Varieties of Continua: From Regions to Points and Back

ISBN-13: 9780198712749 / Angielski / Twarda / 2018 / 224 str.

Geoffrey Hellman; Stewart Shapiro

Opis

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, 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.

Recenzje

Readers interested in a regions-based account of the continuum will find it a temendous resource, and its conclusions are sure to factor into future debates on the foundations of mathematics. Richard T.W. Arthur, Philosophia Mathematica

O autorze

Geoffrey Hellman received his BA and PhD from Harvard (1973). Having published widely in analytic philosophy and philosophy of science, he has, since the 1980s, concentrated on philosophy of quantum mechanics and philosophy and foundations of mathematics. Following the lead of his adviser, Hilary Putnam, Hellman has developed modal-structural interpretations of mathematical theories, including number theory, analysis, and set theory. He has also worked on predicative foundations of arithmetic (with Solomon Feferman) and pluralism in mathematics (with J.L. Bell). In 2007 he was elected as a fellow of the American Academy of Arts and Sciences. 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, and he serves as Distinguished Visiting Professor at the University of Connecticut, and as Professorial Fellow at the University of Oslo. He has contributed to the philosophy of mathematics, philosophy of language, logic, and philosophy of logic, publishing monographs on higher-order logic, structuralism, vagueness, and pluralism in logic.