Zahlenlehre by Justus Grassmann

Justus Grassmann in 1827 published a treatise on the theory of Zahlen . This was a pure theoretical treatment, that means it was an abstract and general investigation of Zahlen. I know this, because Grassmann states this in a footnote in Ausdehnungslehre 1844. He states that this theoretical approach was established in the school system in Stetin from around this date, but was not widely known outside of Stetin.

Now this theory separates quantity of magnitude into 2 forms: whole and discrete, or if you like created in one piece and made up of combined pieces, continuos or contiguos, seamless or set pieces combined to form a whole.

The pure motion of Zahlen deals with ones conscious experience of quantity of magnitude as disposed in Form. It directly uses the Euclidean notions of Gramme, that is a drawn line segment, Schena – schematic or figure, as the abstract and essential notion of Form. As such it links into Plato's notions of form/idea. But to what extent I cannot yet determine.

(As far as I can make out the distinction is subtle, between Zahl as a form of speech that accounts for objects or recounts information(erzaehlen) , and Anzahl the formal concept of a bound quantity, a cardinal label for a quantified form. For example a quaternion as a group of 4, or a centurion as the one man who represents a hundred. Thus Zahlen is the extensive idea of Anzahl, the extension of a definite quantity to an indefinite process of quantifying. Anzahl as a definite quantity has an intensive measure or metron, but is itself a whole that may be used extensively once specified how.)

Now it appears that Justus had analysed the bounded forms or drawn forms(Anzahl) into whole or created complete forms, and Setzen und Verknuepfen forms. This was so that he could make the differential the basis of all number or Anzahl forms, quantities of magnitude. Why conceive this?

Clearly his father(Justus) wanted all his students and Herrmann himself to grasp the fundamental notions that had out fallen from the differential calculus. He did not take the differentials to be some quirks of a method of analysis, but as fundamental, elemental constructs and constructors of our experience of reality. Thus in line with all the latest thinking on calculus, as a more fundamental arithmetic than everyday, commercial and artisan arithmetic he proposed that setzen, Or differences, that is differentials are fundamentally tied together to form wholes, and that in that thing called a moment, presumably an instant in time or an instance of origin both the discrete and the continuous form are interchangeable, and indistinguishable. But only for that moment, after which they dynamically move to their next state.

Setzen, in their most direct referrent are just established entities of a fixed nature, but not specified. They are this general, because Herrmann says they have a wider applicability than just arithmetic and algebra. These set forms can be tied together to create whole forms, thus anticipating topological applications of Euclidean forms to describing forms and surfaces.

Now Justus pulls out the algebraic form of these descriptions of form from the kombinatorischen form, by which he takes a step toward generality. That is to say, if an algebraic description applies to more than one special form, this can only be because those forms have a continuous similarity. We might call it a homeomorphic mapping or even an isomorphic mapping between the similar forms, if we specify a point description of them. However Justus possibly used the natural notion of Anzahl or drawn form to connote continuity. That means that if I can draw forms in the same way, then the algebraic description of those forms would be the same and in that sense the continuity of drawing a form is enough to distinguish similar forms

However, if a form is made up of parts then it is not possible to say two forms have the same algebraic description without first of all specifying that they have the same kombinatorischen structure and rules! Thus to specify an equality or duality between forms , we need to specify the combinatorial system of the elements of the form, and the algebraic system of the continuos form!

The algebraic description is presumably the description of a form as "invented" by Descartes in his Treatise La Geometries.

The dual description that Justus saw as underlying all Zahlen arises because of the differential calculus. Justus realised that differentials are not connected! They are bits of connected things, and to progress with them logically as we do with connected forms, we need to specify how they are connected to wholes. Once that is done, then the arithmetic of the whole is applicable. This arithmetic is the Eudoxian arithmetic in Euclid books 6 and 7, now tied together with the methods of exhaustion and factorisation Euclid also describes.

This notion of Zahlen, as Grassmann states was not widely known, and it predates the work of Dedekind and Cantor by some decades. It places the ausdehnungslehre that is extensive magnitudes, adjugate to the Intensive magnitudes of a form, and foundational to every notion of the differential and integral calculii. This is what Herrmann states.

Now this split into 2 different but equatable descriptions of a form, different in characteristics of continuity and discreteness, goes back even to the Indian mathematicians, but particularly Brahmagupta. Starting with Shunya, that is everything he splits it into fortune and misfortune. The Greeks notion of everything was the whole (Monad) and thus Shunya is the same as the monad philosophically. But now see the power and problem of notation, for monad is universally notated as 1 but Shunya as 0.


The magnitudes and quantities of Shunya are also displaced. Thus 1-1 being a description of the division process subtract 1 is written as if to dual this process with Shunya, whereas the process describes a mental action, a direction of mind. For all such marks 0,1,2,3,4,5,6,7,8,9 are symbols of mental apprehension of quantity within Shunya. All quantities are mentally apprehended within Shunya, as such they form a distinction within Shunya, and the resultant number system discretises Shunya into an unending process of quantification..

What Brahmagupta wanted to convey, in contra distinction to the prevailing Greek monad philosophy, was exactly that both fortune and misfortune are mentally distinguishable in Shunya. Thus if we conceive of one as a mark, we must also conceive of the other as a mark, and the combining of those two specific marks must conceivably return us to just Shunya. The discrete in that moment of combination returning to the continuous background.

This was and is a remarkable Indian cultural triumph over the Hellenistic, Pythagorean style philosophy, and is truely Indian, for fortune and misfortune is equally in the hands of their gods in their cultural conception. While this was philosophically a triumph, commercially it was a bit slow to catch on because of the astrological associations. But over time it became one of the merchants secret weapons of accounting, and it spread far and wide as an accounting practice long before it was taken up formally into " pure" mathematics.
Thus in promoting this philosophical notation of whole magnitudes fortunate or misfortunate, it was necessary for Brahmagupta to specify the combinatorics, very much as Euclid had specified the Eudoxian combinatorics in books 6&7.

Once again I turn to Hamilton for a full mathematical treatment of these aspects of the theoretical basis of form as quantity of magnitude. In his work on couples he takes the magnitude of time, it's elements a as moments and constructs the combinatorial framework for these elements. As he constructs he introduces notation that is helpful for its intended use as a system of arithmetic. However he clearly demonstrates that the ordinal system derivable from his magnitude and it's attributes has to be "elide" iny
To the cardinal system (the label system) to which it is homeomorphic without respect to order. However, remaining in the ordinal system gives the full exposition of the "-" distinction he introduces in notation, as the notion of contra both in effective step and process. Thus very fully we may see why certain puzzling things in cardinal arithmetic must be used to indicate the originating ordinal arithmetic, which in fact we now fully recognise as directed or vector arithmetics. However such a recognition fails to highlight the Grassmanian distinction between the continuity within the magnitude itself as an intensive that is inherent attribute , and the discreteness thay quantification imposes on a magnitude in which continuity has to be combinatorially defined as an extensive property?

We may therefore see that Setzen as standard differences or differentials, correspond to unit quantities of a magnitude, in this arithmetically setting, and the Verknuepfen are the natural way thes units are bound together. Again this is usually determined by the analytical processes that identify these units of distinction.

The quantifying of a form may seem a natural thing to do, but in fact the theoretical basis of doing it requires some clear thinking. This Justus Grassmann was able to do, to prepare his students for a world in which the differential calculus would dominate. It is clear that his thinking was far more fundamental than either Dedkind or Cantor.


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