The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics
Authors: Bell, John L.
Free Preview Explores the concept of the continuous and the infinitesimal
 Combines a history of philosophy with contemporary mathematics
 Gives a comprehensive historical and philosophical overview of the topic
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 About this book

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel.
Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl.
Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are nonstandard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry.
No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.
 About the authors

John L. Bell has been Professor of Philosophy and Adjunct Professor of Mathematics at the University of Western Ontario since 1989. From 196889 he was Lecturer and Senior Lecturer in Mathematics, and Reader in Mathematical Logic, at the London School of Economics. In 1975 he was a Visiting Fellow at the Polish Academy of Sciences, and in 1980 and 1982 at the Mathematics Department of the National University of Singapore. In 1991 he was a Visiting Professor at the Department of Mathematics of the University of Padova, and in 2007 he was a Visiting Directeur de Recherche, CNRS at the Ecole Polytechnique, Paris. In 2009 he was elected a Fellow of the Royal Society of Canada. In 2011 his biography appeared in Canadian Who’s Who. That same year saw the publication by Springer of his Festschrift Vintage Enthusiasms: Essays in Honour of John L. Bell.
He is a member of the Editorial Boards of Philosophia Mathematica, Axiomathes, and the Western Ontario Series in Philosophy of Science.
He has published 11 books and more than 70 papers. The books are with such presses as Oxford, Cambridge, Springer, and NorthHolland: five of these books are in second, third, or fourth printings or editions; two of them have been republished by Dover. They include titles on model theory, mathematical logic, Booleanvalued models of set theory, topos theory, smooth infinitesimal analysis, the axiom of choice, the evolution of mathematical concepts, the continuous and the infinitesimal, intuitionistic set theory, and oppositions and paradoxes His technical papers include titles on model theory, set theory, first and secondorder logic, infinitary languages, large cardinals, incompleteness, Hilbert’s epsilon calculus, the axiom of choice, Zorn’s lemma, Boolean algebras, lattice theory, category and topos theory, type theory, constructive mathematics, quantum logic , and spacetime theory, His work of a more philosophical nature includes papers on category theory in the foundations of mathematics, quantum logic and empiricism, mereology in mathematics, the concept of the infinitesimal, the nature of elementary propositions, the cohesiveness of the continuum, sets and classes as many, the philosophical outlook of Hermann Weyl, Russell’s paradox, the nature of cosmological theories, the infinity of the past and aesthetics in mathematics.
 Table of contents (10 chapters)


The Continuous and the Discrete in Ancient Greece, the Orient, and the European Middle Ages
Pages 339

The Sixteenth and Seventeenth Centuries. The Founding of the Infinitesimal Calculus
Pages 4176

The Eighteenth and Early Nineteenth Centuries: The Age of Continuity
Pages 77104

The Reduction of the Continuous to the Discrete in the Nineteenth and Early Twentieth Centuries
Pages 105146

Dissenting Voices: Divergent Conceptions of the Continuum in the Nineteenth and Early Twentieth Centuries
Pages 147184

Table of contents (10 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics
 Authors

 John L. Bell
 Series Title
 The Western Ontario Series in Philosophy of Science
 Series Volume
 82
 Copyright
 2019
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer Nature Switzerland AG
 eBook ISBN
 9783030187071
 DOI
 10.1007/9783030187071
 Hardcover ISBN
 9783030187064
 Series ISSN
 1566659X
 Edition Number
 1
 Number of Pages
 XVII, 313
 Number of Illustrations
 55 b/w illustrations
 Topics