The moment of creation, or more nearly procreation(Erzeugung!) must be dualed with the whole act of completion by linking things together(Entstehende von dem Verknüpten) Grassmann argues. It is a sustainable notion which means that we do not think in 2 ways, either in a deceptively simplistic way about continuous form or a way that is preferable, and closer to our reality which involves the layout, the settings that need to be in place to link together to form the form; and then the actual act of linking those set pieces together, like a jigsaw puzzle or an Ikea flat pack.
Thus one needs the pieces and the instructions to construct the form, and thus the experience of constructing makes us think differently about the finished form. But Grassmann says no! The set pieces are no different to those pieces in the procreated form, they end up being the same and performing the same role; and to all intents and purposes they were created by procreation, and one just starts with these set pieces as givens! So what really is the difference in the finished product to the procreated one?
At a deeper level Grassmann states these are All Thought Forms, so why think differently about them, to the extent that one fails to make the most fundamental connection: we are thinking and dealing with precisely the same thing exactly at the moment of completion(Enstehende Verknüpften). Thus one description is an analogy of the other more detailed description, and s imilarly the notation follows suit!
This is the point. The complex looking notation is no different to the simpler continuous form of notation. Therefore one should combine the thought processes as one, reduce the level of terminology, and deal with everything as if it was in this moment of creation. The moment after that we have to take into account what has changed!
This is quite revolutionary thinking at the time, but it is essentially Lagrange's viewpoint: look at a situation, analyse it into its pieces; describe the constraints that link those pieces together and treat the analytical model as the continuous or originating circumstance. Go to the next moment after that analysis, and compare and contrast the analytical model with the continuous one.
What if in that new instance they differ? This part of Grassmann's analysis I have not read yet, but presumably we return to the model and alter the pieces and links accordingly. However we have a choice as to when to run the model from: either from the old starting point with modifications or the new with the modifications.
One would hope to " evolve" a better and better analytical model by this approach.
Although we now know that the moment of procreation is in fact a complex series of interlinked events, very much in keeping with Grassmann's assertion. Grassmann is contrasting a continuous created form with that same form constructed piecemeal. However it is again a mark of how deeply he thought about these things , and their fractal applicability, that he so nearly chose the one type of language which automatically and biologically demonstrates his point!
The lesson here is that we do not speak arbitrarily, but as a consequence of these same processes occurring unconsciously within us , when we describe accurately models of empirically founded observations.
The Lagrangian does not seem like a reference frame. That is because it is distributed between the set pieces, ie the identified parameters, and the instructions of how to connect them. The 2 things together give one the necessary framework to construct any form.
In the light of this I wanted to take a preemptive relook st the laws of 2 and 3 Strecken.
In the law of two Strecken there are only 2 pieces , and the relationship does not involve a 3rd party. Because of the way Gibbs baudlerized Grassmann's work it is hard to apprehend the difference. The law of 3strecken involves this 3rd Strecken and the relationship is that the law of 2 Strecken can be labeled by this 3rd Strecken, but only according to the formalism. This formalism Gibbs lays aside and establishes his own.
It is from these initial observations that Grassmann is impelled into a study and analysis that is outlined in the Ausdehnungslehre. It behooves me, therefore to understand the minutest significance of his presentation.
The law of 3 Strecken introduces conjunction of Strecken into the algebra. Grassmann calls it adjacency. The dynamic adjacency defines a Flache, a schematic parallelogram. The third Strecken allows inner and outer products or conjunctions or adjacencies to be related. The significance of this is still a mystery to me, but at this stage it was so for Grassmann . It was only after extensive application and refinement that he realised the significance of the differences. What is quite clear is that the diagram that relates inner and outer products is confused by Gibbs as defining "vector" addition . The addition of Strecken is not the same as vector addition. The third Strecken labels the addition of 2 Strecken, it does not become the resultant vector of the 2 Strecken.
It is natural to see this in Grassmann's diagram, because it echoes Newton's parallelogram rules for Newtonian Vectors, but Grassmann's Strecken are of a different breed to vectors. But, in what grassmann has just exposited is the seed of the idea of dualing the law of 2 Strecken with the third Strecken .
One thing that impresses me about Grassmann's analysis is the clear thinking about the geometrics and mechanics ofspace, and this is based on work he attributes to his father Justus. “““in particular JUSTUS NOT ONLY EStablished a formalism for both direction and notation, within the school system he was establishing, he also wrote penetrating treatises on the nature of things pHYSICAL AND SPIRITUal. In this instance he wrote on the nature of Zahlen. Now Zahlen is derived from an old dutch or Netherlandissh concept, Taalen, and taken up into the common prussian as Anzahlen, Thee notion here is akin to "tally", which on the face of it means to count. But in fact to tally means to Account, and that is a subtly more inclusive notion. As a consequence Justus wrote that to account for an abject or form it was insufficient to just give it a numeral, other marks needed to be included with the account that extended the concept of the Accounting. These other marks would account for the size of an object or it dimensions (Grosse), thus an Anzahl should be a more complex description of an object of which the numeral is just a part.
I have not seen Justus Pape on this point, but Grassmann refers to it as being the source for his extensive notation. To distinguish the different Ausdehnungs Groesse Grassmann uses (1) to denote mumeral marks, and the full printers letter block to denote every other distinctive dimension or size.
The notion of Groesse here is also important, for it is not used to mean the mass of an object, but a metron applied in the analysis of an object. I have established, i hope, the term metron as an arbitrary unit measure of any arbitrary magnitude, the purpose of which is to Quantify the magnitude by a process of mental assent amongst colleagues or even within oneself, that the metron should be the unit of quantity. IN THIS Way one takes what can only be described as an experience and distinguishes it by a conjugation into a quantitative factorisation into 2 reciprocal factors(see the conjugacy schemes of.http://my.opera.com/jehovajah/blog/conjugacy-and-adjugacy-and-the-structure-of-magic.. and posts following), and allowing or adjugates also to be distinguished.
Thus the notion of Groesse is akin to the notion of Dimension as used in the Systeme Internationale Units. Anzahl, by Justus analysis therefore relates to Dimensional evaluation of experience and feeds directly into the notion of Tensors, vectors etc, but is a purer and more straightforward combinatorial description, at least as Grassmann understood his fathers work. Justus was innovative in that he expected differentials to be the basis of all quantitative evaluations, and the supreme computation to be the Calculus (Leibnizian, probably)
The adoption of a lineal combinatorial form to represent this adjugation within the conjugation of a form against shunya is perhaps an innovation we can ascribe to Hermann, but it has many precedents , not least of which is the notions of Bombelli on adjugate numbers. However, it is fair to say that the notion of Form associated with this notation, s opposed to equation, is a Grassmann innovation.
This is what makes Grassmanns arguments so seemingly impenetrable, because the notation he was brought up on, by his father, was very precise both in use and definition, whereas the very same symbols were used rather loosely everywhere else! Grassmann claims that the Combinatorial Theory of his day precisely defines the use of these symbol combinations, but that apparently was not the case, or the Combinatorial theory was largely ignored in terms of the common usage of these symbols(+,.,x,-,÷).
In any case, gollowing his fathers lead, and prior to Gauss publishing his lineal form of the complex notation, Grassmann was already using yje lineal combinatorial form to solve Barycentric problems, as per Moebius, and further refinements to solve Lagrangian mechanical Problems such as the Ebb and Flow of Tides.
Undoubtedly , and Grassmann admits the same, he learned from these great mathematicians of his day, but his curious notion was unique and down to Justus Grassmann's analytical dissection of the core of reality. Before Dedekind, Justus Grassmann had derived a general notion of Quantity, based on Dimension and rooted in Anzahl. Dedekinds "cut" was to be seen as the definition of number by the board of Mathematicians after heated debate. The debate was due to the narrownwess of the definition of Anzahl. As i have shown, there was at least one other way of looking at Anzahlen, and pobably many more. These were all temporarily quashed by he board! many of these notions were preserved in Mechanics and Physics of the day. Later they extended to Chemical notation, but for mathematics the death knoll was tolling. Dedekind's cut had cut too deep. Mathematics was hemorrhaging at the jugular!