The Mathematician's Brain: A Personal Tour Through the Essentials of Mathematics and Some of the Great Minds Behind Them

The Mathematician's Brain poses a provocative question about the world's most brilliant yet eccentric mathematical minds: were they brilliant because of their eccentricities or in spite of them? In this thought-provoking and entertaining book, David Ruelle, the well-known mathematical physicist who helped create chaos theory, gives us a rare insider's account of the celebrated mathematicians he has known-their quirks, oddities, personal tragedies, bad behavior, descents into madness, tragic ends, and the sublime, inexpressible beauty of their most breathtaking mathematical discoveries.

Consider the case of British mathematician Alan Turing. Credited with cracking the German Enigma code during World War II and conceiving of the modern computer, he was convicted of gross indecency for a homosexual affair and died in 1954 after eating a cyanide-laced apple; his death was ruled a suicide, though rumors of assassination still linger. Ruelle holds nothing back in his revealing and deeply personal reflections on Turing and other fellow mathematicians, including Alexander Grothendieck, René Thom, Bernhard Riemann, and Felix Klein. But this book is more than a mathematical tell-all. Each chapter examines an important mathematical idea and the visionary minds behind it. Ruelle meaningfully explores the philosophical issues raised by each, offering insights into the truly unique and creative ways mathematicians think and showing how the mathematical setting is most favorable for asking philosophical questions about meaning, beauty, and the nature of reality.

The Mathematician's Brain takes you inside the world, and heads, of mathematicians. It's a journey you won't soon forget.

"David Ruelle has written an entertaining and thoughtful book on human theorizing in that most abstract science, mathematics. Yet its content has ramifications that extend well into other thought processes." Stephen Smale, Toyota Technological Institute at Chicago

"Fascinating and quite eclectic. Ruelle has a pragmatic approach to discussing philosophical and psychological questions. He is equally pragmatic with regard to ethical and political issues involved in the professional world of the mathematician. As Ruelle repeatedly says, mathematics is a human activity." William Messing, University of Minnesota

"The mathematician David Ruelle is well known for his work on nonlinear dynamics and turbulence, and his new book, The Mathematician's Brain, is a book about mathematics and what it all means....The book's value lies in Mr. Ruelle's description of the curious inner life of mathematicians." David Berlinski, New York Sun

"Ruelle notes the extraordinary amount of context that underwrites mathematical activity." Donal O’Shea, Nature

If mathematicians ran the world, this extraordinary book would be the design document for a whole mathematics curriculum. Students taking geometry, for example, might still learn about the axioms of Euclidean geometry, but the subject would be placed in the context of the more mathematically sophisticated view of geometry as a group of symmetries. This is generally Ruelle's method: take a subject that is covered in elementary math classes and explain its context--the way professional mathematicians see it. So Euclidian geometry leads to projective and affine geometry, illuminated by Felix Klein's concept of the role of symmetry in geometry; and Cartesian coordinates leads to algebraic varieties. Ruelle uses these examples to explain some of the great themes of mathematics, such as generalization of problems, the simultaneous roles of formal axioms and human ideas, translation of problems into new forms to apply different tools, and so on. In the course of these explanations he also articulates ideas about philosophical questions such as the definition of mathematics, and the independent reality of mathematical concepts. Finally, the author shares anecdotes that are a combination of history and gossip about various famous mathematicians. Written with beautiful clarity, this book will give non-mathematicians a feel for how mathematicians think. Yet, for mathematicians, it is raised above "popularization" by the fact that it actually contains mathematical proofs and exercises, rather than merely talking about them. Ruell is one of the creators of chaos theory and is emeritus, mathematical physics, Institut des Hautes Etudes Scientifiques in France, and visiting professor at Rutgers U. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)

David Ruelle is professor emeritus of mathematical physics at the Institut des Hautes Etudes Scientifiques in France and distinguished visiting professor of mathematics at Rutgers University. His books include Chance and Chaos (Princeton).

Preface vii

Chapter 1: Scientific Thinking 1

Chapter 2: What Is Mathematics? 5

Chapter 3: The Erlangen Program 11

Chapter 4: Mathematics and Ideologies 17

Chapter 5: The Unity of Mathematics 23

Chapter 6: A Glimpse into Algebraic Geometry and Arithmetic 29

Chapter 7: A Trip to Nancy with Alexander Grothendieck 34

Chapter 8: Structures 41

Chapter 9: The Computer and the Brain 46

Chapter 10: Mathematical Texts 52

Chapter 11: Honors 57

Chapter 12: Infinity: The Smoke Screen of the Gods 63

Chapter 13: Foundations 68

Chapter 14: Structures and Concept Creation 73

Chapter 15: Turing?s Apple 78

Chapter 16: Mathematical Invention: Psychology and Aesthetics 85

Chapter 17: The Circle Theorem and an Infinite-Dimensional Labyrinth 91

Chapter 18: Mistake! 97

Chapter 19: The Smile of Mona Lisa 103

Chapter 20: Tinkering and the Construction of Mathematical Theories 108

Chapter 21: The Strategy of Mathematical Invention 113

Chapter 22: Mathematical Physics and Emergent Behavior 119

Chapter 23: The Beauty of Mathematics 127

Notes 131

Index 157