The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century. Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart’s history ably demonstrates, mathematics is the social activity in which styles of truth—here, abstraction—become synonymous with ways of knowing.

Proceedings of the 1978 Pisa, Italy, September 4-8, 1978 Conference on the History and Philosophy of Science

During the last decade, scientists working in quantum theory have been engaging in promising new fields such as quantum computation and quantum information processing, and have also been reflecting on the possibilities of nonlinear behavior on the quantum level. These are challenging undertakings because (1) they will result in new solutions to important technical and practical problems that were unsolvable by the classical approaches (for example, quantum computers can calculate problems that are intractable if one uses classical computers); and (2) they open up new 'hard' problems of a fundamental nature that touch the foundation of quantum theory itself (for example, the contradiction between locality and nonlinearity and the interpretation of quantum computing as a universal process).In this book, one can distinguish two main streams of research to approach the just-mentioned problem field: (1) a theoretical structural part, which concentrates on the elaboration of a nonlinear quantum mechanics and the fundamentals of quantum computation; and (2) a theoretical experimental part, which focuses on the theoretical aspects of applications that arise from new technology and novel research perspectives such as quantum optics and quantum cryptography. Particular attention is also paid to the measurement problem, the classical limit and alternative interpretations (such as the hidden measurement approach).

This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand Tarski’s and Gödel’s work, explaining why the problems they discussed are still unsolved. Finally, the book reports on some of the most influential positions in contemporary philosophy of mathematics, i.e., Maddy’s mathematical naturalism and Shapiro’s mathematical structuralism. Last but not least, the book introduces Biancani’s Aristotelian philosophy of mathematics as this is considered important to understand current philosophical issue in the applications of mathematics. One of the main purposes of the book is to stimulate readers to reconsider the Aristotelian position, which disappeared almost completely from the scene in logic and mathematics in the early twentieth century.

In selecting the papers for this volume I have excluded all physics papers proper. I have further omitted all book rev.iews. Instead, I have included two papers not published previously; they are marked by an asterisk (*) in the table of contents. Since many of the papers were occasioned by Symposia or similar gatherings their chronological order is rather accidental. Hence I have tried to group the papers thematically into four parts. Within each part the order of sequence is from the more general to the more special, or from a more popular to a more technical treatment. The same principle has been applied to the sequential order of the parts. The foundational papers on quantum mechanics have been arranged in a somewhat dif ferent manner. Chapters XVI-XIX are concerned with the logic of complementarity while in Chapters XX-XXII a more radical recon ceptualization is carried out. Two of the older papers (Chapters VI and VIII) have been revised to bring them more into line with present terminology. Other papers have been corrected by additions and omissions. Additions are marked by square brackets [ ], while double square brackets [[ II signify omis sions or parts to be omitted. Hence [[A]] [B] means that 'A' should be replaced by 'B'. The heading of one paper (Chapter XX) has been changed to make it more descriptive.

Proceedings and Discussion of the 1968 Salzburg Colloquium in the Philosophy of Science.

The major methodological task for modern economists has been to establish the testability of models. Too often, however, methodological assumptions can make a model virtually impossible to test even under ideal conditions, yet few theorists have examined the requirements and problems of assuring testability in economics. In The Methodology of Economic Model Building, first published in 1989, Lawrence Boland presents the results of a research project that spanned more than twenty years. He examines how economists have applied the philosophy of Karl Popper, relating methodological debates about falsifiability to wider discussions about the truth status of models in natural and social sciences. He concludes that model building in economics reflects more the methodological prescriptions of the economist Paul Samuelson than Popper’s ‘falsificationism’. This title will prove invaluable to both students and researchers, and represents a substantial contribution to debates about the scientific status of economics.