"Each Form is Infinite in Multitude"

If I were to say one thing about my day’s skiing at Sugarloaf and the beauty of the place—just after a fresh 3 inches of snow from the night before—I’d have to say an indefinite number of things. Better, then, to leave it as an idea in my mind.

I asked yesterday, what are the merits of Sam Rickless’ interpretation of the Third Man Argument (TMA) in the Parmenides? The TMA is commonly taken to be an argument to the effect that (in Annas’ words) “if we have even one form, we have infinitely many”. But Rickless points out that the conclusion of the TMA is, strictly, not that there are an infinite number of forms, but rather that each form is infinite: kai\ ou)ke/ti dh\ e(\n e(/kasto/n soi tw=n ei)dw=n e)/stai, a)lla\ a)/peira to\ plh=qoj. (“Each of your forms will no longer be one, but unlimited in multitude”, as Gill and Ryan have it.)

Plato looks to be claiming that each form has multiplicity. But how would this conclusion follow from the TMA? Rickless claims that there is an implicit argument: the TMA establishes an infinite hierarchy of forms, where each form participates in every form that is above it in that hierarchy; but (Parmenides is presuming, and Socrates is granting) participation in more than one form brings with it a corresponding dividedness or multiplicity in the thing that so participates; thus each form would no longer be one and undivided, as Socrates had claimed, but rather plural and divided.

It must be admitted that this way of reading the argument is attractive also, because it makes the TMA similar to well-known arguments of Zeno, which aim to show that something which seemed to be single and coherent in fact breaks up into an infinite composite.

Rickless also claims that his way of construing the TMA fits well his big-picture view of the development of Plato’s thought about forms, according to which Plato in his later dialogues rejects the thesis that forms are ‘pure’, that is, that they cannot have contrary properties.

I wonder if readers of Dissoi Blogoi were as struck by the astuteness and cleverness of Rickless’ point about the TMA, as was I. Nonetheless, I think his interpretation cannot be upheld as a good interpretation of just that contested line—as I shall explain in my next post—and that, as a result, his big-picture interpretation has to be asserted in spite of, not with the assistance of, the TMA.