The notion of multiplication is based on Euclids notion of multipleform(pollapleisios), and on his notion of factorisation into protoarithmoi. These forms used for measurement(katametreesi) and construction are magnitudes or monas perceived as unit. Thus the multiplication notion is based on constructing forms by factors in a building process that requires agency. This notion was exemplified by the use of various gnomons, the right and the curved. Where Grassmann made a difference to the notion of multiplication is in introducing dynamic gnomoms which marked out space without the use of arithmoi. Gradually he perceived that the dynamic gnomon was enshrined already in Euclidean thought and that arithmoi similarly were dynamic. He did not realise the problems with number.
The idea of number is a recent creation, and one would not think it could be so problematical. The notion of magnitude precedes the recent notion of number, and magnitude is regionalised and boundarised and apprehended as "form". Form and magnitude are entwined, but form is bounded. One bounds form by the subjective processing centre, and that is the binding and bounding of magnitude. Unbound magnitude is out there in the "wild" waiting for an individual to lasso and brand. So when a magnitude is bounded and branded it becomes a form that has a name.
The Pythagoreans and all cultures branded forms with marks, in particular a dot. The associated mark forms a one to one correspondence with the form. This link between form and mark has existed for thousands of years, and Euclid carried it over into his teaching material, in the notion of the arithmoi and measurement by s metron. However in analysing space in that way lead to a distinction between the vertex, the edge and the surface and the body of a form. These distinctions possessed the attributes of form, by virtue of being constituents of form.
Time lead to these distinctions becoming entities, and thus disembodied, and embodied in themselves. This enabled the notion of meka, that is length extension, but the idea is disembodied form, magnitude without form. The greeks recognised length, breadth and depth as disemodied entities, magnitude without form.
These disembodied entities were not taken seriously. They were entities seeking to appear in reality embodied within a form. The first form in which this happens is in the surface, the epiphaneia and that is what is taken seriously by Euclid, and enshrined within the notion of arithmos, a complete form. This complete form is what is used to count/ measure and construct larger real forms.
So the dots were associated with a form, an arithmos. Why did the ghostly entities become so significant?
The trigonometric ratios emphasise the edges of the arithmoi on the edges of the right angled triangle.But the arithmos is an undefined form, chosen appropriate to the task. The arithmos for this task is a rod or a cord, a real form whcih has meka length in greater abundance than width and depth. This real metron enables magnitude to be compared in these ratios. We can hold the Pythagorean theorem as true in terms of similar form rather than squares. This is understood using the concept of area, which is a reformulation of arithmoi.
The greater magnitude of the forms around the triangle compared are distinct to comparing the area measured in terms of arithmoi. The area on the edges of the right triangle vary according to which similar figures are placed on them, so that comparison varies. But what does not vary is the fixed magnitudes of the edges, the meka. This magnitude is arrived at by using a thinner and thinner arithmoi, or defining each edge as a unique set of arithmoi units. This arrangement is attempting to abstract an invariant relationship from a dynamic situation. This invariance is sought because it represents the immutable relationships known only by the Gods.
Thus the ghost entities are suitable symbols for the spiritual apprehensions of the gods. Form is the protrusion of the gods realm into the material realm. Nowadays we have an alternative realm to the gods, the subjective realm.
It has been argued by the scholars that the Earliest psychological landscape involved what is called a bicameral mind. That early man did not have an identity in the modern sense, but understood mental events as the voices and workings of other wordly gods. I view the mental workings of an animate as being subject to the accepted premises, and thus the experience is determined but varied by the premises even though the working or process is the same wichever animate is chosen.
Subjective processing is based on iteration of regional signal input to determine an output for that stimulus region. The ghostly entities arise from this subjective process: in order to process this part of the input signal the processing regionalises, and the iterative procedure converges on the output. This convergence itself is sufficient to identify th region even if unbounded.Thus a distinction of perception is generated which may be used by the processing process to identify a region as precisely as any definite boundary mark.
This perceptual distinction is the source of the ghostl entities and the basis of abstract notions.
The abstract notion of length, treated the same way as the notion of bounded form, enables simialr measuement and combinatorial processes to be carried out. However, different constraints apply to different forms, and the ghostly entities introduce new and conflicting constraints, which have to be"ignored" or rationalised away.
The completeness of the dt is analogous to the completenes of a whole form.Such an analogical link does not exist between a dot and an incomplete form .
Sometime in the 1800's Dedkind pioneered the transition from arithmoi to gramme as the definition of a number, and defined the notion number which up until then was undefined but understood as entwined with magnitude. But before this Brahmagupta introduced the notion of misfortunate magnitudes, this lead to an angry response and confusion, and opposition. The concept thus went through many reformulations, misinterpretations and clever manipulations. The consequence of choosing the algebraic ruled definition of negative magnitudes was the introduction of √-1.
The struggle to overcome the revulsion felt for misfortunate numbers, was piqued by the nonsensical magnitude √-1.
Nonetheless, the notion of magnitude was flexible enough to accommodate even this wierd idea, and eventually to locate it on the arc of a circle. What was not able to cope was the notion of number. The development of the notion number really is Dedekind'x Wallis attempted to organise the measuring magnitudes for length on a measuring line. This was a development of Descartes ordinate coordinate teference system, which was a development of the Greek "triangulation" method based on Thales theorem,and developed by Ptolemy.
Dedekind seized on the measuring line concept and developed from it the numberline concept, using the new set theoretic language.This created immediate and profound problems under the banner "number", which was resolved by dodging the form and going for the "defining" properties, the combinatorial behaviours. This was eventually conceived as the set theoretic investigations were developed and refined by ring and group theory.
AN Whitehead, inspired by Grassmann was a prime innovator in establishing the symbolic superstructure of "number" and its behaviours in arithmetic. Magnitude was truly excluded from the mathematical and philosophical research drives and left for the physicists to standardise and develop and establish as dimensions.
For a while the magnitudes did not seem consistent with the complex "numbers" of the "crazy" mathematicians, but some crazy physicists found a use for the arc magnitude in the phase patterns of the electromagnetic signal. This definitely confirmed the complex aspects as magnitudes in the real world..
Suddenly the magnitudes reassert themselves as of fundamental to interacting with reality, where number is some kind of mind twist.
Because number has been so derailed we have had to define magnitudes by entities such as vectors, matices and tensors, within which we have placed numbers to signify what?.
Within the magnitude camp we place within these constructed magnitudes, other magnitudes such as metres, metres per second, kilogrammes etc.
Grassmann's contribution to the concept of magnitude is significant in this regard, but he does not offer a critique of "number" because he takes it as logically sound. However he looked backwards to gain this confidence, not forward, like AN witehead and Russel.