2009 # Formal Proof: Reconciling Correctness and Understanding

**Abstract:** A good proof is a proof that makes us wiser. Manin [41, p. 209].Abstract. Hilbert's concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no pen-and-paper proof…

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“…[56]. Later, Calude and Muller emphasize: 'one cannot prove the correctness of the formal prover itself' [57]. Similarly, MacKenzie observes: 'Indeed, if one was to apply the formal, mechanical notion of proof entirely stringently, might not the software of the automated theorem prover itself have to be verified formally?…”

confidence: 99%

“…[56]. Later, Calude and Muller emphasize: 'one cannot prove the correctness of the formal prover itself' [57]. Similarly, MacKenzie observes: 'Indeed, if one was to apply the formal, mechanical notion of proof entirely stringently, might not the software of the automated theorem prover itself have to be verified formally?…”

confidence: 99%