1 Knowledge of proofs* by Peter Pagin 1. Epistemic constraints on proofs In the main current of contemporary intuitionism, truth is equated with the availability of proof, or verification. 1 The philosophical underpinnings vary somewhat, but anti-realism is a common denominator. Anti-realism can be a basic stance. 2 You can find it incomprehensible e.g. how some numbertheoretic statement 2200nAn could be true in the absence of proof. What could it possibly amount to for every instance A (1), A (2), A (3)… to simply be true, if we could not verify that each of them is true? And this we cannot do for each instance separately. There are also meaning theoretic arguments for intuitionism, or at least against classical logic. I shall return to that issue later. What is a proof, say in mathematics? The concept of a mathematical proof is partly mathe- matical, partly epistemological. A proof is a mathematical construction. We can specify formal properties of proofs. If we are dealing with some formal system S , we can even give an induc- tive definition of the concept proof-in-S . But we need a different conception if we shall under- stand why such a definition is a definition of a predicate which applies precisely to proofs in the general sense. This is analogous to what holds for Tarskian truth definitions. As little as we have an inductive defintion of true-in-L for variable L do we have an inductive defintion of proof-in- S for variable S . In other words, as we cannot define inductively what is common to all true sen- tences we cannot define inductively what is common to all mathematical proofs. The general concept of a proof is something like: a proof is something that establishes the truth (or correctness, or validity) of its conclusion. But what does "establishes" mean here? Does it mean anything more than this: (1) if there is a proof for p , then p But if that were all there is to the general concept of proof, then there would be nothing wrong with saying that the very fact that every instance A (1), A (2), A (3)… is true is a proof for the statement that 2200nAn 3 . Clearly, if every instance is true, then the universal is true. So there