Mathematical Method and Proof

Abstract

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.

Computers in Mathematical Inquiry

“…In Avigad (2006), I addressed one small aspect of mathematical understanding, namely, the process by which we understand the text of an ordinary mathematical proof. I discussed ways in which efforts in formal verification can inform and be informed by a philosophical study of this type of understanding.…”

Proofs as bearers of mathematical knowledge

“…Rav is not the only one who assigns to proofs a role that goes well beyond demonstrating that a theorem is true and why a theorem is true. Avigad (2006) lends support to Rav's central thesis when he says:…”

Informal Proofs and Mathematical Rigour

“…That is, in order to be such, a proof should be either (i) formalisable, or (ii) fully formalised. 1 This view of proof has emerged out of the foundational crisis at the beginning of the 20th century. The foundational concerns brought about the need for longstanding conceptions of mathematical knowledge to be revised, and produced a shift of attention in mathematical epistemology from the actual ways of acquiring mathematical knowledge to the ways of securing the certainty of mathematical results.…”