Diagrams, Visual Imagination, and Continuity in Peirce's Philosophy of Mathematics

Description

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of the very process of the reasoning itself. Thus conceived, one's capacity to diagram their thought reveals a set of characteristics common to ordinary language, visual perception, and necessary mathematical reasoning. The book offers an original synthetic approach that allows tracing the roots of Peirce’s conception of a diagram in certain patterns of interrelation between his semiotics, his pragmaticist philosophy, his logical and mathematical ideas, bits and pieces of his biography, his personal intellectual predispositions, and his scientific practice as an applied mathematician. Preface Contents Chapter 1: Introduction Two Approaches to Diagrams Peirce’s Vision What This Book Is About Chapter 2: Meritocratism, Errors, and The Community of Inquiry Meritocratism and Science Peirce’s Maxim Chapter 3: Logic and Mathematics Self-Interpretation, Conventionality, and The Language of Thought Two Kinds of Minds Fast, Pedestrian, or Both? Chapter 4: Peirce’s Transcendental Deduction and Beyond: Categories, Community, and the Self The Kantian Legacy Peirce’s Deduction Some Consequences of Peirce’s Deduction: The Community of Inquiry and the Self Fixation of Belief Chapter 5: Sign Relation Kinds of Relations Degenerate Relations Categories and Math Chapter 6: One, Two, Three Abstractions, Things, and Signs Modes of Being Chapter 7: Iconicity, Novelty, and Necessity Novelty Necessity Inference as Observation Chapter 8: The General and the Particular Mathematical Diagrams as Icons The General and the Particular Fused Together Chapter 9: Diagrams Between Images and Schemata The Role of Mathematical Cognition in Kant’s 1st Critique Diagrams and Schemata Chapter 10: Existential Graphs Visualized Inferences and Inferential Visuality The Graphs Explained The Graphs Contextualized Chapter 11: Iconicity, Similarity, and Habitual Action Three Formulations of the Maxim What Is Likeness? Chapter 12: Mapping Philosophy: Peirce’s Quincuncial Projection Language and Maps Quincuncial Map The Virtual Chapter 13: L’image-Mouvement, Mathematically Sublime, and the Perception of Totality Synthetic Unity vs. Dynamic Totality The Mathematically Sublime The Architectonic Role of Mathematics Chapter 14: The Metaphysics of Continuity Reality, Generality, and Continuity The Analytic Approach to Continuity: Cantor and Dedekind Continuity and Infinitesimals Chapter 15: Conclusion References