AB +BC = AC <==> AC =AB + BC
I have discussed the Grassmann formalism at length and developed procedural functions or process descriptive terminology. But it strikes me that i am doing this, not because this is eventually what Grassmann developed out of his childhood fascination, but because the simple observation is obscured by a century of vector ruminations, es[ecially Gibbs.
Grassmann's Strecken are not vectors, they are line segments used to indicate a route. Justus Formalism is that the route must always be labelled in alphabetical order. Herman aserved that the supposed sum was always true only if you regarded the strecken as movements in the alphabetical "directions". There is only one alphabetical direction, so even though the movement along that process may involve two oposing directions the "sum" still covered it.
Now these directions were iven in the context of finding length, and just as there were two directions on a straight line there were 2 lengths: the length traveled and the resultant or remaining length, So if the Strecken were in the same direction, then the resultant and the traveled length were the same, if opposing, then the resultant length was different to the traveled length. Thus one needed further instruction to determine which length to give as an answer, or one should give both.
But the Young Grassmann wanted a general law , and this was possible if the sum meant only one thing : the movement along both segments however oriented, in the alphabetical direction. Thus he could begin to see that the formalism freed the notions of length and direction from the strecken. The Strecken ws all about moement according to a formal law. That was the simplest way to view it generally he thought. One would always now need to be conscious of length and direction of movement todetermine the correct answer to questions on those topics, but as far as the "sum" was concerned it was invariant and crystal clear. What he had done was replace a formal convention by a formal process. He had moved from the strecken to the process of describing the whole object about which a question of length may be asked.
Of course he did not understand that at the time, but the insight never left him, because the justus Grassmann formalism never left him. Everything else but the formalism could be arbitrary. There was only one formalism: everything was labeled alphabetically and thus anticlockwise. Now it could have been clockwise, and it could have been clockwise in one diagram and anti clockwise in another; but it was not Justus set the convention that intrigued his son into the discovery of the analytical methods of the Ausdehnungslehre.
Where did Gibbs go wrong, and possibly others? They did not accept the formalism, and could not figure out why it made any difference anyway, and they had their own agenda.
Grassman clearly distinguishes what i have called the law of 2 Strecken, which we have just discussed. H e then went on to propose the law of £ strecken to cove the "product " of Strecken as movements, and these movements were the moving sides of parallelograms and they "produced " parallelograms.
For Grassmann this was a pivotal moment which changed the course of his life. The law of 3 Strecken was the law that convinced him to devote the rest of his life to Physics, and tha analysis of 3d Space :Die Raum.
I have struggled for a great while to understand what was so significant about the Law of £ strecken and to reproduce his result that AB = -BA where a A and B are Factors of a parallelogram not points( so perhaps ab = -ba makes that more clear) I struggled because i did not understand the fundamental role of Justus formalism, neither did Gibbs. When Grassmann said of the law of 3 Strecken that we have to observe the directions of all lines in the law, we have to be watchful to observe al the Strecken directions i did not know that he meant against a formal set of rules ie that Strecken should be labeled in alphabetical order and anticlockwise!
Newton, when compounding two velocities forces or accelerations always gave the directions. Now at first this seems highly reasonable, and natural, but you soon run into relativity problems, and reference problems when you do it that freely in a lare system. The law of action and reaction is partly to blame for this freedon, because we feel able o hop from what is in fact one reference system to another relative one quite freely without concern about whether the 2 reference systems are in any other relationships, which often they are! Lagrange dealth with this apparent relative freedom by setting constraints on the whole system, which meant that everything could be worked ot relative to a single frame of reference , and mysterious forces did not just appear out of a free use of a method without regard. Everything could be accounted for.
By establishing this formalism, Justus enabled Herrmann to see how every kind of magnitude could be related together in one reference frame without confusion. Was this an absolute reference Frame? His mind raced ahead in a grand design of his method that would finally put the capstone on all the branches of Science/knowledge making and Mathematics. But he needed help, and he got none for nearly 17 yers while Gauss was alive. When he died Clifford and Gibbs and Whitehead became very interested in his work. Gibbs completely failed to pick up on the formalism and conflated the law of 2 Strecken with the Law of 3 Strecken, and set Grassmann's program back over nearly 1 and 3/4 centuries. But to be fair, it was Gauss who did the most damage. To imply hat his work was somehow dificult or too philosophical or …was pure chicanery on the part of Gauss and Riemann, who both had doctorates of Philosophy!
I am reading it in German, and it seems clear to me. An Whitehead seemed clear that Grassmann's partition of knowlegemakers into real and formal was a powerfully persuasive one. Peano and others felt he had made soe extremely important distinctions, Saint Venant felt that notion in Grassmann's work were worth plagiarising and passing off as his own. When you put it all together it does not sit as it has been portrayed as a difficult and obscure book!
Grassmann in devoting his life to this insight was nothing if not thorough and he constantly improved, extended, modified and filled out his method of analysis, giving it a thorough workout in the toughest competitions. His solutions were so astoundingly intuitive and right that his peers thought it was witchcraft, or sophistry. Thus began the reputation besmirching. It makes no sense to give him the first prize while accusing him of being obscure!
Grassmann had a method for the Justus formalism of lines, he looked for a similar method for the formalism of points. He chose another process: constructing the bisector line between two points . Because it is in fact a potential line and not just one point, he called it a Schwerpunkt . Two Schwerpunkt crossed in a point which is the centre of the circle going through the three points. Thus the Schwerpunkt for A,B combined with the Schwerpunkt for B,C could be described as the Schwerpunkt for A,C. He just needed to work out how to product Schwerpunkt, which he did, but in the meantime he was encouraged by Moebius Barycentric treatment of points, essentially they were using the same idea. However, Moebius did not go on to devise a viable product, which Grassmann did.
By now Grassmann could see the pattern and he went for it up to n dimensions. His process algebra he developed fully as a lineal algebra of n dimensions. Now, astonishingly, this is not the aim of his effort. His aim was to recommend the tools and methods of analysis he used to build the lineal algebra! Only a few have even attempted to understand this. The lineal algebra is only a model to demonstrate the power of Grassmann,s Analytical Method. It is possibly only Clifford that got this, and he went on to develop other algebras using Grassmann.s analysis.