Preface

Summary

Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.

– Imre Lakatos (1976)

What is the role of proof in mathematics? In recent years we have seen a rebellion against teaching mathematics as a procession of irrefutable logical arguments that build from fundamental principles to universal truths. In this age of computer exploration of the patterns of mathematics, the pendulum has swung so far that some have proclaimed “The Death of Proof” (Horgan 1993). Traditional proofs are disappearing from the high school and even early college curricula.

This loss should not be mourned if what we are giving up is the misconception of mathematics as a formal system without need for scientific exploration, experimentation, and discovery. But there is a very real danger that in our enthusiasm to jettison what is false, we will lose the very essence of mathematics which is inextricably tied to proofs. We are brought back to our initial question: What is the role of proof in mathematics?

Imre Lakatos in Proofs and Refutations (1976) has highlighted one important aspect of this role. His is an ironic insight, for what is proven should be true and therefore not refutable. Lakatos's inspiration was Karl Popper, and the title is adapted from Popper's Conjectures and Refutations (1963).