What is the purpose of Plato's concept of Form?

Question

Concerning this passage from Phaedo:

I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called odd, which is not the same as three? Yet the number three and the number five and half of numbers in general are so constituted, that each of them is odd though not identified with the idea of odd. And in the same way two and four and all the other series of numbers are even, each of them, though not identical with evenness. (104a-b)

In a math.stackexchange question Jon Ericson suggested:

The philosophical point is that there exists an Idea (or Form) called Odd and odd numbers are merely specific instances of Odd. The [numbers] are not, themselves, identical to the concept of Oddness.

I made a remark on this statement:

More to my point, I don't fully understand the point of [the quote].
Why is there a distinction made between odd, in the context of numbers, and some other esoteric and undefined sense of odd? I'm not saying it's not valid, I'm just saying I don't see the use.

I realize that not all philosophers are mathematicians, but I do not believe this is an exclusive question. So, can anyone enlighten me on why Plato made this distinction?

This particular discussion begins (I think) on [103e] and spans on until [105c].

Answer 1

Here's Socrates' lead in to the quote I pulled for my question on Math.SE:

“The fact is,” said he, “in some such cases, that not only the abstract idea itself has a right to the same name through all time, but also something else, which is not the idea, but which always, whenever it exists, has the form of the idea. But perhaps I can make my meaning clearer by some examples. In numbers, the odd must always have the name of odd, must it not?” (103e)

Exactly how the conversation got to this point is a long story, but Plato is introducing the concept that abstract ideas really exist. (In the context of the dialogue, Socrates is trying to reassure his followers that they need not fear death—either their own or his. If abstract ideas exist, then maybe souls do too.) Perhaps the most accessible abstract idea is number. You can have two fingers, two toes and two vats of olive oil. If you add one to any of these, you get three of them. If you do math on your fingers, you know that the results will still hold true for the olive oil. That's a powerful (and very ancient) concept.

What confuses matters these days is that Plato's student, Aristotle, disagreed with Plato about the existence of abstract ideas. Or to put it more correctly, Plato believed they pre-existed before there were people to think about them and Aristotle believed they are human constructs. According to Plato, we recall abstract concepts such as Evenness and Oddness from the time before we were born. According to Aristotle, we create abstractions by observing the world around us.

Obviously, most of us (in the Western world at least) side with Aristotle since our culture has been built on Empiricism. Even so, Plato has has a profound influence—especially when it comes to mathematics.

Answer 2

The point of the doctrine of the forms is to answer three questions:

1. How is universal, necessary knowledge possible?
2. What makes a sentence like `Socrates is white' true?
3. What explains why different objects are members of the same kind?

The doctrine is actually quite elegant in that it offers a unified answer to all three questions. Briefly, here's the significance of each question and how Plato's theory solves it.

Re: 1. It is obvious that there are some truths we know that simply cannot have been otherwise. Think about the pythagorean theorem or a sentence like `All bachelors are unmarried men'. Plato thinks it is obvious these sentences are always true for everyone and at all times and place and that they have to be true. Now he also thinks that every true sentence is made true by something (see q. 2 below), but look, nothing in the visible world around us is universal and necessary and changeless like that. Therefore the things that make sentences like this true must not be part of the visible world around us, it must be part of a different world that we know not by experience, but by pure intellectual insight. And bingo, that's the world of the forms.

Re: 2. Aristotle thinks that a sentence like `Socrates is white' is going to be true if and only if there is some kind of relationship between that sentence and what the world is like--there has to be some object or thing that that sentence resembles or pictures or something. Why he thinks this isn't entirely clear to me, but it isn't entirely crazy. The forms then, are these objects that can fill that explanatory role. `Socrates is white' is true because Socrates is related to this object whiteness in some way. So the theory of the forms again solves an interesting philosophical problem. Of course, in this case the theory creates other problems: Trying to make sense of how Socrates, this individual, visible object is related to this other universal, invisible object whiteness is the biggest difficulty Plato's theory faces.

Re: 3. Again the theory does seem to do some useful explanatory work for Plato in that it helps explain an interesting fact: Socrates and Callias are members of the kind `human being'. What explains their similarity? In short the theory of the forms says that they are the same kind because they are both related to the same object, humanity.

The theory of the forms has really bad problems to avoid too, like the third man argument, the dilemma of participation, and so forth. But it is a really interesting theory and prior to the discovery of these problems (which happens very quickly, Plato's pupil Aristotle knows about them all already) it is a very compelling solution to a complex of difficult problems.