Why is the universe so mathematical? This question, which may never have occurred to you, turns out to be one of the stickiest philosophical conundrums of the day.

The short answer is that no one knows the answer. No one knows why mathematics--an invention inside our heads--so accurately describes the physical world, the independent world outside.

Eugene Wigner, a Nobel laureate in physics, put the question 35 years ago in an essay titled, "The Unreasonable Effectiveness of Mathematics." Since then, no one has been able to advance the ball.

It appears to us that the universe is structured in a deeply mathematical way. Falling bodies fall with predictable acceleration. Eclipses can be accurately forecast centuries in advance. Nuclear power plants generate electricity according to well-known formulas.

But those examples are the tip of the iceberg. In "Nature's Numbers," Ian Stewart presents many more, each charming in its own way. He shows, for example, how the oscillations of fireflies follow the same mathematical rules as the gaits of horses.

"Nature's numbers," he says, are "the deep mathematical regularities that can be detected in natural forms."

In Stewart's view, mathematics is the search for patterns in nature. Mathematics, he says, "is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity." He also tells us that one "function of mathematics is to organize the underlying patterns and regularities in the most satisfying way."

"The most satisfying way"? That's an interesting standard, as it seems largely subjective. Mathematical truths are independent of the human mind and human aesthetics. Or are they?

Stewart devotes one delightful chapter to explaining how the mathematics underlying the acoustics of a Stradivarius violin is the same as the mathematics underlying television.

If you stop to think about it, that is a stunning idea. What's the connection between violins and television? And those mathematical rules that seem to be embedded in reality--where do they come from?

Stewart is able to write about mathematics for general readers. He can make tricky ideas simple, and he explains the math of it with aplomb.

He also has a talent for coming up with comparisons that make things clear. At one point in the book, for example, he discusses an extremely large number: "Written in full," he writs, "it would go 10000 . . . 000, with a very large number of 0s."

How big is that, exactly?

"If all the matter in the universe were turned into paper, and a zero could be inscribed on every electron, there wouldn't be enough of them to hold a tiny fraction of the necessary zeros," Stewart writes.

Now * that* is a big number.

Stewart doesn't have much use for the distinction between pure and applied mathematics, arguing that when you scratch the distinction a little bit, it disappears.

In any case, he says, even if you have only the most pragmatic goals, you still need dreamers as well as practical thinkers.

Directed research alone cannot solve all of the problems we want solved. Sometimes a new approach has to come from where you weren't looking.

"The really important breakthroughs are always unpredictable," he writes. "It is their very unpredictability that makes them important: They change our world in ways we didn't see coming."

But as to the deeper questions that mathematics poses--why is physics so mathematical?--all Stewart can do--all anyone can do--is point them out. If there are answers, they are unknown.

Still, he does offer his best guess about what the answers would look like.

That there are patterns in nature seems incontrovertible. There are numerical patterns, geometric patterns, patterns of movement and, Stewart notes, even patterns in chaos, fractals and complexity, which have only recently been discovered. He gives copious examples of each.

"Mathematics is to nature as Sherlock Holmes is to evidence," Stewart says.

Just as the great detective was able to tease a coherent story from seemingly unrelated facts, so mathematicians are able to deduce the underlying patterns from the facts that nature leaves around.

"Nature is, in its own subtle way, simple," he says.

"However, those simplicities do not present themselves to us directly. Instead, nature leaves clues for the mathematical detectives to puzzle over."

As with many human endeavors, the thrill is probably more in the search than in achieving the goal (though the goal is important, too).

In this thin book, Stewart admirably captures compelling and accessible mathematical ideas along with the pleasure of thinking about them. He writes with clarity and precision.

Those who enjoy this sort of thing will love this book.