Proofs and Refutations by Imre Lakatos is a timeless classic that has never lost its applicability. The book analyzes numerous solutions to mathematical problems and, in the process, raises crucial questions regarding the nature of mathematical discovery and methodology. It takes the form of a debate between a teacher and some students. Lakatos’ work continues to motivate mathematicians and philosophers who want to create a mathematics philosophy that considers both the static and dynamic complexity of mathematical practice. Lakatos demonstrates how mathematics improves through attempts at proof and critiques of these attempts. This book has been reissued for a new audience with a Preface by Paolo Mancosu that was expressly commissioned.