The Mathematician's Brain - 9780691129822

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)