Varieties of continua: From regions to points and back

Two historical episodes form the background to the research presented here: 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 entirely 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, viz. non-standard analysis due to Abraham Robinson, and the more radical smooth infinitesimal analysis based on intuitionistic logic. One goal of the present work is to develop a systematic comparison of these and related including (alternatives constructivist and predicative conceptions), balancing various trade-offs, helping articulate a modern pluralist perspective. A second main goal (pursued in the opening chapters) 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. Although in this project the work has been preceded by various writings, as explained below, it is believed the theories developed here are more streamlined, unified, and comprehensive than others in the contemporary literature. Finally, the book considers various limitations of the systems developed and some of the more striking implications for contemporary philosophy stemming from the pluralism take we our work to support.