The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics

  • John L. Bell

Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 82)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. The Continuous, the Discrete, and the Infinitesimal in the History of Thought

  3. Continuity and Infinitesimals in Today’s Mathematics

    1. Front Matter
      Pages 185-185
    2. John L. Bell
      Pages 187-195
    3. John L. Bell
      Pages 197-207
    4. John L. Bell
      Pages 209-213
  4. Back Matter
    Pages 273-313

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 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.



Founding of the Infinitesimal Calculus Brouwer on Divergent Conceptions of the Continuum Eighteenth century mathematicians Development of the Concepts of the Continuous Development of the concepts of the infinitesimal Autrecourt in Philosophy of Mathematics Veronese On Divergent Conceptions of the Continuum Carnot on Eighteenth Century Mathematics Nicolas Oreme on the continuous and the Discrete Thomas Bradwardine on the continuous and the Discrete Euler on the age of continuity Weyl on Divergent Conceptions of the Continuum

Authors and affiliations

  • John L. Bell
    • 1
  1. 1.Department of PhilosophyUniversity of Western OntarioLondonCanada

Bibliographic information

  • DOI
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Religion and Philosophy
  • Print ISBN 978-3-030-18706-4
  • Online ISBN 978-3-030-18707-1
  • Series Print ISSN 1566-659X
  • Series Online ISSN 2215-1974
  • Buy this book on publisher's site