I wrote about the topic of the symbolic line, but now I want to focus on the abstraction of meaning and procedure.
Grassmanns first shove in the I reaction of his method occurred hn he was habit using himself to the negative sign( and thus the positive) hhe realised that the sign affected notation and perception of magnitude. It also introduced case laws for when a statement was true or not. The change however was in Grassmanns mind. What was a simple line segment suddenly became a magnitude, and the sign changed that magnitude. To appreciate that change he had to abstract and hold on to a noion called orientation. But then he noticed that he also had to abstract and hold onto a notion called length.
Suddenly the line was not just a line. He soon found that a geometric statement using lines was actually a complex procedure involving instructions when to apply the notion of orientation, revealing a new notion of direction and signalling when to use the notion length!.
AB +!BC = AC became a statement that was always applicable to every case because it was to him a set of instructions about two line segments. The number of ways of interpreting it was fascinating, but to keep it simple he just wrote AB + BC = AC for 2 Strecken.
I call it the law of 2 Strecken. Gibbs did not get it , he changed it to the addition rule for 3 vectors. This closed off further insight into Grassmanns mind for Gibbs.
Grassmanns 2 Strecken are two line segments, but they are symbols for infinitely more, the plus sign was the symbol of any act of combination and the = sign was a symbol for a result of whatever kind consistent with the meaning of the 2 Strecken. Thus to teach a closed meaning to this insight is to destroy its purpose.
Later when Grassmann was learning about parallelograms his insight kicked him again. This time he "replaced " one of the sides by the 2 Strecken symbol and found that (AE + ED). AB = ABCD the parallelogram no matter where E was placed inside the parallel lines, but once e went outside the parallel lines The factors of the parallelogram meant ab = –ba in some circumstances. This was used to define when the algebra was exterior or when it was interior. This was the law of 3 Strecken.
Grassmann went on, realising that the line segments could represent other magnitudes! Just as a line was a part of a triple and also a part of a parallelogram it was part of everyother geometrical space. He could represent multiple quantities by a single symbol. He later realised he could represent multiple quantities by a Schwerpunkt, a point with magnitude which could be represented by a line through or attached to the point..
His analysis went deeper over time, always modifying the meaning of the simple symbols he used. He worked to make his ideas more general, more useful. He eventually realised that his method was not confined to geometry but was applicable to all magnitudes in all forms!
He was learning the power of analogically thinking. At each stage his simple pattern enabled him to grasp the essential elements into the form of lines, rows, systems of lineal equations which were governed by an overarching simple structure. Solve that structure and you could write down the form of solutions for a whole system of equations.
He learned this by diligent study and application of others work related to his own. He learned to penetrate to the elements of a subject, ind it's core model like his laws of 2 and 3 Strecken , set up the supporting notation and rigidly apply the rules thus determined.
This was his praxis, and the use of the lineal symbol in his work.
In the einlritung Grassmann takes pains to discuss certain definitions and certain observations, his drift is shaped by his times and beliefs. He thought mathematics consisted in arithmetic, geometry, algebra, combinatorics, number theory etc. he opined that the notions of discrete and continuous were derived from things created whole and things created piecemeal which nevertheless in the end are identical to things created whole. Therefore he felt discrete and continuous was just different aspects of the same experience. He felt that the Zahl, and Zahlenlehre was an exact model of this experiential identity.
But then what of combination theory? He recognised algebra as a model of the combinatorial experience of different magnitudes! Thus algebra modelled the more extensive experiences of reality with offering experiences of magnitudes. His idea was to combine the 2 into a new entity made up of Zahlen and kombinationen . This was an idea Justus his father had briefly posited in 1827" and he had called it Verbindungslehre, Group theory !
Grassmann also felt that there was a division which set geometry apart from the rest of Mathematics. Thus the algebra and the number theory that he felt preceded it lead on to the differential and integral calculus. On the other hand the geometry lagged behind until the 3d geometry of kinematics became important . Then these combined magnitudes cropped up again and again.
Hermann's deep thought was that the continuity of the number theory combined with the combinatorial discreteness would need to be given a thorough theoretical basis, and then they would represent a new type of magnitude he named Ausdehnungsgroesse!
So this was a group theoretic approach to the issue of dealing mathematically with the real life experience of multiple combined magnitudes in a continuous space, and so combining the continuous with the discrete.
Hermann worked on the basics, hoping others would join him and thus transform every aspect of Mathematics as he understood it, to these combinatorial magnitudes. Why?
His own work demonstrated thar these deep forms made many calculation quicker, broke down unnecessary subject boundaries and revealed a deep symmetry in natural " laws". His aim, having started in geometry was to revise physics in 3d so as to reveal the true manifestation of God's handiwork, the ghost in the machine.
These incredible visionary insights drove him to create lineal algebra as a demonstration of the power of his analytical and ynthetic method. This was meant to inspire mathematicians to do the further work and research needed to achieve his visionary goals. This did not happen as he hoped, and this lesson of history is well worth the study.
Today the tGrassmanns are being recognised for a fundamental paradigm shift in mathematics and physics. But it is fair to say that they are still not clearly understood. Hermanns lineal algebra is usually confused with the linear algebras of today, they are related as archetype to instance, it is the archetypical nature of his original work which has still to be recognised. His method generalises to and will generalise to all systems that consist in a combinatorial process.