RUNNING NUMBERS

THE POWER AND RELEVANCE OF MATH AND THE LIMITATIONS OF CERTAINTY

Author: By Robert March

Date: SUNDAY, January 25, 1998

Page: F1

Section: Books

Science journalism can be a frustrating craft. On Monday it's a cloned sheep, Wednesday a Mars rock, Friday the top quark -- all subjects that no amount of schooling could have adequately prepared you for. Even if you get the facts right, the scientist is likely to gripe that you missed the whole point, while your editor rarely gives you enough space to explain it adequately. Still, there are a few practitioners who somehow manage to do the job superbly, and K. C. Cole of the Los Angeles Times is one of the best.

It helps that she works for one of the few big-city dailies that can afford to give science the space it deserves, and Cole's reputation rests on long articles or even series that explore a complex or controversial area of science in depth. Still, there are some topics for which any periodical is too constricting a format: Only a book will do, and that is the origin of this slim but informative volume.

Cole begins with a simple and familiar example, compound interest. To a mathematician, it's a form of ``exponential growth,'' a concept with relevance to everything from population and resources to the creation of the universe. She ends with an examination of the mathematics of symmetry, which guides modern physicists to the hidden structure of matter.

In between, Cole touches on a number of interesting topics, and one of the best chapters is on the problem of size. A human being on the scale of an ant would constantly be cold and hungry, while one that overtowered the World Trade Center would quickly collapse into a bundle of broken bones. The title of the book comes from one of these observations. On the scale of the solar system, gravity rules, and gravity strongly prefers spheres. A teacup the size of Jupiter would quickly collapse into this simple, universal shape. Yet to a bacterium, gravity is barely noticeable: The impacts and adhesion of molecules and atoms rule in its domain.

On the subatomic scale, the very laws of nature change in ways that our intuition can scarcely grasp. Cole comes up with some neat analogies that help illustrate what Albert Einstein called the ``spookiness'' of the quantum theory. This reviewer earns his living teaching that theory to non-scientists, and found material here that will show up in next term's lectures.

Cole does not confine herself to the natural sciences. In the realm of human relations, she shows how mathematical thinking might lead us to fairer voting schemes and less acrimonious divorce proceedings. It can also demonstrate that the Golden Rule may still be the best strategy, even when you're dealing with a bunch of selfish rotters.

She examines the problems faced by scientists serving as expert witnesses in trials, noting that this role is complicated by a clash of different standards of proof. She concludes that there is no easy way out of this dilemma: A scientist can afford to defer judgment, while judge and jury must do the best they can with whatever evidence is at hand, however inconclusive it may be in scientific terms.

Human factors are also important in the tangled area of risk assessment. Most of us know, at least on a rational level, that modern air travel has such a good safety record that the drive to the airport is probably the most dangerous stage of a trip. Yet once you are strapped into that airline seat, your fate is out of your hands, and Cole is sympathetic with the palpitations and sweaty palms that this realization engenders.

In her final chapter, Cole introduces an important neglected figure of 20th-century science, German mathematician Emmy Noether. Back in the 1920s, Noether discovered a key connection between mathematics and physics. ``Noether's Theorem'' concerns conservation laws, those peculiar rules that enable scientists to tie down a process of change by showing that something has stayed constant.

Noether showed that every conservation law is a consequence of a ``symmetry'' in nature. Energy conservation, for example, arises from a symmetry called the uniformity of time: Given the same circumstances, the same thing will happen, whether it is now, an hour from now, or a billion years in the future. It took a couple of generations for the import of this to sink in, but today theoretical physicists exploring a new area usually start by examining the symmetries of the situation.

Yet Noether remains obscure, in part because a woman of her era faced enormous career obstacles. When she was turned down for a position at Gottingen University because of her gender, the celebrated mathematician David Hilbert expostulated that ``we are a university, not a bathing establishment.'' And she was unlucky in other ways. The rise of Hitler obliged her to flee Germany and start anew in America; she was a mathematician whose greatest contribution had been to physics; and finally she died relatively young, before the importance of her work was fully appreciated.

Cole's work shows that she has learned a great deal about how scientists think and work. Science progresses through the combat of ideas and the formation of consensus, a process that is not entirely logical. If you are hankering after certainty, she realizes, you are better advised to seek it in religion: The stock-in-trade of science is not certainty but doubt. All scientific knowledge is conditional, and a mature scientist who has not had to abandon a ``truth'' learned in grad school has probably been working in a pretty dead area.

Cole reminds us that the mathematician Kurt Godel, more than 60 years ago, devised a mathematical proof that even in mathematics, not everything that is true can be proved. Mr. Spock may have been just a bit too logical to make a first-rate scientist.