Confronting Ideals of Proof with the Ways of Proving of the Research Mathematician

Abstract

In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the "axiomatic conception" of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books. Along with Cellucci, Rav, Grattan-Guinness and Grosholz, we deplore this view of mathematical proof, and favour instead the "analytic conception" of mathematical proof, where the axiomatic proof, when it exists at all, is only the core of a proof. An analytic proof solves a problem, by making hypotheses and using a mixture of deductive moves and induction (loosely construed to include diagrams, etc.) to present a solution to the problem. This implies that proofs are not always finite, that it might involve much more than axioms and straight logical inferences from these deductions and a proof can always be questioned. Moreover, this is where a lot of the interesting conceptual work of mathematics takes place. We view proofs as communicative acts made within the mathematical community which ensures correctness through application, context and standards of rigor.