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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. 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) 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 When you think about it, it seems obvious: The only mathematical ideas that human beings can have are ideas that the human brain allows. We know a lot about what human ideas are like from research in Cognitive Science. Most ideas are unconscious, and that is no less true of the mathematical ones. Abstract ideas, for the most part, arise via conceptual metaphor-a mechanism for projecting embodied (that is, sensory-motor) reasoning to abstract reasoning. This book argues that conceptual metaphor plays a central, defining role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms: transfinite numbers, points at infinity, infinitesimals, and so on. Even the real numbers, the imaginary numbers, trigonometry, and calculus are based on metaphorical ideas coming out of the way we function in the everyday physical world. This book is about mathematical ideas, about what mathematics means-and why. The authors believe that understanding the metaphors implicit in mathematics will make mathematics make more sense. Moreover, understanding mathematical ideas and how they arise from our bodies and brains will make it clear that the brain's mathematics is mathematics, the only mathematics we know or can know. -- Publisher description "When you think about it, it seems obvious: The only mathematical ideas that human beings can have are ideas that the human brain allows. We know a lot about what human ideas are like from research in Cognitive Science. Most ideas are unconscious, and that is no less true of the mathematical ones. Abstract ideas, for the most part, arise via conceptual metaphor-a mechanism for projecting embodied (that is, sensory-motor) reasoning to abstract reasoning. This book argues that conceptual metaphor plays a central, defining role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms: transfinite numbers, points at infinity, infinitesimals, and so on. Even the real numbers, the imaginary numbers, trigonometry, and calculus are based on metaphorical ideas coming out of the way we function in the everyday physical world. This book is about mathematical ideas, about what mathematics means-and why. The authors believe that understanding the metaphors implicit in mathematics will make mathematics make more sense. Moreover, understanding mathematical ideas and how they arise from our bodies and brains will make it clear that the brain's mathematics is mathematics, the only mathematics we know or can know."--Rabat de la jaquette Renowned linguist George Lakoff pairs with psychologist Rafael Nuez in the first book to provide a serious study of the cognitive science of mathematical ideas. 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

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.

"This book is about mathematical ideas, about what mathematics means - and why. It is concerned not just with which theorems are true, but with what theorems mean and why they are true by virtue of what they mean. And it provides an answer to one of the deepest problems of the philosophy of mathematics: how a being with a finite brain and mind can comprehend infinity."--BOOK JACKET. THIS BOOK ASKS A CENTRAL QUESTION: What is the cognitive structure of sophisticated mathematical ideas?