ON THE FIRST PART OF PLATO'S PARMENIDES. 17 the discernment of infinite numerical relations in the mani- fold, is a mere delusion if either of the two great conflicting views, that extension is a mere continuum or that it is a mere aggregate of purely discrete units be accepted. Neither Eleaticism nor Pythagoreanism provides us with a satisfac- tory theory of the relation between the curve as a continuous and single qualitative datum and the numerical infinity of discrete positions which can be taken upon it. Eleaticism, by affirming continuity pure and simple, had burked the problem ; Pythagoreanism, by attempting to treat the posi- tions as themselves extended and as constituent parts of the curve, had fallen into the absurdities of the indefinite regress. And the source of error has been the same in both cases. The thing and the Idea have been treated as if they were both existences of the same order, as if each in fact was a thing in the sensible world. The reason why the earlier philo- sophies made this mistake again is that they failed to distin- guish conceptual from perceptual extension. The real problem before Plato is therefore to provide for this distinction, and it has been elaborately shown by M. Milhaud that it is here that the theory of "ideal numbers" cornes in. The general lines of the solution can indeed be divined from the hypotheses of the latter part of the Par- menides itself. We can there see that Plato is affording us hints of a view according to which the Idea is at once one, that is to say a unique form of qualitative existence, and many, that is dependent in some way upon a quantitative law of the inter-relation of the indefinitely manifold. The full understanding of his view has however, as is well known, to be sought in the Philebus and Timaus, as read in the light of what Aristotle has to say about the ideal numbers. Thanks to the insight of M. Milhaud the meaning of these ideal numbers can be said to be no longer doubtful. The ideal number is a quantitative law by which a unique quality is determined. And, as its character of a number shows, the relations in question are those between positions in space, and the unique qualities are qualities of extension as actually perceived by the senses. 1 The ideal number is, in fact, precisely what we know now as the equation to a curve or surface. Two points in connexion with it are specially note- worthy. The first is that the numbers, unlike those of arithmetic, are incapable of addition to one another. This is betranise the numerical formulae are rules for the constitution 1 1 do not mean that nothing but what is spatial can be counted, but that Plato pretty .certainly held that it is so. 2
