Imre Lakatos (1922–74) was one of the twentieth century's most prominent philosophers of science and mathematics, best known for his theory of the methodology of proof and refutation in mathematics.

• Preface to this edition Paolo Mancosu; Editors' preface; Acknowledgments; Author's introduction; Part I: 1. A problem and a conjecture; 2. A proof; 3. Criticism of the proof by counterexamples which are local but not global; 4. Criticism of the conjecture by global counterexamples; 5. Criticism of the proof-analysis by counterexamples which are global but not local. The problem of rigour; 6. Return to criticism of the proof by counterexamples which are local but not global. The problem of content; 7. The problem of content revisited; 8. Concept-formation; 9. How criticism may turn mathematical truth into logical truth; Part II: Editors' introduction; Appendix 1. Another case-study in the method of proofs and refutations; Appendix 2. The deductivist versus the heuristic approach; Bibliography; Index of names; Index of subjects.

'For anyone interested in mathematics who has not encountered the work of the late Imre Lakatos before, this book is a treasure; and those who know well the famous dialogue, first published in 1963–4 in the British Journal for the Philosophy of Science, that forms the greater part of this book, will be eager to read the supplementary material … the book, as it stands, is rich and stimulating, and, unlike most writings on the philosophy of mathematics, succeeds in making excellent use of detailed observations about mathematics as it is actually practised.' Michael Dummett, Nature