I have formely explained hw i got the wrong end of the stick with regard to the a priori nature of multiplication. It is rapidly becoming clearer now that before duplication,triplication..multiplication there is no other thing to add or subtract together. Also the very act of aggregation is a vector process and deeply rooted in relativistic rotation of individaul parts. Thus we must be given multiple parts before we can aggregate.
Now in the pythagorean philosophy the multip;e parts ari8se from the monad, as being meros within the monad, and are called henads according as they themselves ar particulate in form. Thus enads lie within henads which themselves form a larger henad…
Thus meros aor a part is strongly associated with an original and transformative division process. We see this miracle in cell mitosis and misosis. But we also see it in the fragmentation of a rock, the shatterin of a plate etc.
My subjective experience is therefore a unified whole constituted of parts or a collection of partds divided off grom some "whole", whether objectively or subjectively. Thus division is an a priori notion to multiplication.
Multiplication arises when the parts that a whole is divided into are the same or identical. then the parts can form multiples of each othe, but not of the whole. Similarly the whole if it has a set of congruent others can be used to form multiples. These multiples are products.
A product is a form arrived at by aggregating multiples ,that is bringing them in contact with each other, either by overlapping or just barely touching. That form is called a pollapleisisos, and is a form made up of recognised multiple other usually minor forms.
Now these minor forms do not have to be identical, and so any form can be "covered by a suitable collection of smaller forms, each distinct subform could be considered as a distinct monad.
However, Euclid is next concerned with accurate subforms and approximate subforms of a product . These subforms are the arithmoi, that Euclid defines in terms of accurate and appoximate, proto meanin a prototype subform, perfect, and a few other distinctions. In the course of defining these products aggregation rules for addition and subtraction naturally arise, particularly in the case of differing monads related by scale.
Arithmoi are subforms specifically designed to :cover, measure and form by filling a given shape or form. Therefore they constitute a set of measuring tools , flexibly used to account, model and analyse a form with a view to building or replicating that form for some greater purpose. For pythagoreans the purpose was theurgical in nature and intention, and it was a life long purpose of developing the person toward a higher and better state of just about everything. In the main though, the karmic aspect of theurgy may have been significant.
There is therefore no real sense of developing an axiomatic system in Euclid's Stoikeioon, more a development of the logos and kairos of the God's, whom they wished to emulate in wisdom, knowledge and purity of life and thought.
Later interpreters have mistaken the religious rigour, which is a moral purity for so called mathematical deductive rigour. Amongst those who seek purity even their words must be pure. to assure this pure words had to be defined and used. By thus habit many besides Pythagoreans have found the principles of harmony much more speedily.
However, the use of obscure notions is not edifying. Therefore Pythagoreans start with common notions. In this way every one has a chance to learn of the pythagorean philosophy. Pythagoras, after all was a public teacher.