What is a proof? One answer, common especially to some logicians, is that a proof is a finite configuration of signs in an inductively defined class of sign-configurations of an elementary formal system. This is the formalist's conception of proof, the conception on which one can encode proofs in the natural numbers. Then for any elementary formal system in which one can do the amount of arithmetic needed to arithmetize proof, and metamathematics generally, one can prove Gödel's incompleteness theorems. Gödel's theorems are often described as showing that proof in mathematics, in the formalist's sense, is not the same as truth, or that syntax is not the same as semantics. On this view “proof” is always relative to a given formal system. One might ask whether there is some common feature that these different purely formal ‘proofs’ share, a feature by virtue of which we are prepared to call them ‘proofs’.

The formalist's conception of proof is quite alien to many working mathematicians. The working mathematician's conception of proof is not nearly so precise and well delineated. In fact, just what a proof is on the latter conception is not so clear, except that it is not or is not only, what a strict formalist says it is. In mathematical practice proofs may involve many informal components, a kind of rigor that is independent of complete formalization, and some kind of “meaning” or semantic content.