Die Mannigfähltigkeit der Herr Grassmann is often presented mysteriously. Well instead of Manifold, read Mosaic..
Now we are talking in a common language!
We have to embed a mosaic in something to keep its pieces all together. We simply specify where we want it, and the artisan begins to construct the mosaic piece by piece fixing each piece by a bit of cement or mud or grout, laying it out like a bricklayer.
Now the mosaic brick comes in all shapes and sizes, the thinnest are used to lay out a line, the flattest to laminate an area( Arithmos, mosaic) the bulkiest to construct raised reliefs in the mosaic..
The thinnest one is a line, it sets out a direction to be drawn in other extended pieces. The individual line piece has to be multiple for a mosaic.
Let's take a flat piece that is shaped like a diamond, it has 4 edges, 4 potential orientations for each face. Once an orientation is set it is set for the whole mosaic,monlyncurved pieces can change the direction of patterning. Thatbsingle piece is the Monas all others are extending its pattern through out plain surface space,. This is a surface manifold made up of many many copies of the Monas, thus making a multiple form.a Vielenfach.
In solid space determines orientations in the multiply extended mosaic form..
Their is a system of reference computation mfor any tile in this mosaic. That is called an exterior algebra.. But for any point within the founding tile, that has to be found by a means called an interior algebra.
Now I can hand you over to the prof who will take you through a modernised treatment of Grassmann's Lineal( not linear!) algebra.
Follow the course to get a good grounding.
This is not Hermann's Ausdehnungslehre, but Roberts revision of it. Hermann goes deeper and sets out an analytical method which Clifford utilises as the basis of his Clifford algebras.
One other point, Gibbs totally mixes Grassmanns lineal algebra up, so the prof here uses Gibbs approach. Grassmann has 2 lineal "laws": the law of 2 Strecken AB + BC;
And the law of 3 ( or the third) Strecken AB.AD + BC.AD. I have tried to understand Grassmann's insight or instructions from the German and it is eluding me directly because I do not grt the use of " statt". But I intuitively know he means that in a parallelogram one can take any side and replace it by an arbitrary combination of 2 lines ( straight) so the 3 sides form a triangle. This is an application of the law of 2 Strecken. Then using the other side of the parallelogram as a "multiplicand " create a construction from the pieces so formed!.
Herrmann defines " multiplication" in this case as 2 dynamic sides rubbing past each other to form a parallelogram. This is a dynamic construction which he always relates back to an idea his dad was proposing. Fortunately Micel Rideau has written a great paper on Justus early ideas on dynamic constructionist geometry which helps to understand yhis reference. What Justus did not note was the truism of the first law.
The first law is where many go wrong. It is a truism, not a truth . AC I a label for the 2 Strecken combination AB + BC. The Strecken AC is not equateable clearly with this combination except in one specific case! Grassmann discusses this case and his insight that to keep things simple let AC always refer to the 2 Strecken combination!. We, following Gibbs have accepted the strange and straining equation of a third line with 2!
Thus when Hermann introduces a third line, I and everyone else I have read disconnect from his insight! Taking the 2 line combination I can immediately see it as the corners of a parallelogram. Understanding that this 2 line combination has since Euclid always referred in general to a parallelogram gave Justus and Hermann an identification with a constructivist approach to multiplication!
The technical problem is that multiplication is misunderstood! Hamilton treated of the issue but points out that multiplication is an arithmetic concept, usually taught by rote and not understood. In Algebra one has to recognise and apprehend the multiple form as the basis for the rote multiplication. The multiple form is therefore not multiplication! It is a fundamental combination!
Thus the combinatorial nature of " geometrical" ideas are what underpins the arithmetic notions of addition and multiplication. These are constructivist activities! Justus Grassmann was at pains to lay out the dialectic foundations of geometry as clearly as possible, to show what was constructed and what was decided by logic or interpretation. In this way he hoped to teach youngsters to discern for themselves fact from fiction.
By paying attention to this distinction, pointed out by Schilling in an argument against Kants notion of a largely " discovered" basis to " mathematics" (the technical word is "discursive"). Schilling took the view that mathematics was a largely constructed discipline. As a consequence the axiomatic approach to. Mathematics alluded to by Kant, took shape when Ficht demonstrated the fundamental axioms that constructed algebraically all the rules of Arithmetic. Geometry then was subsumed into algebrs.
One of the clearest expositions of this is Hamilton's Science of Pure Time. The main difficulty Justus had,after pinning everything down rigidly was multiplication. So hermanns insight that multiplication was not an arithmetic process logically constructed by repeated addition( a very persuasive argument) but a fundamental product of the geometry of 2 dy namic lines is difficult to grasp through years of miseducation. Of course The ancient Greeks would have recognised it straight away.
Multiplication is a misleading concept.
Factorisation is what we fundamentally mean by it, and that relates directly to construction of multiple forms either by extending a monad to cover a space, or equally to breaking a monad into smaller monads, thus constructing an interior multiple form . This is where the interior and exterior product distinction derives from.
The issues of sign and non commutativity and anti commutativity arise in a Grassmann analysis because of the strict proposional dialectic. Why so stat? Partly because it was a faith among Prussian philosopher that logic was fundamental to truth and congruency and consistency. Thus every statement must either be a premise or logically and soundly connected to some premise. The hope was to demonstrate logically the constructivist approach as true or the discursive approach as true. That was then! Now, due to Goedel in particular, we know logic cannot deliver. We always need a larger set of propositions to demonstrate consistency within a subset.
One could say that that was intuitively obvious, from the very pragmatic nature of the adjectives true and consistent, but back then they would have argued the hind legs off a donkey!
These kinds of arguments bear testimony to the extremely autistic nature of those we revere as geniuses! Nevertheless it has left us with a kind of compact largely consistent body of knowledge, analysis and dun thesis which we call mathematics, but it is split into 2, and many of the " deep" theorems are arcane, or deliberately obscure.
Most of us accept from arithmetic that AB = BA, but Grassmann wrote AB = –BA. He found this to be the case both for combination of the 2 Strecken when used as a label, but also for the Strecken a and the Strecken b considered as factors of a parallelogram ab. ab was found to represent –ba. This I where my German fails me. This odd and confounding result– Grassmann could hardly believe his eyes for several months I think he implies comes about due to the strict application of sign and the strict denotation of the symbols. Everything is fixed with only one interpretation at this level of analysis. So a rubbing against b is not identical to b rubbing against a no matter what the geometry looks like! We commonly make that judgement. In fact in Euclid, when it is not being misleadingly translated as axioms tha is precisely what they are called: koine ennoia common judgements or interpretations or inner "rotations.".
If we remove these common notions and proceed only according to our strict definitions we apparently arrive at ab being identical to –ba. When I try to get this result I always have to fudge it either logically or geometrically. I therefore suspect that this is a fundamental mistake agrassmann made on the basis of a mistake his father made in his dialectic approach.
Now there is no problem if it is set out as an axiom. What is a problem at the moment is deriving it from the set out axioms as Grassmann intimates. He does say you have to be extremely careful with the sign designation, and that I do not yet grasp.
However, beautifully I see the Euclid rules for parallelograms and parallel lines being used to explain a distributive( think) law of parallelograms on related bases. That is fantastic enough without even getting the anticommutative( ab = – ba) axiom.