Platonic Socratic theory of Form/Idea as taught by Euclid is an analysis of the sphere, and indeed the relationships between spherical forms as they overlap, their spherical surface from total inclusion to total exclusion. In short, how bubbles connect and disconnect.
The notion of a sphere is analysed down to a seemeioon, and from the seemeioon several special forms and relationships are distinguished as a useful superstructure for synthesis of any form using distinguished seemeioon.
Here the distinction between the pragmatic seemeioon and the formal or analytical one can be made.
The theory of form/ idea requires a set of tools and procedures used in the analysis and synthesis. To formally explain these tools requires that the theory of Form/ Idea be tautological or self reflexive. To hide, obscure or avoid considering this fact theoreticians ether hand wave some convenient axioms into place as givens,( that is axles around which their demonstration will swivel!), or they simply demand or require you to assent to certain understandings, that is they postulate or itemise as a requirement.
It is perhaps hard to see how thy got away with it for so long, but these behaviours are enshrined in traditions, and are overlaid with some traditional benefits, not the least of which is societal acceptability! Indeed so jealous are the guardians of their society of their societal cohesiveness that they will require some sunthemata of their postulants( those begging to join) and seal the agreement with some sumbola.
The Academic system today still follows this ancient Aristotelian tradition, extolling entry requirements to any course as a way to maintain standards.
Regardless, Euclid makes pedagogical demands on his students , which of course results in a social norm being imposed by ability. The demands are constructional : you must be able to draw a straight line between points; you must be able to extend that straight line as far as necessary; you must be able to extend straight lines so they cross one another or meet , however far away that is: you must be able to draw a circle( in a plain surface set out for drawing) as big as is needed.
The final requirement is a assent to the notion that all orthogonal " knees" are identical . This in fact can be proved, but the Stoikeioon is an introductory course, and Euclid left the proof to a later stage or level of the course. It is in fact a theorem of Thales, which he reputedly brought back from Egypt .
The motion of identity or identicalness ( iso) is used to support common understandings or judgements about formal combinatorial relationships, but the final procedure invoked is a physical, pragmatic lifting and rotating of the rigid form and placing it on top of the putative identical form.
This physicality, which provides visual and kinaesthetic confirmation of "fit", that is contiguity of boundary, from which flows without definition collinearity of arbitrary lines , and also the notion of " Artios" and it's conjugate "perisos" which are fully introduced in Book 7 with the mosaic concept otherwise called Arithmos.
Arithmos allows us to interact with form by processes of factorisation, counting, fractionalisation, and synthesis of multiforms. This along with Book 6 represents Euclids teaching on the Eudoxian theory of proportionality, which Eudoxus derived analytically from his deep study of the sphere., the goal of the Stoikeioon.
The pragmatic seemeioon represents this formal setting in everyday affairs, thus attesting to Playo's theory of form through the intriguing question? Which is real: the pragmatic or the formal? Each man must decide for himself.
The pragmatic seemeioon is construct able where the formal one is not. Every formal seemeioon is in fact represented by a pragmatic one. Thus the Grassmann's in tackling this issue , struggling to make a university level course accessible to primary school kids, as well as to justify their resolutions gave such a fine reworking of Euclid's Stoikeioon that we have a full algebraic representation of it. The groundwork of "rigour" laid by Justus, enabled Herrmann and Robert to develop a consistent Formenlehre. But it was Herrmann who cracked the formal difficulties and inconsistencies of his Fathers approach. This he did by a rigorous re analysis of the problem based on his fathers framework..
If a Vedic scholar would recognise the Ausdehnungslehre as a philosophical Guna, and the lineal algebra as a Ganitas Algrbra, they would have a deep appreciation of Hermann Grassmanns intuitive thinking!
It is quite beautiful to read their thinking, as Robert saw an opportunity to build his own ideas on his Brothers. Roberts ideas are principally laid out in Herrmanns 1862 reworking of his 1864 Ausdehnungslehre. The surviving principle seems to be Keine Abweichung! That is invariance of result no matter what the form.robert published several more books on his version of the Ausdehnungslehre, in honour of his fathers pioneering work in what later became group and ring theory.
There were others working in this field at the time, most notably Hamilton, and to an unknown secret extent Gauss, but Hamilton acknowledged the superiority of Hermanns work, not realising perhaps that it was virtually ignored in Prussia!.
The reason I wrote this post is because Hermmann's lineal algebraic rules use the notion of parallel exclusively to define the lineal algebra. This constitutes a second decomposition of space into parallel and intersecting planes, contiguous with Euclid/Eudoxus decomposition into concentric and MULTI centred spheres. The order of priority is, for a given set of tools, points or seemeioons are required to synthesise spheres and spheres are require to synthesise circular planes and circular planes are required to synthesise straight lines, and all of these forms are constructed from "iso" " points" that is dual seemeioon. By intersecting spherical surfaces precisely in these dual points/ seemeioon.
All thes decompositions require 2 arbitrary seemeioon as the starting point for synthesis using the given tools.
Once one accepts these 2 kinds of decomposition, Archimedes realised there are a whole host of other decomposition based on the dynamic combination of these which lead to the varying forms of vortices, vortex surfaces like the cone and spirals, all of which I have called the Shunyasutras. In a more western terminology I have also called them vorticular space. Dynamic situations require shunyasutras to describe them, and this is what Brahmagupta was describing, not a perpetual mobile!. Brahmagupta had actually isolated what we call the roots of unity in the sphere, and what I have called the children of Shunya.
The choice, by Hermann to found the Algebra of the seemeioon on the bisection construction, and so the Schwerpunkt as a point midway between the 2 summe point, has intended or unintended consequences when we move up to the lineal algebra. The lineal algebra requires the parallelogram to underpin the process of combination called producting,
'multiplication", factorisation etc. As a consequence the midpoint theorem for triancles surfaces as the lineal equivalent to the addition of points, using parallel lines in addition to bisection to establish the schwerpunkt relations in lineal algebras..
The importance of this is that a complete decomposition of the plane is possible using parallel line constructions. In fact Norman"s course on Wildlinear algebra demonstrates this fully.
I must confess that i do balk at the use of the word multiplication anf producting when describing these group and ring theoretical actions. While i can except them in the course of the Grassmann struggling to reconfigure the logical base of geometry along the lines of Arithmetic, a pythagorean initiative if ever there was one, i find them retarding and confusing i the light of the full concept of group and ring theory we have today.
These are activities or sequential processes of synthesis that we engage in to construct a resultant form/idea. Consequently, since we construct from "elements" sequential combinatorial aspects play a vital role. Thus in general i prefer to rename these processes as combinatorial processes of synthesis.
Thus Hermann noted that one combinatorial process of constructing a triangle from 3 points enshrined the law of 2 Strecken. That a + sign is used in that law disitinguishes it from the law of 3 Strecken in the construction of a parallelogram , where he found that he could enshrine a distributive rule for parallelograms on the same base using the x or , to indicate a different construction process to the triangle which has the + associated with it. Thus the + symbolises a triangle construction process and the x or . symbolises a parallelogram construction process.
It took a while for Hermann to realise that he had in fact stumbled on a greater algebra, a process algebra in which the construction or combinatorial process between the elements and the resultant is key. He describes his exhilaration at realising this in his 1844 Vorrede.
Today Group and Ring theory carry forward his vision, often without refeering to the work of the Grassmanns and particularly Hermann. But in context, it is the result of the Euclidean standard of pedagogy, apprehended somewhat similarly by Leibniz and Newton, that all philosophical musings should be structured veinally, that is usig Isos, and even religiously using Isis. This seems to have been misinterpreted as an axiomatic development. a dream held since the 1500's, that everything can be explained using a few principles. I say misinterpreted because often scholars get their postulates mixed up with their propositions and do not distinguish between Aithema, ennoia and oroi as they should. If they did hey would have to admit that it takes ore than a few principles to explain everything! a lot of defining has to take place and a lot of careful systematic demonstration, proportioning and dialectic{logical consenting} has to occur!
Many others are creditied with the development of the field of ring and group theory, but i think that the Grassmann and Hamilton are fundamental to the present shape of the subject boundary.
The Children of Shunya 