After young Socrates
proposes the theory of Forms, Parmenides and Zeno are described as paying "the closest attention" to him, "and often looked at one another, and smiled as if in admiration of [Socrates]." The impression given by that detail, and the subsequent questioning, is that Socrates' theory is one they both have discussed -- and thus a theory whose problems are well known to them.
Parmenides takes over the questioning of Socrates, to explore the difficulties of the theory of Forms -- but along the way, he illuminates what the Forms must be like if they do in fact exist.
The first difficulty Parmenides raises is whether all things end up having Forms on Socrates' model -- not just things like The Good or Justice, but whether there is a form of Man that is apart from the many men; Socrates says there must be. What, then, about trivial things, like mud or hair? Socrates is unsure as to whether such things merit a Form. Parmenides puts his hesitancy down to his youth:
Soc: I am afraid that there would be an absurdity
in assuming any idea of them, although I sometimes get disturbed,
and begin to think that there is nothing without an idea; but then
again, when I have taken up this position, I run away, because I am
afraid that I may fall into a bottomless pit of nonsense, and perish;
and so I return to the ideas of which I was just now speaking, and
occupy myself with them.
Par: Yes, Socrates, said Parmenides; that is because you are still young;
the time will come, if I am not mistaken, when philosophy will have
a firmer grasp of you, and then you will not despise even the meanest
things; at your age, you are too much disposed to regard opinions
of men.
This point may seem trivial, but it is not. The Forms must be vast in number if they are real, because they must embrace all sorts of likenesses. It is not just great and important ideas that have Forms, but all ideas that we would use in discussions of the things in the world.
This leads to another problem: in what way can a single Form be participated in by all these many things? Socrates proposes that it is like the way that all of us participate in the same day; the "Day" isn't anywhere in particular, but somehow everywhere, and we are all participating in it. Parmenides proposes an analogy that he claims is fair (though it is not, as we'll see) to having a big sailcloth draped over everyone: then, everyone under the sail participates in being under the sail, but it is common to all.
The point of disanalogy is that the day can't be divided into physical parts like the sail can.* Once Socrates accepts the analogy for discussion, Parmenides immediately uses that point to prove that the Forms can't in fact be like a sail. For if they were, then each person would have only a part of the idea captured by the Form, and not the whole.
Thus, if all men are participating in the Form of Man, we would have to say that each one was only part of a Man; and, worse, that your part was different from mine, so that we couldn't really say that we participated in "the same thing" at all. The whole idea of the Form is that it is what is alike in two things that make it proper to discuss them as being the same. The Form thus can't have parts, but must exist as a unity (a 'simple,' in later terminology, meaning an indivisible).
So the idea is not just that "each equal thing, if possessing some small portion of equality less than absolute equality" still must "be equal to some other thing by virtue of that portion only." The idea is that the Form itself either is or is not participated in by the individual that is (or isn't) equal.
Now that is a problem given where we began, although Parmenides doesn't bring it out here. Zeno's account of motion was that you can't get from White to Not-White because you'd have to be two contrary things at once. Socrates' proposed solution was that a thing (Aristotle will call this kind of thing a 'substrate') that can be either white or not-white is what makes the motion from white to not-white. Thus, White doesn't have to admit of its contrary; rather, the substrate, which could have been the one or the other, begins admitting of ('participating in') the contrary Form.
Yet Parmenides has just shown that the Form must be a simple unity, and that participating in it therefore means participating in it fully because the Form is indivisible. So to participate in Whiteness is to have the whole of Whiteness; and participating in Not-Whiteness would mean having the whole of that present. The logical contradiction doesn't end up being escapable in this way (a problem also for Aristotle, whose account in the Physics 1&2 depends on just this move.)
The last problem I'll treat today is better known by its Aristotelian name
'the Third Man argument.' Parmenides is raising the same problem as an objection to the Forms.
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?
If the Form of Largeness embraces all the large things, doesn't it seem large itself? If so, then there must be another Form that embraces the whole set of large things, plus Largeness as well. Yet won't that set seem larger (being, after all, the whole previous set plus one more big thing)? Then there must be another Form that embraces everything Largeness embraced, plus Largeness, plus the form that embraced the rest.
Aristotle's treatment of the Third Man argument takes it as a serious objection to separate Forms (this is in the Metaphysics). Aristotle doesn't admit of separate forms for the most part, excepting the Unmoved Movers (of whom there were several for Aristotle; later thinkers reduced them to one, God). Socrates has a simpler answer: since these are ideas, they don't admit of the problem in the first place. You can think about "largeness" all you want without thinking a large thought; thoughts aren't 'large' in even an analogous way to the physical things that are large. You can think about all the men you know, and try to identify a thought that approaches something like the Form of Man; but it won't be a man, it'll be a thought.
Socrates thus thinks that the problem Parmenides is trying to raise here is a non-starter. But Parmenides has more to say about it, which we'll get to next time.
* Except according to metaphysics that treat time as a kind of dimensional space (e.g. spacetime), in which case dividing the 'time' makes exactly as much sense as dividing the sail; but Socrates is still OK even on such a metaphysics, because everyone is equally present in each part of the fourth-dimensional 'day.' The fourth dimension is a single dimension, so all the third-dimensional parts of each of us would all be fully present in this time line extended from dawn to dusk.
