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 earlytwentieth 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 non-standard 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.

 

John L. Bell has been Professor of Philosophy and Adjunct Professor of Mathematics at the University of Western Ontario since 1989. From 1968-89 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, CNRSat the Ecole Polytechnique, Paris. In 2009 he was elected a Fellow of the Royal Society of Canada. In 2011 his biography appeared in Canadian Whos Who. That same year saw the publication by Springer of his FestschriftVintage Enthusiasms: Essays in Honour of John  L. Bell. 

 He is a member of the Editorial Boards ofPhilosophia Mathematica, Axiomathes,and theWestern 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 North-Holland: 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, Boolean-valued 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 second-order logic, infinitary languages, large cardinals, incompleteness, Hilberts epsilon calculus, the axiom of choice, Zorns lemma, Boolean algebras, lattice theory, category and topos theory, type theory, constructive mathematics, quantum logic , and space-time 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, Russells paradox, the nature of cosmological theories, the infinity of the past and aesthetics in mathematics.

 

Part I: The Continuous, the Discrete, and the Infinitesimal in the History of Thought.- Chapter 1. The Continuous and the Discrete in Ancient Greece, the Orient, and the European Middle Ages.- Chapter 2. The 16th and 17th Centuries: The Founding of the Infinitesimal Calculus.- Chapter 3. The 18th and Early 19th Centuries: The Age of Continuity.- Chapter 4.  The Reduction of the Continuous to the Discrete in the 19th and early 20th Centuries.- Chapter 5.  Dissenting Voices: Divergent Conceptions of the Continuum in the 19th and Early 20th Centuries.- Part II: Continuity and Infinitesimals in Todays Mathematics.- Chapter 6.  Topology.- Chapter 7.  Category/Topos Theory.- Chapter 8.  Nonstandard Analysis.- Chapter 9. The Constructive and Intuitionistic Continua.- Chapter 10. Smooth Infiniteimal Analysis/Synthetic Geometry