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. Contents Acknowledgments Preface Introduction: Why Cognitive Science Matters to Mathematics Part 1. 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 2. Algebra, Logic, and Sets 5. Essence and Algebra 6. Boole's Metaphor: Classes and Symbolic Logic 7. Sets and Hypersets Part 3. The Embodiment of Infinity 8. The Basic Metaphor of Infinity 9. Real Numbers and Limits 10. Transfinite Numbers 11. Infinitesimals Part 4. 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 5. Implications for the Philosophy of Mathematics 15. The Theory of Embodied Mathematics 16. The Philosophy of Embodied Mathematics Part 6. e^{pi * i} + 1 = 0. 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. e^{pi * i} + 1 = 0 - How the Fundamental Ideas of Classical Mathematics Fit Together References Index A Note on the Text