Book Review : The World According to Mathematics

Mathematics and the Unexpected by Ivar Ekeland (Chicago: $19.95; 146 pages)

The changes that science has wrought over the last several centuries have been pervasive and spectacular. First we think of technology: airplanes, electricity, telephones, computers, radio, antibiotics and the like. It is an impressive and virtually endless list.

But the changes do not stop there. Modern science has also engendered a revolution in the way we look at the world. Since Isaac Newton showed that the laws of mechanics and physics could be expressed mathematically--a stunning achievement--we have adopted the view that mathematics can describe most things.

This view is not nearly as well founded as classical mechanics, but it has a powerful hold on thought. Economists, for example, are unwilling to give up their effort to describe and predict human behavior mathematically though their efforts have met with virtually no success.

This failure should not be surprising, for it turns out that not even classical mechanics--the paradigm of a mathematically predictable system--is as successful as it is cracked up to be.

Unanswered Questions

“The most basic questions have gone unanswered since Newton’s time,” Ivar Ekeland writes in “Mathematics and the Unexpected,” a delightful examination of the limits of mathematics in describing the world. “What is the Earth’s true trajectory? Is it gradually nearing the sun, and will it finally be swallowed up? Or is it slowly drifting away, finally to escape into the cosmic vacuum? No one knows.”

This shortcoming is partly a result of computational inefficiency. Since every object in the universe exerts a gravitational influence on every other one--including the Earth--there are simply too many factors to consider in working out the exact trajectory of the Earth around the sun.

But that analysis implies that if only we could build faster computers we could eventually solve the problem. Not so, says Ekeland. The problem is so complicated that it is inherently unsolvable.

“In this century, the development of digital computers has vastly expanded the range and accuracy of predictions in celestial mechanics,” he writes. “But the frontier of feasibility is still there, even though it has been pushed farther away. There are a great many computations that cannot be performed now or in any foreseeable future.”

A Matter of Data

The same is true of economics, though economists will tell you that it’s just a matter of gathering more data and refining their formulas and they will be able to make accurate forecasts. To repeat, “There are a great many computations that cannot be performed now or in any foreseeable future.”

Put aside the social sciences. The limits of mathematics in the natural sciences have become increasingly apparent throughout this century. In the last couple of decades, these limitations have been addressed systematically, starting with the “catastrophe theory” of Rene Thom, a controversial subject on which Ekeland sheds considerable light.

A catastrophe is a major change in a system brought about by a minor change in the starting conditions. This is what makes the weather so unpredictable, though the physical laws that govern it are perfectly known. Nature is deterministic but random.

This is the point that Newton and his intellectual descendants have missed. The future cannot be predicted from perfect knowledge about the present, even if we had the perfect knowledge, which we don’t have and aren’t likely to have. Ekeland writes: “Like the queen of England, determinism reigns but does not govern.”

This does not mean that the scientific enterprise should throw up its hands in despair. Even if perfect knowledge is not attainable, much can be achieved short of that, including a qualitative understanding of the world separate from a quantitative one.

Doesn’t Champion Theory

Though Ekeland explains these ideas very well, he is not an unabashed champion of catastrophe theory. He notes that it, too, has limitations and that it has not lived up to the hopes of its supporters. “In the 20 years of its existence, there has not been a single undisputed success of catastrophe theory in the field of experimental science, that is, an undisputed fact that could be explained more adequately by catastrophe theory than by any other means,” Ekeland coolly observes.

Nonetheless, he argues, the approach has much to commend it, even if it does not accomplish everything. Ekeland places catastrophe theory between determinism and chaos and argues that this is a fruitful realm of science and all thought.

Ekeland is a mathematician at the Univesity of Paris-Dauphine, and his book was first published in French as “Le calcul, l’imprevu. “ It won the 1984 Jean Rostand prize for scientific writing directed to laymen. Now Ekeland himself has translated it into English and provided a most-welcome addition to the literature on this subject. There is much packed into this short book.

This is not the first time in history that the dreams of mathematics have foundered on reality. The Pythagoreans believed that the world was numbers, and they tried to keep it a secret when they found out that it wasn’t.

In our own time, the same secret is again getting out.