Mathematics and the Unexpected

Mathematicians have always stressed elegance and economy in formulating proofs. Ekeland fulfills both these criteria in this exposition of themes drawn from contemporary mathematics - even providing anidiomatic English translation from his own French original. His purpose is to describe how mathematicians have dealt with time; in so doing, he draws upon the celebrated 19th-century mathematician Henri Poincare, and several contemporaries: Rene Thom of catastrophe theory fame, and others associated with the dynamics of chaos. As background, Ekeland describes the historic attempts to predict the positions of the planets, which culminated in Kepler's laws of planetary motion. Later, Newton was able to derive the laws from his own development of the law of universal gravitation and calculus. In Kepler's and Newton's formulations, time can be read forward and backward: the laws are symmetric with respect to time; the universe is a deterministic clockwork. By the 19th century, however, it was clear that the values derived from Kepler's laws were only crude approximations. It took Poincare to show that even with the better and better approximations, certain orbits could not be computed. Thus, determinism was gone. In its wake contemporary mathematicians have seen that "initial conditions" can lead to extraordinary future states - the repeated "patterns" of chaos, for example, and curves with "strange attractors" and self-replicating geometries. In other instances, the conditions that define "dissipative structures" can lead to values that give rise to the "cusps" of catastrophe theory. Ekeland believes catastrophe theory to be important but limited, unwisely generalized by social and behavioral sciences, in both these mathematical developments, time loses its eternal character and becomes a one-way street - time gains an arrowhead. To conclude these heady observations, Ekeland contrasts the concept of time in the Iliad with the Odyssey, compares Thorn's uses of time with Proust, and discourses on evolution, Stephen Jay Gould, and Hieronymous Bosch. For readers of philosophical and mathematical bent, an elequent expression of new ideas. (Kirkus Reviews)