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 can match, but an essential component of mathematics -the insight and understanding -seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding.

What are the ultimate limits to computational techniques: verifier theory and unverifiability

“…[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?…”