PYTHAGORAS TO PLATO instance, can be grouped into the class of human beings Ideas in some way resemble classes: participation in an Idea can be assimilated to membership of a class The diYculty in identifying Ideas with classes arises again over the Principle of Self-Predication The class of men is not a man and we cannot say in general that the class of Fs is F However, it seems at Wrst sight as if there are, indeed, some classes that are members of themselves, such as the class of classes But just as Plato was to Wnd that the Principle of SelfPredication led him into serious problems, so modern philosophers discovered that if one was allowed total freedom to form classes of classes one would be led into paradoxes Most notorious is the paradox of the class of all classes that are not members of themselves Bertrand Russell pointed out that if this class is a member of itself it is not a member of itself, and if it is not a member of itself then it is a member of itself It is no accident that Russell’s paradox bears a striking resemblance to Plato’s self-criticism in the Parmenides Paradigms It has more than once been suggested that Platonic Ideas might be looked on as paradigms or standards: the relation between individuals and Ideas might be thought to be similar to that between metre-long objects and the Standard Metre by which the metre length was formerly deWned.29 This notion Wts well the way in which for Plato particulars imitate or resemble Ideas: to be a metre long was, precisely, to resemble the Standard Metre, and if two things were each a metre long it was in virtue of their common resemblance to the paradigm However, such paradigms fail the Principle of Sublimity: the Standard Metre was not in heaven but in Paris Concrete universals Philosophers have sometimes toyed with the notion that in a sentence such as ‘Water is Xuid’ the word ‘water’ is to be treated as the name of a single scattered object, the aqueous portion of the world, made up of puddles, rivers, lakes, and so on This would give a clear sense to Plato’s principle that particulars participate in Ideas: this particular bottle of water is quite literally a part of all-the-water-in-the-world Moreover, water is undoubtedly water, and nothing that is not water is really and truly water This notion also suits Plato’s preference (not often shared by his commentators) for referring to Ideas by a concrete mode of speech (e.g 29 The idea originated with Wittgenstein See P T Geach, ‘The Third Man Again’, in R E Allen (ed.), Studies in Plato’s Metaphysics (London: Routledge & Kegan Paul, 1965) 55