Print Page - The Ausdehnungslehre of Hermann Grassmann 1844 reprinted in 1877

From time to time I publish the commentary before the post, usually because I am reworking the translation.

If you feel inspired to share your insights from reading the translation, or even working on your own translation please do so, because then we all learn!

In the first section , the extending magnitude and the first chapter Hermann takes the reader in the stadium position through his adventure to find the well spring of Mathematical creativity and expression. In this manner he hopes to give the reader the maximum freedom and mastery over his experience of Hermanns researches! Hermanns researches by reason of rigour, and by the nature of his necessary abstract approach is unlikely to suit many of his intended audience, and very likely to cause established mathematicians to recoil in Dread! Thus he elects to do the hardworking on his own offering up the results to his audience for scrutiny as propositions, demonstrations and the like in an easily accessible form.

Hermann in setting about his work in this way hoped not to tax or tire the reader unduly by pages of abstract concepts thoughts and reasoning, and in particular makes a point to use long familiar concepts. In fact even his phrasing and adjective use was designed to be as accessible as possible. Thus we often get long, wordy adjectival phrases attempting to communicate important distinctions.

But the most important aspect of the 3 writings plus this first chapter is the gradual revelation of the expertise as it is put into practice! Here we find the clearest summary of it: that the worst possible approximation to a solution or description is taken as a starting point and then by a process of reasoning and experimentation a process called a dialectic , that situation is resolved into a better and better solution! In the course of do doing certain labels play a vital role as keepers. These keepers enable the refining process to tautologically anchor to a label and gradually transform from bad to good to better and eventually the best representation of the situation circumstance or process.

The labels are varied, from letters to every typeface in the printers block. Some labels are signs some are names some are symbols, and all combine to express the idea or resolution sought or under discussion.

The use of other relevant doctrines particularly the Combinatorial doctrine is mployed for method,notstion, terminology and Analogy and inspiration, and by these means and methods an expertise is established based on concrete developed and developing resultd.

The nature of his material is that it is dynamic and ever changing, but yet certain invariant patterns inevitably appear. It is these that form the fundamental basis of his application of his expertise, developing and as it develops, to the dynamism of real space and time .

Certain concepts of creating, being and existing , coming into being and dissolving from being are deeply established so that the reader knows that what is demonstrated is fully considered and not half baked.

Because of this approach, the work is an introductory philosophical text, which enlightens any reader who engages with it. The important attitude is to believe that it is intended to be crystal clear, and consequently the sudden realisations one obtains while reading are likely to be cathartic of many years of wrong headed thinking and teaching!

At the end of the day everything he says is empirically testable by the simplest of Geometries. Thus the works of Euclid, Apollonius, Archimedes etc are brought into light as examples of clear empirically based, dynamically testable thinking. This type of dialectic differs immensely from the linguistic grammar based logic which we are presented with through our education system.

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction of the simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§13 The pure expertise like way, to treat the Doctrine of the extending magnitude, the pure expertise would be, that we concord with the artform, how it in the Induction is pursued from the labels out., which labels, by this expertise, lay to ground ( as a basis) all individually developed entities.

Alone,

to not tire out the reader through continued forth abstractions , and at the same time therethrough

to set him in the ( stadium) Stands ,

to move himself with greater freedom and self standing quality ,

that we tie up besides familiar entities,

I tie up over everything else, by considering the deriving of new labels alongsides the Geometry;

our expertise builds a representation of the Geometry Basis

Therefore in the entity I lay the abstract label to ground (as a basis), every time, by considering the deriving of the enduring qualities, which qualities build a representation of the content of this expertise , without myself, to shore it up, thereby considering each upon some random one in "the geometry demonstrating" truth ;

thusly I still hold out the expertise concording to its content, completely pure and independent from the Geometry.•)

Therehere I tie up besides the Creating whole enity of the line to gain something in the vicinity of the extending magnitude.

Here it is a creating whole Point, which point takes aside ( to scrutinise) differing positions in continuous succession ; and the totality of the points , over which the creating whole point is going by scrutinising this everyway varying ( in the totality), builds a representation , the line.

The points of a line appear thuslywith,
essentially as differing entities, and become also as such besigned ( with differing printers block types);

therefore, how the Like is adhered alongside to the differing entity always at the same moment in time ( imagine wholly over in a subordinate sense) , thus also here the differing points appear as differing positions of one and the same creating whole point.

Upon like manner we reach our expertise to the extending ( magnitude), only if we set here the "intercommunicating label-like" entities in place of the there "inter-stepping space-like" Relatings .

Initially in place of "the point", that brands, "of the special place", we set here "the element", whereunder we everyway stand the special entity badly way off apprehended as differing from the other special entity. And indeed we lay by considering "the element" in the abstract expertise no half baked other content.

Therehere, It, the conversation here, can not be half baked therefrom , what conversation were usually set for a special entity, because this entity is central-like –

because it is plainly badly way off the special entity, without all real content—,

or in which relating the one from the other is differing –

because it is plainly in a bad way appointed as a differing entity, without that one random real content , in relating onto which it is a differing entity,

By This label of "the element" is our expertise of a common design with the Doctrine of the combination ( combinatorial theory), and therehere also the besigning scheme of the elements ( through differing printers block type) are both of a common design with the combinatorial doctrine.

The differing elements could now at the same time become apprehended as Differing Condition/Status markers of the same created whole element , and this abstract Differing quality of the condition/ status markers is it, which of the differing place quality inter communicates. The overgoing of the creating element out of a condition/status marker in one other we name "a varying"/ "transforming" of the same , and this abstract transforming of the element inter communicates therefore to the "place transforming " or the kinematic moving of the point in the Geometry.

Now, how in the geometry, through the forward kinematic moving of a point, immediately nearby a line roots and rises up,

and for the first time ,
in which entity one under throws/ subjugates the achieved representation onto the new representation due to the kinematic moving ,
space-like representations of higher steps/ ranks/stages can root and rise up,

thusly roots and rise up also in our expertise through continuous transforming of the creating whole element immediately nearby the extending representation of former/ prior step/ stage/ rank.

The result of the until here developing Summarily Grabbing together report, the definition, we can set down :

Quote

" Under an extending representation of former / prior step/ stage / rank, in which a creating whole element by continuous Varying over travels, we everyway stand the totality of the elements, "

And in particular we name the creating whole element in its former condition/ status marker the Beginning element, in its latter condition marker the End element.

Out of this label output results itself thuslike, that to each extending representation a "running into against set "entity relates, which entity the same elements enholds( holds within) but in turned around rooting and rising up manner/ cognisance, thus therefore,that name- like becomes the Beginning element of one the End element of the other. Or , more concordingly expressed, if through a varying out of a b comes to be, thus is the running into agains set varying the varying, through which out of b a comes to be, and the representation relating to an extending representation "running into against set" entity is the indicated entity, which through the running into against set varyings in turned around succession goes henceforward, wherein at the same time lies, that the running into against set being is a side changing entity.

Footnotes

•In the Induction (Nr.16) I have shown, how by considering the presentation of each an expertise and in particular the presentation of the mathematical one, two developing rank arrays grip in one another ; from which the one delivers the Material, that brands, "the complete rank array of the truths", which builds a representation of the central- like content of the Expertise; while the other to the reader should give the mastery over the material . That former developing rank array now is it, which I have to hold out completely iindependent of the Geometry , while I have placed the greatest Freedom before me by considering the latter array commensurate to my goal .
•• the difference lies only in the artform, how in both expertises the thought patterns come to be achieved out of the element: in the Doctrine of the combination specifically through direct knitting together, therefore discrete, but here through continuous creating whole.

Commentary on §13

Once again I can say I am profoundly surprised by this section.

I have some redacting to do but I hope it is clear that Hermann has a different take on interacting with space. Firstly it is Newtonian, and that particular conception of Newtons called Fluxionic. This is not surprising in the least as LaPlace, LaGrange and indeed Euler were " students" of Newton. However, Berkley attacked the Fluxionic method on the basis that it was as unfounded as religion, that is if one is accepted by the scientific, and to him atheistic community, then so ought the other!

However, as piercing as his argumentation was it was nevertheless based on misreadings, mis-teachings and misunderstandings of Newton's Astrological Principles. Berkeleys criticisms apparently goaded Mathematicians into wrong headed number concepts .

It was said that Newton had a dream, as of God, in which the Fluxions were revealed. In that dream a glowing point moved to trace out a line, and that line itself moved to trace out a surface; and that surface itself moved to generate a solid, and that solid itself moved to trace the locii of a moving object.

The story related, that from then onwards Newton pursued these Fluxions to derive his theory of the calculus, that is the Fluxions.

It is quite clear that this dream influence LaGrane and LaPlace and Euler and was the basis of their concpt of Mechanics.

I on the other hand was fed the Leibnizian view of Mechanics in which motion replaces Fluxions and hard billiard balls and thn elastic billiard balls replace the Dynmics of Fluxions!. The differential Calculus thereof was made up of discrete minutiae called infinitesimals.. They were arrived at by analytical abstraction whereas Newton was given a synthetical Geometry to found the theory of Fluxions and Fluents upon.

The analytical Leibnizian approach actually was initiated in he School founded by Cartesian scholars, and developed by the Brilliant work of Leibniz, whose notation by and large is the standard notation in integral and differential Calculus. On the other hand the Synthetical approach of Fluxions was adopted only by British scholars who found in it a national pride of Newtonian genius . However the French, Kant and Euler also like De Moivre found Newtons conception to be to their liking and thus much of the Mathematics of the Ecole is based on Newtonian principles thoroughly revised and extended by French Scholars and engineers.

Hermann was aware of this and he called these Approaches , or rather I have interpreted Weise as Cognisance! Of course I could not justifiably maintain a constant translation for so psychological a word, and I have ranged across the vocabulary appropriate to that concept in context. Thus the cognisance is exhibited in a certain style or manner of proceeding, thinking, speaking snd notating..

Thus the 2 cognisances resulted in 2 competing versions of calculus. It is this historical opposition and competition that dialectically has given us the modern Calculus. It is also a fascinating case study of how dialectical reasoning is naturally embedded in historical movements both progressive and regressive. This was a constant Hegelian theme, which he had much to say about. Hegel advised that we identify and choose the progressive processes of the dialectical process.

There is much to comment on in this section and I will do so in several posts , but Hermann here builds on the work of his 3 writings in an ever more fractal and detailed way which he is at pains to make absolutely clear to the reader in the Stands!

Commentary on §13 continued

The fractal structure of thought is well documented by Hegel's writings. This carries over into Hermanns structuring of his work.. The fractal is based on a repeated pattern of 3 at several levels . The general pattern is : comparison, contrast and conclude.

Of course you need two items to compare and Hermann chooses the GegenSatz format. To make a comparison one must have 2 statements to compare, and Hegel starts his discourse with a historical narrative. . Hermann in the Vorrede does the same.

Then Hermann gives a formal induction covering the whole topic once again, but differently , in fact in a philosophical format establishing propositions. The third concluding treatment is the general Doctrine of the thought pattern. On e again he covers the whole topc but this time he focuses on elements that can go forward into the later works . Readers, from their vantage point are conducted backwards to previousvandvother ideas in order to draw out the resolving conclusions from the comparison and contrasting.in the discussions.

We see clearly how by carefully revisiting each aspect of the case in focus how the proposition is carefully revised redacted and updated : it is gradually resolved into the best expression of the conception.

A fractal is a broken of piece ( fractus) or a pattern of such pieces. It is usually formed by recursion or more simply reiteration, repeating the process over and over on the same material , but with a slight difference every time. It is that slight difference which takes " almost" self I liar patterns and transforms them into a fractal synthesis or analysis.or some combination.of both.

This revisiting of the same material oncept or definition in order to slightly adjust its meaning in use is called Tautology. It is a fundamental aspect of reasoning apprehension and dialectical development. Yet some frown upon it as sophistry or bad practice! You need to understand tautology as part of self reflexive thinking, this is easier for languages that retain self reflexive verbs..

With that fractal process in mind we can move into the Details of §13

Commentary§13 continued

The Induction is a piece of work!

I recommend reading and retreading it meditatively. It is a true hypnotic induction piece of writing so do not be surprised if you find ideas popping into your mind from just about everywhere!

So given its " general" approach( rather mind blowing nd mind bending presentation) we have to touch base somewhere.. We have been on a wild horse ride across un familiar terrain, or a white water rapid ride through echoing canyons and now it's time to " tie up" the horses reins , or "tie up" the boat to some moorings and lay down some firm footings to the ground, plant our feet firmly to ground and set up a base camp, as a basis for more specific examination of the Locle or local terrain.

I hope you appreciate the similes or metaphors because Hermann not only uses thrm explicitly and directly, but his whole writing style and word choice is suffused with them, along with word puns and other literary devices. Many times even his grammatical constructions and sentence structure reflect or convey the relationship he is discussing, or the " connection" he is making or expressing.

The expression is the thing. It not only conveys a semantic meaning it also conveys a grammatical design. That grammatical design often and deliberately becomes the mathematical synthetic and product design.

You and I need to ponder that deeply.

I have rather deliberately translated the word Begriffe as Label ( or sometimes handle) wherever it has occurred. This was because it is absolutely foundational to his approach, and his Förderung, or promoted mindset. The word itself etymologically derives from griefen that is "to grip ". The participle " Be" underpins " bei" which sounds like " by" but I translate as "by considering". This translation serves 2 purposes: one it deals with the inner space of the reader, it implies the outer spatial relationship of the reader and author to the subject under consideration.

Thus Begriffe I like to think of as the handle I use to grasp something, some notion, some concept, some idea.. Thus it intimates the notion or idea so labelled, but sometimes it refers simply to the handle that will be used to attach to some yet nebulous concept. It is thus areal thing that refers and relates to many abstract immaterial or none concrete ideas and concepts. The real thing by which we grasp the un realities of thought!

So here Hermann states that the pure way is to work from these labels out wards!
Pure because he values the highest objectivity in his thinking and results. The subjective and objective processing conundrum I have dealt with in the Fractal Foundations threads, but here, in a romantic age, it was believed and hoped one could really divorce the Two. Surprisingly Hegel confounded this conception and thus for Hegel, the pure way involved not deluding oneself any longer about these " absolute " distinctions which are not so. Hermann is of the same mind, as he demonstrates in the Induction. So pure here refers to the purity of the dialectical process he has expressed prior to this section.

As a consequence he refers us back in the footnote to the last and quite difficult section in the Induction, where he describes the 2 rank arrays. I have to admit I do not fully grasp this description, and here he succinctly clarifies it a bit more. So I will be revisiting and reviewing that section with a view to clarifying the translation.

However, I do grasp that these 2 arrays are a piece, they "grip " in one another. That essential nd central notion is the real message of these arrays: they will not be separated, like Siamese twins! How these arrays are realised is the whole "point" of the Work! The Transformation or Varying or Developing of one into the other is what the labelling is all about. It is the labels that we expertly use to ground real truths of space and time , set out in or populating an array, and it is the freedom we then have to vary that array toward some appropriate goal that is the power and the purpose of this whole entire work.

In a commentary on the induction translations I dealt with the use of the labels " algebraic" and " combinatorial".

It is worth pointing out here that Algrbra has had a number of meaning changes over the period since Hetmann wrote the Ausdehnungslehre. However the meaning I shall be using in this thread is simply nd straightforwardly symbolic arithmetic.

We need a cold shower sometimes to wake up and realise how we get hoodwinked by others misconceptions. Symbolic arithmetic is nothing different to ordinary arithmetic. We need to wake up and smell the BS! The Arabic numerals are symbols.

We could and indeed have used many symbols in the past to perform so called arithmetic. The roman and Greek numerals are a case in point. The Arabic numerals were mo different to these.

As a child you are taught to believe that you could not do sums easily with these symbols and thus Arabic ones are the best! In fact this is far from true. The ancient Chinese used a stick symbol in various patterns which are in fact a base 10 system. The Ancient Sumerians used a base 60 system, all of them are advanced civilisations, capable of Astrological thought and apprehension,

The true benefit of the Indian system is their absolute mastery of long intense calculations, often done entirely within their heads apparently. This so impressed other merchants that they abandoned their traditional systems to gain the bartering advantage of the Indian superlative system. This was not well received by the Greek loving West who opposed its creeping insurgence vehemently, as a bishop to India found to his cost!

Although we now adopt the Indian system of numerals we never fully adopted the Vedic Ganitas and Sutras tHat underpin it. In particular we disdained the absolutely fundamental secret of the Indians ability to calculate so well! They used their gingers and thumbs and toes!

Even now the west cannot accept the plain fact that neurological programming is the fundamental consequence of using your digits in this way. People who count on their gingers are laughed at in the west, whereas they are all trained to begin there in the Vedic school systems. Contrast that with say a Chinese savant who can calculate at blinding speeds on an abacus!

We are hypocritical . The abacus is no better or worse than our fingers and toes, and an adept, or savant can indeed out perform any mentalist calculator using his fingers and toes to reorder and retain the out put result of the calculation. This is neurological programming at its finest.

There are many other advantages to the Indian 10 fingered Shunya is everything system , which the Vedic schools exploit and explore, promoting confident , swift, accurate and able computations on the spot!

Thus we should be aware that symbolic arithmetic is not some abstract weird thinking, but instead it is the foundation on which all numeral systems are designed and based. These basic properties by which we implement a design derive as much from the behaviours of real objects as from aesthetic design patterns we might like to promote. In contrast to the symbols , counting is our true vocal and emotional response to these dynamic patterns and pattern transformations in space. We can go further and incorporate dance position and poses, and indeed many movements within Indian dance styles encode such counting patterns.

The point is that counting will occur any and everywhere in our response to and expression of dynamic geometries, but it should not obscure or define the dynamic Geometries per se. The dynamic Spaciometry should promote the design of numeral systems in keeping with the synthetic and product design constraints and protocols established by empirical evaluations and experimentation.

Commentary on§ 13 continued.

In 1844 when Hermann finished the Manuscript he sent it to Gauss on the advice of Möbius who found the work beyond his capabilities . After a while a busy Gauss returned the manuscript advising a rewrite using more familiar less original terms. He had not been able to devote timee to learning Hermanns new terminology .

Later in 1853 Gauss directed Riemann to speak on the topic of the hypothetical basis of Geometty in his 1853 Habilitations speech. This is the topic of the next few sentences of §13 in which Hermanns explains his contribution to the growing debate in 1844.
Riemann (http://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.html)

Quote

To complete his Habilitation Riemann had to give a lecture. He prepared three lectures, two on electricity and one on geometry. Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry. Riemann's lecture Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854, became a classic of mathematics.

There were two parts to Riemann's lecture. In the first part he posed the problem of how to define an n-dimensional space and ended up giving a definition of what today we call a Riemannian space. Freudenthal writes in [1]:-

It possesses shortest lines, now called geodesics, which resemble ordinary straight lines. In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras' theorem.

In fact the main point of this part of Riemann's lecture was the definition of the curvature tensor. The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in. He asked what the dimension of real space was and what geometry described real space. The lecture was too far ahead of its time to be appreciated by most scientists of that time. Monastyrsky writes in [6]:-

Among Riemann's audience, only Gauss was able to appreciate the depth of Riemann's thoughts. ... The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented.

It was not fully understood until sixty years later. Freudenthal writes in [1]:-

The general theory of relativity splendidly justified his work. In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data.

So this brilliant work entitled Riemann to begin to lecture. However [6]:-

Not long before, in September, he read a report "On the Laws of the Distribution of Static Electricity" at a session of the Göttingen Society of Scientific researchers and Physicians. In a letter to his father, Riemann recalled, among other things, "the fact that I spoke at a scientific meeting was useful for my lectures". In October he set to work on his lectures on partial differential equations. Riemann's letters to his dearly-loved father were full of recollections about the difficulties he encountered. Although only eight students attended the lectures, Riemann was completely happy. Gradually he overcame his natural shyness and established a rapport with his audience.

Gauss's chair at Göttingen was filled by Dirichlet in 1855. At this time there was an attempt to get Riemann a personal chair but this failed. Two years later, however, he was appointed as professor and in the same year, 1857, another of his masterpieces was published. The paper Theory of abelian functions was the result of work carried out over several years and contained in a lecture course he gave to three people in 1855-56. One of the three was Dedekind who was able to make the beauty of Riemann's lectures available by publishing the material after Riemann's early death.

Whatever Riemann presented in his lecture was hardly more subtle and far reaching than Hermanns conception. And yer Gauss raved about his protegé's lecture some 10 years later after reading the Susdehnungslehre!

To be fair Geometry was in a very bad plight and was being thoroughly analysed for its weaknesses by many great and disappointed philosophers and Geometricians, Justus Grassmann amongst them as a lowly cleric and district School Inspectir. But I still find it a curious tale that Gauss ignored one and promotes the other without any compunction!

Be that as it may the labelling and ideas are related but clearly different. Hermann singlehandedly created the Lineal Algebra from his conception, while Riemann as brilliant as he was lauded to be merely posed the questions..

I have redacted the section of the translation to hopefully make this point clearer from the text..

Commentary on §13 continued.

This is even more genius than I could imagine !

The point is first addressed.
There is much I have written about the point but nothing like this!

Hermann does not start with Geometry or even Astrology . He starts with thought patterns represented by labels.

But to ground these thought patterns he binds them to a well known geometrical label.

However the geometrical label has no influence on the thought pattern!

Well of course it does, but not in the expected or traditional way or cognisance!

In a way it is precisely like saying: I am going to change the referrent of your most fundamental labels before your eyes, so that when you speak in your everyday language you will be saying something with an entirely new semantic!mor another analogy would be : you speak your language in such a way that there is a direct intercommunicating , a 1 to 1 transliteration into another language, for example griffen is 1 to 1 mapped ont gripping.

But this is also a serviceable " translation" going beyond the direct transliteration!

Because Hermann has separated geometry as a real , concrete space-like experience, his labels, and indeed all labels are forml, independent and thought- like experiences that relate to our subjective modelling or copying of the real entities. But how well we label and how good our modelling is depend on our skill sets and life experiences. This he sums up in the term Cognisance. Allied to our subjective cognisance is our expertise . It is this combination of Cognisance snd expertise that makes it possible to build weird and wonderful mental representations of real behaviours:Subjective processes representing real objective processes.

In an analogous way or by simile we may anthropomorphise certain natural processes, that is we may speak of water as if it had legs and could voluntarily run down hill to the sea!

This is th kind of place Hermann has brought us to, where what we label here has an intercommunicating equivalent there. What is loosed here on earth is loosed also in heaven where God oversees everything.

So we start with creating elements. Our thought patterns perceive or we perceive through thinking that entities come into ontological existence and then may exist or pass out of existence, leaving only a memory as a built representation within us. In that process Hermann has identified nd thus labelled a class of entities called Creating Elemnts. As you study the Induction you experience how these elements tke shape and receive their formal labelling.

Without contradiction then Hermann can only start his process on the a priori foundational primitives that bring everything into real existence and thus apprehendable status! But in so doing he cannot define such primitives within his system. He has to relate to a prior system of definitions , or a larger system beyond the one he is specifying now.

Thus he cn specify the line segment, but not the point in his real system model. His model thus is founded on the line segment.

We will see how this enables him to model the most general transformational processes by the permutation processes explored in the Doctine of Combinations; how permutations can be directly linked to translation, rotation, and anti translation and contra rotation in space. We will also see how permutations can have no real objective space like nehaviour yet be subjectively useful as stages in our mental journeying to space-like solutions. In this category I specify the notion of reflection in a mirror.

Commentary on §13 continued.

The remarkable process that Hermann now goes through is important for its honesty and clarity.
The creating whole entity will initially be called " the point" and the context will be the line.
Its creative role consists in taking to one side and scrutinising each point in space.. Thus the creating point behaves precisely like a Human judge or a computer programme that tests every point.!

The inescapable involvement of human or computer agency is acknowledged from the outset. As a consequence of this activity, by going over all the points , the line appears. This line is not necessarily a straight line! The line that appears is dependent on this process of scrutiny and what constraints or conditions are placed on the output result.

Thus the dynamic motion in Newtons dream is replaced by a continuous process of input, scrutiny output as a point in the line or not, making plain that any model is engendered by human / computer activity.

However what follows after this derivation of the label point, which is independent of a geometrical description of the points of space (The line so constructed is also free of any specific geometry, except that it is labelled as a line due to it being differing to a point) is a typical design or construction model Hermsnn uses time and again.

But now, this becomes the " Mould" for creating all manner of representations of continuously extending magnitudes.

It is vital for his method that labels are " badly fitting" abstract terms or labels. Thereby much inane chatter is avoided. The focus is then on how to achieve the output result by synthesis.

The process involves this individual scrutiny in a continuous , successive manner, of every point in some totality of points. Thus I may constrain or place a condition on the totality of points which the process may scrutinise.

The process of scrutiny itself implies that some condition or status is bing tested for, and so once again conditions and constraints are vital to achieving an output result that is a line, or an arc, or a spiral or in some cases a "random " scatter - like appearance. These are all possible outputs of a fractal generator app!

A duality or dual cognisance always exists for a process output: the output may be considered as a collection of differing results or positions that fulfill a system of constraints, or they may be considered as the successive ( and thus differing) results or positions of the creating whole entity. This duality pervades all our thought patterns and Hermann acknowledges and utilises it.

Many of us are taught to remove all duality, all tautology all analogy in our pursuit of some thing called "the Truth " ; whereas the Truth is : duality and Tautology and fractal structure are all we can ultimately derive!

Now , surprisingly, the bad fit of the abstract contentless label makes it a prime example of a combinatorial label!

This surprising connection is not explicitly stated, even in modern Group theoretic discourses and discussions. And yet now that it is here blatantly expressed it is obviously " true!".

Every course on group theory it has been my misfortune to endure has obscured this combinatorial simplicity in implicit examples of the " way it could go" . Terms like Homologous and Homotopy obscure this simple combinatorial fact: we interact with space in fundamental sequence patterns, and we can do no other, no matter how random the output result may appear to others.

Thus ultimately all our calculations are constrained by the principles of Combinatorial Doctrine! Justus Grassmann, Abel and Galois were perhaps amongst some of the pioneers exploring this early Ring Theoretical behaviour.

Because of the confusion about multiplication, the combinatorial properties and principles were obscured by multiplication tables . However Cayley and others gradually clarified these tables as fundamental product primitives and started the Matrix notation.

Sequences, permutations of sequences and combinations of labels within sequences are fundamental to our synthetical structures we call Algebra, Arithmetic , and Geometry.

So now realise that Newton pioneered all of this in his work on infinite series and the Binomial series expansion. In a very real sense all our calculation schemes are based on finite ring theoretical structures and their attendant group theories.

Commentary on§13 continued
Lancelot Hogben intouced me to permutations when I was an adolescent determined to become a mathematician of renown! Of course he useda pack of gaming cards to expound on the subject, and forever made cards an enduring interest! Later I was introduced to the permutation tree method , which I alo taught with as little understanding as I was taught it . Later my own research with letter symbols rather than cards established a direct link to the tabular form for multiplication and bracket expansion.

Despite these experiences these observations were in no way flagged up as fundamental. They were of interest but that was it.

Oh I did think it rather wonderful that God had so ordained the behaviours of the world or Cosmos that they were first possible, then measurably probable, and finally statistically inevitable( stochastic processes). The concept of Random being equally likely made no sense to me as a child, and still makes no sense, which is why I soon realised it was a human designed artifice!

At university I was introduced to the Axiomatic approach, in Analysis, and then the Group theoretic approach in Algebra. Neither of which were particularly attractive, but the analytical treatment took pole position because I encountered it first. Group theory I dismissed as so much insane babbling!

So where were the lectures on Combinatorics? Confusingly buried in a semi computational subject called number theory which seemed to be attempting to ape its more respected academic counterpart Analysis.

I went to university to study " mathematics" and ended up as a casualty in a war zone engendered by the subject boundary wars! Fortunately for me I found David Hilberts book on the foundation of Mathematics and read that over the course of 3 years, so my university education was not a total waste of time and effort!

Oh yes and I learned how to programme a computer in an elective course!

Combinatorics thus figured little in any formal way as fundamental. Even Hilbert in his section on Galois did not elaborate on the ring structure he was investigating for the quadratics, not that I would have grasped it back then.

Lancelot Hogben stands out as one of my most informative sources with Hilbert Next. But I have to say that Hermann and Norman have both been vital to my putting everything in some sensibly constructed order.

in all my years of exploring combinatorics I have never thought of the combinatorics of a sequence of oriented line segments, even when I was precisely using the notion to construct polynomial rotations and the Newtonian Triples, as well as to critique Hamiltons Quaternion 8 group!

There has to be something wrong with mathematics if this fundamental observation is not taught at an advanced level even when it really belongs in a primary level curriculum .

Commentary on §13 continued.

I have skirted around the many ground breaking propositions or promoted viewpoints in this section, mainly because they are so deep that I would end up writing extensively on topics hich Hermann fully plans to extend on in the rest of the work. However I cannot stress enough that Hermann is not reiterating what one is commonly taught in mathmatics even today.

Pay attention to the labels and symbols and how they are used Rhetorically. Have in mind that subjective and objective experiences and viewpoints are constantly employed explored and contrasted . Realise that the labels are a formal abstract that brands, "virtually empty" entity waiting to be populated with true empirical data which comes from the real empirically observed and geoetrical space we call reality. Understnd that every label is held independent from the geometrical Doctrine and is derived by a scientific scrutiny of empirical behaviours snd properties of space-like objects that are related to a geometrical label .

That last point is crucial. One may start with the geometrical point, which thus makes that thing a real entity to be explained, or one may start with a label " point" and describe what it is Designed or defined or constrained to do, or how it is to behave. However once we go down this route we must specify precisely what it can and can not do , what it can or can not be, and what it's principal functions and interactions are.

If you have done any objective programming, that is a design coding system called object oriented, you will recognise the task of specifying how the " object" may or may not interact or relate to other bits of code within the grater code body of a programme or application.. This is the kind of discipline and constraint Hermann worked under..

But now, as coders know, once you have designed and implemented how that piece of cide will run,it now can be used for any analogous operation! In fact, as the actual elements in the hardware are bistable floppy circuits, what you designed to behave like a point will actually accept any data encoded within that precise data structure. So if a code a square function/ procedure to square any double length byte, it will square any data encoded in that format, whether it is a numeral or a chRacter!

Thus Hermann states: if we go down to the level of the bistable circuit , that is the fundamental element of the hardware we can make that whatever we want/ physically can and the exact same process will run it and produce the analogous output.

This is a Functional description or expression of behaviour. The function is fixed, the input and output are free, up to certain conditions or constraints of the function.

So the creating whole element labelled "the point" will out put a line, labelled "a line" it will out put a surface, labelled "a surface " it will output a space.... All of course subject to the constraints. And since the process involves scrutinising every "point ", line etc it naturally is related to every permutation of every Zustände or method of recording the condition and status of a point, line etc, within the totality of these condition / status markers.

You can feel why the labels must be so Abstract, that is devoid of real content! While this initially feels so weird,eventually, as in programming or coding, you realise that you can code for a general behaviour by coding for a specific behaviour and then adjusting to give it the greater generality. This is the greatest Freedom Hermann is referring to, and the way two arrays grip in one another, one specific the other extremely general based on that specific one..

As a consequence of linking to the combinatorial Doctrine Hermann has to have a beginning and end " point" or as it is branded " a special or specific position".

The final property he discusses in the context now of beginnings and endings is the contra processes that reverse the out put relative to the original process. Again there is this duality of opposingly running through each other processes which nevertheless use precisely the same element in reverse order.

Remember now, as Hermann points out, any permutation of symbols or labels can be interpreted as a variation of position or a transformation of position. Thus the permutations now encode translation, rotation, reflection etc ,in these extended magnitude structure creating processes.

These transformations are viewed as Sidechanging ones!

The translation of §14 is again throwing up deeper pedagogical issues ! It does not at all go as I initially anticipated.

When I was first introduced to " algebra" at primary school level, it was called geometry. Consequently after learning some basic geometrical formulae and demonstrations at primary level I somehow grasped the adhoc jumbled and confusing introduction of Algrbra at secondary level.

At that stage it was all a,b, apples , oranges,x,y and let x be thus and such!!

To the question 'what is x, what does it "stand for"(Zustand)?' several replies were given! Initially " an unknown" number, later quantity; a variable(?); an empty "box " in which we can place anything. To a 12 year old this was not apprehensible! I picked the empty box idea.

Later when we were introduced to equations, and the term "an expression", I already knew how to duck the brain frying effect of this confusion of terms!. Oh yes the word " term" crept in there too! I had an English language grammatical apprehension of that word.

My point is, my elementary teachers and later I found my university teachers knew how to use the symbols, per se, but we're incapable of explaining or expounding on the labels or names which were attached to these symbols!

One of the causes for this was the originators of these terms were French and German. The French were very fluid and expressive thinkers, proud of their achievements and their mindset on the world, society and historical movement. They were the originators of Revolutionary thinking,Liberty, equality and Fraternity or all! Thus their philosophy and mathematical expression reflects that.

The Prussians were more rigidly structures, and feared the revolutionary chaos on their borders! But the younger Prusdians were liberated by it, drawn to its exciting possibilities. Most were educated by French thinkers. However the Prusdian expression was deeply explorative and connected, based on homiletics and liturgical exposition of biblical texts. Philosophers generally left no sentiment unexplored and so their expositions were weighty, tautological and full of case by case arguments. The same text or symbol thus could generate multiple meanings related by the same textual form. Thus Prussian philosophers used symbols as a kind of variable expression whose meaning has to be determined by and which is constrained by context.

The difference between equating( equalité ) and likening( gleichung) is sometimes profound. The elevation of an equation to a symbolic expression in France , liberated it from any common meaning to a specific " mathematique", whereas in Prussia , the writing of an expression symbolically was still a rhetorical shorthand. Consequently the symbolic form were exegetically treatable: they were mired into common sensibilities and circumstance, case by case.

Hermann therefore takes this second route. He treats the label of the creating whole element as an exegetical exercise, in which the reader is brought to an intimate apprehension of the feelings and sensibilities of that element!

What is it like for a creating whole element to vary? According to Hermann it is as agonising as giving birth!

With that analogy in mind, and the general background of these differing cultural conceptions one may grasp how the variable concept permeates modern physical theory, while the abstract symbolic expression dominates Mathematicl thinking. Mathematicians do not suffer through variables as Theoretical physicists do. In fact they only have a hard time when they come to explain what they are doing ! Unfortunately we have lowed them to hide behind the mantle of God, asserting that they are s
Raking in Gods tongue!

That is why to most ordinary people mathematicians speak gibberish.

Translating §14 has brought this pointedly to the fore. Hermann strives to be crystal clear, but our mindsets make us confused about what his exegesis is about. It is about what we are fundamentally thinking when we describe a formal meta geometry or a subjective Spaciometry that models empirical behaviours of partial objects as they transform. What we really do is cherry pick from an infinite set of possible outcomes( output results). Whatever we cherry pick is determined somewhat by constraints.

Constraints are supposed to be objective: but everything we do or select is ultimately subjective! The discipline therefore is to be autisticlly rigorous! If we choose a whimsical set of constraints, and we design the product knitting and the synthetic and analytic knitting of these elemental magnitudes, then we must watch ourselves strictly so we do not fudge the results! We have to let the rules play out as they were designed and implanted!

The confusion that many Mathmaticians felt when an automatic system comes up eith an unexpected result is profound. I realised this when computer programming in 1973 using Algol60. I spent many weeks being feverishly ill trying to get my programmes to work as expected! I eventually embraced the programming paradigms and let go of the subjective mathematical intuitions I had built thus far. I could admit to the sloppiness of the way I had been allowed to think!

Rigour was a new and cleansing baptism of fire back then, and it has been a foundation of my meditation ever since.

Proceed slowly to gain the maximum benefit from Hermanns exegesis.

I have had a problem with Normans promotion of Quadrance since I understood wht it was!

I suddenly understood where my difficulty lies. Normn bases his development on some abstract thought pattern he calls a rational type or a rational number. Both I and Hermann base our undertaking on magnitude, a general experience of space.

In particular I have learned by Hermann about the intenive and extensive magnitude experiences. I was thereby willing to accept the bounded line segment as the best symbolic expression of these aspects of magnitude in both it's continuous and discrete expression.

But the line segment is only an aspect of a higher level/ step representation of a extensive/ extending magnitude. . In particular the flat or planar extending magnitude is the primary magnitude we experience as magnitude both extensive and intensive, do why pick the line segmnt as a primitive?

The reason is subjective, we can draw a line segment, either as an arc or a straight(ish) line. This has meaning for us, connected with motor muscle activity and thus with proprioceptive functions that occur within our deeper unconscious processing. This is why the point as a physical object and a line are indissolubly connected. Thus a line can be associated with 2 points or it can be associated with planar figures. It is the most generative symbol we apprehend.

Length and orientation and direction are also attributable to the properties of a line, but do not define a line! Rather a line segment defines these properties!
Thus a line segment is the foundational symbol allied to both a point and a plane .

Here Norman gives his clearest explanation thus far

http://youtu.be/YfY7RyAzHPE
http://www.youtube.com/watch?v=YfY7RyAzHPE

The use of quadratic versions of quadratic forms(!) cannot be apprehended without the " functional" role that the line takes in ordering magnitudes in space. The line segment thus becomes an arbitrary Metron or Monas.

Realising this the difficulty with roots or surds is not a problem. The number theoretical difficulty is not Pythagorean in origin. The incommensurability of the units as Metrons is not only embraced but utilised in the distinction of Protos Arithmos. The protoi Arithmoi are the first or base Arithmoi for an accounting system . Thus within a square, the perimeter can be accounted by one system, the diagonals have to be accounted by another .

In attempting to place all the Arithmoi on a number line the difficulty of reducing all to one fundamental unit system becomes dramatically illustrated. To say that the Pythagoreans were upset by this is one of those perennial myths . It was the Pythagoreans that distinguished the difference between monads.

The difficulty for the number line disappears if the Pythagorean theorem is used as the fundamental organising principle. Thus the use of "squares" is the secret of the Pythagorean "unit". This means surfaces are the fundamental form in geometry. The line segment however is a " secret" encoding system the Pythagoreans used to record important ratios.

The surds form the ratio segments that encode a spiral form called the spiral of Theodorus.

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction ofthe simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§14 The extending magnitude representation then, only comes to seem as a simple entity , if the varyings, which varyings the creating whole element outwardly suffers, are continuously becoming set like one another ,

So that therefore,
if
through one varying, out of an element a another element b henceforward is going , both which relate to that simple extending magnitude representation,
then next ,
through a like varying, out of b an element c of the same extending magnitude representation becomes created whole,

then also In addition this like quality indeed comes to be having a place to find!

If a and b come to be apprehended as continuously alongside one another bordering elements, this like quality by considering the continuous creating whole should by passages there find a place.

We can name such a varying a fundamental varying, through which out of one element of a continuous thought pattern a nearby alongside bordering entity come to be created whole, and then we can come to be declaring :

Quote

"the simple extending magnitude representation let be such a representation which through continuous continuing forth of the same fundamental varying goes henceforward"

Now In the same sense , in which the varyings become like oneanother set, we also become able to like set the "therethrough created whole" representation , and in this Sense, specifically that: the "through like varyings" entity comes to be self like set upon the same "cognisance created whole" entity , we name the simple extending magnitude representation of the former step "an extending magnitude" or "an extending (magnitude) of the former step" or "a line segment".

Therefore the simple extending magnitude representation comes to be related to the "extending magnitude" , if we look away from the elements, which elements the former enholds, and hold fast only the artform of the creating whole Activity;

and then, while two extending magnitude representations can become oneanother like set only if they the same elements enhold,

Then thusly two "extending magnitudes" are already created whole on like manner ( that brands, through the same varyings) , even if also 'they are created whole without the same elements to enhold.

At the last The totality of all line segments, which through continuing forth of the same And of the"running into against set" fundamental varying of the same are createable whole, we name a System. ( or a field of study) of former step.

All The line segments related to the same system of former step therefore become created whole through continuing forth, either of the same fundamental varying, or of the "running into against set" fundamental varying.

Before we go over the knitting of the line segments, we want to everyway alongside show the set down labels in the previous § through the application to Geometry.

The like quality of the varying cognisance comes to be here through like quality of the direction everyway representing;

therehere, the unending direct line represents itself here as system of former step , as simple "extending of former step", the bounded direct line. What there would be named as a like artform, appears here as parallel, and the parallelism features like case way its two parallel sides as parallelism in the same and in the" rinning into against set" sense. We can holdfast The name of the line segment in intercommunicant sense for the geometry ,

and therefore here under " like line segments", such bounded lines we can everyway stand, which line segmnts have like direction and length.

Footnotes
• the abstract assigning of this originating concrete Nomenclature requires wholly no legal finishing permission, there the names of the Abstract originating all have concrete assigning .
•• I present now the expression "field of study " before the expression " System", which many times in other peoples sensibilities Is like used
••• Thusly This distinguishing is important for the Geometry , that it not least would be by consideration carried to the associate teaming of the geometrical statements and demonstrations , if one this distinction through simple Nomenclature fixes, where to I would be pleased to thump forward some type of expression around " like running" and "against running" .

Commentary §14

This was very difficult to piece out, and I am still working on the footnotes,

The if, then then also structure took some finding, but it was important to show how Hermann considers every case.

The important statement about overlooking fundamental elements is important to understand. While Hermann synthesises from the ground up he also synthesises from the top down. So when a constructive or logical difficulty occurs by an Aristotelian analysis, he can look beyond to the higher goal and impose conditions from the top down. This is the detail of the Hegelian Dialectic process.

You do have to think about this carefully. Is he constructing in a manner that is basically unsound? Or is it pragmatically a brilliant solution to move forward?

The two § 13 and 14 are a detailed setting out of his method, using labels. The interpretation in a geometrical sense is but one interpretation! Therefore one has to attempt to grasp the generalities he uses as labels. Here he concludes that his definition or work through of the line segment label is safe enough to use for geometry providing certain distinctions are fixed and held firm, and these seem to be parallelism and direct lines. However the generality of his set up allows me to establish arc segmnt , circular arc segmnts , both concentric and around different centres but with same radius as geometrical candidates.

Again note how direction, that is orientation and translation are his general varying models / labels.

When he states his aim to make the extensive representation simple by setting all varyings equal or better like, he ie establishing an equally segmented geometry in every orientation. . But the building of such systems have to find a place in reality. So a crystal may grow along to axes but if it is to grow in thethirdaxis it has to be the planar crystalline form that is extending in this 3rd direction. No one has seen a single row of molecules forming a third axis!

This is the physical meaning of subjugation: a plane is subjugate to a three dimensional art form or its development, so it is practical not to get hooked up on the axes and to focus on the " higher" step form.

Commentary §§13-14

Well if you are feeling spaced out by these 2 paragraphs, join the club!

I think this morning I have a better insight on them than I did last night, when my brain refused even to read the translations!

The labels are really where we start, but by way of introduction Hermann starts with the " point " label. But almost immediately he replaces that with the representational extending form! This is the most general idea or form he can express.

This form actually has 3 interchangeable expressions. : the extending magnitude is the mst general metrical one, but this relates to the previous or prior representation as being an extending( magnitude) of the first step! . Thus an n- dimensional extending magnitude is an extending of the first step from an n-1 dimensional extending magnitude..

If you recognise this as a recursive definition you will be on the right track.

The recursive definition starts with the third expression of the representational extending magnitude that is the line segment. Hermanns definition starts with the line segment and builds from there.

And we know how the line segment is generated from the creating element the point.

Because of this general structure to his recursive definition he has to ensure that the next level variation " finds a place" in a real system. How do we check on that?

His answer is to look away from the elements that are varyin and look at the developing form. From this one can determine the next step variation, if it exists.

Thus for a cube, many of us would think: that can't happen. But mathematicians try to explain the hyper cube as casting a shadow that is 3 dimensional. While this is an ingenious attempt it is not happening, it is not real. What is real is a cube moving in space! Thus we can trace out the form such a shape makes, and it is this form that is hermanns guide to the next level.

There are several other alternatives: a cube rotating in space, or a cube expanding in space. The requirement for orthogonality would perhaps restrict these suggestions, but as you can read Hermann applies constraints to fit the representational extending magnitude into a real situation, not to limit the choices and freedom we have to imagine or design next level extending magnitudes.. Orthogonality is not the same as independent!

So before going on to the general notion of addition and subtraction, Hermann provides a demonstration of how to fit the general definition to a specific interpretation: in this case geometry. He shows how his general definition is applicable to geometry by use of parallelism in any and all oientations( directions) . Length is an afterthought:?it arises from the bound nature of the line segment. The fundamental system is undefined as to length, merely being remarked as " endless", and it is the line.

Notice how the line is built up from countless line segments, not the other way round.

As I pointed out earlier parallelism is one property that fulfills the definitions, the circular arc is another.

In passing, the simple extending magnitude representation is intercommunicant with the notion of a " uniform" representation. Indeed the word uniform means formed by the same measure in every orientation, and that measure is defined as unit,unity or Monas. All philosophers start with this uniform space or tablet and develop from there. I do not think one starts from the Fractal, non uniform space and attempts to build from there!

In the Fractal Foundations thread I attempted to do just that , and perhaps that is why I perceive the circle as the fundamental unit not the square. With the circle, sphere or circular arc you cannot hide away the fundamental discontinuities in real space , or rather the need for many scales of circles to cover space! It is fractal all the way down!

Again this is advanced in the sense it refers to work in the 1862 complete version of the Ausdehnungslehre, but this lecturer has captured something of the Förderung that Hermann uses to develop his outputs from. The expressions on the board are general rules, whimsical and seemingly arbitrary but it shows how these expressions arise out of our psychology, our interaction with space, the system which naturally fits the topic under discussion!

The calculus argument is simply an approximation argument , what Newton called the fluent of the Fluxions in space. These fluents all derive from the binomial series expansion and are all approximations.

http://youtu.be/rtEk1at3sVk
http://www.youtube.com/watch?v=rtEk1at3sVk

The idea that these things "occur naturally" is Maths speak. Hermann demonstrates how much philosophical background preparation or a priori preparation is required to get this to happen. That is what Ausdehnungslehre 1844 is principally about! What mindset or Förderung makes all these expressions easy and simple to derive in a lecture, or teaching system? F

inding these cognisances were Hermanns life's work. His aim was singlehandedly to do all the hardworking, sweat the details, wander in the dark, take all the risks to find the best way to put the reader in the stadium position! As the observer the reader has greater freedom greater overview, and can feel like the lord and master of what he surveys! But that is only(!) because Hermann has worked to put him in that position..

The reason is historical , and relates to the Humboldt educational reforms. Hermann wanted to help his Prussian nation and people to become self actualised individuals, so they could take their place on the worlds platforms equal to the French and hopefully surpassing them.

What does that mean for us today?

We can benefit in the stadium position, but someone or some group needs to continue to do the hardwork, to use the method outlined in the induction to further apply his method and insights and heuristic style and label and product design goals and constraints.

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction of the simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§15. If the continuous creating whole of the line segment comes to be thought interrupted within, broken underfoot in its waygoing , then around hereconcording, continuing forward again to come to Be, thusly appears the complete line segment as knitting of 2 line segments, which continuously close themselves up besides one another , and from which the one appears as continuing forth of the other

Both the line segments, which the limbs of this knitting build a represention of are " in the same sense" created whole(§8), and the output result is the knitting from the beginning element of the former to the end element of the latter, if both continuously in one another are " laid" ,that brands, are so presented that the end element of the first at the same moment is the beginning element for the second.

Forward running we besigned the line segment from the beginning element $\alpha$ ( compare with fig.2) to the end element $\beta$ with [ $\alpha$ $\beta$ ], [ $\alpha$ $\beta$ ] and [ $\beta$ $\gamma$ ] are in the same sense created,
Thus is [ $\alpha$ $\gamma$ ] the output result of the above indicated to one side knitting, if [ $\alpha$ $\beta$ ] and [ $\beta$ $\gamma$ ] are the limbs.

We have already (§8) concordingly demonstrated above, that this knitting, there it represents the unioning of The" in like sense created whole" magnitudes, as Addition, and their intercommunicant analytical unioning as subtraction apprehended must become,

And therehere all rules of these knitting artforms for it empowers.

We have here still only the centrally acting assignings concordingly to demonstrate, which the negative magnitude upon our field of study achieves. Specifically , around the initial , the assigning of the subtraction Side by side like showing to make to us , thusly we can thereout that

[ $\alpha$ $\beta$ ] + [ $\beta$ $\gamma$ ] = [ $\alpha$ $\gamma$ ] exists,
Thus directly [ $\alpha$ $\beta$ ] and [ $\beta$ $\gamma$ ] in like sense created are, we can the conclusion draw, that plainly thusly generally

[ $\alpha$ $\beta$ ] = [ $\alpha$ $\gamma$ ] – [ $\beta$ $\gamma$ ] exists( compare fig.2)
Therefore, that brands, if we now to us the In the subtraction usual nomenclature serve up,

Quote

" the remainder is, if minuend and subtrahend with their end element one lays onto one another, the line segment is from the beginning element of the Minuend to the element of the Subtrahend"

One sets in the latter format $\alpha$ and $\beta$ identical, thusly one outwardly holds
[ $\alpha$ $\alpha$ ] = [ $\alpha$ $\gamma$ ] – [ $\alpha$ $\gamma$ ]

That brands, like Null!

Further is everyway pleasing of the label of the negative line segment.

(–[ $\alpha$ $\beta$ ]) = 0–[ $\alpha$ $\beta$ ] = [ $\beta$ $\beta$ ]–[ $\alpha$ $\beta$ ]= [ $\beta$ $\alpha$ ]

That brands, the line segment [ $\beta$ $\alpha$ ], which to an other [ $\alpha$ $\beta$ ] its label concording to (§13) running into against set is, appears also in its relating to the Addition and subtraction as the running into against set magnitude to that one.

There at last now. a + (–b) = a –b exists.,

Thusly one has , if [ $\alpha$ $\gamma$ ] and [ $\gamma$ $\beta$ ] in the running into against set sense created whole are

[ $\alpha$ $\gamma$ ] + [ $\gamma$ $\beta$ ] = [ $\alpha$ $\gamma$ ] +(–[ $\beta$ $\gamma$ ]) = [ $\alpha$ $\gamma$ ] – [ $\beta$ $\gamma$ ] = [ $\alpha$ $\beta$ ]

That brands, if also both line segments are created whole in the running into against set sense, their Sum is the line segment from the beginning element of the first to the end element of the second continuously laid besides them.

And we can this result with the above grabbing together summary, declare

Quote

" if one knits together 2 like-artformed line segments continuously, that brands so knitted that the end element of the former becomes the beginning element of the latter, thusly the Sum of both is the line segment from the beginning element of the former to the end element of latter.";

And in which entity it as Sum is besigned, thusly it should lay therein expressed, that all rules of addition and subtraction for this knitting style empower.

Still I want to herebesides have a Following to close, which for the wider development, abundantly fruitful is, specifically that , if the bounding elements of a line segment in the same System themselves both around a like line segment vary , then the lying between the 2 new bounding elements line segment to the former is like.

In practice let the originating line segment be [ $\alpha\beta$ ] ( compare fig3) and [ $\alpha\alpha$ '] = [ $\beta\beta$ ']

Thusly is to show, that if all named elements are related to the same system
[ $\alpha'\beta'$ ] = [ $\alpha\beta$ ] be.

But it is [ $\alpha'\beta'$ ] = [ $\alpha'\alpha$ ] + [ $\alpha\beta$ ] + [ $\beta\beta'$ ]
concording to the definition of the Sum,

And there
[ $\alpha'\alpha$ ] = –[ $\alpha\alpha'$ ] = –[ $\beta\beta'$ ] exists

Thusly heaves up themselves [ $\alpha'\alpha$ ] and [ $\beta\beta'$ ] by considering the Addition, and it is really by working through
[ $\alpha'\beta'$ ] = [ $\alpha\beta$ ]

Footnotes
•This besigning of the line segment is Only for a forward runnining line segment. The enduring besigning of the same through its boundong elements can firstly become everyway standing, if we the knitting of the elements have learned to Recognise to be( see the second section § 99)
The besigning [ $\alpha\beta$ ] is chosen in the Ausdehnungslehre from 1862 for the product of both elements $\alpha$ and $\beta$ , which if [ $\alpha$ and $\beta$ are points, represents the line part between $\alpha$ and $\beta$ , whererom the line segment therethrough distinguishes itself, that in these ones only Length and direction, but in those ones at the same time, the position and unending-like, direct line comes to be held fixed, to which the line part is relating.

Therefore it is here around, thusly more therebesides, to firmly hold, that the besigning of the line segment through $\alpha\beta$ only a forward running Aide is, the material measured besigning $\beta-\alpha$ can concording to the principle of representing first in §99 com to be given. (1877)
•• above all, compare here §7

Commentary on §15

I must direct the reader to the induction as indeed Hermann constantly does. The labels are not what they might seem to anyone partially versed in the Mathematical arts. In particular the diagrams reinforce the direct line interpretation, but the treatment is equally applicable to circular arc segments and the endless-like circle to which they relate.

In particular do not regard the = sign as equating, but rather as likening. Many of the labels are not only assigned as representations, according to a general principle of representation, or representational theory, but also have a symbolic status as a likeness..

Hermann draws a distinction between this constrained useage here of the labels to the freer use we may adopt after §99. Hamiltonnalo in his theory of couples took a logically sustainable path to support the way we use algebraic symbols andcarithmetic ones, that is quite freely and often very loosely specified. This freedom is not safe if it's basis is not understood or comprehended. In addition the free use of analystical knitting will lead to negative assignments per se and that has to be justified in the context of line segments and elements of line segments.

Try to establish that [ $\alpha\beta$ ] = (–[ $\beta\alpha$ ]) to grasp the procedural or logical difficulty that we step over without thought.

The difficulty is not in space or our apprehension of a real extensive magnitude, but in our rules of label manipulation.

When the problem or construction task is "visible" we can move spatial objects relative to each other deftly and correctly, but when it is "invisible ", and not possible for us to reorient the construction we can imagine what the result would be. However, sometimes , even quite often, we reverse the spatial orientations of the result, because we overlooked the effect of changing relative positions, or worse still changing labelling. Here both Hermann andvHamilton draw attention to the constraints an "Expert" must consider to get the correct result.

We are taught to be very blasé about such concordances, but when you build a house or send a manned spaceship to the moon and back, reversing the resultant of a calculation or specific solution has disastrous consequences.

It just dawned on me, after considering Hermanns use of and reference of the figures , that the difference in sense between " ist" and " sei" is the difference between objective being and existence and subjective being and existence . Thus "sei" : let it be, may be , imagine it being , contrasts with " ist" : it is, palpably is, right in front of you being!

Thus (– $\alpha\beta$ ) is imagined where $\beta\alpha$ is not, as long as everything is running in reverse. Thus $\beta$ becomes the beginning element and $\alpha$ the end element, and we mentally have changed our cognisance of the one objective entity.

That one objective entity is thereby in Superpostion!

Superposition is thus a cognisance we have of an objective entity in which we imagine more than one state/ status exists, but which we can only glimpse unpredictably.

Thus all static situations are in superposition. Dynamic forces, motions, deformations may well be occurring beyond our ken within that object, but in such a way and at such a rate of change that all we perceive is the equilibrium state.

Positing such a state of affairs is in keeping with Hermanns and Justus religious convictions that Geist or God's spirit or mind is active in material reality in a way we must study to perceive, or isolate. The use of labels and symbols expertly and consistently can help the thinker, the meditator, the purchaser of Nature as profound objects of worth( diamonds, crystals, rocks) to keep track of where their imaginations are roaming.

Beyond that intensely personal use, symbols and labels serve to obscure and confuse. Thus to advance that Mathematics is the language of God or Nature is perverse. Such intense runs of symbols are nothing more than a ball ache! What is useful is when an interpreter comes and explains not the symbols only but the imagination and adventure they help to facilitate.

In a very real sense, a fractal image, sculpture or Chladni resonance pattern are interpreted by symbols and vice versa, if the encoding is expertly done and the automatic machine can decode it correctly .

Tautology is fundamental to our thinking and evolution of thought.

It is self confirming, and rather than begin logically inept( Aristotelian school of thought) it is in fact very confirming and convicting. Those languages that retain the reflexive participles of verbs are well aware of this self affirming nature of the tautological statement. The word itself comes from the greek ,tautes, taute,, which is/was a particularly troublesome version of the word auto ,aute, autes meaning "self". Thus self referencing was given its own participle in agreed. In English there was no equivalent, butbinmaddition Aristotles autistic brain could not handle it!

Most of us understand the role of emphasis and it's importance in convicting and developing confidence, but for Aristotle the truth is not made truer by emphasis! For most others of us emphasis and reemphasis does confirm, reaffirm and convict again the truth of an issue.

I use truth as an undefined notion here, as I do not actually usebthebconcept myself. I prefer the complartive " true". Such a comparison admits andcevenndemandsvtautology, for one thing may be truer to a standard than another, yet both are true to the standard!

$\alpha\beta$ = $\alpha\beta$ + 0 = $\alpha\beta$ + ( $\beta\alpha$ - $\beta\alpha$ )

Now $\beta\alpha$ we know Hermannmhas not yet deconstructed into its elemental structures. But if we look at his format we would have to describe it as "forward running. ". The issue then is the confusion about the term" forward". However if we say that $\beta\alpha$ is "running into against" then we have a sensible Definition.

This definition means we can say
$\beta\alpha$ = - $\alpha\beta$ .

When we substitute that into the above likening string we get

$\alpha\beta$ = $\alpha\beta$ - $\alpha\beta$ - $\beta\alpha$ = $\alpha\alpha$ - $\beta\alpha$ = 0- $\beta\alpha$ =- $\beta\alpha$

Thus a tautology confirms the substitution we have made, the decision regarding the definition of $\beta\alpha$ as being consistent with the procedures we are establishing for the labels.

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction of the simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§16. I now take to my side to scrutinise, in order to reach around to the knittings of differing artformed line segments , to nearby me, 2 differing artformed fundamental varyings
And let the first fundamental varying ( or the running into against set entity of it ) whimsically continue forth an element,
And then the thus varied Element in the second manner of varying likecasely whimsically stride forth
Thusly I come to be capable therethrough to create whole out of an Element an unending like crowd of new elements
And the totality of the thusly creatable whole Elements I name a System of second Step.

Further I take a third fundamental varying to my side for scrutiny,
which varying does not once again guide out Beginning elements from that second step System to an element of this System of second step
And which varying I besign of its behalf as independent from both those first ones, and let one whimsical element of that system of second step this third varying( or the running in against set entity of it) whimsically continue forth,

thusly comes to be the totality of thusly createable elements in order to build a representation of a system of third step
And there is no restriction set to this manner of creating whole concording to the label
thusly I come to be capable upon this method to stride forth to Systems of whimsically higher steps .
Hereby is it important to hold firmly
That all upon this manner created whole elements, are not permitting to become apprehended as by othermeans already given entities , rather as "springing out of a source like" created whole entities
And that they therehere all,
sofar as they are created whole "springing out of a source like" through differing varyings ,
also seem outwardly as differing, their label concording .

Thereagainst, is once again clear
That
Concording to the entity the elements one time are created,
they appear off from there as givens,
and their differing quality or identical quality can not by another come to be to decide upon, as if one goes back onto the original creating whole event.

Before I now go over to our personal task, specifically to the knitting of differing varying manners, want I the manifesting through the Geometrical trackings to come to aid .
It is specifically clear
That the system of second step to the plane inter-communicates,
and the plane therethrough created comes to be thought of
That all points of a direct line concording to a new, not enholden in it direction ( or concording to its running in against entity) themselves kinematically move forth , whereby then plainly the totality of the thusly creatable points builds a representation of unending- like plane .

It appears thuslywith the plane as a totality from parallel lines which all a given Direct entity intersect

And it is outwardly seeable
That, there these parallel lines do not cut themselves,
and also the "springing out of a source like" direct entities not yet a second time are hit, all upon that manner created points from one another are Differing , and thuslywith the analogy a completed entity is.

Thusly so one reaches to the complete un end-like space, as to the System of third step, if one the points of the plane concording to a new, not in the plane lying direction ( or its running into against set entity) kinematically move forth

And further can the geometry not stride forth,
while the abstract expertise knows no boundary!

Footnote
• how sometimes in the doctrine of space, all points are already originally given through the prior set out Space .

http://philsci-archive.pitt.edu/2155/1/Method.pdf

http://www.math.rutgers.edu/~cherlin/History/Papers1999/kijewski.html
Commentary on §17
I was going to start by referring to Newtons Dream of a dynamical point, but I cannot find the reference I have so clear in my minds eye. It is likely that recent scholarship has made that particular presentation inaccurate .

My research has shown that the philosophy of Rational Mechanics had progressed up the academic slope to become considered by Natural Philosophers who were basically theological philosophers. A treatise on rational mechanics was kicking around but it seems not to have been definitive.

Newton read widely influential authors, Huygens, Wren to garner speculUationsand to inspire his own explorations. The chief principles of rational Mechanics were formulated by Aristotle and then Archimedes, followed by Nicolaus and others this principle was that of continued motion in a straight line.

It was this motion, or dynamic that Newton extended. Into his idea of locii traced in space. But then he used an inductive argument to describe surfaces traced out by motions and finally spaces traced out by moving surfaces.

Whether these dynamic principles are in the rational Mechanics Of his times, or whether he vouchsafed them for Rational mechanics of his time I cannot find out, but it is clear that these notions are key to both Newtons and Hermanns construction of a representation of space.

Commentary on §17
The main points to observe is the inductive synthesis of the Systems. Each step is an inductive step in an inductive process.
The main constraint is the varying ( a) . The creative element is undefined but is constrained by a' the system step(s) is the count of the number of varyings thus
s:=#{ a}
Adding a different varing b
s:=#{a,b} =#{a}+#{b} = #{b}+s= #{b,s}

It is this last symbolic arrangement which is inductive.
On the face value it always gives the tally 2 but s can be any valid tally and so s is transformed by the process of tallying.

This is an infinite counting " loop" or an iterative definition.
Consider z:= z+ c
We know this does not make arithmetic sense , but it makes combinatorial sense! The := sign is one I learned only from computer syntax and it thus was described as a computational operator. Really it is a combinatorial marker, allowing the process on RH side to be subjugate to the label on the LHS. It is o ly inside a for loop that its significance appears clearly.

Since Hermann is describing an inductive process he stops the reader here to point out that each varying creates an element for the next step! Thus the process never has more than 2 " statuses" :the creating element, the created element output as a result of the varying.

However, the " combinatorial tally" of varyings increases at each creation of an element. The" element" become increasingly more " complex" combinatorially with each varying.
We can notate this combinatorial tally using subscripts
s_n+1= s_n+ 1 for n= 0,1,... s₀=0
The subscripts or superscripts or even index marks do the counting.

It is this use of subscripts or superscripts that hide the combinatorial nature of our labels.
Wallis called them potentials or exponentials in his algebra. Thus they became powers or exponents rather than combinatorial tally markers. They soon became lost in the fog of logarithms and lost their simple combinatorial association.

However the move toward these subscripts super becoming logarithms was accompanied by a fresh insight into the organising power of these combinatorial markers. They took on a role in a topic later called combinatorics , but very much obscured within larger subject bases.

Sequences and series appear as if out of nowhere, their combinatorial origins down played and their combinatorial significance From geometry and later topology , completely adumbrated.

As a child I learned to count, add and multiply, but never to combine the counted objects into combinatorial series. 'I explores sequences without realising the combinatorial structure I was imbibing. Learned disconnected rules about dimensions and units without realising a combinatorial doctrine was being imposed, or that one even existed.

The work that Justus did not only makes that clear it illuminates a murky basis to all so called "mathematics". It connects simple counting to increasingly mor complex counting driven by the increasing complexity of varyings in the objects counted .

Once you really get that you will understand the post I am writing for V9 about the origins of the Arithmoi in the Stoikeia. The Arithmos is a combinatorial mosaic. The Greek ' ar" means connection in a rhythmical pattern. This pattern is everywhere and it is fractal. Even in our disease arthritis it appears referring to the connecting combination of bones we call joints.

Combinatorics sits at the very roots of our rational systems : of thought, mathematics,architectural design,physical chemical and biological expertises, and technological innovations from mechanics to engineering to computer manufacture and design,to software design and implementation. The list goes on.

We cannot really progress much further without acknowledging the simple crowd doctrine Hermann and adjusts use. There are end-like crowds and unend-like crowds, thus we have bounded and unbounded, or unbound able crowds.

These crowds are crowds of any object Or entity.

Now let us consider the geometrical instance of these crowds. In that case the creating element may usefully be identified as a creating point, that is, an entity that creates points!
Thus first varying is a whimsically chosen orientation and the " point" moves only in that orientation obeying the variation constraints. Such a combination of orientation and motion is called a directed motion or a Direction. . The creating point under this direction creates a Line or Line segment.

The line segment is an end-like crowd of points, but it has no Finite tally! Thus a finite crowd is perhaps a misleading descriptor of it.

In addition let us also consider the motion that creates an arc segment or a circle. In this case the varying constricts the creating point to a fixed measure from some central point . That point is mysterious until we develop the second step system.

In the second step system a second varying causes any created point from the first step that obeys its command to move . This creates in fact an entity that consists of directed line segments or circular arc segments that are parallel ( or concentric ) and non intersecting. Or these lines and arcs may also be unend-like . No matter which, they consist of an un" countable" crowd of created points.

Now it is clear that the running into against set direction or motion constraint on the circular arc will produce similar results.. Thus the system of the second step is a finely ruled surface called a plane, consisting in either parallel lines or concentric circles that intersect the system of the first step.

If the system of the first step was a circular arc, then the pattern becomes more intricate. The parallel lines no longer occur. Directed lines that intersect at 1 pont called the centre radiate through the arc or circle;

or arcsegments or circles of the same constraint as the first pass through this point called the centre;

or arcs from one central point (a point which is the centre for these arcs) as concentric circles of expanding size intersect this circle in 2 points or just one, of which there are 2 such points called points of tangency;

Or indeed this may be the case for every point created in the original arc or circle.

Such a surface is indeed a " plane" surface and clearly has more complex potential than the first type of surface. But they too are an uncountable crowd of creating elements.

From these types of crowds of created elements consisting of so ordered and arranged point creating elements we may proceed to ever more complex but finely ruled lines and / or arcs in the system of the third step. Such a system usually called space is thus neither empty nor in fact simple, but rather full of every potentiality of motion.

Now Hermann draws an oft misquoted distinction. " geometry" can go no higher than step 3, but his expertise knows no bounds!

Geometry, my friends is not space!

It is clear from the outset that these systems are interpretable by Geometry, but in his day Geometry was restricted to 3 orthogonal axes. Lagrange in fact felt that this was the nature of space also despite pursuing these notions as generalised coordinates. But then everybody believed space was Euclidean, the new nonEuclidean spaces were fictional, and Newton was a Divinely inspired Messenger from god. His laws were immutable because they were the Laws of God and Nature!

For a theologian this is not too far a conclusion to leap to. However it confuses empirical data with "Rational" opinion. In those days Rational meant inspired by the word(Logos) of God. Today scientists and atheists have commandeered the word to attack any religious or superstitious belief system! What poetic justice is that?! Such systems are Irrational while the Rational system remains curiously undefined! For to say it is the "scientific method " is to fall into the greatest state of gullibility ever created!

Any professional scientist will soon disabuse you of the fairy tale nature of the so called scientific method!

So it is then, that some would have you believe that space is an empty nothing, that it consists of only 3 possible " dimensions" and it is predominantly ruled by straight lines. Finally that all forces or impulses act only in straight lines!

If you have a moment read both Örsted and Ampères accounts of their empirical research findings!

Hallo Jehovajah,

Thanks for the link Normans Video on Higher Data Structure.

During my seasons holiday I have worked on Data Structures from a software developers point of view by developing some packages for abstract data structures for example I have a generic list package, a package for manipulation of bit arrays and digits. So I have started implementing the basics of mathematics from a software developers point of view.

Instead of a mathematical prove I have to convince my Ada-Compile, that my code is correct Ada-Code and I have to watch my programm if it behaves as expected.
May be I find some time to present the results on my page on Ada-Data-Structures.

As usal the holidays are over and I have to concentrate on my daily bussiness.

Hermann
P.S Here is the actual specification of my linked list:
(The generic parameter may also be a linked list.)

generic
type Value_Type is private;
package Primitive_List_Generic is

type Value_Array_Type is array (Integer range <>) of Value_Type;

type List_Type is limited private;

function Is_Empty (List : List_Type) return Boolean;
function Length (List : List_Type) return Integer;

procedure Append (List : in out List_Type; Value : Value_Type);
procedure Append (List : in out List_Type; Value_Array : Value_Array_Type);

Empty_List : exception;

function First (List : List_Type) return Integer;
function Last (List : List_Type) return Integer;

function Convert_To_Array (List : List_Type) return Value_Array_Type;

Out_Of_Range : exception;
Undefined_Element : exception;
function Get_Slice (List : List_Type; Start, Stop : Integer) return Value_Array_Type;

type Element_Type is private;

private

type Element_Access_Type is access Element_Type;

type List_Type is record
Number_Of_Elements : Integer := 0;
First_Element : Element_Access_Type;
Last_Element : Element_Access_Type;
end record;

type Element_Type is record
Value : Value_Type;
Next : Element_Access_Type;
end record;

end Primitive_List_Generic;

Thankyou Hermann. You have helped me decide what I am going to do with some old papers and studies I have to sort through from the loft!

I am currently, as you are, in a bind. I would like to progress my work on the Grassmanns translations but I have to prepare for my family so things are ticking over very slowly. However I am heartened by your communications and interests. These constructivist processes are crucial to understanding the Grassmanns state of mind.

The Formenlehre or doctrine of thought patterns is so broad that it is easily missed and segmented into differing subject boundaries. The rise of combinatorial studies has been a welcome but mysterious adjunct to Mathematical doctrine! The historical provenance of this collection of studies is often obscured in topological, mathematical and computational subjects. I believe the Grassmanns indeed pioneered a Key or set of keys to understanding how we can think and model patterns in our sensory experience.

The connection to language and particularly Sanskrit is so fundamental that mathematicians are blinded to it. This blindness arises out of Aristotelian subject boundary wars! This is why Hegel, the Grassmanns and the Prussian Renaisance was so powerful. Aristotle was debunked by this time! Sir William Rowan Hamilton to Newton were greatly enamoured of Aristotle, sir William Gilbert was not! Constructivism arises by overthrowing Aristotelian Platonism and sourcing back to Pythagorean schools of thought. Through these sources comes the treasures of all past great civilisations and language cultures!

So for me computational science is tha natural home for so called Mathematics. In fact it may well be the modern conceptualisation of Hermanns tentative suggestion: " Formenlehre" .

The stumbling block has been the shifting mis identification of Algebra!
Historically it really arises as the Indian method of generalisation. There it has a long and perfected tradition based on constructivist ideology. It was not called Al Jibr which was an epithet given to it by Al Khwarzimi! It has many Sanskrit names for each of its departments and applications but let us say Gita and Sutra are Sanskrit indicators that we are involved with this system of thought patterning and expression.

So Al Jibr has a vernacular meaning: " mind fluff!" . This is a classical joke! Scholars from Islam found this one of the most difficult processes to apprehend! This was because you had to become devotee to an Indian guru to be properly trained in all these things. Outsider: the Greeks, the Arabs, Christians did not want to give up their belief systems to learn this stuff! And yet it was clearly a higher learning and facility.

Bombelli is perhaps the most influential but least famous westerner who demonstrated the engineering, constructivist power of this thought system, but there are many artists and engineers who guided by the Pythagoreans filtered this system into Western thinking including Wallis, and by default Newyon, De Moivre and Cotes.

The Greek model was the Arithmoi, or do we are told. But in fact this was the Pythagorean school of thought redacted by Plato! Little is truly known about Pythagoras in the west, but of course there are Indian Buddhist traditions about all our great heroes including Jesus!

I mention this only to encourage you Hermann in a great tradition of language study and implementation derived from Sanskrit pioneers. Your computational explorations are language process explorations and this is what drives you, me The Grassmanns and NJWildberger!

http://m.youtube.com/watch?v=a8Ufs4lowc4

http://m.youtube.com/watch?v=sW_IkMQEAw
Norman reveals the heart of the inductive method of defining an Arithmetic.
For those not familiar with Justus Grassmann this was the key stumbling block in his Logicsl Doctrine of Mathematics, explored in his work for Nature Clients and Physicists! .
At the moment I am not clear if this is how Hermann solved his Fathers difficulty by utilising an inductive methodology to define everything as Norman illustrates, but if it is it certainly is a testament to sophisticated understanding of thought processing or thought patterning everywhere in evidence in the Ausdehnungslehre of 1844.

We can clearly see that Algebra is a symbolic Arithmetic, and that Geometry is an allied explanatory system to this exploration of inductive thought patterning. The Begriff or labelling/ handle designing becomes a separate but important co dependent expertise and practice to put these Notions into a useful and efficient aid to combinatorial process that underpins calculus: that is direct counting using calculii or calculation.

The geometrical interpretation serves as a reference guide only. Soon the thought patterning itself sparks off remarkable analogous thinking that unifies widely separated subject areas that rely on counting expertise!

Is Geometric Algebra the Grassmann Formenlehre?

As wonderful as GA and its derivatives are I do not yet think so! The Methodology and Systemology of the AusdehnungsLehre is probably better evolving into the area now called Homotopy.

The lineal Symbol was chosen by Grassmann not as a line, but a representation of both intenive and Extensive magnitude. Thus his methodology can be applied to spaciometric topologies or to spaciometric metaphors of intensive experiences . That is : the topologies we choose to locate in our experiential continuum both inside and outside the boundary of our skin or proprioceptive sensors.

For induction or recursion read fractal iteration .
Z= Z+C is Mandelbrots famous definition of " almost similarity" . This inductive thought pattern underpins all Fractal geometry so called and speaks to the dynamical nature of space-time or the aether.

To go from there to Spinors( Twistors) and n-dimenional Fourier reference frames is perhaps a surprising connection to some, but reflects the triumph of the Grassmanns labours and research into putting human philosophy onto a "scientific " or rather Hegelian basis!
I do not think Justus subscribed to Hegelian philosophy, but Hermann certainly embraced it and probablynRobert did too to a lesser extent since he had his own agenda to promote.

Do not get too worried bout the technical detail! As many researchers will tell,you after years of labour" Hermann has already sweated the hard stuff!"

One of the main readons I am diving in to the technical detail and background of the Ausdehnungslehre is because historically Hermanns plans to produce a second Volume that I have dubbed the Schwenkenlwhre were thwarted by events. Yet much of the material survives in a reworked form in the 1862 redaction produced in collaboration with a sympathetic editor who nevertheless imposed his own slant on the finished product : his brother Rovert!

I have written before about this collaboration but suffice it to say Robert saved Hetmann and the family nme fom obscurity ! Hemanns reprint of the 1844 version in 1877 with intense Anotation additional appendices and back references is an attempt to correct this after the material became critically significant. Fom these notes and the sketch of the Schwrnknlehre I hope to come to understand Hermanns original thought patterns with regard to arc segments as lineal symbols .

My suspicion is that the general theory especially the Elementar presentation extends simply to this arc segment description of rotationlal extensive magnitudes but not so intuitively to rotational intensive magnitudes( whatever they may be in our experiential continuum) ! I suspect that spiral forms fulfill this rotational intensive magnitude concpt, thus requiring both the arc and the lineal segments to describe.

In my opinion thebarcvnd lineal segments together form the fundmental basis for representing spacetime / the aether, both extensively and intensively . It is possible thereby to model an inherent computational consciousness in spaciometric dynamics.

http://youtu.be/oNqSzzycRhw
http://m.youtube.com/watch?v=oNqSzzycRhw

This series of videos is interesting because it reflects the application of the Mechanical principles as derived in classical times particularly by LaGrange and D'Alembert . Thus it is directly the kind of systems Hermann was studying while applying his Förderung!
The matrix presentation is conceptually derived fom Cayley and others, but that contribution would not have been so well received had not Justus Grassmann tackled the combinatorial problem of symbolic arithmetic!

The combinatorial problem is how to organise the large number of equations that describe any mechanical system on the page! These considerations apply to any arranging of symbols on the page. We see that the summation sigma becomes useful to organise these procedural combinations. Basically these summations are arrays of staus/condition markers.

These arrays as combinations appear at all levels of organisation, and Justus particularly derives it from studying the presentation of mineralogy taxonomies and classifications.

So the arrays are fundamental to the description. Hermann chose to use summation notation, Cayley decided the arrays themselves were more accessible.

For presentation the matrix notation is a great boon but the content of the presentation was worked out before Justus and Hermann and Cayley. However it is Justus and Hermann and all Ring and Group theorists(Lee, Abel) who highlight the combinatorial layout of the symbols on the page

Hermann took it to be alongside a geometrical derivation in association , but later realised it was a combinatorial methodology for mathematical thinking about Foms! The line segmnt then became symbolic for a directed magnitude of any form or intensity.

When bishop Berkely asked what a Fluxion was he did not intend to elucidate the answer but rather to defend the Faith. Amongst the faithful he caused a clerical split: those who supported ate British view and therefore Newton, and those who supported the European view and therefore DesCartes school . That school was revolutionised by the French and so a godless mechanical philosophy based on rational logic and logical propositional mechanics / mathematics of the astrologers was championed in Europe . The clergy was thus under pressure to defend the traditional view of the faith.
Newtonnwas seen as the head of this particular snake and he was ruthlessly attacked in his own time. Despite the French, and the British and Kant defending and promoting Newtons concepts it still took 600 years for him to be recognised as the father of modern Astronomy.

So Fluxions were confused with European infinitesimals by some Leibnizians , whereas the Newtonian infinitesimal was not a number at all but a moment in time.
A Fluxion was a dynamic quantity which had a temporal property, and It was represented by a line in dynamic flux. .

We might well regard Herakleitos as the progenitor of this dynamic conception, but it is perhaps the Pythagorean school that codified the dynamics of the line segment as part of a rotational system. However the western redacters of their work imposed a static sense to their philosophical discourse in the Stokeia. Newtonnwho read the original derived the pure dynamics of the Greek thought and carried this through into his own works.

It is in his papers on motion that he introduces the infinitesimal time stamp so characteristic of his notion of Fluxions.

It is certain that Newton intuitively worked in a lineal,algebraic way, but he despised revealing his surmisings in algebraic notation, something Wallis pleaded with him to do.

So it should be no surprise that the Ausdehnungs größe are Fluxions. Certainly by the time Justus had presented his combinatorial findings to the Nature and Mathematics lovers societies , the dynàmic in nature as symbolised on the page were being set as a true foundation .

Hermann Grassmann was inculcated with these ideas and viewed the Strecke as a jostling entity. These dynàmic entities arevNewtonian fluxions

Fluents are Newtons intensive magnitude or quantity associated with a Fluxion. So for example displacement is a Fluxion in a dynamic system . The fluent is the velocity or speed of the displacement.
Or mass change isbanfluxion in a pressure dynamic, then density variation becomes a fluent in fluid dynamics alongside rotationnshearingband general stressing of a fluid volume.

If a Fluxion is combined with a Fluxion the result is a Fluxion . Hermanns line segment process and notation highlighted this lineal algebra

The product of fluxions was dealt with by Newton as resulting in a Fluxion of a different magnitude . Thus a linealnfluxion becomes an areal Fluxion when in product with another Fluxion. However it was Hermann that dotted the I 's and crossed the t!s here, and really exponded on the role of product design .

Several fundamental products were systematised in his process or method .

Newton dealt with the fluent of a product of Fluxions. This often causes debate amongst those who use the differential geometry the dynamic differential is different . The Fluxions modified by its temporal property not its spatial one..
Thus if a,b are fluents of X,Y then bX + aY is the fluent of the Fluxion product XY .

The the product of fluents is discussed in higher order fluents , but suffice it to say that the product of fluents is a vanishingly small quantity. Whereas we effectively ignore it, in point of fact we fractalise it by applying the same product rule. The extended expressions so obtained of course are an infinite process of smaller ans smaller fractions. As such a process the principle of exhaustion may be justifiably applied and that is why we truncate it at the first 2 terms of the fluent.

We are not determining the area of the supposed parallelogram in determining the fluent .,we are determining the intensity of the fluxionnvariation or change and that is mostly in the gnomon to the parallelogram not in the corner or verticial form . This proportional relationship is why we can use the fractalnexpressionnover and over with certainty , something the proponents of infinitesimals in Europe failed to grasp.

http://m.youtube.com/watch?v=-wokVaaGFgA

In this remarkable series Norman is at last able to present the general beauty of his rational trigonometry in the context of lineal algebra.
It has profound implications for teaching so called mathematics.
There is one misconception he holds on to and that is angles. He misconceives what an angle is, or rather he deals with the general nd pervasive misconception of the term " angle" .

The arc length Ø is dealt with both by the Pythagorean school, Sir Robert Coates, Euler and indeed Newton and DeMoivre. It is a corespondence between the diameter( radius) of a disc and how far its centre moves along the diameter as it rolls on a flat surface. It is a ratio between non homogenous magnitudes where the eternal curve is compared irrevocably to a Rectilineal line by a process of point to point matching. The point on the curve is called the tangent point the Rectilineal line through this point and only through this singleton point is called the tangent line.

In terms of infinite processes this is an unimaginable definition! In pragmatic terms we accept it from a diagram as existing as drawn. There is no logical connection between the drawing nd the statement of definition. The drawing is always perisos( approximate) the definition artios( perfectly fitting)

So the arc as a magnitude is quantified by sub arcs, but the usefulness of these sub arcs depends on a correspondence or ratio( logos) that can be developed in an analogos( proportional) way.

Thus what Norman makes explicit is that ratio and proportion theory Eudoxus sets out in books 5 and 6 of Euclids Stoikeia.

While Norman rightly points out the infinite nature of certain ratios when reduced to a single standard form, it is the attitude that asserts we can ignore pragmatics that is questionable, and indeed misleading.

Eulers and Cotes theorem involve the cosine and sine Ratios, not infinite expressions of the same distinguished not as polynomials, or even power series , but as complete functions!

But here Norman demonstrates how the doctrine of extensive magnitudes leads to algebraic simplicity,uniformity and applicability to multiple dimensions AS directions.

It is the notion of orientations as dimensions that makes lineal algebra so powerful . In addition the direction of travel in a given orientation is fundamental to a specific conception of Dimendion in space. Dimension in general relies upon metrons by which quantification processes are established. So we have dimensions of space, mass and time, and dimensionless quantities like angle which are ratios. When I draw an arc length, I identify a dimensionless quantity called annngle or a corner. It is dimensionless precisely because it is encoded by a ratio, and recorded in ratio tables of approximate "Values" or results of a division process.

The tables have always been exact(artios) or approximate( perisos) results of a difference process called division. The power series expressions of this difference process or division provide a systematic way to find these differing values, and should not be used to replace the underlying ratios.

In the modern world of driving nd the McIntyre world of gears arc length is important, but for construction the sine tables have been specifically calculated to help engineers pragmatically. It is only through the discoveries of men like Ōrsted, Ampère Faraday and Boscovich that the long held belief in the vorticular motion of natural powers has established the applicability of the sine ratios to these invisible yet powerfully manifested phenomena.

As part of this approach to mathematical description it is imperative to understand that circular arcs are legitimate extensible magnitudes that describe rotational dynamics in an algebraic way set out in the Ausdehnungslehre

http://youtu.be/f81crFAj5UE

http://m.youtube.com/watch?v=f81crFAj5UE

Here Norman clearly sets out the inductive step that underpins the Ausdehnungslehre .
The first point is historically the bits and pieces lay scattered throughout geometrical experience. What Justus Grassmann did was to align the whole onto a consistent combinatorial basis. It is this combinatorial presentation that allows induction to flourish in defining more complex stages( stüfen) .

It is these stages that are inductive which allow iteration or recursion to be inherent in the construction of notions, notations and processes and procedures.

Thus we embed, by the combinatorial method identified by Justus, but not invented by him, the fundamental concept of Fractal Geometry : almost self similarity at every stage!

When Benoit pulled together his intuition based on visual forms he repeated the same practices of the ancients but more boldly and bravely than his teachers, who had become trapped by convention and dread of getting the answer wrong!

The ancients appreciated that which could be exactly described and that which was forever approximate. They did not fear the approximate, but of course they venerated the exact. But the real dynamic mechanical world relied on applying exact concepts approximately, intuitively and iteratively. The principle of exhaustion deals with that issue, and thus with fractal geometry as a pragmatic expertise, obscured by exact aesthetic rhetoric, and defensible logics( systems of debating or argumentation to defend a presentation).

What is truth? One hopes that logic would capture it, but must realise that logic itself is not truth!
One would hope logic would be true, but realise that it is as bent as the foundations it is presented upon!

So, as impressive as Aristotles linguistic and grammatical rules approach is, it is as flawed as the persons who utilise it in isolation from their ensemble of senses. The Pythagoreans worshipped the Musai, they venerated all the gifts and arts and senses they identified in any gifted individual willing to develop their gift into an expertise.

The method of training was not by examination but by Koan, by overcoming a task identified as truly given to you by the Musai. You were welcomed into the seniority of the Pythagoreans not by some written exam pass but by being clearly capable, knowledgeable and expert in the gift given you by the Musai. Then as recognised Mathematikos you could discourse with the other gifted Mathematikoi and mutually enhance one another's expertise.

Fractal geometry offers that opportunity, in a way the old Academic subject boundaries doesnot. It clings on to life and dynamism, not dry dead bones however beautifully arranged.

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Fractal Math, Chaos Theory & Research => Mathematics => Topic started by: jehovajah on March 04, 2015, 12:03:01 PM