I had been going along, presuming that the Third Man Argument (TMA) was a vicious regress argument—that it was meant to show, in the words of Julia Annas, that, “if we have even one form, we have infinitely many.”
So naturally I was astonished when a reader of Dissoi Blogoi, Sam Rickless (UCSD), wrote in to say that the text says something very different:
The last line of Annas's rendition is: "If we have even one form, we have infinitely many." But the actual text reads as follows: "Each of your forms will no longer be one, but infinitely many." The claim that each form (of largeness) is infinitely many does not follow directly from the claim that there are infinitely many forms of largeness. This is a puzzle.You may consult his original comment in its entirety, here .
The text reads: “And each of your Forms, then, will no longer be one, but rather an unlimited multitude. ( 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.) Was Rickless right, and had everyone simply been misunderstanding the text, reading into it what they presumed was there, but wasn’t?
Yet when I looked more carefully at the context of the TMA, I concluded, to my satisfaction, that the argument should be understood as Annas had it, because ‘each of your Forms’ means, in effect, ‘what is implied by the One Over Many argument, in each of its applications’. See my post, What Does the Text Say?
And then it appeared to me that Rickless was making a claim about what the text said, by making a claim about what the text did not say. It was Rickless, it seemed, who was reading into the text something that was not there. This was so in his original post:
The claim that each form (of largeness) is infinitely many does not follow directly from the claim that there are infinitely many forms of largeness.
But the words ‘of largeness’ do not occur in the text! It is Rickless' gloss.
And the same thing turned up in his 1998 Phil Review paper:
Parmenides then concludes that each Form of Largeness “will no longer be one, but unlimited in multitude”. (520)
But there is an additional problem for the standard answer. For at the conclusion of the argument, Parmenides says not only that each Form of Largeness “will no longer be one”, but also that each Form of Largeness will be “unlimited in multitude”, that is, infinitely many. But how are we to understand the statement that each Form of Largeness is infinitely many? (521)
Let’s begin with (M), the claim that each Form of Largeness is many. (522)
It may be true that Rickless' interpretation is consistent with the text of the TMA, but it's not true that his interpretation gets special support from the language of the text ("But the actual text reads as follows", he wrote), or pays attention to that language, in a way that the usual interpretation does not.