The Theory of Stretchy Thingys

[jehovajah]

part of the figures at the back

[jehovajah]

I have made several searches online to see if anyone has posted a copy of these geometric drawings to no avail.

http://www.forgottenbooks.org/
I had forgotten about this resource. If you type in Grassmann you will be able to find readable copies of Hermann and Robert's works.

You will notice that Robert published his own version of the Ausdehnunglehre after Hermann's death . You will also find the 1862 redaction of Hermann's ideas, and see it is a completely different book! It shows the intense mathematical rigour of his brother Robert. That is not to say that Hermann was not rigorous, but Hermann was certainly more innovative and flexible.

Roberts own philosophical viewpoint dominates the redacted version, which was done at a time when Hermann had come to the end of his tether and resigned himself to obscurity. So Robert was able to edit the new version with little real resistance from Hermann. It was only on the relative success of this second book, really only evident after Gauss and Riemann died, that Hermann's hope revived! He then reasserted his authorship by republishing his original book unaltered and heavily annotated with references to the redacted version, under his name.

I have concentrated on the 1844 version because I hate what mathematicians do to relatively simple ideas! We code it in symbols and make the symbols pretty. Then we try to convince others that this makes everything perfectly clear!

One cannot do mathematics without meditating, and one cannot explain mathematics without sitting beside the interested person and chatting and demonstrating.

Norman in effect does all this. His symbolic work is always behind him. To paraphrase  Newton, he is the main show, not his algebra!

This project is really of interest to me because I can read Grassmann's own words and grasp his thought process because he made it accessible.

I have emailed 2 resources to see if they can get me a copy of these geometric drawings.

I would appreciate it if anyone can get hold of a copy of the drawing on page 324 on a fold out sheet. Thanks, especially if we can publish it here as part of the research.

Plus anyone who can translate any part of the Ausdehnungslehre and publish it in this thread would be most welcome to.

I know that Kannenberg has done an excellent translation, but that is not what I am after. Hermann speaks authentically and philosophically not so much mathematically. His words resonate and radiate beyond mathematics. I would like to capture your response to those words after so long a time!

[jehovajah]

This quote is from the Encyclopedia of Scientific Biography link in a post above.

Other Mathematical Works. Despite the long neglect of his ideas, Grassmann was always convinced of their importance. In several works he attempted to show how the theory of quaternions and invariant theory (then called modern algebra) can be understood on the basis of the Ausdehnungs-lehre. Still more important, however, are his writings on the “lineal” generation of algebraic entities, in which he also draws on his theory. This group of publications deals, for example, with the theory of constructing points of algebraic curves and surfaces by simply drawing straight lines and planes through given points, as well as with the determination of intersection points of known straight lines.
As early as 1721 Maclaurin had demonstrated that given the three points a, b, c and the two straight lines A, B of general position in the plane, the locus of the third vertices of all triangles the first two vertices of which lie, respectively, on A and B and the sides of which pass, correspondingly, through a, b, c is a conic section. In terms of the calculus of points, this statement means that the mixed outer product (a × AbB × c) vanishes. Grassmann made the important discovery that in this way every plane algebraic curve C can be generated lineally. As a result, if C is of order n it can be described by setting equal to zero an outer product in which, in addition to symbols for certain fixed points and straight lines, the expression for the variable point x of C appears n times. A cubic can be expressed, accordingly, as (xaA) · (xbB) · (xcC)= 0. This signifies that the locus of the point x of the plane is a cubic, if the line connecting x with three fixed points a, b, c cuts the three fixed straight lines A, B, C in three collinear points. Moreover, one can obtain every plane cubic in this manner. Grassmann was thus able to refute Plöcker’s assertion that curves higher than the second degree could be conceived only in terms of coordinate geometry. In writings collected in volume II, part I, of the Werke, Grassmann considered, in particular, the lineal generations of plane cubics and quartics, as well as of third-degree spatial surfaces. (One of these generations bears his name.) He demonstrated that by setting equal to zero the products he designated as planimetric or stereometric, all these generations could be obtained from the Ausdehnungslehre.
A large portion of the Ausdehnungslehre is devoted to analysis. Grassmann treats functions of n real variables as functions of extensive quantities of a base domain . Since he introduced a metric into in the form of the inner product, he as able to derive Taylor expansions, remainder formulas, and other items. His most important studies in analysis concern Pfaff’s problem-that is, the theory of the integration of a Pfaffian equation
This question had interested leading nineteenthcentury mathematicians both before and after Grassmann, especially PfafF and Jacobi. Grassmann contributed the following important theorem: If one calls k the class of ω -that is, the minimum number of variables into which ω can be transformed-then, when k = 2h, ω, can be transformed into the normal form
and, when k = 2h- 1, into
p · (dzn + zn+1 dz1 +… + z2n-1 dzn-1),
where p is a function of z1… z2n-1. Even these results, however, which appeared in the 1862 edition of the Ausdehnungslehre and surpassed Jacobi’s achievements, obviously did not attract much attention. Recognition had to await their translation into the more customary language of analysis by F. Engel in his commentary on Grassmann’s works.
The calculus of differential forms, which is based on Grassmann’s outer multiplication, occupies a firm position in modern analysis. This calculus has enabled mathematicians to develop differential geometry in an elegant manner, as is particularly evident in the work of E. Cartan.

The whole article is worth a read. For example, Mõbius role in Grassmann's life is made clearer. In the past I have speculated on the role of Gauss and Riemann in influencing the progress of Hermann's work. I found documented evidence of Roberts influence, and this article gives evidence of several others. I find it hard to swallow the line that Grassmann is too difficult to understand! I think that documented evidence shows that established figures just "took agin" him because he would not conform! In today's spin doctoring speech he was bringing mathematics into disrepute!

That said, because of the times certain metric norms are incorporated in his Ausdehnungslehre that could usefully be revised , and by this I mean the sign rules. These rules are fundamental to calculus in that they underpin aggregation and disaggregation, but they also underpin half turns and whole turns around a centre of rotation.

The difference between lineal and linear is also intimated in this article, and that is a fundamental distinction. I also suspected Jakob Steiner was a pupil of Justus Grassmann and that seems increasingly likely.

The " militaristic" or structured nature of Prussian society also supports the concept of the definition of a mathematical function from a social metaphor.

Hermann, like his Father was a hard worker. To have produced so much is testament to the difference that avoiding television and video games can make to ones productive life. On the other hand, most people would not work in this way even in that era, so it is not conclusive that modern entertainment is cause.

[jehovajah]

AUSDEHBUNGSLEHRE

The second Impression, with regard to text not changed all over.

Print layout by Otto Wigand 1878
Leipzig

Foreword to the first Impression

If I designate the work , this first part which I am handing over to a public readership ,as a by-product of a new disciplin in Mathematics, then such a heading can only be justified by the way the work itself has to have been given to come into being)( justification of such a claim can only come to be given by the work itself )
In support of which, I myself every other justification therefore submit( hammer home!) in describing how I reached these conclusions laid out below. Step by step, I go over the way exactly as much as that is do-able , so that all that initiated this new discipline is presented.
(to bring  to the manifestation of it.)

[jehovajah]

Ausdehnunglehre continued
Grappling with the negative in Geometry gave me the first spur onwards. I made it my habit to apprehend the Strecken ( line segments) AB and BA as magnitudes that sit in opposition to one another, from whence thus proceeds onward that, if A,B,C are three points of a direct line then it is always the case that AB + BC = AC! Wholly so even if AB and BC are drawn in the same mannere, as also when they are drawn in an opposing manner: that means when C lies between A and B. ( drawn in a manner set against one another)

[jehovajah]

Ausdehnungslehre

In the cases of the last two, AB and BC were not simply apprehended as lengths but their directions were applied as an equal attribute  to them. Through which even, Making it possible that they were perceivable as set against one another!

And thus the difference drew attention to itself ( urges itself on ones attention) the distinction between the sum of lengths and the sum of such  Strecken, in which the direction was firmly applied as of equal importance.!

From here out the uncritical support of these ideas itself gave  rise to notions that the last used terminology of the sum , the last used handles  was not just simply for the case of the last mentioned sum where the Strecken were directed either in the same manner or in a manner set against each other , but also for every Other case set down unswervingly. This can be demonstrated on the simplest cases in which the Law that  AB + BC = AC is true, then it may  be  vigorously held as true even if A, B,C are not laying in a direct Line !

[jehovajah]

Commentary:
I found this passage about the law of 2 Strecken and 3 points difficult to get a sense of the verb tenses, and more importantly what Grassmnn was thinking. It all depends on Förderung which was difficult to grasp by me, not having read a lot of pre 20th century  literature in German.

However I see clearly the mechnism of the terminology or rather the naive simplicity which is that for 3 points a stretch from A to B  followed by a stretch from B to C brings you to the same point C as a direct stretch from A to C.

This notion is intuitively clear, but it hardly seems mathematical or algebraic or numerical. This is why it was overlooked. Hamilton in his paper on the theory of couples missed this point. Or this stretch. It is the more remarkable because Hamilton caught the notion of a step. And he understood the necessity of assigning complex notations to accord these steps with the then arithmetical norms..

Assignments like this are fundamentally representative theory. But Hamilton missed the geometrical representation of steps by a single resolving line. He missed the general notion of a sum of any 2 Strecken, because he was fixated on Numbers not Geometry.

This particular passage , on the face of it looks like a definition. But logically speaking Förderung means it was purely a bit of wishful thinking! As such it stands naked and vulnerable to being dismissed as nonsense.

What it actuall represents is Grassmnn going ith a notion even when his normal thinking is screaming at him, this is nonsense. He has to vigorously defend his process not to his colleagues but to himself. This is why he defines a law or uses law terminology where none exists. . Here he creates a general law or rule by fiat, or of 2 cases where the "law" can be unequivocally demonstrated. It is the law that makes AC equal to the sum of other 2 Strecken when the 3 points are not collinear.

Importantly, he analyses why we might intuitively dismiss the law: because we do not regard the Strecken we regard the units of length!

Hermann learned to view the lines as symbols with 2 inherent but undefined properties: magnitude and direction.

In fact reviewing this conception which he fostered( Förderung) we find finer instinctive attributes of the line symbol. We also see the logical confusion possible by those who do not use force of will to promote this concept over the normal inculcated thinking about summation in geometry.

The notion of a Strecke as a symbol is also novel. It requires one to not see a Strecke as a construction line between 2 points, and thus of a fixed length period! It requires the concept of a dynamic experience bounded by these 2 points. The concept is enriched by more and more applications, so it becomes difficult to define it by a single example. However, it seems to me that Grassmann meditated on this Strecke and thereby found it was symbolic of so many experiences and processes.
Förderung is crucial to this development , because it describes his meditative approach to the symbolic nature of a well known element of geometry, the segmented line.

[jehovajah]

It has struck me how close the notions of Förderung and axiom are. The notions of postulation , a begging of privileges , belies the inherent coercion barely concealed in our systems of merit and advancement.

And this is all in the name of the gods truth. For in the past, truth was not known, except it be the actions or words of a god that were witnessed by those who observe. And for many, that a god or fate should survive a victor was trial enough!

But later more sophisticated ways were adopted, in which a contender willingly submit, in acknowledgement of the alpha male or female. Thus falsely assuring them of right and truth, only to be dismayed by a victorious contender!

When , in ancient Greece philosophers chose to debate publicly on many issues, the victor was he or she who won the heart of the crowd. Thus it became admissible to attack the debater's character, irrespective of the merit of his ideas! To avoid such castigations rules of logic were formed in the schools of rhetoric. Whether they were all adhered to I do not know, but ad Hominem was included in these rules as justifiable in the real situations of court hearings and trials of veracity.

It was however preferred to advance an idea systematically and based on clear and obvious connections, or previous convincing demonstrations. . The ground of such an approach was a Förderung, a postulate, a proposition in which one is begged to suspend judgment and be open to the arrangement set forth..

Quite often, these postulates would be persuasive analogies or metaphors, which mimicking an accepted experience would then be used to justify some undiscovered connection. The veracity of the connection was supposed to justify the means by which it was arrived at.

However these days, getting a correct answer does not necessarily justify the method. Other criteria are used to justify methods and much stricter protocols are used. Thus the Förderung has changed over time to make a more rigorous platform on which to develop theoretical and hypothetical notions into the category of "truth" or rather a " true theorem".

In the end it is we who decide by consensus which experiences of individuals we will value and which we will denigrate. Thus we do not do so on any intrinsic truth value but merely on our consensual considered opinion, our consensual Förderung!

[jehovajah]

Good news.

The New York Library has responded and i may soon have digital images of those Grassmann constructions and geometrical figures.

In the meantime , play with this in the new year, and have a good one!
http://home.comcast.net/~trochoid/TroWithMesh.html

[jehovajah]

https://mail.google.com/mail/u/0/s/?view=att&th=14349c87686811e9&attid=0.1&disp=attd&safe=1&zw

The geometric images have arrived today . Thanks NYPL !

[jehovajah]

Initial examination of the diagrams reveals the deep neatness formalism of my youth. The horizontal line was the initial reference. All angles, and thus Strecken at varying angles were measured relative to this initial horizontal. All measurements or compass displacements were measured along this initial horizontal. Thus to switch Strecken meant to switch which Strecke was the initial horizontal. Consequently the angle measure also reverses. What was measured anti clockwise would then have to be measure clockwise to maintain the same relative position. The negative sign in Grassmann's factorisation into biStrecken relates to this fundamental anti rotation.

It is initially not clear if Grassmann did perceive the inner product as I say. The diagrams show 2 types of projection( in fact 3 including the circular in figure 1) and those are parallel  and perspective. The vertical projection is evident as a vertical construction. For example in figure one I believe Grassmann starts with a Strecke ABE and draws the arc BCD. Then Using E I think he marks off the intersections C and D to construct a rhombus. The vertical line CD is a chord to the arc, the shorter diagonal of a rhombus which is orthogonal to the longer diagonal at F

This I think was an initial construction of an arbitrary parallelogram in which the angle at a is arbitrary, and the sides are in parallel pairs. Which can now be arbitrarily extended. This arbitrary Rhombus was perhaps the seed of his biStrecken concept.

A perusal of the index/ contents( inhalt) at the back reveals Grassmann conceived of 5 or 6 products! The inner and outer are only 2 of a much larger concet of combinatorial product. However, it seems Grassmann started with these initial ideas of the sum( which in group theoretic c terms he made closed by representation theory! ) and the combinatorial or constructional product which was not closed but in fact extended the algebra of the sum of Strecken to an exterior or enveloping algebra of the sum of BiStrecken. In this conception it becomes important that anti commutativity is accepted as a Förderung to keep the summation intuitive.

So e1e2 + e1e2 = 2(e1e2) = 2e1e2 = e12e2 = (e1e2)2
But e1e2 +  e2e1 = 0

This zero comes about through the anti rotations summing to 0. Relating this to figure 1 it is as if the construction of the Rhombus shows the 2 parts as triangles  which indeed have their altitude pointing in opposite directions.

Further thought intimates that the triangle is the fundamental construction and constructor. So although multiplication is most clear in constructing a parallelogram it is the triangular parts or pieces he is focussing on as primary constructors and primary Metrons.

Up until now I was focussing on the triangular summation and the parallelogram multiplication. But I feel that Grassmann was looking at the triangles in all cases!

[jehovajah]

Ausdehnungslehre Vorrede page Iv, new paragraph

Thus, with this ( axiom) the first step toward an Analytical method was taken, which in the process of  following I was journeying to  this new branch of Mathematics, which is layed out before ( you) here. But in no way whatsoever did I realise then what a fruitful and  rich field of study I had reached. Much more that result ( the axiom or law) appeared to me to be of little observational worth, until I combined it precisely as it is with a related idea. In the idea I specifically follow the labels ( handles) for products  in geometry how my father Had apprehended  them, and this made me realise that the Product of 2 adjacent sides was not only to be considered   as the rectangle, but over arching that was the  parallelogram, in which the 2 jostling against one another  sides are understood to generate the form in the same manner, even if one once again no longer focuses on the product of Lengths  but rather on both Strecken with their direction held in equal importance .

[jehovajah]

Ausdehnungslehre

Now, within this idea, this realisation, I brought together in combination the previous set out Sum ( rule idea) with these labels( handles) for the product : this itself gave rise to the most astonishing Harmony! Even if I specifically, replaced according to  the SENSE of the aforementioned rule the nominated( perceived) Sum of any two Strecken with a third Strecke, within the same plane, (but) within the sense of this (new) idea of multiplication in the plane.(in order to multiply in this idea of a plane  layout sense )
I multiply the pieces each individual with This same Strecke , and the products associated positive or negative values carefully observed i add them together,and so show that in both cases every time  the same result I have to obtain, that which I got to  before and that which I go toward from here.

[jehovajah]

Commentary/Kommentar


I have to say that this is the most important paragraphs in the Vorrede, and the one I have wrestled with the most.
Ausfallendste for example is more like serendipitous than surprising, because Hermann is clearly thinking geometrically, and this perhaps in meditation on his dad's paper opened in front of hom! SI what caught his eye and his imagination? How does he go from he rectangle as the product in geometry( what is that?) to the parallelogram as the generalised product to multiplication in the plane by planar Strecken?

And why was his unworthy insight into Strecken summation suddenly cataclysmically important?

I have meditated on this section many times in different ways, but now the diagrams give me guidance I so orely needed. Experience and research have helped me junk misconceptions, including" Algebra" , and focused me on symbolic arithmetic and it's geometrical justification. And what I learned and relearned, and learned when first introduced to geometry was the fundamental role of the triangle.

I think that Grassmann went from the product in a parallelogram to the products in a triangle!. But this was only logically possible if the sum of Strecken was considered to obey his previously unworthy axiom or rule for representation of Strecken summation.

Multiplication in the plane  had to be based on the sum of 2 ( basis )Strecken  in the plane. The 2 basis Strecken formed the fundamental basic unit of the plane, the bivector or biStrecken. And all Strecken were some linear combination of these basis Strecken as in a reference frame. Any enclosed planar form was now measured by these fundamental bi Strecken forms.

The mixture of linear variables with lineal Strecken was just there, on the page, provided you used the first rule of summation! The geometrical temptation is just so strong that one is forgiven or not properly analysing what one is tempted to do!

There was only one issue. A parallelogram consisted of 2 triangles essentially in opposition! How was this to be accommodated?

The point is Grassmann chose to accommodate this essential opposition rather than define it away.

What in fact was going on is analogous symbolic representation. But this time the objects are totally subjective! A line was used to symbolise some objective magnitude, but here Grassmann uses a line to symbolise a subjective experience. The basis Strecken symbolised a reference experience, the generalised Strecke was generated by these experiences, and the result was that the individual was now in a position to count or measure space or an analogous spatial experience.

Arithmetic was not being done as an variable calculus, it was being done as a subjective process of manipulating and synthesising space itself, that is as a subjective experience.

Somehow, these handles gave direct access to manipulating space itself , and this realisation just drops out of this way of thinking about construction; and summation is embedde in that synthesis.

[jehovajah]

Commentary

Grassmanns sentence structure is complex but this is because it is hypnagogic. Where he picked up this style is a matter of research, but it is a style that establishes" truth" where doubt could exist,, it is not propositional but coercive. It is not the style of a postulant either, but rather that of a sophist who places ideas into others minds!
Thus the statement about replacing the 2 Strecken by a third Strecke in the context of defining the geometric products has insistent commands: the 2 Strecken have to be selected according to the sense previously given to us as a sum of two Strecken .
Thus the Förderung regarding the rule for 3 points joined by 2 Strecken turns into a Law, not to be disobeyed.
But further, this law is applied to substaniatiate the sense of multiplying in the plane, thus the third Strecke Must lie in the plane of the 3 points.

This third Strecke was created or established to multiply! So he multiplies each individual of the pieces by it in the page layout sense , in the plane, but being careful to observe the positive or negative value of the pieces before adding.

This positive or negative value can only be defined by the 3 points! If the third Strecke is positive it means the other 2 Strecken are positive. But to multiply each Strecke has to push on the other to form the parallelogram. This is a Newtonian idea. If the Strecken are parallel they cannot be multiplied in this sense.so the Strecken have to be adjacent and pushing to form a parallelogram. In that case certain adjacent Strecken have to have there positive or negative values reversed in order to push out a parallelogram.

It is also possible to extend a Strecke so it pushes out a new "exterior" parallelogram . In fact by extending and jostling the whole plane can be tessellated by parallelograms exterior to the original, some of different sign value!

What remains unclear to me is precisely how Hermann chose any 2 Strecken and which pieces he individually multiplies to form the products whose sign values he so carefully observes before adding!