Is the third man argument truly a refutation?

[This post assumes that you know what the third man argument is]

I'm not so sure. For example, the form of a triangle which we'll call triangle A can be one of which other triangles say, triangle B, C, and D, conform to by sharing some sort of chracteristic or characteristics.

Supposedly the problem is that "the form" is also a triangle, and must conform to a new form of a triangle, and so forth in an infinite regress. But why can't the first form of the triangle and the "new form" be exactly the same? Why can't we say that not just B, C, and D, but also A conforms to the new form... A! It would seem to me that the form of the triangle is simply the example triangle we most want to visualize in our heads, that has certain characteristics that other different looking triangles nonetheless share. The example triangle can conform to itself; I don't think it has to conform to a different looking triangle.

True, this requires one to assume without justification (other than intuition I guess) what the "ideal form" is. But I think Plato's point is simply that we draw from some form when identifying something, even if we don't know exactly what that form is. He's not saying the ideal triangle in our head must be 5 inches tall or anything; he just makes the general point of how we draw from a form in our heads. Maybe I'm thinking about a triangle with base of 3 inches and height of 2 inches in my head that I'm comparing an image to when determining if it's a triangle; sure, it's arbitrary, but a person is free to do that if he wants   Perhaps I'm not correct on what Plato truly meant?

I don't really have a position on whether or not the forms are actually correct; but I don't think the third man argument necessarily refutes it.

I think A is actually not necessarily a triangle if it is to be a form. What ever A is, it is a label which in some way encapsulates the set of all conforming objects. Taking this into account, A cannot just be a triangle, since A was initially chosen to be a form representing a multiplicity of triangles based on certain characteristics. If A is one of those triangles then it is not a form, since it has other properties (like side length, which you mentioned) which the objects conforming to it do not have.

We simply cannot directly put an ideal on equal footing with the objects it represents. I can go into detail about how mathematicians have attempted overcome this issue if you'd like; an ideal of a ring is actually one construction that comes to mind.

Edit: After rereading your post, it seems that you are not concerned with self-referential objects; i.e., letting A conform to A. However, as in all instances of self-referentials, there is no way to define A and not get a contradiction.

Maybe I am just not conceptualizing what is really meant by a form. Why can't the form of a triangle just be a closed object with 3 sides (and sure, let's say it is 5 inches on the base and height; obviously the specific length is not the defining characteristic), that all other triangles no matter their size conform to based on having 3 sides? Even the form from which the examples are drawn from can serve as a conforming object too, since it also has 3 sides.

Perhaps the answer is that if the characteristic of "3 sides" is that general, it means that triangles B, C, or D, could just as easily serve as the ideal form, and so there is no reason to assume that it's not actually A, B, and C conforming to D, or A, B, and D conforming to C, and so on?

Does Plato imply that there has to be something special about this ideal form that serves some function that its subordinate triangles would be incapable of serving themselves? I got the feeling that Plato was ok with just making an assumption about what the ideal triangle that we used was.

Obviously, if the problem is that the ideal rectangle itself must be justified, you've got the infinite regress problem, but the same could be said for justifying anything -- not just the theory of forms, but any theory must rest on an assumption anyway.

Ah, the Quest for the Holy Grail.  Yes, it all started with those pesky Greeks.

Philip Mirowski labored mightly in 1986, seeking to use Group Theory in order to reinvent the Theory of Value in Modern Economics.

Indeed, lots of material in the book (below) addresses the same questions with which you appear to be grappling.

http://www.amazon.com/Reconstruction-Economic-Theory-Recent-Thought/dp/0898382114/ref=sr_1_1?s=books&ie=UTF8&qid=1365193051&sr=1-1&keywords=philip+mirowski+the+reconstruction+of+economics

But the algorithm in Amazon that makes this $30 book so ridiculously overpriced on Amazon is probably also grappling with that same Form versus Substance argument from Aristotle.  So Buyer Beware.

Another option is Richard Rorty.  He makes great sense of the Greeks, and most everyone who came after.  Lots of incisive short essays, in his 3 Volume work.

Best of Luck.  

[1] The ideal triangle is distinguished in that we use it to represent all triangles. It could very well be that we envision a specific triangle for this purpose, but as I argued in my first post the triangle itself is necessarily different from the form simply because we are capable of gathering a different sort of information from a form than we can from a triangle.

For the sake of greater rigor, let us call the form some letter, F, and the figure we envision the "ideal". If the ideal conforms to F then we can get F back through only perceiving some characteristics of the ideal. Conversely, when considering all the triangles which the ideal represents, one must specify which characteristics are to be observed, and this specification makes up the form F.

The process of conceptualizing a number of objects when given an ideal defines a form. The form is necessarily different from the ideal because we could also use the ideal to represent polygons, a set of objects which includes all triangles, or we could focus in on additional properties of the ideal to make yet another form.

[2] I have a feeling that my response here addresses this too. However, I want to make certain that I realize every proposition rests on assumptions.

Yawn.  We are all Post-Modernist (PO-MO) now.

Feel free to rejoin the human race, anywhere in the past 113 years.

I am happy to report that I have a much better grasp of what "forms" really are. They need to be able to explain, for example, every other triangle, so just being some triangle at some specific length won't do that. A 2 by 3 triangle would have parts that were not essential to being a triangle for instance, because they do not need to be that particular length. So it seems like a form does have to be this strange, abstract nothingness if it exists at all.

One interesting point Bertrand Russel made was that there are certain sentences we can understand even if we can give no instances of them, like "a number that humans will never think of." It seems then that understanding this sentence/criterion relies on something more abstract, rather than some concrete instance that makes us think of that statement. But who knows, it's always possible I could be misunderstanding something.

I thank repossession for his contribution a little more than that of zborg's.

Ok, that tells me what people think, and that's cool. But I'm not trying to be a psychologist here. I want to think about whatever is satisfying. But of course, people being satisfied is not your priority -- nope, they have to do everything your way, because then we will all discover some ultimate truth and live happily ever after.

I read this discussion sometime ago and was curious what this argument was about. I ended up writing a blog about it (used to do that more often), see http://loekbergman.nl/ThirdManArgument. I started with the question where the sudden gap between man and idea of man came from.

I have to thank zborg for bringing this thread of you to my attention. It was interesting to do. :-)

Elubas, how many of Plato's dialogues have you spent time with? I don't mean to overly complicate the picture, but I don't think that the standard account of the forms coming from analytic scholars like Vlastos, Irwin, and Russell is very satisfying. That is, analytic philosophy people tend to "aristotelianize" Plato. They go in and logic chop Socrates' positions, and call it a day. But since Plato wrote dialogues and not treatises, it seems that a different kind of orientation is required for reading the dialogues--one that pays attention to the drama of the dialogue, and sees the arguments in light of the action. I think if you work through the Republic, Phaedo, Meno, and Parmenides this way, you will have a very different view of the forms. 

I'd be happy to make more detailed case for this position if your interested. In any event I'd be interested to hear what you think, or if you think I've misrepresented your position. 

I've read a fair amount of such dialogues, although not particularly recently. I do feel like, especially with the original few posts I made here, I was getting something wrong about Plato, even if I wasn't entirely sure what.

Nowadays I feel like the third man could simply be answered by saying, it's not clear why a "form" needs to, or even "ought" to, instantiate any sort of property (including itself). Concrete objects instantiate properties; not abstract things. Otherwise we would just look at them as another concrete object and there would be no point in distinguishing between "abstract" and "concrete." But then Plato was a pretty smart guy and maybe, indeed, I don't know precisely what he means by a "form."

Absolutely I'd be interested in what you think.

The way I see a form (or idea) is that it is a set of well defined properties. Some properties might put limits on itself (Rick Smits is big for a human being, but not for an elephant) where other properties might display a big variation. You can compare it to DNA, where 99% of the DNA of all humans is exactly the same. It is the 1% that creates all differences. When you see a DNA string that is around 95% the same, then are you most likely looking at the DNA of a chimp.

I see it as an explanation how the world can create and recreate objects of different species or figures (like triangles) all the time. Every year he saw equal yet different plants growing from seeds that look the same for us. It is amazing indeed. Ideas are for Plato required to explain the natural ordering of things and to explain how it can recreate itself. Ideas are the alpha and omega of the natural ordering.

That is why they do not instantiate properties in my opinion, but they can set limits to the concrete properties of any object that are concretizations of them.