Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[jehovajah]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[jehovajah]

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

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

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

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

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

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

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

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

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

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

[hermann]

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


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

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

Hallo Jehovajah,

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

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

Hermann

[jehovajah]

Thanks for the interest and contributions  to the thread.

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

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

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

[jehovajah]

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

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

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

[jehovajah]

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

<a href="http://www.youtube.com/v/jPwo1XsKKXg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/jPwo1XsKKXg&rel=1&fs=1&hd=1</a> the royal society channel

<a href="http://www.youtube.com/v/jPwo1XsKKXg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/jPwo1XsKKXg&rel=1&fs=1&hd=1</a>

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

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

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

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

[jehovajah]

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

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

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

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

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

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

[jehovajah]

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

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

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

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

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

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

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

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

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

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

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

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

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

[jehovajah]

Ausdehnungslehre 1844
Induction

A: deriving the Label/ handle of the pure Mathematic

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

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

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

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

Ausdehnungslehre 1844

Induction

A; deriving the labels/ handles of the extensive magnitudes study.
Deriving of the notation of the pure Mathematics

1.The supreme partitionIng of all the Expertises is the one into Real and Formal. From the former ones the experiential continuum  constructs an impression in thought , as what is self evident undeniable to the thought process; and which expertises have their veracity in the super conccording of the thought process with that  experiential continuum.

The latter on the other hand are they which have their contents  self established  by thought and which have their veracity in the super concording of the thought process under them.

Thought is only in relation on top of an experiential continuum, what to it has become impassable and through thought has been constructed into an impression, But this experiential continuum is for the real expertises a self evident one, the outer presentation of the thought that stands by it, while on the other hand the formal expertises the one which self exists by thought , what now further establishes itself over against a second thought act  as  an experiential continuum!

Now even if the veracity , importantly, rests with the super accord of the thought with the experiential continuum, then especially so the veracity of the formal expertises rests with the super accord of the seond thought act with the "established through the first thought act "experiential continuum, thus in the super accord of both thought acts.

Demonstration in the formal expertises thus does not go over the thought itself,over far out into another Sphere of experience, rather it in the combination of different thought acts governs its presentation. Therefore the fundamental grounds of the real expertises are fundamental empirical statements , while the foundations of the formal expertises are only definitions! Fomal expertises are not permitted to make fundamental empirical statements as real expertises are; they must build their structures on foundations of definitions.

Footnote

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

[jehovajah]

Commentary

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

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

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

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

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

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

[jehovajah]

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

[jehovajah]

Ausdehnungslehre 1844 Induction.

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

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

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

[jehovajah]

This is Spooky!

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

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

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

Get on with the translation then should I?

[jehovajah]

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

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

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

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

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

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

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

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

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

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

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