[72] Plato, Parmenid. pp. 137 C, 142 B.
[73] This part of the argument is the extreme of dialectic subtlety, p. 139 C-D-E.
First demonstration ends in an assemblage of negative conclusions. Reductio ad Absurdum, of the assumption — Unum non Multa.
All these are impossibilities (concludes Plato). We must therefore go back upon the fundamental principle from which we took our departure, in order to see whether we shall not obtain, on a second trial, any different result.[74]
[74] Plato, Parmenid. p. 142 A.
Here then is a piece of dialectic, put together with ingenuity, showing that everything can be denied, and that nothing can be affirmed of the subject — Unum. All this follows, if you concede the first step, that Unum is not Multa. If Unum be said to have any other attribute except that of being Unum, it would become at once Multa. It cannot even be declared to be either the same with itself, or different from any thing else; because Idem and Diversum are distinct natures from Unum, and if added to it would convert it into Multa.[75] Nay it cannot even be affirmed to be itself: it cannot be named or enunciated: if all predicates are denied, the subject is denied along with them: the subject is nothing but the sum total of its predicates — and when they are all withdrawn, no subject remains. As far as I can understand the bearing of this self-contradictory demonstration, it appears a reductio ad absurdum of the proposition — Unum is not Multa. Now Unum which is not Multa designates the Αὐτὸ-Ἓν or Unum Ideale; which Plato himself affirmed, and which Aristotle impugned.[76] If this be what is meant, the dialogue Parmenides would present here, as in other places, a statement of difficulties understood by Plato as attaching to his own doctrines.
[75] This is the main point of Demonstration 1, and is stated pp. 139 D, 140 A, compared with p. 137 C.
[76] Aristot. Metaph. A. 987, b. 20; A. 992, a. 8; B. 1001, a. 27; I. 1053, b. 18. Some ancient expositors thought that the purpose of Plato in the Parmenides was to demonstrate this Αὐτὸ-Ἓν; see Schol. ad Aristot. Metaph. p. 786, a. 10, Brandis.
It is not easy to find any common bearing between the demonstrations given in this dialogue respecting Ἓν and Πολλὰ — and the observations which Plato makes in the Philêbus upon Ἓν and Πολλά. Would he mean to include the demonstrations which we read in the Parmenides, in the category of what he calls in Philêbus “childish, easy, and irrational debates on that vexed question?” (Plato, Philêbus, p. 14 D). Hardly: for they are at any rate most elaborate as well as ingenious and suggestive. Yet neither do they suit the description which he gives in Philêbus of the genuine, serious, and difficult debates on the same question.