From Regions to Points and Back
Author: Geoffrey Hellman,Stewart Shapiro
Publisher: Oxford University Press
View: 7546Varieties 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, entirely punctiform frameworks for analysis and geometry found in modern texts (stemming from the work of Bolzano, Cauchy, Weierstrass, Dedekind, Cantor, et al.). The second is the mid-to-late-twentieth century revival of pre-limit methods in analysis and geometry using infinitesimals, non-standard analysis due to Abraham Robinson, and the more radical smooth infinitesimal analysis based on intuitionistic logic. Hellman and Shapiro develop a systematic comparison of these and related alternatives (including constructivist and predicative conceptions), balancing various trade-offs, helping articulate a modern pluralist perspective. A second main goal of the book is to develop thoroughgoing regions-based theories of classical continua that are mathematically equivalent (inter-reducible) to the currently standard, punctiform accounts of modern texts. The theories developed by Hellman and Shapiro offer a more streamlined, unified and comprehensive study than others in the contemporary literature.