Grassmann creates a combinatorial problem because he assumes "multiplication", without combinatorial precedent. Thus he has to go back and fix it so it works combinatorially and in line with other mathematics. Thus the "multiplication" of Strecken is an odd procedure, cobbled together to mainly preserve a suggestive notational form. Because he stuck to his guns we have the dot and cross product today, with odd behaviour but looking cool!:lol:
The dynamic product, or Grassmann product is a new departure in combinatorics. Euclid's approach ws through the multiple form, and all other mathematicians have followed suit ever since.
The multiple form, for a product, is obtained first by division. The division that is "regular" means the form is divided into smaller equal parts. Equal means quite literally that the parts laid down on each other match. Irregular division means that the parts are not equal when laid down on each other: some are smaller some are larger. Thus we see naturally the Eudoxian trichotomy arises involved with division of a form.
Most mathematicians then concentrate on regular division, as Euclid did. This is not to avoid irregular, but to establish laws as simply as possible which may or may not hold for the irregular sitution. The form itself , on regular division provides the notion of multiplicative factors, which we all learn as multiplication tables. But some factors are prime. That means the factors as a form cannot be rearranged (by transltion and rotation) into another form which itself has factors. These prime factors are like "natural "units" for a form, so they are called "proto" by the Greeks.
The reduction of form into "proto" types that are factors of all other forms was an essential "scientific" discovery. It explained how a universal unit could exist and yet forms could be so "fixed" in their formation. Thus the only thing that could form a crystal with a diamond shape would be a "proto" form with that shape as a factor, multiplied over and over. "
Proto" forms are types of what we call "Quanta" in Quantum Physics. The fact that prime numbers can only be factored by a unit was of great significance to Pythagoreans. To modern science its significance is that basic building blocks of our reality will come in the form of "strings" of a more fundamental building block. Theses "strings" do not remain"straight, but curl and spiral into convoluted forms due to the rotational and translational motion field, the Shunya field.
The way the "curling" of these "proto" forms is negated, is by the simultaneous construction of the strings over wide contiguous area. The advancing edge of that construction necessarily shows itself as curved, the wider the area of construction, because the rate of formation remains constant in all directions. Initially, however, the strings are governed by the interaction between the force configurations on the "proto" form and regular polygons/ polyhedra form.
This is the notion of multiple form that Grassmann started with: a dynamic multiple form, not a static one. In this form the "proto" factors grow relative to each other. They are "dynamically adjacent". The product of the "multiplication" of these factors is the crystal parallelogram!
Grassmann had good reason to think this way, especially as his Father Justus was doing major research into this area of crystallography. How different this is, is shown by the fact that we look for an "answer" for a multiplication of factors. One does not look gor the form, bu for a cipher. This is cipherism, and it permeates all of our mathematical thinking, like a disease, like some form of bacterial infection.
The product of a combination is a form
At this stage Grassmann chose the parallelogram to be his form. He decides to define multiplication dynamically: the product of any dynamically adjacent sides is a parallelogram
The job is done. We do not need to concern ourselves with a number, one looks for a form. Similarly with the law of 2 Streaks " we look for a resultant form as a line.
Thus a + b = some form(a line in the plane providing a and b are streaks in the plane)
ab = some form( a parallelogram in the plane providing a and b are streaks in the plane).
Here is where Grassmann is misunderstood. In vector maths we usually define a + b = c where c is another vector not a line. Because a line and a vector are symbolically the same this seems a natural extension, but it obscures the fact that the ab is the parallelogram, which is not a vector but a form. It leads us to attempt to vectorise all forms and to miss the simple combinatorics, and finally to exclude points from our analysis.
As you have seen, Grassmann is a bit tricky to pin down just on a few sentences, and the differences are subtle and so missable. Grassmann seeks to Analyse what he believes to be geometrical space: the point, the line, the polygons, the polyhedra; and to do so in a new way that draws on all of mathematics, not a new foundation to mathematics. That others, like myself have taken it to explore the roots of mathematics was not his initial intention. His intention was ANALYSIS.
Gibbs attempted to modify it, Clifford attempted to modify it, Peano attempted to modify it and Hamilton recognised that it formed the basis of his thinking and work that Warren had been doing on triples . An Whitehead attempted to utilise it to set the foundations of mathematics on a secure footing, Bertrand Russel used it to discover his paradoxes. The wide ranging effect of Grassmann's insight is testimony to how fundamental it was. So it was not a branch, but a root of the mathematical tree.
As a root it naturally gives rise to offshoots, of which the various vector algebras are some, analytical geometry is anothr and Theoretical Physics and mechanics is another.
The 2 basic laws Grassmann himself modified, but they have to be understood as modifications to the core ideas, an thus not confused with the core ideas. That is the only way to determine if these ideas are valid for a universal algebra.
Grassmann decided To combine the 2 definitions to give the law of 3 streaks. The principle rason, i think is because the law of 2 streaks defines direction in a line in all circumstances without + and – needin to be considered too carefully. But the convention of order intimates the contra signs. Thus a + b = -[b + a]. This is of course clearer when you realise the + now stands for the join. Thus AB + BC means the 2 Streaks are joined at B, whereas BA + BC are not joined, they diverge from B.
BA*BC is exactly the form of the product
AC + BC is a meet point, where they converge. All these combinations are inherent in the law of 2 streaks, but only the join associates to length and so keeps the + sign for the join. The diverge gets the product sign for the commonpoint and the converge i do not yet know.
So BA*BC is -AB*BC, but But BA*BC = -(CB*AB)