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

Author: J L Bell
Publisher: Cham : Springer, 2019.
Series: University of Western Ontario series in philosophy of science, v. 82.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Bell, John L.
Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics.
Cham : Springer, ©2019
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: J L Bell
ISBN: 3030187071 9783030187071
OCLC Number: 1119620195
Notes: Tracing the Idea of Cohesiveness: Aristotle, Veronese, Brentano
Description: 1 online resource (320 pages)
Contents: Intro; Preface; Introduction; Contents; 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; 1.1 Ancient Greece; The Presocratics; The Method of Exhaustion; Plato; Aristotle; Epicurus; The Stoics and Others; 1.2 Oriental and Islamic Views; China; India; Islamic Thought; 1.3 The Philosophy of the Continuum in Medieval Europe; Chapter 2: The Sixteenth and Seventeenth Centuries. The Founding of the Infinitesimal Calculus; 2.1 The Sixteenth Century From Stevin to KeplerGalileo and Cavalieri; 2.2 The 17th Century; The Cartesian Philosophy; Infinitesimals and Indivisibles; Barrow and the Differential Triangle; Newton; Leibniz; Supporters and Critics of Leibniz; Bayle; Chapter 3: The Eighteenth and Early Nineteenth Centuries: The Age of Continuity; 3.1 The Mathematicians; Euler; 3.2 From DÁlembert to Carnot; 3.3 The Philosophers; Berkeley; Hume; Kant; Hegel; Chapter 4: The Reduction of the Continuous to the Discrete in the Nineteenth and Early Twentieth Centuries; 4.1 Bolzano and Cauchy; 4.2 Riemann; 4.3 Weierstrass and Dedekind 4.4 Cantor4.5 Russell; 4.6 Hobsonś Choice; Chapter 5: Dissenting Voices: Divergent Conceptions of the Continuum in the Nineteenth and Early Twentieth Centuries; 5.1 Du Bois-Reymond; 5.2 Veronese; 5.3 Brentano; 5.4 Peirce; 5.5 Poincaré; 5.6 Brouwer; 5.7 Weyl; Part II: Continuity and Infinitesimals in Todayś Mathematics; Chapter 6: Topology; 6.1 Topological Spaces; 6.2 Manifolds; Chapter 7: Category/Topos Theory; 7.1 Categories and Functors; 7.2 Pointless Topology; 7.3 Sheaves and Toposes; Chapter 8: Nonstandard Analysis; Chapter 9: The Continuum in Constructive and Intuitionistic Mathematics 9.1 The Constructive Real Number Line9.2 Constructive Meaning of the Logical Operators; 9.3 Order on the Constructive Reals; 9.4 Algebraic Operations on the Constructive Reals; 9.5 Convergence of Sequences and Completeness of the Constructive Reals; 9.6 Functions on the Constructive Reals; 9.7 Axiomatizing the Constructive Reals; 9.8 The Intuitionistic Continuum; 9.9 An Intuitionistic Theory of Infinitesimals; Chapter 10: Smooth Infinitesimal Analysis/Synthetic Differential Geometry; 10.1 Smooth Worlds; 10.2 Elementary Differential Geometry in a Smooth World 10.3 The Calculus in Smooth Infinitesimal Analysis10.4 The Internal Logic of a Smooth World Is Intuitionistic; 10.5 Smooth Infinitesimal Analysis as an Axiomatic Theory. Consequences for the Continuum; 10.6 Cohesiveness of the Continuum and Its Subsets in SIA; 10.7 Comparing the Smooth and Dedekind Real Lines in SIA; 10.8 Nonstandard Analysis in SIA; 10.9 Contrasting Nonstandard Analysis with Smooth Infinitesimal Analysis; 10.10 Smooth Infinitesimal Analysis and Physics; 10.11 Relating Sets and Smooth Spaces; Appendices; Appendix A: The Cohesiveness of Continua
Series Title: University of Western Ontario series in philosophy of science, v. 82.
Responsibility: John L. Bell.

Abstract:

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.

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