Why Proof? A Historian’s Perspective

Abstract

The history of mathematics provides evidence that proofs let mathematicians distinguish between true results and merely plausible ones; that the careful formulation of arguments allows them to see how individual mathematical results relate to broader mathematical ideas; and that the process of proving teaches logical reasoning. This paper presents key historical examples, including the origin of logical proof in Greek geometry, Aristotle’s proof-based model for science, the classical uses of visual demonstration, the developing power of abstraction and symbolism, the role of the principles of optimisation and symmetry, the changing standards of rigour between the algorithmic calculus of the eighteenth century and the proof-based version of the nineteenth, the discovery of non-Euclidean geometry, the modern reciprocal influences between philosophy and proof-based mathematics, and the current importance to society of understanding logic. We conclude that observing and teaching this history also helps us teach proof and proving.