BOOK REVIEW : Mathematician’s Journey Into the Regions of Chaos

Does God Play Dice? The Mathematics of Chaos by Ian Stewart (Basil Blackwell: $19.95; 317 pages)

“Does God Play Dice?” is a book about chaos, and my reactions to it were chaotic. Sometimes I liked it and sometimes I didn’t. Sometimes I thought there was a lot there, and other times I was sure there wasn’t. At the end, I saw the value of it. The last chapter was the best.

But I’m getting ahead of myself.

Chaos is a hot topic in many sciences these days. The weather, the stock market, the course of an epidemic -- they all exhibit chaos. They and many other phenomena are unpredictable. Small, almost imperceptible changes in initial conditions produce wildly different results.

A couple of years ago I bought a videotape of fractals (the structured irregularity of the natural world) produced at the Cornell National Supercomputer Facility. The camera zooms in on very pretty pictures generated by a computer, showing their complex internal structures in bright colors. Electronic music plays in the background.

I look at that tape every so often and think, “So what?” They are intriguing pictures, but what do they show?

Stewart, a mathematician at the University of Warwick in England and a frequent writer on mathematics for general readers, is puzzled by the same question. His answer is that chaos theory offers something in understanding some problems, but it is not the universal panacea that many of its notices imply.

“Chaos has many lessons to teach us,” he writes. “Its prime message is a general one: ‘Don’t jump to conclusions.’ . . . Sometimes it works. . . . Sometimes it doesn’t.”

For readers who enjoyed James Gleick’s “Chaos”--a masterful introduction to the subject--Stewart’s book may be just the ticket. It takes the next step or two, both deeper and broader.

Stewart’s book is more rigorous and more difficult than Gleick’s, though it should be within the range of interested, determined readers. Don’t think of it as dry, though. Stewart writes with an engaging, cheeky tone.

The two books cover much the same ground: Edward Lorenz and his “strange attractors,” Mitchell Feigenbaum and the physics of turbulence, Benoit Mandelbrot and fractal geometry. But while Gleick looked at these people and their work from the outside, Stewart writes about them from the inside. Gleick is a science writer; Stewart is a scientist.

He places the development of chaos theory in historical perspective. Over the last few centuries, science has made incomparable progress, using the inexorable laws of physics as the model for knowledge. We have a deep emotional commitment to those laws and that model.

But only certain kinds of problems lend themselves to classical solutions of that sort. Balls rolling down inclined planes are one thing, but they may be only a small subset of the questions we want to investigate. They are the easy problems; we now want to tackle the hard ones.

Researchers have by now studied many phenomena that appear chaotic, and they are devising ways for handling the unpredictability of such systems. The question that remains is whether chaos is inherent in nature or whether if we just had more facts, we’d see the pattern, right before our eyes.

The title of Stewart’s book is taken from Albert Einstein’s famous remark about quantum uncertainty in a letter to Max Born: “You believe in a God who plays dice, and I in complete law and order.”

This is the same dilemma we now face about chaos. Is it built in or will it all be explained away some day when we know more? Stewart writes: “Either God is playing dice, or He’s playing a deeper game that we have yet to fathom. I agree with Einstein. I like the second idea--the deeper game which we don’t understand yet--a lot more.”This theological theme recurs periodically. “Whatever it is that God deals in, it’s not explicit formulas,” Stewart says. “God’s got an analogue computer as versatile as the entire universe to play with--in fact it is the entire universe--and He finds little fascination in formulas designed for pencil and paper.”

Even if chaos theory turns out not to be as powerful as its adherents once thought, Stewart is undeterred. He doubts that it will help predict the weather, but he thinks it will be useful in other fields. Besides, he says, in research, the hunt is usually more interesting than the answer.

“The brightest ray of light that chaos sheds focuses on the nature of complexity,” Stewart concludes. “We now know that simple equations can have simple solutions--or complex ones. Complex equations can have complex solutions--or simple ones. . . . Where will the torch of chaos lead us? We cannot tell.”

I will look at the fractal videotape with new eyes. Having read the book, I now understand the movie.