VARIETIES OF CONTINUA C

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.