TITLE: Is P=NP relevant to finding proofs of everyday mathematical propositions?
QUESTION [33 upvotes]: Disclaimer: I don't know a whole lot about complexity theory beyond, say, a good undergrad class.
With increasing frequency I seem to be encountering claims by complexity theorists that, in the unlikely event that P=NP were proved and an algorithm with reasonable constants found, mathematicians wouldn't bother trying to prove things anymore beause we could just use our P-time algorithm to search for proofs.  Usually this is part of an argument for why all mathematicians and logicians should care a lot about P=?=NP.
I think most of these claims are exaggerations of the first full paragraph on page 8 of the Cook's problem description for the Clay Institute (which itself is stated in a completely reasonable and unexaggerated manner).
However, it's quite clear from the Clay Institute description that P=NP is relevant only to classes of problems, parameterized by some integer $n$, for which we have already proved all three of the following:

the question is not independent of our chosen axioms ($T\vdash \phi\vee T\vdash \neg\phi$)
any proof of the proposition must have size at most polynomial in $n$
any proof of the negation of the proposition must have size at most polynomial in $n$

This way we know there's a proof of either the proposition or its negation, and the search problem for the one that does exist falls inside NP, so we can dovetail the two searches and stop when one of them succeeds.
This puzzles me.  Most of the propositions mathematicians care about don't come in integer-parameterized classes, let alone classes with known proof-size bounds.  Usually they come in classes of size 1 with no knowledge of proof-size.  Is there some trick for turning the sorts of results mathematicians care about into these integer-parameterized-polynomially-bounded classes?
Example: how would you do this for the question of whether or not CH is independent of ZFC?
Cook and Reckhow's JSL article The Relative Efficiency of Propositional Proof Systems  (which seems to be the starting point for the literature) actually mentions that if you take the problem class to consist of all propositions in some proof system (such as first-order predicate calculus), take the length of the proposition as the parameter, and take the question to be "is it entailed by the axioms", then at the time the paper was published (1979) no real-world proof system was known to have the desired property, and a few were known not to have the desired property.
I suppose I am being slightly lazy here, since the study of which problems have this property is a whole subfield with plenty of literature I could read, but really I'm only interested in whether or not that subfield's positive-results-to-date justify the claims I've been hearing lately.  A reference to a paper containing the "trick" above would be fine as an answer.

REPLY [4 votes]: Others have already given good answers, but I'd like to point out that the constraint of "reasonable constants" is potentially much more stringent than most people think.  For example, in the real world, space tends to be more expensive than time.  So suppose that there is some proof of the Riemann hypothesis that is just barely small enough that we can just barely implement a computer program that, when run using all the world's computer resources, can just barely step through the entire proof and verify it within an acceptable human time-frame.  Chances are, we would not have enough resources to write down such a proof explicitly; the computers would probably be constantly reusing space and erasing stuff that had been used in checking earlier stages of the proof but that would no longer be needed.  The plan that Andreas Blass outlined, of writing down explicitly the length of the proof in unary and running our magic algorithm on this input, would not be feasible in this scenario.
Anyway, the point is that in today's world, one should not think of a proof as something that takes just a few million characters to write down explicitly.  Although it is an important observation that it would be revolutionary to discover an algorithm for searching for proofs that is as efficient as verifying them, we should recognize that the claim that a proof of P = NP would achieve this is a slight exaggeration.