Intentional Gaps In Mathematical Proofs

A central question in the philosophy of mathematics is: How do mathematicians know that mathematical propositions are true? The standard answer is that a mathematician knows that a proposition is true because she knows a proof of that proposition. Of course, this answer leaves a couple of important things unexplained. First, how does the mathematician know that the axioms that her proof appeals to (e.g., the axiom of choice) are true? Second, how does the mathematician know that the rules of inference that her proof appeals to (e.g., the law of the excluded middle) are truth preserving? A lot of work in the philosophy of mathematics has gone into trying to give answers to these two questions (see, e.g., Maddy 1988). In this paper, I argue that even if we had completely satisfactory answers to these two questions, we would still not have a completely satisfactory explanation of how mathematicians actually know on the basis of proof that mathematical propositions are true. The reason is that mathematicians often intentionally leave gaps in their proofs.

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