987 a 9-27. Down to the Italian philosophers and with the exception of them the rest have spoken more reasonably about these principles, except that, as we said, they do indeed use two principles, and the one of these, whence is motion, some regard as one and others as twofold. The Pythagoreans, however, while they in similar manner assume two first principles, add this which is peculiar to themselves: that they do not think that the finite and the infinite and the one are certain other things by nature, such as fire or earth or any other such thing, but the infinite itself and unity itself are [Page 139] the essence of the things of which they are predicated, and so they make number the essence of all things. So they taught after this manner about them, and began to discourse and to define what being is, but they made it altogether too simple a matter. For they made their definitions superficially, and to whatever first the definition might apply, this they thought to be the essence of the matter ; as if one should say that twofold and two were the same, because the twofold subsists in the two. But undoubtedly the two and the twofold are not the same; otherwise the one will be many – a consequence which even they would not draw. So much then may be learned from the earlier philosophers and from their successors.
That twofold and two ness do not subsist in each other, and therefore are distinct is not established by contradiction as Aristotle would on the face of it have us believe " for then the one would be many" does not actually demonstrate a contradiction. However, since the translation is unsafe in terms of the notion of arithmos, and the platonic understanding of arithmos differs from Aristotles exposition of it, and I daresay from the Pythagorean schools traditions of it, it is of course not easy to establish the problem from this quote alone.
Suffice it to say, that the texts in translation have been sufficiently contaminated by the notion called "number" as to make that the inevitable conclusion of the meaning of arithmos. Whereas I say to you dear reader, that if you would but refer to the notion we call a mosaic on each and every occasion you find the word arithmos that you would then understand better every subtlety of the notion from " Pythagoras" to Plato.
We have the equally important cultural influence of the shrines to the Muses in which the word Mousika is ascribed a higher ranking than mathematikos. We do not see the term Geometry as. Muse inspired field of expression, which is anomalous to the claim that Plato's Acadeny welcomed lovers of Geometry as an entrance requirement. This is most probably a fiction of early Hellenistic Christians eager to promote Pythagoras but disentangle him from the growing Jesus mythologies.
The Italian pythagoreans greatly influenced the young and questing Plato with their principles and practice, which Plato gave the name "formalism or Idealism" . He was also mightily impressed by the teaching method called a Konan style, setting the student a problem that leads to his enlightenment by inspiration of the Muses.
What can be said about geometry has to be said in terms of Gematria, and though measuring the land was a surveyors activity, it was of a lower order to measuring the stars, the job of an Astrologer. In the Pythagorean tradition the Astrologer was a master of the intricacies of measurement and calculation, and was denoted as Mathematikos. Thus Mathematikos refers to an accomplished astrologer who could calculate the positions of stars and planets.
The fundamental methods of this calculation and measurement were called Gematria. The related geometry was the crossover of these skills in application to land survey . That the two terms derive from the same origin is to be expected, because the civilisations that promoted the skills of an Astrologer also required the astrologer to survey the lands of the Empire. We have a very modern example in Gauss's career, and for that matter Euler.
So without straining too hard we may appreciate that the confluence of arithmos and Mathematikos and Gematria is complete in the shrines of the Muses. As causes of learning therefore the Muses take preeminence in inspiring notions and thus the mosaic takes preeminence to the Gematria and thus to the astrology and the geometry which rely on Gematria,
Now the notion of a mosaic in Pythagorean times was a decorative floor pattern made out of coloured pebbles which honoured the Muses, and displayed aspirations or inspirations, thus these mosaics were highly abstract inspirational forms celebrating harmony rhythm, poetry dance and movement. These are the referent to Arithmoi.
Aristotles blunder is to accept Plato's theory of Form/Ideas and thus in a sense accept the reality of these forms as independent of human conscious/unconscious interaction, but ever since Hume we have understood that knowledge is generated by experience, and thus these forms derive from our collective experience and are stored, in encoded form of languages, in our brain and cultural interactions, chiefly language. Therefore we form the abstract notions by abstracting from common cultural notions and experiences. Thus twofold does subsist in two, but we abstract twofold to use it for a distinguished purpose. That distinction still subsists in two , but for clarity's sake we deemphasise it in it source and emphasise it in its application
We now understand that most of our fundamental ideas are tautological in precisely this way.
As for one being many, the later Neoplatonists took the monad to that Aristotelian extreme by creating the Henads. These were all precursors to and in fact fractal notions of reality.
Of course I have also blundered by implying Aristotle decided to accept what I claim he did in mistake. One is of course free to choose what one wants to accept, and for all I know Hume may not be the best opinion to access, but the point is emphasised by this piece of hyprbole, and so it remains in title.
Another word for mosaic Is psepipedos, (epipedos) which in this context literally refers to a decorated floor. Both terms inform the philosophy of form/ideas exposited by Euclid, on Plato's behalf,
There is one further point that Plato playfully puts , and that is how alpha and alpha can be equated or dualed with beta.
That opens up another can of worms!