Dissertation committee:
Διονύσιος Αναπολιτάνος, Ομότιμος Καθηγητής, Τμήμα Ιστορίας και Φιλοσοφίας της Επιστήμης, ΕΚΠΑ,
Μιχαήλ Ανούσης, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου
Ιωάννης Εμμανουήλ, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
Στυλιανός Νεγρεπόντης, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
Athanase Papadopoulos, Directeur de Recherche, Institut de Recherche Mathématique Avancée, Université de Strasburg et CNRS
Δέσποινα Πόταρη, Καθηγήτρια, Τμήμα Μαθηματικών, ΕΚΠΑ,
Βασιλική Φαρμάκη, Καθηγήτρια, Τμήμα Μαθηματικών, ΕΚΠΑ
Summary:
In the present work we aim to obtain the true understanding of the nature of the Receptacle as presented in the Timaeus 48e-49a and the reason that Plato felt obliged to introduce a third entity beyond the intelligible Beings and the sensible entities.
We must, however, in beginning our fresh account of the Universe
make more distinctions than we did before;
for whereas then we distinguished two Forms (τότε μὲν γὰρ δύο εἴδη διειλόμεθα),
we must now declare another third kind (τρίτον ἄλλο γένος).
For our former exposition those two were sufficient,
one of them being assumed as a Model Form (παραδείγματος εἶδος),
intelligible and ever uniformly existent, (νοητὸν καὶ ἀεὶ κατὰ ταὐτὰ ὄν),
and the second as the model's Copy (μίμημα δὲ παραδείγματος),
subject to becoming and visible. (γένεσιν ἔχον καὶ ὁρατόν)
A third kind we did not at that time distinguish, considering that those two were sufficient;
but now the argument seems to compel us (ἔοικεν εἰσαναγκάζειν) to try to reveal by words
a Form that is baffling and obscure (χαλεπὸν καὶ ἀμυδρὸν εἶδος).
What essential property, then, are we to conceive it to possess? This in particular,—
that it should be the receptacle, and as it were the nurse, of all Becoming
(πάσης εἶναι γενέσεως ὑποδοχὴν αὐτὴν οἷον τιθήνην).
Yet true though this statement is, we must needs describe it more plainly. 48e2-49a7
Timaeus is Plato’s creation story and description of the physical world. In the middle of the dialogue, Timaeus 48b2, the discourse of the universe is interrupted and a “new beginning” is essessential for describing and explainind the physical world. In the passage 48a2-53c3 Plato introduces a new, third kind, “the receptacle of all coming-to-be”, alongside with the twofold distinction of Forms and the sensibles. The question raised is how could there be such a thing, what is it and why it must be introduced at all. What intriged our interest the most is a bold and, at first sight, unexpeted geometrisation (53c4-58c) following the receptacle passage (48e2-53c3). Plato seems to be following the greek physiology tradition and speaks of the existence of four elements, fire, air, water and earth but by exploiting the theory of the regular solids, as presented in the XIII Book of Euclid’s Elements, he associates the four elements with the canonical solids. It is fair to state that both the reason why Plato felt compelled to introduce the Third Kind and its nature have so far eluded understanding and remained a mystery up to this point, and this certainly not because of any lack in efforts. We were surprised when studying other scholars that the receptacle passage was taken in isolation from the rest of the Timaeus. There has been considerable discussion in a pure philoshophic aspect by the previous scholars about whether the receptacle is thought as matter, or space (in 52a8 the third kind is reffered as “chora”), and whether it is possible to think of it coherently as having both of those roles or not.
In the present work we aim to obtain a new and, we believe, definitive understanding of the nature of the Receptacle and of the reason that Plato felt obliged (ἔοικεν εἰσαναγκάζειν) to introduce it. The research is based on S.Negrepontis’ anthyphairetic interpretation of Platonic philosophy according to which Plato’s geometry and in particular the notion of anthyphairesis is the most crusial concept in understanding Plato’s philosophy. In this regard the intelligible Being is the philosophical analogue of a dyad in periodic anthyphairesis and a sensible body, participating in the intelligible, is described, as already seen in pre- Timaeus dialogues, mainly in Politeia 475-480, as an initial finite segment of the infinite intelligible anthyphairesis. The serious problem that results from the built-in intelligible anthyphairetic infinity, is that this infinity leads to the formation of infinite kosmoi, a problem akin to the intelligible Third Man Argument; it has not been realised that the sole reason for the necessity of introducing the Third Kind in the Timaeus is the necessity to avoid infinite kosmoi. At this point it must be emphasized that the intelligibility is neutral on this problem, its only necessary condition being that the intelligible must retain the control of the sensibles, whether these sensibles live in infinitely many kosmoi or in one kosmos.
Plato attempts to deal with the problem, by eliminating the infinite multitude of anthyphairetic remainders, replacing them by the mathematically equivalent tight double inequalities and generalized side and diameter numbers (the “convergents” of modern continued fractions), which are vividly and aptly described, in order to represent every sensible body, as a dyad consisting of Content and Receptacle, but he realizes that such arithmetical methods, while they work perfectly well for the geometric anthyphairetic remainders, fail to make sense for the intelligible philosophical analogues of the geometric anthyphairetic remainders.
It is precisely because of this differentiation that Plato finds it necessary to introduce the geometric Third Kind; it consists of the four primary bodies, each identified with one of the surfaces of the four (minus the 12hedron) canonical solids, primary earth PE/cube, primary water PW/20hedron, primary air PA/8hedron, primary fire PF/pyramid, studied in the Theaetetean Book XIII of Euclid’s Elements.
