Mathematics and the Unexpected

by Ivar Ekeland

Book Overview

"Not the least unexpected thing about Mathematics and the Unexpected is that a real mathematician should write not just a literate work, but a literary one."-Ian Stewart, New Scientist"In this brief, elegant treatise, assessable to anyone who likes to think, Ivar Ekelund explains some philosophical implications of recent mathematics. He examines randomness, the geometry involved in making predictions, and why general trends are easy to project (it...

• Format: Hardcover
• Language: English
• ISBN: 0226199894
• ISBN13: 9780226199894
• Release Date: January 1988
• Publisher: University of Chicago Press
• Length: 146 Pages
• Weight: 0.70 lbs.

Customer Reviews

A Literate Popularization of Some Surprising Results

This fascinating little book -- originally published in French with the subtitle "The Figures of Time from Kepler to Thom" -- deals generally with discoveries related to dynamic systems. The first chapter recounts the history of results related to the granddaddy of dynamic systems: celestial spheres and their orbits. It's a gripping account that places Kepler's and Newton's breakthroughs in context, so that their importance is all the more appreciated. By the end of the 19th century, scientists believed they could accurately predict the location of any planet at any point in time, past or future.But as Ekeland so well conveys, the haughty hopes of scientism would soon be dashed. The second chapter describes the difficulty -- nay, the impossibility -- of making the required calculations to any accuracy. This is perhaps the most startling part of the book. Even if you have a math and physics education, you may not be aware of the results of Henri Poincaré, who showed among other things that there is no closed-form equation or non-divergent series for expressing the positions of three orbiting bodies in space (the so-called "three body problem"). Ekeland also describes Poincaré's qualitative results, which demonstrated the fractal character of planetary orbits. He then goes on to describe Lorenz's discovery that dynamic systems modeled by even a few simple differential equations exhibit chaotic behavior: a small perturbation in the initial conditions can lead to a vastly different outcome over time. This result re-enforces Ekeland's point that dynamic systems cannot be simulated computationally with any accuracy.The third chapter describes the results of René Thom's catastrophe theory, which explains why a small change to the configuration of a system can cause it to change states rather dramatically (Thom calls such shifts catastrophes). Ekeland is careful to point out that catastrophe theory applies only to dissipative systems, that is, systems in which stable equilibria are reached due to the dissipation of energy (usually through friction). He also uses a somewhat skeptical tone in describing the more far-fetched areas to which catastrophe theory has been applied by Thom and its other proponents. Nonetheless, catastrophe theory is certainly interesting and does make intuitive sense.This book is an all-too-rare literate popularization of mathematical ideas. Indeed, literary references abound, especially in the final chapter, which is a long analogy between dynamic systems and Homer's epics The Iliad and The Odyssey. Although I found this analogy a bit strained, I highly recommend the book's first three chapters. In addition to clearly describing a wide variety of mathematical results (using relatively few equations and copius illustrative figures), it also contains a number of surprising little gems. The foreword by Felix Browder is also excellent, putting the rest of the book into appropriate context. Finally, Ekeland has included two fascinatin

Deceptive Simplicity and Powerful Insight

Occasionally you encounter a book about a subject that you know reasonably well, or least think you do, only to discover that you had little understanding after all. When I first read Ivar Ekeland's "Mathematics and the Unexpected", I was absolutely startled. I knew some physics and math, and I had always assumed that differential equations and physics accurately described nature. Differential equations and physics seemed to go together like bread and butter. Ekeland quickly showed me that I had not really thought about the way differential equations are used in physics.Ekeland has written a book about mathematics, not a mathematics book. "Mathematics and the Unexpected" does not require familiarity with advanced mathematics and celestial mechanics, and yet Ekeland discusses both topics. The thoughtful reader, rather than becoming confused, will come away with insight into some of the most exciting work in mathematics in the last thirty years. I suspect, however, that the reader knowledgeable of more advanced mathematics will be even more appreciative of Ekeland's achievement.This is not a new book. Ekeland received the Jean Rostand Prize in 1984 for this outstanding work of scientific popularization. He discusses classical determinism, impossible calculations, deterministic but random systems, unstable but stable systems, and dissipative dynamic systems as he positions us to understand catastrophe theory. In the 1980's catastrophe theory was more controversial as some early proponents had either applied the theory inappropriately or exaggerated its capability. Ekeland concludes with a thoughful critique of catastrophe theory.I encourage anyone interested in pursuing more recent works (popular or technical) on chaos theory, fractals, or nonlinear dynamics to first read "Mathematics and the Unexpected". Ivar Ekeland has produced "a cultured text with the rare combination of deceptive simplicity and powerful insight" that provides a solid foundation for exploring many contemporary topics in mathematics. The phrase in quotes is from the London Times. This is definitely a five-star book.

The Best Popular Math Book I Have Read

As a mathematician who doesn't generally enjoy popular math books, I found Mathematics and the Unexpected to be quite pleasant reading. Instead of being a treatise on the enjoyment of puzzles, I think it conveyed much of how mathematics is integral towards human understanding of the world around us. I have recommended it to my non-mathematician friends, to the point that my copy is on permanent loan.