Truth, Provability and Mathematical Realism
Sometimes I think it's unfortunate that advanced logic and metamathematics usually presuppose various mathematical truths. For instance, in discussions on mathematical realism I've heard people arguing that by the first incompleteness theorem, mathematical truth can't be identified with provability in a formal deductive system. For, those people argue, the first incompleteness theorem proves that for any reasonable formalized system of mathematics, there is a true arithmetical sentence G that is unprovable in the system.
But if this is what Gödel's theorem says, it seems to me that in discussions about mathematical realism one shouldn't take it for granted that the theorem holds.
Compare this argument for the existence of numbers:
There are infinitely many prime numbers.
Therefore: there are numbers.
This is a perfectly good argument. But one shouldn't say the premise must hold because Euclid proved it. Euclid did prove it, but his proof presupposes that there is at least one prime number, that if numbers n and m exist, then so does n*m+1, etc. So when the existence of numbers is at issue, Euclid's result can't be taken for granted.
Similarly, the usual proofs of Gödel's theorem are up to their ears in mathematical realism, assuming e.g. that there are non-trivial numerical properties and functions that can or cannot be represented in various theories. (The same holds for other results in metamathematics, in particular the categoricity proof for second order Peano Arithmetic seems to presuppose that the (unique) standard model of arithmetic exists.)
It would be nice if one could isolate a purely logical result from Gödel's proof that does not rest on any mathematical assumptions. Perhaps one could still show that any consistent extension of Robinson Arithmetic is incomplete, but I'm not sure about this. Then in discussions about mathematical realism we'd at least know that if we identify truth with provability, we must accept truth-value gaps in mathematics. (Which doesn't sound too bad, given that many mathematicians and philosophers believe that the Continuum Hypothesis is neither true nor false.)
Hi Wolfgang,
I think G?del's proof can contribute to an argument in favor of mathematical realism if you allow two things:
1) It's an argument AGAINST one of mathematical realism's major rivals, namely against identifying mathematical truth with derivability or provability of some sorts rather than a direct argument FOR mathematical realism.
2) We need the additional premise that all props of elementary arithmetic are bivalent.
Under these conditions, there is no undue presupposition, it's a reductio argument. Suppose we identified mathematical truth with L-derivability from axiom set A, such that L and A are strong enough to yield elementary arithmetic. It would then be L-derivable from A (and thus, by our own definition, true) via G?del's proof that there is an arithmetical proposition G such that neither G nor not G are L-derivable from A. So G is not bivalent, which contradicts our premise.
As regards the justification for the premise of bivalence, we should keep in mind that G is a statement about positive whole numbers, not about some eccentric set-theoretic apparitions. Here the intuition of bivalence is very strong: For all we know, e.g., the hypothesis that every even number (greater than 2) is the sum of two primes might end up being neither true nor false. For this is one statement that has so far neither been proven nor confuted, and could very well be one of those for which neither proof nor falsification is possible. You have to be an extremely tough formalist to bite this bullet and concede that, under these circumstances, there simply is no fact of the matter that determines whether all the even numbers from 4 on are sums of two primes or not. (Haskell B. Curry would be an example of a formalist that tough.)
You can of course, still be a mathematical anti-realist and keep bivalence. E.g., you can just keep the standard referential semantics for mathematical statements and say that numbers, sets and the like don't exist and thus that "There is an even prime number" is false and "There is no greatest prime number" is true (but, from the realist perspective, for the wrong reason). Field is such an anti-realist. Therefore, the argument above need not bother him. Other arguments do, but that's a different story altogether.
Cheers from old Bielefeld
Torsten