Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.

Conjectures and Refutations is one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. It provides one of the clearest and most accessible statements of the fundamental idea that guided his work: not only our knowledge, but our aims and our standards, grow through an unending process of trial and error.

This book offers an excursion through the developmental area of research mathematics. It presents some 40 papers, published between the 1870s and the 1970s, on proofs of the Cantor-Bernstein theorem and the related Bernstein division theorem. While the emphasis is placed on providing accurate proofs, similar to the originals, the discussion is broadened to include aspects that pertain to the methodology of the development of mathematics and to the philosophy of mathematics. Works of prominent mathematicians and logicians are reviewed, including Cantor, Dedekind, Schröder, Bernstein, Borel, Zermelo, Poincaré, Russell, Peano, the Königs, Hausdorff, Sierpinski, Tarski, Banach, Brouwer and several others mainly of the Polish and the Dutch schools. In its attempt to present a diachronic narrative of one mathematical topic, the book resembles Lakatos’ celebrated book Proofs and Refutations. Indeed, some of the observations made by Lakatos are corroborated herein. The analogy between the two books is clearly anything but superficial, as the present book also offers new theoretical insights into the methodology of the development of mathematics (proof-processing), with implications for the historiography of mathematics.

The work that helped to determine Paul Feyerabend's fame and notoriety, Against Method, stemmed from Imre Lakatos's challenge: "In 1970 Imre cornered me at a party. 'Paul,' he said, 'you have such strange ideas. Why don't you write them down? I shall write a reply, we publish the whole thing and I promise you—we shall have a lot of fun.' " Although Lakatos died before he could write his reply, For and Against Method reconstructs his original counter-arguments from lectures and correspondence previously unpublished in English, allowing us to enjoy the "fun" two of this century's most eminent philosophers had, matching their wits and ideas on the subject of the scientific method. For and Against Method opens with an imaginary dialogue between Lakatos and Feyerabend, which Matteo Motterlini has constructed, based on their published works, to synthesize their positions and arguments. Part one presents the transcripts of the last lectures on method that Lakatos delivered. Part two, Feyerabend's response, consists of a previously published essay on anarchism, which began the attack on Lakatos's position that Feyerabend later continued in Against Method. The third and longest section consists of the correspondence Lakatos and Feyerabend exchanged on method and many other issues and ideas, as well as the events of their daily lives, between 1968 and Lakatos's death in 1974. The delight Lakatos and Feyerabend took in philosophical debate, and the relish with which they sparred, come to life again in For and Against Method, making it essential and lively reading for anyone interested in these two fascinating and controversial thinkers and their immense contributions to philosophy of science. "The writings in this volume are of considerable intellectual importance, and will be of great interest to anyone concerned with the development of the philosophical views of Lakatos and Feyerabend, or indeed with the development of philosophy of science in general during this crucial period."—Donald Gillies, British Journal for the Philosophy of Science (on the Italian edition) "A stimulating exchange of letters between two philosophical entertainers."—Tariq Ali, The Independent Imre Lakatos (1922-1974) was professor of logic at the London School of Economics. He was the author of Proofs and Refutations and the two-volume Philosophical Papers. Paul Feyerabend (1924-1994) was educated in Europe and held numerous teaching posts throughout his career. Among his books are Against Method; Science in a Free Society; Farewell to Reason; and Killing Time: The Autobiography of Paul Feyerabend, the last published by the University of Chicago Press.

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.