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Varieties of Continua: From Regions to Points and Back,Used
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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, pointbased 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 midtolatetwentieth century revival of prelimit methods in analysis and geometry using infinitesimals including nonstandard 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 tradeoffs, 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 regionsbased theories of classical continua, including Euclidean and nonEuclidean geometries, that are mathematically equivalent (interreducible) to the currently standard, pointbased accounts in mainstream mathematics.
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