BOOK REVIEW / SCIENCE : Journey in Search of the Right Metaphor : EMBLEMS OF MIND: The Inner Life of Music and Mathematics b<i> y Edward Rothstein</i> , Times Books, $25, 263 pages

Music is ethereal and mathematics is concrete. Music is art, while mathematics is science. Nonetheless there is a deep connection between these seemingly very different endeavors.

Everyone knows that Einstein played the violin, and he was hardly the only scientist with a musical turn of mind. This is no accident. Music and mathematics are among the very few fields in which there are true child prodigies. (Chess is another.)

But the exact nature of the connection is very hard to say. What--and where--is the link?

Trying to find it is the task that Edward Rothstein sets for himself in “Emblems of Mind,” a profound book that has much to say about both music and mathematics. No matter if the answer ultimately eludes him. The journey itself is worth the trip.

My copy of the book is now festooned with marginal notes, question marks, underlinings and responses. That in itself is a mark of a very rewarding book. I was engaged, though sometimes baffled.

In a sense, Rothstein has made an effort to unify his own intellectual life. He is chief music critic of the New York Times, but he studied advanced mathematical analysis before getting a doctorate from the Committee on Social Thought at the University of Chicago. He brings wide learning and insight to his subjects.

Rothstein starts by arguing that mathematics is not nearly as abstract and mechanical as many people believe. Style is part and parcel of the field. Often, the same fact can be proved in many different ways, each with its own “style,” each evoking a different emotional response, which, after all, is what music does.

Rothstein says that when he studied mathematics, he “had the same sense one has when confronting an artwork whose powers seem more mysterious than those of simple propositions and whose influence can be longer lasting than any other experience of our senses.”

Mathematics, he says, “is not so much a science as an art.”

Rothstein speaks frequently of the beauty and elegance of mathematics and its sublime ability to find powerful patterns in nature and in thought.

“The beauty is in the simplicity of the reasoning when faced with the complexity of expected calculations,” he says.

On the other hand, music is highly organized and structured. As a result of this organization, “Music commands time; it determines whether time seems to rush forward or hang suspended; it determines when time is to become knotted, turning back on itself.”

For Rothstein, the touchstone of both music and mathematics is that they are ways of knowing truth, a point he elaborates at some length. But then again, many other things are also ways of knowing truth--all of the sciences, for example, and even some of the social sciences.

Yet no one argues that biology is like music or that paleontology or sociology are like music. So it can’t just be the search for truth that unites music and mathematics.

The heart of the matter, Rothstein says, is that music and mathematics are both “projects of abstract exploration” that are guided by “analogy and transformation.” He makes much of the importance of metaphor and quotes Aristotle, who said, “A good metaphor implies the intuitive perception of similarity in dissimilar things.”

Mathematicians and musicians strive to find the right metaphors. They look for the right patterns, the right analogies and the right abstractions.

Ultimately, Rothstein argues, music and mathematics are about knowing, and they provide models for how we know anything at all. “We begin with objects that look dissimilar. We compare, find patterns with what we already know. We step back and create abstractions, laws, systems, using transformations, mappings and metaphors.

“This is how mathematics grows increasingly abstract and powerful; it is how music obtains much of its power, with grand structures growing out of small details. This form of comprehension underlies much of Western thought.”

Maybe so, but at the end of the book Rothstein reverts to poetry to make his points, conceding, in effect, that the ineffable remains ineffable.

In some sense, the connection between music and mathematics cannot be expressed in words. It can only be pointed to.