David Ruelle’s stated purpose in writing The Mathematician’s Brain is to present a view of mathematics and mathematicians which will be of interest to those both within and without the profession. He has succeeded admirably in this regard: The Mathematician’s Brain is both fascinating and intellectually challenging, and is written in a style disarmingly direct and collegial. One minute Ruelle is discussing the influence of computers on mathematical practice, the next minute relating how the Russians managed to practice anti-Semitism even within so apparently ethnicity-free a subject as mathematics. Throughout, he is more concerned with presenting an interesting take on a number of ideas than in delivering the last word on any particular topic, but his discursive journeys are so interesting that you are often sorry to see them end so soon.
Ruelle attempts to supply both 1) a flavor of what mathematicians do, and what it means to think mathematically, and 2) some insider’s tales of the mathematical world and the individuals who make it up. There's not that much actual mathematics included, and the short examples presented are there to illustrate larger points. One unintended effect of including even these "simple" examples, however, is a demonstration that the definition of simplicity is relative to one's expertise and usual habits of thought.
Not surprisingly in a book which ranges so widely and takes so informal an approach, there’s material within to offend as well as to delight. For instance, I found the chapter on Alan Turing both terminally clueless as to the implications of being a gay man in the Britain of the early 20th century, and trivial in its overall question. Who knows or cares if personal strangeness causes mathematical genius or vice versa? Most likely it’s a variant of the old third variable problem: both the genius and the strangeness are caused by slightly different "wiring" of the mental apparatus. Not that I can prove this, but it’s as good a speculation as any.
But people who seek to offend no one are frequently dull, one charge which may not be laid upon this author. To put it in terms of tennis commentators, Ruelle is more John McEnroe than Cliff Drysdale, and if he occasionally causes offence he more than makes up for it with the quality and originality of his thoughts. Not all chapters will appeal to everyone, but the material covered is so varied that there's almost certain to be something to interest anyone concerned with math, philosophy, or the history of science. I read most of this volume in one evening, and finally set it down with the feeling that I just spent a lengthy train ride sharing a compartment with a garrulous and somewhat eccentric but fascinating companion.
David Ruelle obtained his PhD in Physics from the Université Libre de Bruxelles in 1959 and has worked at ETH Zurich, the Institute for Advanced Studies in Princeton, NJ, and since 1964 at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France. He has published seven previous books, including Chance and Chaos (Princeton: Princeton University Press, 1991).
Sarah Boslaugh (firstname.lastname@example.org) is a Performance Review Analyst for BJC HealthCare and an Adjunct Instructor in the Washington University School of Medicine, both in St. Louis, MO. Her books include An Intermediate Guide to SPSS Programming: Using Syntax for Data Management (Sage, 2004), Secondary Data Sources for Public Health: A Practical Guide (Cambridge, 2007), and Statistics in a Nutshell (O'Reilly, forthcoming), and she is Editor-in-Chief of The Encyclopedia of Epidemiology (Sage, forthcoming).
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