I apologize for the lateness of my reply. Life intrudes from time to time.
radical said:
But Hilbert was a formalist who claimed that mathematics was just a game of symbols and that they didn't have real properties. The symbols had artificial properties, the one's Hilbert himself assigned to them.
I haven't said anything about formalism just as such, though it was bound to come up at some point. Of the major positions in the metaphysics in mathematics, this seems to be the only one that nominalists can adopt without some serious wrenching.
radical said:
Idk if Godel, who did destroy Hilbert's program, can be applied to nominalism in that way. I mean, perhaps he can, but I think it would take more than 4 paragraphs to demonstrate the similarities of Hilbert and nominalism, and then demonstrate the similar Godel statement that refutes them both.
Oh, sure. I was doing no more than sketching an argument that tends to float around discussions like this one. Books have been written about this subject; since the space limit in a post on these boards is 5,000
characters, it's pretty difficult to actually flesh out the full argument. I'd have to make something like 70 posts or so, and I don't have time for that.
But the argument itself is fairly well represented in modern philosophy. Most philosophers who work in this area are willing to accept a weaker version of anti-nominalism, in which abstract objects exist, but are reducible to physical ones. However, it seems to me that no good argument for
that position has so far been able to defend against a nominalist challenge. Godel's argument does so definitively, IMO, though of course if it is successful, it establishes a much stronger position. Godel thought he had done more than merely destroy Hilbert's program. He thought he had proved Platonism with regard to the objects of formal systems essentially correct, and I think he's right. I don't really see any way around this result (although, I did just recently acquire a book whose author purports to show otherwise. Haven't read it yet).
radical said:
I think I also may conceive of "nominalism" different than you do. I think of nominalism as denying the existence of abstract objects to varying degrees.
Well, I would quibble over the phrase "to varying degrees." Nominalists deny the existence of abstract objects
tout court. A nominalist cannot accept that there exist abstract objects which are reducible to physical ones, for example. The terms "Nominalism" and "Platonism" carry with them a connotation of absolute-ness. If you think there's a single abstract object, no matter how remote, you cannot be a nominalist. Calling oneself a nominalist "only with regard to x" is kind-of like calling oneself a racist "only with regard to x," moral opprobrium aside.
radical said:
I actually agree with the nominalist for the grandest of abstract objects, largely due to my denial of Platonic universal forms. However, I don't deny form exists, I just limit it to the actual instances. Both you and I have the form of man, but my form "man" and your form "man" are not the same "man" nor is it an extension of some great "manness" that descends in two incomplete ways or something like that.
Then how do you get around the argument which Plato himself makes for forms? We may both be men, but the properties we have in common are likely fairly tenuous, and strained even further once we consider all men. We both have (presumably) two arms and two legs, but not all men do. We both (presumably) have male genitalia, but so do other male animals. What is it that we actually share in common? The question becomes even more accute for other things like triangles, money, lakes, art, games, etc. There are no properties in common betwen all members of the class, and yet, we recognize those members rather easily. What are we doing, if we're not apprehending something like a form?
Note that I'm not saying there isn't an answer. I am saying that the argument has some legs, and it shouldn't simply be dismissed.
radical said:
For the example of removing all hydrogen from the universe by turning it into helium, I would say that the form "hydrogen" no longer exists. Because it doesn't. Where is the form?
I'm not sure whether you mean that question rhetorically or not. If so, then I don't really see an argument here--just a kind of incredulous stare. Plato has posted an argument for the existence of forms, and platonists (like Frege and Godel) have updated that argument with some important and forceful points. If the incredulous stare is the only thing that can be brought against the argument, there doesn't seem to be a good reason to accept your position.
If you do not mean the question rhetorically, then it's basically an unfair question. Forms don't exist in space; they were never posited to do so.
