Knowledge of proofs

If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch's paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett's meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost.I am much indebted for comments to Lars Bergström, Per Martin-Löf, Wlodek Rabinowicz, Fredrik Stjernberg, Dag Westerståhl and Tim Williamson. I owe even more to the many seminars about truth and meaning, led by Dag Prawitz, at the philosophy department of Stockholm University. These were especially intense in the mideighties, when I was a graduate student.