Posted here with corrections
In Grassmanns toolkit everything ifts together. Strecken as construction lines between points reveal a formal notational structure, that is a Begriffe. This literally means a way to grip things that are as slippery as distinctions and Ideas/ Forms. The notation literally can have a one to one onto correspondence ith each idea , distinction or form. This means one can represent the essential process of experiencing, distinguishing, comparing, recognising, representing and manipulating the aspects of an actual or imagined interaction with space by a careful selection of symbols.
We do this all the time, so naturally and in several neural representation systems that we forget or ignore it as a fundamental process. We regard it through the smokescreen of language , not realising that our languages are cultural constructions. Thus if we build a crystal clear language or jargon in which every symbol has a precise meaning, we can actually use it to explore and model inner and outer experiences in relation to a Form/ Idea. This is precisely and painstakingly what the Grassmanns have done in their Ausdehnungslehre series,
In the course of doing this Hermann revisited every fundamental notion in science, really deconstructing Euclid abd resynthesising his Stoikeioon. This was Justus stated goal. Hermann got interested in it at a stage when Justus had implemented school trials of many of his ideas and reformulations, picking out an inner structure that Justus could not apprehend because he was the pathfinder! Finally Robert extended the analysis to the foundation of philosophy itself! He called this approach Formenlehre, and really engaged with the Platonic philosophy or theory of Forms and Ideas.
That Euclid should serve to unite a constructive approach to geometry, and an empirical approach to philosophy says to me that the Stoikeioon is not a book about geometry or mathematics. It is a book about the philosophy of Form/ ideas.
Hermann literally was between these 2 seminal path finding researchers. He did not live in the same household as his father and brother, but with his uncle. This was a child adoption arrangement, but not an abandonment by any means. Hermann seems to have always known who his biological father was. But he lived with and was brought up in his uncles family due to infertility it seems.
Hermann's absorption of Euclid was thus mostly unconscious and lifelong. It was not as a classical training in Euclid, but as a result of an innovative programme of restructuring geometrical education that he imbibed Euclidean forms. Much the same happened in my education. When geometry was taught , it was not to understand Euclid it was as a subject called Geometry. It was not until much much later in life that I actually began to look at the Greek text!
What that does is dissociate the student from the originator. The student never actually knows hat the original thought was. This is why I read Grassmann's own words, in German or Prussian .
The second point is, by the time you do get to study the original the prejudice has already been imposed, so the original thought is almost indiscernible!
The advantage that those who have a classics education have is that through learning the actual language they can " rinse" their brains of prejudicial ideas. The more languages they learn to meditate in, the clearer their conceits become.. But then they face the problem of dissociation: they no longer think the same as those who are brain washed differently!
Any way. Newton read the original Greek text or a Latin copy. Thus the quality of his thinking and insight was suffused and informed by Platonic ideals. Hamilton read Euclid in he Greek and was similarly motivated. The Grassmanns had access to the notions of geometry associated to Euclid. Justus clearly had experience of parts of the text, but I do not know yet which parts or books.
Hermann was a great linguist and that makes me think he had access to the texts either directly or through deconstructed copies in the form of lexicons. Someway, somehow, the ideas and Forms of Euclid suffused all the above mentioned individuals.
Thus when Newton established his " vector" algebra based on the parallelogram , it was not in isolation. Hooke, Huygens and all the geometers knew precisely what he meant. But when Grassmann similarly deconstructed and resynthesized the geometry of the parallelogram, suddenly he is talking " obscurely"?
The only difference between Hermann and Newton is the subject boundary called Algebra. In Newton's day algebra was not a subject, it was included within general Rhetoric and Reasoning. Reason is derived from ratio and proportion, logos and Kairos in Greek, and thus not considered part of the mathematical subject boundary. It was and still is within the Philosophy subject boundary.
Thus we have this academic compartmental approach blocking understanding for millennia! The shake up and mixing of the Humboldt reforms , creating the mix of primary and higher academic teachers , alongside the philosophical argument about the structure and nature of mathematics and the sciences engendered a school of thought that knowledge is constructed, not discoursed! Discoursed means several things all at the same time: but the essential idea is that you have to run round all over the place discovering through inquiry, discussion and debate what may be called divine or spiritual Truths, which you intuitively know to be true!
The constructionists basically say, truth is not the criteria, fact is. It is true because it is fact, that is because it has been constructed!
Thus Justus enters the fray , utilising the growing conception of ring and group theory as expounded in crystallography. He actually makes a fundamental contribution to Verbindungslehre as it was called before it came to be dominated by others who renamed it KôrperLehrer!.
To get to the point , an analytical approach to geometry in order to reconstruct it on a systematic, logical and congruent basis revealed a repeated combinatorial structure in every geometry: the geometry of the line, the geometry of the plane, and in the geometry of the Raume , that is 3d geometry. This structure was deliberately based on Arithmetic, because it was believed arithmetic was logically pure and unsullied.
Justus analysis showed however that this was not the case, and he made several suggestions as to how to put it right. Some of which were conceptually confused. He could not resolve certain logical difficulties without bringing in the observer as a crucial part. This actually meant that general rules were subject to the individual assenting to a consensus. In those days they believed that the consensus could get it right , even though they were agitating against an old consensus which they felt was wrong!
So Grassmann J, H and R were looking at the ring or group structure of various geometries and finding connections between them. When Hermmann looked at the triangle he clearly picked out an additive Algebra! He was looking at geometry and he could see an Algebra. It was only manifest when the correct Notation was used.
Hmm interesting but not earth shattering until he noticed the same thing when he was looking at the Geometry of the Quadrilateral. Again, looking at geometry , with the appropriate Notation revealed an Algebra, but this time it was multiplicative! And in addition it connected with the triangular algebra of addition to produce a distributive rule of combination!
This was so intriguing that he began to explore it and found that the " Algrbra" held true. Testing it a bit harder he put in the metrics of length and it still held true. It was when he put in the notion of direction that his world turned upside down! The factors , if you could call them that in yhe analogy did not commute, but instead required the sign that denoted direction to change.
After waiting sometime to get over the shock and general unease at his conclusions he tested them over and over and found them to be logically consistent. He then decided to devote his life's work to exploring these geometric Algebras. There was much work to be done, many gaps in the algebras to explore, but his hard work and dedication to detail, following the strict guidelines of his father seemed to be being rewarded handsomely. He entered and won mathematical competitions to the astonishment of all around him! He found independent confirmation of his ideas in other researchers work. He read nd digested Lagrangian Celestial. Mechanics with astonishing ease because his insight into the geometrical algebra suddenly made it simpler clearer, more symmetrical and beautiful!
Intacling the problem of Ebb and Flow he discovered not only the nascent hyperbolic geometry, but his insight revealed its fundamental algebra. The algenpbra of Newtonian vectors, as understood by Lagrange was suddenly placed before him, and he could clearly see the parallelogram and in this case the rectangular parallelogram geometric algebra.
Newton and thus Lagrange had fully worked this algenpbra out. In fact most researchers like Huygens, Leibniz Hooke for example were fully conversant with it. But it was called " algebra" only loosely. It was to Newton his own private cogitation by which he mentally manipulated ideas and relationships to gain insights and find solutions. Although Newton was highly organised and structured in his thinking, he did not see that that was important enough to publish.
Bombelli is probably the first author in this period to write a book mostly about Algebra. Descartes is the next author,of renown, but he called it geometry. De Fermat popularised this geometric algebra, but it was Wallis that, drawing upon Euclid and Barrow wrote the first real modern book on Cartesian Geometric Algebra with his great insight. He pestered Newton for his algebraic notations, because he believed that through studying the algebraic reasonings of genius students could benefit and emulate and surpass. Thus Wallis's work was the standard for Algebra for a long time. And it was always a geometric algebra.
The Grassmanns were different. They studied the Algebras of the geometries, not the algebra of the geometers!
The algebra of the geometries as I explained above required appropriate notation or terminology to distinguish. Thus Hermann struggled to find for each geometry that appropriate notation hat made its appearance manifest, or clear, visible(Anschaung). This was like making ghosts or spirits visible. It was like making subjective notions, ideas and forms visible. It was giving form to an invisible structure of formal thinking , revealing how it followed similar and analogous patterns in all the geometries.
So when Hermann came upon the projective geometries, especially the Newtonian decomposition of " vectors" as forces or velocities, he recognised hat it was am algebra that applied like an algorithm to any description of physical situations.. He might have petered out at this stage, because essentially he was going to be repeating Newtons Classic Principia. He would have brought little that was new to the discussion. However, his strong notion of geometric algebra in a given geometry lead him to look for addition, multiplication analogues in the projective geometry. He was able to bypass the detail and see the product, and how the parallelogram formed the general product( product here means the constructed form, which Hermann long ago had convinced himself was an analogue of multiplication). But then he found the inner product and with it division!
In the parallelogram geometry Hermann had been able to see that addition and multiplication algebras existed. He therefore knew that if he could solve for the parallelogram he could solve any problem that could be reduced to its form. However, because there was no sense of division, the algebra of the parallelogram geometry was incomplete. In fact it was full of holes. Herrmann was already considerably blessed with simplifications due to his earlier discoveries. His reasoning and equation formulation was already considerably shorter and smoother than his contemporaries, simply by using this geometric algebra through appropriate notation as his page layout. What I mean by this is that mathematical notation is set out on the page. If you organise it carefully you can make beneficial use of that layout. To speed up and streamline calculation.
The organisation of the calculation on the page has always been a pedagogical concern. It was clear that " neatness" helped in solving problems and performing calculations accurately.mrows and columns have always been a significant part of the mathematical fidcipline since Babylonian script was invented abd tablets of information recorded. But the grucial geometric structure of a page layout is derived not from cuneiform, but from Mosaics or Arithmoi. An Arithmos is not just a mosaic it is the fundamental of geometry itself! As a fundament of geometry it is used to record Astrology( astro-merry,astronomy astro-logia). Thus these mosaics become fundamental and obscured organising principles. Onto such a mosaic a geometrical form may be drawn, the mosaic then representing an epipedos epiphaneia, a so calle flat surface, but in fact often a very colourful abstract art form we now call a mosaic.
"On such a platform geometry can be done !" mused Pythagoras. And of course he was right. Geometry is always done on an embedded mosaic. Because we have lost sight of this se do not understand Arithmos, Arithmoi, and how forms can be shown to be equal without some other notion like length or area or volume. For Greek mathematics these fundamental notions are embedded in the mosaic. They are the literal structure of space .
So the layout on the page was also in hermanns mind and when he discovered the Type he called the inner product he states" but this notation also places another product ( to the exterior product as he soon calls it) TO THE side! ( zur Seite), by this I think he means that alongside the exterior product one must also, on the page work out the interior product. The reason is that when both these are done they are a proportion or ratio( by dividing one into the other a fraction) that has a valid value. This value he realised uniquely replaces the angle in parallelogram geometry and provides proration and division into the algebraic tool box!
He knew this to be true because he had done the work in his Ebb and Flow paper In which he had used the hyperbolic sin and cosine to create a sinle multiple form involving a Strecken and an exponential function. At the time he had taken it as evidence of the parallelogram multiplicative distribution rule, what I called the law of 3 Strecken. But now his insight into the inner product made him realise it was based on the projected Strecken ( obviously!) and not just any Strecken and the angle between them. The angle was not the important ratio of magnitude . A more general ratio of magnitude was involved and that was that of the outer and inner product!
Angle has long been one of those unquestioned measuring algorithms. But it needed to be questioned, and Newton, Cotes and DeMoivre did question it, particularly as it did not concur with astronomical practice. Astronomers use areas and always have. Someone, ome teacher converted the arc into the angle notion and created a serious problem. Or rather later generations misread the ymbol for arc and mistakenly developed the notion of angle. Cotes apparently suggested the Arian as an arc n measure which could easily be used astronomically and geometrically for mundane land measure. The Greeks used Chords, not angles. Each chord has an associated arc, and the 2 together form the connecting link to orbital motions in the heavens. Thus the straight line measure of spread is precisely what the sine and cosine tables record in ratio form. We give a precise ratio of cord to diameter to measure a circular arc. We may never be able to measure the arc directly but we can construct it and through that solve for the triangle ANzd for the arc by approximation..
Grassmann had now got a complete ring in his parallelogram Algrbra. But more than that he could see how to generalise it to any form made up of facets like a crystal.. This is what he means by n dimensional Algrbra! Any complex form made up of parallelograms is n dimensional, depending on how many facets were distinguishable. Each of these facets he called a space, and worked out, purely using his algebraic toolbox for parallelograms how to construct a crystal, and how to distinguish crystal forms.
This is where he found and corrected a mistake made by his father. The fact that he could see it clearly was testament to the powerful tools he had created for describing the Algrbra of geometris!
There was yet more research to do, but he wanted to publish his irst volume and his results to create a stir and get others involved in the research which was now an extensive and rich field of study. That plan and hope backfired on him disastrously, and almost extinguished his belief in his system. That is why when his brother offered , demanded that he redact and republish his work, insisting that it was not to be left idle!, he was willing to give in to his brother's views of his work. Later, much encouraged by the response he reasserts his own view republishing his unredacted original with annotations.
This is the background to the Grassmanns contribution to a revolution in science.