Although this is a brief synopsis of this mornings thoughts, i have to write about the extensive and persuasive influence of the Ausdehningslehre on the conception of reality. I am mightily impressed by the facility it gives the description and discussion of geometrical relationships.
I began by reading a description of gear theory written in so called vector form. The ausdehnungslehre is so evident in this type of description that it is staggering to see it referred to as a vector treatment. Were it not for my apprehension of the influence the Ausdehnungslehre had on Gibbs et al , i would perhaps not have appreciated the connection. Subsequent to this point is the influence on Hamilton. As one of the major competitiors in the formulation of a general algebra describing physical relations it is undoubted that Hamilton knew of Grassmann' work shortly after he published his own on quaternions, but apart from competing i have not researched any collaboration.
The use of Grassmann/Hamilton algebra, especially in the modern notation, allows many similar and homogenous relationships to be surmised by form alone, but also by feel of the calculus involved in the formulation. Thus part way through the reading of the gear theory i recognised the demonstration i had seen of the trochoid family of curves. Though the trochoid treatment had been specific to a geometric diagram, never the less it was a general enough treatment to see the general gear theory forms i was reading about this morning.
Remarkably, without the clutter of explicit reference frames and explicit quantities the relationships involved had been clearly set down and established. Marvellous!
Grassman Ausdehnungslehre i have proposed is a crossover between language and geometry, by which one may precisely describe a geometrical relationship and from which one may then implement any number of equations linked to specific reference frames. I see in this a demonstration of my assertion that
Geometry leads to algebra which in turn leads to arithmetic.
I coined the phrase spaciometry because i did not want to be tied down by common notions of geometry. Spaciometry has always been defined in a dynamic space, not a atatic one. The consequence of this is that so called euclidean geometry was distinguished from Riemannian geometry. However i now see this as a false distinction, based on a false notion about Euclidean geometry. Euclidean geometry was never static!
Much of what Newton was able to divine was from a dynamic geometry, as was what Riemann's insights. Postulate 5 as described by translations of euclid do not allow lines to mee if they are parallel, yet in our own experience parallel line do meet at the vanishing point of perspective. Thus we have a pragmatic, parallaxian view of geometry and an ideal one which does not accord with experience in some minor but far reaching details. Allowing postulate 5 to be varied not only gives us Riemannian geometries but other strange geometries which have apragmatic applicability.
The origin of static geometry is the pythagorean school of thought that put " arithmoi" as the apriori subject or set of principles that governed geometry, music,science and rationality. We have simplistically translated this as number, but that is not the meaning of arithmoi. The meaning is a special set of geometric forms. Thus pyt hagoras and his school looked for relationships that could be described by this special set of geometric forms, and found them everywhere in there environment. The belief, engendered by his revelation from the muse that these relationships were wholly integral, to be harmoniuos was where he had a leadership difficulty. Having pronounced on the matter, it was found not to be true. It was found that not everything could be described in monadic forms.
What the pythagoreans had found was not properly described until Benoit Mandelbrot. They had found the rough nature of all things, the approximate nature of the constructions and relations in reality, the fractal geometry.
Pythagoras had prescribed a dynamic role for "number" and music,but a static role for geometry. However that is a misinterpretation. At the core of reality is a dynamism, and that is represented by these dynamic conformal arithmoi. There very dynamism enabled invariance to be apprehended. Geometry was a record of the invariance derived from these dynamic arithmoi.
It was mot until Leibniz and Newton that this dynamic nature of the arithmoi was appreciated and used to describe the dynamism of space. Dynamic arithmoi conflate into the real number system eventually under the leadership of wallis Dedekind and Cantor, but by then a rigid and non pythagorean view of arithmoi had taken hold of the concept of "number". Arithmoi was lost in translation, mistranslated.
The introduction of negative numbers was the biggest upset. The indian cipherism did not dismiss arithmoi, it merely pointed out a collection of rules and tricks that used the invariant relations between arithmoi, based on modularity and periodicity, to speed up and regularise calculation.. They were called sutras and were gathered into the vedic mathematics we know about today.
The vedic mathematics was of such facility to calculation for astronomers and merchants that it soon took pride of place, and cipherism was born. brahmagupta drew into the mix the philosophy of shunya, and advised on contrapositive numbers. But he did more than just discuss contrapositives, he looked at the whole relationship to rotation, the movement of stars and direction. Thus he and the Indian astronomers also furthered the notion of arithmoi in terms of the special relationships within the circle and right triangle. I coined the phrase shunyasutra to describe these forms which were combinations of curved boundaries, before i fully realised the aptness of the name.
Thus shunyasutras are dynamic forms that reveal the invariant geometries of space in a different way to arithmoi.
The upshot of this train of thought and exposition is that special geometric forms underlie the notion of " number" and therefore the notion of counting number is therefore different to the geometric specialities. For example, whereas 1 will always denote a monad or unit, it does not characterise its size. However 2 does characterize size, magnitude, scale, plethora, and form and relationship. 3 denotes these things, but with regard to form there are at least 3 spatial arrangments that are contiguous, and many more if you include rotational symmetries etc. Thus to state 3 is to only begin to describe what is meant spatiallqy and relationally. Thus, adjectively i may use three in a sentence structure that specifies more precisely the relational structures of the subject matter eg ,3 dance steps,that sequentially move the feet into the subsequent orientation….
Counting off the numeral names is numbering, but it is a rote learning mnemonic, as well as a measuring tool for many other things besides spatial geometrical relations. It is used to order time. Thus a set of geometrical relationships and forms, dynamic in nature are utilised to describe temporal or periodical events, by there very dynamic nature. Hamilton begins his analysis of number at this point, clearly eschewing the investigation of the geometrical basis in his preface to couples. Grassmann starts with the geometry, but assumes the numerical structures as well founded, especially on the basis of Mathematician's analysis and exposition. Both therefore fail to deal adequately with the notion of arithmoi and number at its origin, but both give cogent and powerful foundation to the necessity of extemporising the relationships between space and time sequences.
My concluding point is that if one returns to pythagorean arithmoi one has, in Grassmann in particular,but especially by Hamilton an incontrevertible basis for the geometrical underpinnings of all calculations, and thus a topos situs, as Leibniz once wished for. It is Grassmann who provides the calculus by reworking the common notion of Euclid into the dynamic geometry it always was, and it is Hamilton who then makes sense of the Brahmaguptan shunyasutras as dynamic relationships that embody rotation of arithmoi.
Thus Pythagoras Euclid and Grassmann and Hamilton provide us with a calculus of space based on rotation and extension and relative comparisons, which sit within what Benoit Mandelbrot described as a fractal geometry.