Varieties of Continua From Regions to Points and Back

Varieties of Continua explores the development of the idea of the continuous. Hellman and Shapiro begin with two historical episodes. The first is the remarkably rapid transition in the course of the nineteenth century from the ancient Aristotelian view, that a true continuum cannot be composed of points, to the now standard, point-based frameworks for analysis and geometry found in modern mainstream mathematics (stemming from the work of Bolzano, Cauchy, Weierstrass, Dedekind, Cantor, et al.). The second is the mid-tolate-twentieth century revival of pre-limit methods in analysis and geometry using infinitesimals including non-standard analysis (due to Abraham Robinson), and the more radical smooth infinitesimal analysis that uses intuitionistic logic. Hellman and Shapiro present a systematic comparison of these and related alternatives (including constructivist and predicative conceptions), weighing various trade-offs, helping articulate a modern pluralist perspective, and articulate a modern pluralist perspective on continuity. The main creative work of the book is the development of rigorous regions-based theories of classical continua, including Euclidean and non-Euclidean geometries, that are mathematically equivalent (inter-reducible) to the currently standard, point-based accounts in mainstream mathematics.

Geoffrey Hellman, University of Minnesota,Stewart Shapiro, Ohio State University

Stewart Shapiro received an MA in mathematics in 1975, and a PhD in philosophy in 1978, both from the State University of New York at Buffalo. He is currently the O'Donnell Professor of Philosophy at the Ohio State University. He specializes in philosophy of mathematics, logic, philosophy of logic, and philosophy of language.

Geoffrey Hellman focussed on philosophy of language and general issues in philosophy of science in the early part of his career. Since the mid-70's he has concentrated on philosophy of physics, especially quantum mechanics, and philosophy and foundations of mathematics. He has also worked on topics in the philosophy of logic, with contributions on the Quine-Carnap debates on the nature of logical truth, on the communication problem between intuitionism and classicism in mathematics, and on pluralism in logic and mathematics.