Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being

Description

This book is about mathematical ideas, about what mathematics means-and why. Abstract ideas, for the most part, arise via conceptual metaphor-metaphorical ideas projecting from the way we function in the everyday physical world. Where Mathematics Comes From argues that conceptual metaphor plays a central role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms. »Mathematics is seen as the epitome of precision, manifested in the use of symbols in calculation and in formal proofs. Symbols are, of course, just symbols, not ideas. The intellectual content of mathematics lies in its ideas, not in the symbols themselves. In short, the intellectual content of mathematics does not lie where the mathematical rigor can be most easily seen—namely, in the symbols. Rather, it lies in human ideas. But mathematics by itself does not and cannot empirically study human ideas; human cognition is simply not its subject matter. It is up to cognitive science and the neurosciences to do what mathematics itself cannot do—namely, apply the science of mind to human mathematical ideas. That is the purpose of this book. One might think that the nature of mathematical ideas is a simple and obviOus matter, that such ideas are just what mathematicians have consciously taken them to be. From that perspective, the commonplace formal symbols do as good a job as any at characterizing the nature and structure of those ideas. If that were true, nothing more would need to be said. But those of us who study the nature of concepts within cognitive science know, from research in that field, that the study of human ideas is not so simple. Human ideas are, to a large extent, grounded in sensory-motor experience. Abstract human ideas make use of precisely formulatable cognitive mechanisms such as conceptual metaphors that import modes of reasoning from sensory-motor experience. It is always an empirical question just what human ideas are like, mathematical or not. The central question we ask is this: How can cognitive science bring systematic scientific rigor to the realm of human mathematical ideas, which lies outside the rigor of mathematics itself? Our job is to help make precise what mathematics itself cannot—the nature of mathematical ideas.« (from the preface) Cover Contents Acknowledgments Preface Introduction: Why Cognitive Science Matters to Mathematics Part I THE EMBODIMENT OF BASIC ARITHMETIC 1 The Brain’s Innate Arithmetic 2 A Brief Introduction to the Cognitive Science of the Embodied Mind 3 Embodied Arithmetic: The Grounding Metaphors 4 Where Do the Laws of Arithmetic Come From? Part II ALGEBRA, LOGIC, AND SETS 5 Essence and Algebra 6 Boole’s Metaphor: Classes and Symbolic Logic 7 Sets and Hypersets Part III THE EMBODIMENT OF INFINITY 8 The Basic Metaphor of Infinity 9 Real Numbers and Limits 10 Transfinite Numbers 11 Infinitesimals Part IV BANNING SPACE AND MOTION: THE DISCRETIZATION PROGRAM THAT SHAPED MODERN MATHEMATICS 12 Points and the Continuum 13 Continuity for Numbers: The Triumph of Dedekind’s Metaphors 14 Calculus Without Space or Motion: Weierstrass’s Metaphorical Masterpiece Le trou normand: A CLASSIC PARADOX OF INFINITY Part V IMPLICATIONS FOR THE PHILOSOPHY OF MATHEMATICS 15 The Theory of Embodied Mathematics 16 The Philosophy of Embodied Mathematics Part VI A CASE STUDY OF THE COGNITIVE STRUCTURE OF CLASSICAL MATHEMATICS Case Study 1. Analytic Geometry and Trigonometry Case Study 2. What Is e? Case Study 3. What Is i? Case Study 4. How the Fundamental Ideas of Classical Mathematics Fit Together References Index