Mathematics in Kant's Critical Philosophy: Reflections on Mathematical Practice (Studies in Philosophy (New York, N.Y.).)

First published in 2003. Routledge is an imprint of Taylor & Francis, an informa company. Cover 1 Title 4 Copyright 5 Dedication 6 Contents 8 Preface 12 Introduction 16 Part 1. Euclid: The Role of the Euclidean Diagram in the Elements 22 §1.0. Euclid: An Introduction 22 §1.1. Characterizing the Euclidean Diagram: Plane Geometry 24 §1.1.1. The Definitions 26 §1.1.2. The Postulates 30 §1.1.3. The Common Notions 32 §1.2. Reading the Euclidean Diagram: Warranting Implicit Assumptions 34 §1.2.1. The Incidence Axioms 35 §1.2.2. The Betweenness Axioms 36 §1.2.3. The Congruence Axioms 38 §1.2.4. The Continuity Axioms 39 §1.2.5. The Parallelism Axiom 41 §1.3. Countering Common Objections to the Euclidean Method 41 §1.4. Characterizing and Reading the Euclidean Diagram: the Arithmetic Books 47 §1.5. The Role of the Euclidean Diagram in Indirect Proof: A Special Case of “Pre-Formal” Demonstration 49 §1.6. Euclid: Conclusion 51 Part 2. Wolff: The Elementa and Early Modern Mathematical Practice 54 §2.0. Wolff: An Introduction 54 §2.1. Euclid’s Elements in the Early Modern Period 57 §2.2. Geometry and Arithmetic in Wolff's Elementa and Other Early Modern Textbooks 62 §2.2.1. Wolff's Mathematical Method 62 §2.2.2. Elementa Geometriae 65 §2.2.3. Elementa Arithmeticae 69 §2.2.4. Early Modern Foundational Views: Descartes and Barrow on the Relationship Between Geometry and Arithmetic 70 §2.3. Algebra and Analysis in Wolff’s Elementa: The Tools that Relate Arithmetic and Geometry in the Early Modern Period 75 §2.3.1. Viète and Descartes: The Beginnings of Analytic Geometry 76 §2.3.2. The Construction of Equations as a Standard Topic in Seventeenth and Eighteenth Century Textbooks 82 §2.3.3. Constructing Equations in Wolff's Elementa 86 §2.4. Wolff: Conclusion 100 Part 3. Kant: Mathematics in the Critique of Pure Reason 104 §3.0. Kant: An Introduction 104 §3.1. Pure and Empirical Intuitions 106 §3.1.1. What is a “Pure” Intuition? 106 §3.1.2. Two Types of Mathematical Demonstration 109 §3.1.3. Pure Intuition and the Synthetic A Priority of Mathematical Judgments 115 §3.1.4. Objections to the Role of Pure Intuition in Mathematical Demonstration 119 §3.2. The “Schematism” 122 §3.3. Algebraic Cognition 128 §3.3.1. Kant on the “Symbolic Construction” of Mathematical Concepts 128 §3.3.2. Shared Assumptions: The Prevailing View of Kant on “Symbolic Construction” 130 §3.3.3. “Symbolic Construction”: A New Reading 136 §3.4. Kant: Conclusion 144 Endnotes 148 References 182 Index 188 Pt. 1. Euclid: The Role Of The Euclidean Diagram In The Elements. 1.0. Euclid: An Introduction. 1.1. Characterizing The Euclidean Diagram: Plane Geometry. 1.2. Reading The Euclidean Diagram: Warranting Implicit Assumptions. 1.3. Countering Common Objections To The Euclidean Method. 1.4. Characterizing And Reading The Euclidean Diagram: The Arithmetic Books. 1.5. The Role Of The Euclidean Diagram In Indirect Proof: A Special Case Of Pre-formal Demonstration. 1.6. Euclid: Conclusion -- Pt. 2. Wolff: The Elementa And Early Modern Mathematical Practice. 2.0. Wolff: An Introduction. 2.1. Euclid's Elements In The Early Modern Period. 2.2. Geometry And Arithmetic In Wolff's Elementa And Other Early Modern Textbooks. 2.3. Algebra And Analysis In Wolff's Elementa: The Tools That Relate Arithmetic And Geometry In The Early Modern Period. 2.4. Wolff: Conclusion -- Pt. 3. Kant: Mathematics In The Critique Of Pure Reason. 3.0. Kant: An Introduction. 3.1. Pure And Empirical Intuitions. 3.2. The Schematism 3.3. Algebraic Cognition. 3.4. Kant: Conclusion. Lisa A. Shabel. Includes Bibliographical References (p. 169-174) And Index. Mathematics in Kant's Critical Philosophy provides a much needed reading (and re-reading) of Kant's theory of the construction of mathematical concepts through a fully contextualized analysis. In this work Lisa Shabel convincingly argues that it is only through an understanding of the relevant eighteenth century mathematics textbooks, and the related mathematical practice, can the material and context necessary for a successful interpretation of Kant's philosophy be provided. This is borne out through sustained readings of Euclid and Woolf in particular, which, when brought together with Kant's work, allows for the elucidation of several key issues and the reinterpretation of many hitherto opaque and long debated passages. Contemporary discussion of Euclid's Elements, possibly the most widely read mathematics text in all of history, emphasizes what are perceived to be gaps in the reasoning employed therein.