In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to an intuitionist conception would (...) be incommensurable, that is, that some classical statements, possessing meanings which cannot be preserved in the intuitionist language, would become unintelligible. The vague but intriguing thesis is then tentatively advanced that Kuhnian revolutions are even historically possible, in the sense that only what we might call ‘accidental’ historical factors may have prevented mathematics from undergoing just such a Kuhnian revolution in the early years of the twentieth century. (shrink)

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. (shrink)

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. (shrink)

During the last three decades, reflections on the growth of scientific knowledge have inspired historians, sociologists, and some philosophers to contend that scientific objectivity is a myth. In this book, Kitcher attempts to resurrect the notions of objectivity and progress in science by identifying both the limitations of idealized treatments of growth of knowledge and the overreactions to philosophical idealizations. Recognizing that science is done not by logically omniscient subjects working in isolation, but by people with a variety of personal (...) and social interests, who cooperate and compete with one another, he argues that, nonetheless, we may conceive the growth of science as a process in which both our vision of nature and our ways of learning more about nature improve. Offering a detailed picture of the advancement of science, he sets a new agenda for the philosophy of science and for other "science studies" disciplines. (shrink)

Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could (...) be applied to mathematics as well as science. Michael Grove declared that revolutions never occur in mathematics, while Joseph Dauben argued that there have been mathematical revolutions and gave some examples. This book is the first comprehensive examination of the question. It reprints the original papers of Grove, Dauben, and Mehrtens, together with additional chapters giving their current views. To this are added new contributions from nine further experts in the history of mathematics, who each discuss an important episode and consider whether it was a revolution. The whole question of mathematical revolutions is thus examined comprehensively and from a variety of perspectives. This thought-provoking volume will interest mathematicians, philosophers, and historians alike. (shrink)

Structural realists of nearly all stripes endorse the structural continuity claim. Roughly speaking, this is the claim that the structure of successful scientific theories survives theory change because it has latched on to the structure of the world. In this paper I elaborate, elucidate and modify the structural continuity claim and its associated argument. I do so without presupposing a particular conception of structure that favours this or that kind of structural realism. Instead I focus on how structural realists can (...) best account for the neutrally formulated historical facts. The result, I hope, crystallises some of the shared commitments, desiderata and limits of structural realists. (shrink)

A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.

Over the last two decades, the debate over scientific realism has been dominated by two arguments that pull in contrary directions: the 'no miracle' argument and the 'pessimistic induction'. The latter suggests that the historical record destroys the realist's belief in an explanatory connection between truthlikeness and genuine empirical success. This paper analyzes the structure of the 'pessimistic induction', presents a move--the divide et impera move--that neutralizes it, and motivates a substantive yet realistic version of scientific realism. This move is (...) also compared with Worrall's and Kitcher's recent reactions to the 'pessimistic induction'. (shrink)

(Statement of Responsibility) by Lewis Carroll ; with illustrations in colors.

Originally, the expression “Teichmüller theory” referred to the theory that Oswald Teichmüller developed on deformations and on moduli spaces of marked Riemann surfaces. This theory is not an isolated field in mathematics. At different stages of its development, it received strong impetuses from analysis, geometry, and algebraic topology, and it had a major impact on other fields, including low-dimensional topology, algebraic topology, hyperbolic geometry, geometric group theory, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, theoretical (...) physics, and there are certainly others. Of course, the impacts on these various fields are not equally important, but in some cases (namely, low-dimensional topology, algebraic geometry, and physics) the impact was crucial. At the same time, Teichmüller theory established important connections between the fields mentioned. This, in part, is a consequence of the diversity and the richness of the structure that Teichmüller space itself carries. From a more subjective point of view, the result of pondering on these connections and applications demonstrates the unity of mathematics. The aim of this paper is to survey the origin of Teichmüller theory and the development of its early major ideas. (shrink)

Philosophy of Science, Volume 63, Issue Supplement. Proceedings of the 1996 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers (Sep.