Proofs and Refutations

Proofs and Refutations: The Logic of Mathematical Discovery is a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour. At the end of the Introduction, Lakatos explains that his purpose is to challenge formalism in mathematics, and to show that informal mathematics grows by a logic of "proofs and refutations". The 1976 book Proofs and Refutations is based on the first three chapters of his 1961 four-chapter doctoral thesis Essays in the Logic of Mathematical Discovery. But its first chapter is Lakatos's own revision of its chapter 1 that was first published as Proofs and Refutations in four parts in 1963–4 in the British Journal for the Philosophy of Science. Many important logical ideas are explained in the book. For example, the difference between a counterexample to a lemma (a so-called 'local counterexample') and a counterexample to the specific conjecture under attack (a 'global counterexample' to the Euler characteristic, in this case) is discussed. Lakatos argues for a different kind of textbook, one that uses heuristic style. To the critics that say such a textbook would be too long, he replies: 'The answer to this pedestrian argument is: let us try.' The book includes two appendices. In the first, Lakatos gives examples of the heuristic process in mathematical discovery. In the second, he contrasts the deductivist and heuristic approaches and provides heuristic analysis of some 'proof generated' concepts, including uniform convergence, bounded variation, and the Carathéodory definition of a measurable set. The pupils in the book are named after letters of the Greek alphabet.