Proofs and Refutations

Proofs and Refutations 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".

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 they 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.

Though the book is written as a narrative, an actual method of investigation, that of "proofs and refutations", is developed. In Appendix I, Lakatos summarizes this method by the following list of stages:

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