# Being Precise About Randomness

Just when you think you know all the answers they may change all the questions

BOSTON — The Jungles of Randomness

By Ivars Peterson

Wiley, 239 pp., $24.95

Here's a quick arithmetic quiz: 2+2=4, right? Does it also equal 3? Probably not. But you can't prove it using mathematical logic. The structure of arithmetic is too random to prove that statement - or an infinitude of other arithmetical statements - through rigorous use of the relevant mathematical axioms.

No one has yet proved that the axioms mathematicians use every day are consistent. Rather than being a logical necessity, mathematicians' confidence in their work is more like the statistical "certainty" that flipped coins are as likely to come up heads as they are tails.

That's a taste of the cold water Ivars Peterson's book, "The Jungles of Randomness," throws on the romantic notion that mathematics guarantees us a safe place in a universe dominated by random chance and coincidence. Math does provide some relatively civilized areas. But the frontier guards must be constantly vigilant. It's a "jungle" out there. And subtle errors are ever ready to creep into mathematicians' tidy formulations. They also can mess up the sense of order we try to impose on our everyday world.

Take the stock market for example. There's no proof that its movements are anything other than random. A string of purely random numbers will show trends and patterns. So too, would a purely random stock market show trends and patterns that investors can use to construct sophisticated investment schemes. Yet a random event - an unexpected war, natural disaster, or financial disruption half a world away - can torpedo those calculations overnight.

Court cases can turn on the human tendency to perceive patterns where none may exist. Does a cluster of ailments near a major industrial plant point to pollution as their cause? Or is the unusual high rate of incidence within the statistically expected range of variations for the ailment? Juries are not always persuaded by the lack of statistical evidence to pin the blame on a polluter.

And as Peterson points out, the mathematics may itself be flawed. This can catch out research scientists. They often test their data against computer generated sets of random numbers to see if they have a genuine effect or just a chance occurrence of the phenomenon they are investigating. No one has yet come up with a foolproof random number program. This can trick scientists into publishing nonsense.

It's hard to define the term "random." What seem to be purely chance happenings on one level may have deeper hidden causes. Peterson shows how to search for some of these. With four other math books behind him and as math and physics editor for Science News, he has honed his explanatory skills finely. He is a readable guide through the tangles of probability and random chance. He shows how we are taming these through practical mathematical applications.

This is not an easy book. Readers have to think along with the author. More illustration would have helped. Yet it will give some insight into one of the most fruitful areas where math meets practical living. And, yes, you can count on the fact that 2+2=4 and nothing else in ordinary arithmetic. That's overwhelmingly the most probable outcome of the calculation.

* Robert C. Cowen writes about science for the Monitor.