Richard Dedekind and the Structuralist Transformation of Modern Mathematics

University Of California-Riverside, Riverside CA

Investigators

Abstract

This research project in the history of mathematics is supported by the Science, Technology and Society program at NSF and jointly by the SBE and MPS directorates by way of a Dear Colleague Letter, Research on Mathematical and Physical Sciences and Society. One of main tasks in the philosophy of science is to study the significance of revolutionary scientific developments such as the birth of theoretical mathematics in Ancient Greece, the rise of experimental natural science in the early modern period, and the emer¬gence of relativity theory and quantum mechanics in the twentieth century. Another case in point is the radical transformation of mathematics that took place in the nineteenth and early twentieth cen¬turies, a shift from the study of "quantity" as it had been under¬stood for millennia to the much more general study of "relational structures" (groups, rings, fields, geometric manifolds, topological spaces, etc.). A central figure in the transformation in mathematics noted above was Richard Dedekind, who is generally recognized as one of the most influential mathematicians of the nineteenth century, as well as one of the most important contributors to algebra, number theory, and the foundations of mathematics of all time. Yet there is no book-length treatment of Dedekind and his work. The main goal of this project is to engage in research that will eventually result in such a book. The book will have a special focus on the "structuralist" approach to mathe¬matics that Dedekind, more than anyone else, helped to inaugurate. The proposed research will build on the results of several historians and philosophers of mathe¬matics. It will be to synthesize the insights already achieved, and it will be to expand the corresponding investigations into new sub-fields of mathematics. In addition, the project will assess the conceptual tools commonly used in charac¬terizing Dedekind's works and sharpen those tools. Finally, because the centrality and the continuing relevance of them remain underestimated, another goal will be to make Dedekind's ideas fruit¬ful by relating them directly to current philo¬sophy of mathe¬mat¬ics.