When i was first introiced to ring theory and frou theory, it was a bewildering undergrafuate experience. The axiomatic system was introduced without much fuss and was totally unfamiliar. The properties of a ring or a group were not related to combinatorics, Kombinationslehre. That is the downfall of any curriculum that introduces this university level introduction.
Kombinatinslehre is the gematria of structuring form, A form is divides and the divisions recombined. How they are recombined is the subject of combination theory.
The use of + and * derives from Euclidean combination theory, and that is based on the common inferences, the gnomons, right and curved, and the notion of dual.
However Euclid is interestes in more than just combinatorial rules, he is also interested in attributes of narious "schemas", and relationships between distinctions in Schema and between schema.
The schematics refer to the epihaneia(the plane), but the Stereos are also extensively analysed, compared, contrasted distinguished and combined, related and aprehended. The aim is Theurgical, not numerical, or geometrical. Gematria is theurgical and vocational. The application of the person to the spatial interaction is complete.
The notion of collinearity is not proveable. It has to be assigned, or attributed . Thus it is a fundamental attribute for points and several corolloraies and consequences have to be drawn from its attribution. Menelaus Theorem or corollary is just such a arrangement of dependent or deducible notions.
Thus we hold that the property of collineaarity holds precisely when certain conditions hold. We "tick off" those conditions and thus declare collinearity.
The whole of mathematics is ridden with these recursive relations. That is we observe something and define it. Its validity is determined directly by sensory verification and we subjectively define it as a property. We then draw dductions from that property sothat we can see planly that the property implies the particular consequence. We now have a tautology which self justifies, and the consequence becomes interchangeable with the observed property. We then pose a "problem" to our friends and colleagues, as much for social advancement as for intellectual stimulation, in which we belabour the consequence and challenge our colleagues to explain some consequence of the consequence. Of course, once the correct definitions and observations are known, this becomes a straightforward task, but the challenge has become something else, a social tool of oppression or distinction, y which one may earn nothing more than an accolade. but with accolades come flatteries and possible social advantage.
A similar process avails itself of those who have some other talent that those in power choose to predilect. Thus an industrious worker who has laboured to provide for many is valud less than some lazy but brainy prodigy.
Let us understand the tricks we play on one another, for if a thing is observed and defined to be so does not make it so for all "time" and in all circumstances. Thus Pascal and Papus theorem does not hold for non commutative rings over the projective plane. And what are such things anyway?
Plato redacted Pythagoras to fit his conceptions, Euclid derived from the platonic concepts a collection of inferences and a system of organisation that exposited Pythagorean ideals, and which worshipped the God Isis. Thus eerything which did not contribute to this "dual", this dividing like a vein was pared away. And yet this sparseness still had to provide pragmatic guidance o artisans. Thus deductive logic and inductive logic, based on Kairos, proportinality were employed in the service of the dual, Isis.
Eudoxus demonstrated that the dual was not enough. The triple comparators less dual and greater were necessary to avoid foundering on he rock of exactitude. Because of his reasoning, Eudoxus was not held in high esteem in some quarters, because he saved Pythagorean monism from being disproven as "incorrect", by challenging the very notion of "correctness", Eudoxus really demonstrated that everything is relative to our own subjective processes.
Consequently Euclid's exposition of Pythagorean theurgy, and the gematria that Pythagoras espoused is just one of all possible theurgies and gematrias
We have no real understanding of the term equal as it is used to represnt "isos" or dual. For example my cherry tree exhibits in its branching pattern a representation of Isis. But not all plants divide into duals in their branching, neither do all veins . The flower buds on my cherry tree burst open to reveal 5 stems with a flower at each head. The monad had produced a quintuplet!
Careful study of plant branching shows that it it may appear dual. but it is potentially multipotent. What does this mean? Itmeans that when we say that we have identical twins that identity is only one of a number of identites we have experienced. Thus we have identical triplets etc. The greater the number the grater the variation in identicality.
So now, what if instead of basing "equality" on duality, we entertained a more subtle notion, basing it on triple or quadruple? We would compare three or four things before declaring "equality", because the notion is dependent on what the structure of identicality is perceived as. Euclid chosse 2 as the structure of identicality, and it was a common inference that if 2 things fitted on each other they were dual/equal. but think for one minute, what if we needed 3 things to fit on each other before we declared them triple/equal. Euclid took up the common practice of comparing 2, but it also enshrined the common notion of Isis, the Egyptian god that was worshipped and respected, and it supported the Philosophy taught by Pythagoras that the monad split into the Dyad, and the dyad became the trias etc. Thus the gematria was not just for commerce and measurement, but also for theurgy. The dual/ equality that underpins our modern thinking derives from this theurgy.
In modern treatments of Euclid the "geometry" is allied to "the number theory", without understanding that the relationship is not in these modern terms, it is in terms of the interaction with space, how we engage with, manipulate and apprehend space and spatial relations and magnitude of intensity. It is about the relative rotational motion of which we through our subjective processing are a fundamental, fractal part. The circle and the sphere are ur most powerful tools for engaging in that Theurgical experience, but its true form is acknowleged as Vorticular.
It is the vortex, the fractal patern f vortices that we strive to apprehend.