Ask a philosopher what a proof is, and you're likely to get an answer emphasizing one or another regimentation of that notion in terms of a finite sequence of formalized statements, each of which is either an axiom or is derived from an axiom by certain inference rules. (We can call this the formal conception of proof.) Ask a mathematician what a proof is, and you will probably get a different-looking answer. Instead of stressing a particular regimented notion of proof, the answer the mathematician will give is likely to presuppose that: (i) proofs are arguments designed to persuade the (mathematical) community to accept the corresponding mathematical results; and (ii) proofs (often—although not always) are to enable us to understand why these results hold. Moreover,...

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