First Thoughts About Hilbert
Today I've been reading Hilbert. I must admit that I don't really understand his view on the foundations of mathematics. It seems to me that he always confuses truth with consistency. For example, he writes in his "New Grounding":
If we can produce [a consistency proof of formalised mathematics], then we can say that mathematical statements are in fact incontestable and ultimate truths.
Obviously, Hilbert uses "true" in a very unusual way here: Both ZFC + the Continuum Hypothesis and ZFC + its negation are consistent. Hence, on Hilberts account, both CH and its negation are "incontestable and ultimate truths".
What does Hilbert mean when he speaks of truth? A nice interpretation was given by Frege in his "Grundlagen der Geometrie II". Here, Frege suggests that whenever Hilbert asserts a mathematical statement p, or equivalently claims that p is true, then what he really means is that p is true in all models of some unnamed axiomatic system which Hilbert says "defines" the non-logical constants in p. Hence "truth" is a relative matter: CH is an ultimate truth relative to ZFC + CH, but not relative to ZFC + Not-CH.
A problem with Frege's structuralist reading, as with every structuralist account of mathematics, is that there may not be any model satisfying the relevant axioms. This does not appear to bother Hilbert. So maybe his point of view is a kind of syntactic structuralism: p is true relative to a given theory T iff p is provable from T. Or, even better: p is true relative to a given theory T and a given deductive system S iff p is provable in S from T.
I don't think there's anything wrong with this view, except the misleading usage of "true". If this is what Hilbert actually means, I wonder why Gödel's incompleteness theorems are said to have been a devastating blow to his position.
Maybe the problem with the first incompleteness theorem is that Hilbert also wanted to defend the application of classical logic in mathematics, whereas Gödel's theorem proves that for some arithmetical statement p and theory T, neither p nor not-p is true relative to T. But does this really endanger the application of classical logic in mathematics? I don't think so. After all, the proof theory encoded in Gödel's proof is classical. Only in talking explicitly about mathematical "truth", the incompleteness matters.
What's the problem with the second incompleteness theorem? Hilbert certainly wasn't silly enough to think that the fact that a theory can prove its own consistency means anything. Any inconsistent theory can do that. So the problem must be that Gödel's theorem also proves that no theory that is strictly weaker than a given arithemtic theory (of the proper sort) can prove the consistency of that theory, whereas Hilbert thought that the metamathematical, "contentual" reasoning has to take place in such a strictly weaker theory. This makes some sense, for the alternatives are 1) to leave metamathematics entirely informal or 2) to use a rather strong mathematical theory in metamathematics. Both of these options will be of no help to anybody who is seriously uncertain about the consistency of the theory in question; for example, there is very probably some kind of informal "proof" of the consistency of naive set theory.
But then, why is consistency at all so important for Hilbert? On the deductivist point of view I just sketched, it scarcely matters if a theory is consistent of not. Of course, we do not intuitively believe that every arithmetical statement is true, and so if the Peano axioms turned out to be inconsistent, Hilbert can't claim that an arithmetical statement p is true iff it follows from the Peano axioms. But why should he claim this? Couldn't he just say that our informal notion of arithemtic truth has to be spelled out as truth relative to some theory of roughly the same kind as Peano arithmetic? He needn't pick any particular theory once and for all.