Kant's Mathematical World: Mathematics, Cognition, and Experience

Description

Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to its 'arithmetization' in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other. Cover Half-title Title page Copyright information Dedication Contents Preface and Acknowledgments 1 Introduction: Mathematics and the World of Experience 1.1 Kant and the Theory of Magnitudes 1.2 Mathematics Then and Now 1.3 Mathematics in Kant's Theoretical Philosophy 1.4 Kantian Transformations 1.5 Kant's Theory of Magnitudes and Kant Interpretation 1.6 Overview of the Work Part I Mathematics, Magnitudes, and the Conditions of Experience 2 Space, Time, and Mathematics in the Critique of Pure Reason 2.1 Introduction 2.2 Mathematics in the Transcendental Aesthetic and the Axioms 2.3 Space and Time in the Transcendental Aesthetic and the Axioms 2.4 Determination, Construction, and Mathematics 2.5 Determination, Construction, and Magnitudes in the Axioms 2.6 The Possibility of Mathematics Revisited 2.7 Mathematics in the Transcendental Aesthetic and the Axioms Revisited 2.8 Two Objections and Kant's Formulation of the Axioms Principle 2.9 Conclusion 3 Magnitudes, Mathematics, and Experience in the Axioms of Intuition 3.1 Introduction 3.2 The Magnitude Argument 3.3 Kant on the Definition of Concepts 3.4 Kant's Definition of the Concept of Magnitude 3.4.1 The Concept of Magnitude and Categorial Unity 3.4.2 The Concept of Magnitude and Noncategorial Unity 3.4.3 Explaining the Concept of Magnitude without Synthetic Unity 3.4.4 The Content of the Concept of Magnitude 3.5 Quantum and Quantitas 3.6 Quanta and Quantitas in the Schematism 3.7 Summary 4 Extensive and Intensive Magnitudes and Continuity 4.1 Extensive and Intensive Magnitudes 4.2 The Extensive Magnitude Argument 4.3 The Categories of Quantity and the Mereology of Magnitudes 4.4 The Extensive Magnitude Regress Problem 4.5 Two Attempts to Solve the Extensive Magnitude Regress Problem 4.5.1 Indeterminate Quanta within Boundaries 4.5.2 The Aesthetic Estimation of Magnitude 4.6 The Solution to the Extensive Magnitude Regress Problem 4.6.1 The Indeterminate Representation of Quanta 4.6.2 Continuity and Successive Synthesis 4.6.3 Indeterminate Magnitudes within Boundaries Revisited 4.7 Continuous Synthesis and the Categories 5 Conceptual and Intuitive Representation: Singularity, Continuity, and Concreteness 5.1 Introduction 5.2 A Quick Solution and an Alternative 5.3 The Generality of Conceptual Representation 5.3.1 Concepts as a Mode of Representing Generality 5.3.2 Generality and the Relations among Concepts 5.4 The Singularity of Intuition Explained and Defended 5.4.1 Intuition as a Mode of Representing Singularly 5.4.2 Intuition as a Mode of Representing and as Represented 5.4.3 Representing Singularity and the Represented Singular 5.5 Three Implications of This Interpretation of Singularity 5.5.1 Singularity and Mathematical Cognition 5.5.2 Singularity and the Current Kantian Nonconceptualism Debate 5.5.3 Singularity and the Extensive Magnitude Regress 5.6 Abstract and Concrete Representations and Objects 5.6.1 Kant on the Concrete and the Abstract 5.6.2 From Representing in concreto to Concrete Objects of Representation 5.6.3 Singularity, Representing in concreto, and Mathematical Objects 5.7 The Concreteness and Abstractness of Quanta and Quantitas Interlude The Greek Mathematical Tradition as Background to Kant 6 Euclid, the Euclidean Mathematical Tradition, and the Theory of Magnitudes 6.1 Introduction 6.2 Organization of the Elements 6.3 The Deductive Structure of the Elements 6.4 Magnitude and the Euclidean Theory of Proportions 6.5 The Definitions of Sameness of Ratio and Similarity 6.6 Euclid's Definition of Number and the Euclidean Theory of Proportions 6.7 The Euclidean Geometrical Tradition 6.8 Magnitudes and Mathematics in Kant's Immediate Predecessors 6.9 Conclusion Part II Kant's Theory of Magnitudes, Intuition, and Measurement 7 Kant's Reworking of the Theory of Magnitudes: Homogeneity and the Role of Intuition 7.1 Introduction 7.2 Kant's Rethinking of Magnitude and Homogeneity 7.3 Strict Logical Homogeneity and Magnitude Homogeneity 7.4 Strict Homogeneity and the Limits of Conceptual Representation 7.5 The Role of Intuition: Kant contra Leibniz on the Identity of Indiscernibles 7.6 The Relation between Kantian Homogeneity and Euclidean Homogeneity 7.7 The Categories, Intuition, and the Part-Whole Relation of Magnitudes 7.8 The Composition of Magnitudes and Intuition 7.9 The Role of Strict Homogeneity in Representing Magnitudes: Clarifications 7.10 Conclusion 8 Kant's Revision of the Metaphysics of Quantity 8.1 Introduction 8.2 Leibniz on Identity, Quality, and Quantity 8.3 Quantity and Quality in the Metaphysics of Wolff and Baumgarten 8.4 Kant's New Understanding of Quality in Relation to Quanta and Quantitas 8.5 The Qualitas and Quantitas of Quanta 8.6 Conclusions 9 From Mereology to Mathematics 9.1 The Gap between Kantian and Euclidean Magnitudes 9.2 Euclidean Presuppositions: Aliquot Measurement 9.3 Sameness of Ratio Revisited 9.4 Euclid's General Theory of Pure Concrete Measurement 9.5 From Mereology to Measurement to Mathematics: Equality 9.6 The General Theory of Measurement in the Euclidean Tradition 9.7 Kant on Equality and the General Theory of Measurement 9.7.1 Kant on Comparative Size Relations of Continuous Magnitudes 9.7.2 Kant on Composition and Multiplication of Continuous Magnitudes 9.7.3 Kant on Comparative Size Relations among Collections 9.7.4 Kant on Composition and Multiplication of Collections 9.8 The Place of Equality in Kant's Account of Human Cognition 9.9 Conclusion 10 Concluding Remarks Bibliography Index