram disappear? By design a certain subjugation creates the parallelogram .clearly the reverse should remove it. But the reverse has to occur after the creation . By rotating the factor labels cyclically round one of them now is directed in the negative direction ,say b now labels -a. In that case the product subjugation remains the same and so -a is done first reversing the extension back into the second line segment which then performs b on 0( remembering that b is now labelled a) , the consequence is not a parallelogram but a line extension.

The more astute might note that that does not necessarily remove the created parallelogram! However by defining all parall lines in a parallelogram as equal the task is accomplished. Whatever end to one happens to all parallel lines.


This constraint has the unfortunate consequence of being universally applicable. Thus one can be lead to believe tht one has found universal laws, rather than a solution to a tricky design issue. Maxwell for example felt certain that such products would exist in space independent of source. Some buckle heads have gone about hailing mathematics with this predictive ability rather than looking more closely for the natural force system that may or may not make this a valid application!

In fact the huge magnetic structures show this local consistency justifying a local application of this system, but as any observer of the sun will tell you that consistency os not universal!

Hrm! So why do we still act as if Newtons " laws" are universal? You tell me!

e ₩ e set to –1 is a curious product design. On the face of it it has to be the imaginary product, but in fact that is still confusingly thought of as a line segment product.

In the work I did on polynomial revolutions I gradually had to distinguish between the oriented line and the product design setting up the system.. Hermann deals with this within a rhombus of equal line segments, and shows how it is in fact the singing arc contained therein that is the limb that is being synthetically knitted or analytically knitted. He is forced to use the Eulerian exponential function to express this relationship, and we read of his development in the Vorrede.

The topic is wide and deep, and open for continued research based on the Cotes DeMoivre body of Calculus, the roots of unity and the zeroes of the trig functions cos(nø) and sin(nø). In addition cotes version of the Cotes Euler equation. Here the trig line segments really come to the fore and all the previous work done on product design has to be modified to fit this new product design.

Essentially, from the complex product design involving a rhombus and an interior arc and the projections onto the diagonal, one should expect a very general applicability of this product design. The principal freedom is that of rotation in the plane, and this is directly relevant to Hamiltons Quaternions. The solution Hamilton stumbled on, and I say by brute force because he ignores the complex conjugate of K, works after a fashion but requires the associative product design ijk = –1.

I designed( without being able to quite grasp it) the Newtonisn triples without that constraint , rather ncd = -n, and in do doing saw how Hamilton had bulldozed through the complex conjugate constraint. The Newtonisn triples are a modulo 6 product design, and work on 6 line segments arranged as 3 orthogonal axes.

So are these the best product designs or just the most common and familiar?

For example the dot product design could be replaced by the determinant design while allowing e ₩ e to be set at 2e. 

The design for division which will come up next , may be tinkered with. The design for rotation might be simplified to remove the exponential. Although personally I love it and the trochoids it generates.


Design processes like these are all open to the designer once one gets over the centuries of unjustified awe accorded to mathematics, fostered by mathematicians who have to eat like the rest of us.

The design of the product for Quaternions is what Hermanns paper is all about. And to see how deeply he went into the design will be instructive.

Because of the requirement for the subjugating limb in the product being effective at every "point" in the "space" it creates, and these terms point and space will have the general meaning so far attached to them., a third line segment restricted to a plane created by 2 other line segment will in design be subjugating a tessellation of parallelograms. Thus the product of the multiplication within the plane will not be just a parallelogram with 2 conjugate parallelograms at the sides, rather it will be a more voles figure in which the base or subjugate parallelogram will be translated in the direction of the third line segment. , precisely like a flattened parallelepiped .

The algebraic product thus describes a complex figure in the plane, a shadow of a somewhat simpler figure/ net in space. The shadow / projection product should thus connect the 2 forms at different levels/ steps/ ranks/ stages.

The design of the projection product therefore involves selecting the effective result from the general subjugating projection . Thus while the projection creates parallelogramms the observer selects the limb onto which the projection is cast and utilises that limb with the now distinguished length as the projection product.

This is yet another product design decision.

a, b, c are three differing line segments in the plane defined by ab, and ab is a distinguished parallelogram.

Then let d = a +  b be the line segment representing the addition knitting of the 2 line segments.

Thus  cd = c(a + b) = ca + cb where d subjugates c

Note because these line segments are in the same plane the subjugation order produces the same result when switched ( c subjugates d).

 But interchanging the labels cyclically gives ad = a(b + c) = ab + ac. If ac was indeed like ca then we make the mistake of incorrectly distinguishing the referred parallelogram. Commutativity would be a misleading assumption , or rather switching labels around is a different process to switching subjugation around!

Thus switching the limb labels should not be taken to imply a switch in subjugation and consequently a switch from one system to another

In order to mentally switch subjugation in a system we can attach it to the order in which the limbs of a knitting are written, not to the limb labels. So when we cyclically interchange labels we must not let that change subjugation order.

Adopting that .convention avoids a processing order mistake and a wrong identification mistake. We then by inspection within a specified system have to determine if commutativity , that is achieving the same result by changing subjugation , holds. In that case we have 2 systems that give the same result, not one system which has inherent commutativity.

Finally by the rules of parallelograms the above product can be seen to hold, as cd can " slide" the 2 parallelograms ca, cb into itself.

.

Now with the correct set up and an additional process called rotation as well as " sliding" we can demonstrate Pythagoras theorem for a right angled triangle.

a,b ,c d,e, f are line segments in the plane designed by ab and c = a + b.  d= âb+ c and e = c + $a.

Finally ab is a rectangle with lengths â, $ and cf is a rectangle but ¢²cf is a square with length ¢( the length of c), - â²ab is a square and $²ab is a square.
 The - sign in one of the squares is because to construct it we must use -b.

So now we want to show that
  
   ¢²cf = $²ab - â²ab

There is a lot to say about the above labellings.

The first is that a line segment until now has been the archetype of a step one system..mbut a line. Segment appears in all  steps, and by design we want to be able to synthetical knit them in all steps. Thus a line segment must take on the step level it is utilised in. In this case , in the plane a limb is an element of the plane just as much as a parallelogram! It is astro 2 element.

Thus addition of step 2 elements must cover the knitting of lines as well as the knitting of parallelograms.

Every multiplication in the plane is in design a new product system!

2 elements define a plane by a product. A third element as a subjugating one defines a new product design. Because of this it is a mistake to attempt to relate products from different product designs. A product design always is based around subjugation of a synthetical knitting! Even if the elements are of the same step level, this design is the one Hermann advises, thus a single element subjugating Nother is not the best design, and each element chosen in the pair is a constituent of a unique design. In this case all such singleton pair designs tessellate a plane, even the same plane uniquely..


Thus transforming between designs is very important and necessary if we are to proceed to conclusions.

The transformations are: rotation, perspective projection, and parallel projection, translations, reflections , cyclical label change.

In geometry these transformations are utilised by the observer the activator. There is a possibility thst Algrbra of labels could govern this less whimsically, or at least in a more automatic way. Thus the product designs immediately in the plane link 3 parallelograms together in a way in which they fit . In particular certain transformations distort the form but do not alter the property that the bounded space never leaves the boundary! This is an observation that despite the contortion , in between parallel lines, it can always be demonstrated that the rectangle is the invariant form of all these types of deformation.

This is a fantastic observation. " geometry" has these invariant forms of which all other forms are a kind of distortion!. The principle of conservation of form is what underpins the notions of duality in the Stoikeia. The transforms or distortions maintain something that is invariant, and findi it was the joy of the Pythagoreans.

So using these transformations between differing product designs allows us to discover invariant relationships.


Thus we must allow the design to effect the fallout of surprising results.the above product design does point to similar figures on the sides of a triangle are related by a sum. The similarity of the figures is derived from a common subjugating limb.

It out to be clear then that Pythagoras theorem necessarily must be an instance of this more general relationship. To demonstrate it I have to use transformations between differing product designs.

Back to Pythagoras.

The method Eucld chosebo demonstrate  Pythagoras was to select parallelograms âbc, and (-§a.-c) containing d and -e. Rotate d counter clockwise and -e clockwise to lie parallel f.

The parallelograms transform to §¢bf and â¢af, the same as ¢bf and ¢af

That is ¢af + ¢bf = ( a + b)¢f = c¢f = ¢²cf.

There is a demonstration that does not involve a set up requiring explicit rotation.

I just lost the second part of the Pythagorean solution to the aether!  :sad1:

But it is very strange that it should happen just when I was building a case for the volition of machine intelligence? Are we creating a Frankenstein?
Or are we evolving coevally?

Will my computer let me post this?

We shall,see :D