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The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer

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The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer THE FACT OF MODERN MATHEMATICS: GEOMETRY, LOGIC, AND CONCEPT FORMATION IN KANT AND CASSIRER by Jeremy Heis B.A., Michigan State University, 1999 Submitted to the Graduate Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2007 UNIVERSITY OF PITTSBURGH COLLEGE OF ARTS AND SCIENCES This dissertation was presented by Jeremy Heis It was defended on September 5, 2007 and approved by Jeremy Avigad, Associate Professor, Philosophy, Carnegie Mellon University Stephen Engstrom, Associate Professor, Philosophy, University of Pittsburgh Anil Gupta, Distinguished Professor, Philosophy, University of Pittsburgh Kenneth Manders, Associate Professor, Philosophy, University of Pittsburgh Thomas Ricketts, Professor, Philosophy, University of Pittsburgh Dissertation Advisor: Mark Wilson, Professor, Philosophy, University of Pittsburgh ii Copyright © by Jeremy Heis 2007 iii THE FACT OF MODERN MATHEMATICS: GEOMETRY, LOGIC, AND CONCEPT FORMATION IN KANT AND CASSIRER Jeremy Heis, PhD University of Pittsburgh, 2007 It is now commonly accepted that any adequate history of late nineteenth and early twentieth century philosophy—and thus of the origins of analytic philosophy—must take seriously the role of Neo-Kantianism and Kant interpretation in the period. This dissertation is a contribution to our understanding of this interesting but poorly understood stage in the history of philosophy. Kant’s theory of the concepts, postulates, and proofs of geometry was informed by philosophical reflection on diagram-based geometry in the Greek synthetic tradition. However, even before the widespread acceptance of non-Euclidean geometry, the projective revolution in nineteenth century geometry eliminated diagrams from proofs and introduced “ideal” elements that could not be given a straightforward interpretation in empirical space. A Kantian like the very early Russell felt forced to regard the ideal elements as convenient fictions. The Marburg Neo-Kantians—the philosophical school that included Ernst Cassirer (1874-1945)—thought that philosophy, as “transcendental logic,” needed to take the results of established pure mathematics as a “fact,” not a fiction. Cassirer therefore updates Kant by rejecting the “Transcendental Aesthetic” and by using elements in Richard Dedekind’s foundations of arithmetic to rework Kant’s idea that the geometrical method is the “construction of concepts.” He further argues that geometry is “synthetic” because it progresses when mathematicians introduce new structures iv (like the complex projective plane) that are not contained in the old structures, but unify them under a new point-of-view. This new “Kantian” theory of modern mathematics, Cassirer argues, is inconsistent with the traditional theory of concept formation by abstraction. Drawing on earlier Neo-Kantian interpretations, Cassirer argues that Kant’s theory of concepts as rules undermines the traditional theory of concept formation, and he gives a “transcendental” defense of the new logic of Frege and Russell. (In an appendix, I discuss the contemporaneous accounts of concept formation in Gottlob Frege and Hermann Lotze.) v TABLE OF CONTENTS Acknowledgements....................................................................................................................x Citations and Translations.................................................................................................... xiii Abbreviations .........................................................................................................................xiv INTRODUCTION.....................................................................................................................1 CHAPTER 1 Kant’s Philosophy of Geometry and the Ancient, Diagram-Based Model..5 1. Pure Intuition and Geometrical Concepts....................................................................10 2. Pure Intuition and Geometrical Postulates ..................................................................19 3. Pure Intuition and Geometrical Proofs........................................................................27 4. The ‘Logical’ Interpretation of Pure Intuition in Geometry.........................................33 5. Pure Intuition and the Objects of Geometry................................................................42 Mathematical Appendix, Continued From note 43.....................................................49 CHAPTER 2 The Projective Revolution in Nineteenth Century Geometry......................51 1. Two Geometrical Revolutions, Two Philosophical Projects........................................51 2. A Case Study: Jakob Steiner’s Projective Geometry and Points at Infinity..................53 3. A Case Study: Poncelet’s Projective Geometry and Imaginary Points.........................58 4. Projective Unification.................................................................................................61 5. Diagrams in Projective Geometry...............................................................................67 6. The Philosophical Challenge Posed by Mathematical Conceptual Innovation .............73 vi CHAPTER 3 Mathematics and the Logic of Concept Formation in Cassirer’s Substanzbegriff und Funktionsbegriff .....................................................................................84 1. Cassirer’s Philosophy of Mathematics in the Context of Late Nineteenth Century German Work in Logic .....................................................................................................85 2. The ‘Aristotelian’-abstractionist model of the concept................................................92 3. The Lotze Objection to the traditional A-a model of the concept ................................95 4. Cassirer’s Dedekindian Philosophy of Arithmetic.....................................................102 5. Cassirer’s Criticism of Frege’s and Russell’s Definition of Number .........................113 CHAPTER 4 Kant’s Theory of Concept Formation.........................................................123 1. Introduction..............................................................................................................125 2. Kant’s Logic and the Traditional Model of Concept Formation.................................129 3. The Circularity Objection to the Traditional Model ..................................................136 4. Neo-Kantians and Kant’s Theory of Concepts as Rules ............................................141 5. Kant on Mathematical Concepts...............................................................................146 6. Conclusion ...............................................................................................................152 CHAPTER 5 Transcendental and Formal Logic in Kant and the Marburg Neo-Kantians .....................................................................................................................156 1. Kant’s Distinction Between Transcendental and Formal Logic.................................162 2. Potential Difficulties with Kant’s Distinction ...........................................................169 3. Hermann Cohen’s Rejection of the Distinction.........................................................176 4. Paul Natorp’s Rejection of Formal Logic, Old and New ...........................................182 5. Walter Kinkel on the Insignificance of Kant’s Logic.................................................188 6. Cassirer on Transcendental and Formal Logic ..........................................................195 vii CHAPTER 6 Cassirer’s Philosophy of Geometry.............................................................210 1. The Marburg Neo-Kantian School and the Fact of Modern Mathematics..................212 2. Kant’s Mathematical Method and Cassirer’s Dedekindian Paradigm ........................216 3. Kant’s Mathematical Method and a Wider Conceptual Setting for an Object............220 4. The Representation of Space ....................................................................................227 5. The Essential Applicability of Mathematics and a New Problem for a Kantian.........231 6. Conclusion ...............................................................................................................235 APPENDIX to Chapter 3: Frege, Lotze, and Boole: Concept Formation and the ‘Functional’ View of the Concept.........................................................................................239 A.1 Frege’s Project in Begriffsschrift............................................................................244 A.2 Frege’s Theory of Concept Formation ...................................................................254 A.3 Sluga and Dummett on Frege’s Principle of the Priority of Judgment ....................260 A.4 Frege’s Criticism of the Boolean Theory of Concept Formation ............................264 A.5 Hermann Lotze’s Theory of Concepts in his Logik (1874) ....................................272 A.6 Lotze’s Theory of Mathematical Concepts.............................................................287 A.7 Lotze’s Criticism of the Boolean Theory of Concept Formation ............................292 A.8 The Historical Relationship between Frege’s and Lotze’s Criticisms of the Booleans . ..............................................................................................................................296 A.9 How Lotzean was Frege?.......................................................................................300
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