Proofs and Refutations

A study of how mathematics grows, written mostly as a classroom dialogue between a teacher and students named after Greek letters. It developed from Lakatos's doctoral thesis, appeared as articles in the British Journal for the Philosophy of Science, and was published as a book after his death, edited by John Worrall and Elie Zahar. The class examines Euler's conjecture that for any polyhedron the number of vertices minus edges plus faces equals two. A proof is offered, and the students produce counterexamples such as a cube with a cube-shaped hole inside it. They respond in different ways: monster-barring, which redefines polyhedron to exclude the counterexample; exception-barring; and lemma-incorporation, which uses the counterexample to locate a hidden assumption and improve both proof and theorem. Footnotes trace the real history of the debate. Lakatos argues that mathematics is quasi-empirical and fallible, growing through criticism rather than by deduction from certain axioms. Later material applies the method to Cauchy's work in analysis.

It shows mathematics as a human, argumentative activity and is one of the most enjoyable books in the philosophy of mathematics, needing only school geometry.