Parmenides' objections to Plato's theory of the forms?

Question

What is being argued against Plato's theory of the forms in this passage (this is a dialogue):

Well, said Parmenides, and what do you say of another question? What question?

I imagine that the way in which you are led to assume one idea of each kind is as follows:—You see a number of great objects, and when you look at them there seems to you to be one and the same idea (or nature) in them all; hence you conceive of greatness as one.

Very true, said Socrates.

And if you go on and allow your mind in like manner to embrace in one view the idea of greatness and of great things which are not the idea, and to compare them, will not another greatness arise, which will appear to be the source of all these?

It would seem so.

Then another idea of greatness now comes into view over and above absolute greatness, and the individuals which partake of it; and then another, over and above all these, by virtue of which they will all be great, and so each idea instead of being one will be infinitely multiplied.

Answer 1

The passage is from Parmenides, 132a-b. The line of argument is reasonably clear but I am less clear about the soundness of Parmenides' critique.

Parmenides' statement of Socrates' argument for the Forms

1. When a number of things - particulars - have a common predicate ('good', 'beautiful', 'square' for instance), there is a single, determinate property denoted by this predicate, a property in which they 'participate' (methexis) - e.g. Parmenides, 129a.

2. This property is equates to - is identical with - a Form. And so we have the Form of the Beautiful, &c.

3. However, the Forms are self-predicating. That is to say that e.g. the Form of the Beautiful is not only the source of beauty in particulars but is itself beautiful.

4. But then, by virtue of what are (a) the Form of the Beautiful and (b) the beautiful particulars both beautiful ? In effect the Form becomes another particular and a further Form (c) has to be assumed in which both the Form of the Beautiful and the beautiful particulars share a common property of beauty.

5. This triggers a regress (cf. Republic, X.597c) since a yet further common property has to be assumed to be present in (a), (b) and (c) - and hence another Form (d).

Is Parmenides' critique sound ?

I question what seems to be a vital assumption that Parmenides makes. I could query self-predication and Plato's commitment to it but this would be a brave manoeuvre in face of Phaedo 102b-c and elsewhere. However, there appears to be a safer reply to Parmenides, who assumes that the Form of the Beautiful (say) cannot be a Form by virtue of its own intrinsic Phaedo 102b-character but must be so by reason of a further Form that links it to the particulars which participate in it.

I write subject to correction but I cannot see that Plato/ Socrates has to accept this assumption by virtue of the logic of the theory of Forms.

Reading

N. B. Booth, 'Assumptions Involved in the Third Man Argument', Phronesis, Vol. 3, No. 2 (1958), pp. 146-149.