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Table of 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; Tracing the Idea of Cohesiveness: Aristotle, Veronese, Brentano
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; Tracing the Idea of Cohesiveness: Aristotle, Veronese, Brentano