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Septembre / Octobre 2000 Shortcuts in Proof

How can we know if a proof is correct? People often imagine that it suffices for a mathematician to make the effort to read it carefully, line by line, after having taken note of the definitions and known results that might be of use. If certain questions are unresolved, as to be sure some are, we would known precisely which ones, and the work of the researcher would consist uniquely of resolving these classified enigmas. Certainty would reign throughout mathematics. This situation, if it were true, would make mathematics categorically the opposite of physics. In physics, nothing can definitively establish a theory, which is nothing more than a hypothesis that can always be called into question by additional evidence. We know thus that one cannot prove a physical law, which is a general assertion based on observations and experiments which are no more than particular assertions. Logicians support the idealized vision of mathematics by maintaining that since the beginning of the century codified systems for writing proofs, called formal systems, have been available. If a proof is written with one of these systems a computer can verify its correctness without human intervention. The correctness of a formal proof, say the logicians, is mechanically verifiable and requires no intelligence. Actually, a multitude of reasons complicate this ideal tableau and smudge the excessively beautiful image of a mathematical science without moods, without disputes, without errors, where unanimity holds on every question and where discords can be settled with computers as arbiters.