BOOK REVIEW / SCIENCE : Question of Consciousness Unanswered in Elegant Effort : SHADOWS OF THE MIND / A Search for the Missing Science of Consciousness <i> by Roger Penrose</i> ;Oxford University Press $25, 457 pages

How do our brains, which are physical objects, give rise to our thoughts, which are not?

This is a simple statement of the mind-body problem, which has bedeviled Western thought since Rene Descartes inflicted it on the world nearly four centuries ago. The question is easy to ask and can be understood by anyone. But answering it turns out to be among the most profound and difficult dilemmas ever posed.

Many have struggled with the question, but so far no satisfactory and satisfying answer has been proposed.

In recent decades, the staggering growth in the power and speed of computers has focused renewed attention on Descartes’ question. For computers are “electronic brains,” and in certain well-defined areas--chess, for example--they can outperform all but a handful of humans. Does this mean they are thinking? Does this mean they have consciousness? Or that someday they will?

The scientific view is that there is nothing in our heads but flesh and blood and that every thought, every emotion, every dream, every nuance is merely the way we experience a certain arrangement of chemicals in our brains. To be sure, science today is far from having a full understanding of those arrangements of chemicals, but we know the laws that govern them, and ultimately we will figure it out and be able to replicate a fully functioning brain.

The non-scientific view says that no completely physical explanation will ever be found because there is something else in our heads--something non-physical--that enables this three-pound hunk of matter, our brains, to think and be conscious. Call it the soul , if you will, or the elan vital , or whatever (it doesn’t have to be religious), but what we call mind, in this view, is not completely explicable by the brain and never will be.

Five years ago, Roger Penrose, the Rouse Ball professor of mathematics at Oxford and one of the world’s most distinguished philosophers of mathematics and physics, wrote a book called ‘The Emperor’s New Mind,” in which he proposed a middle ground between the two views.

According to Penrose, the workings of the brain and the mind are indeed ultimately explicable by the methods of science. But the laws of chemistry and physics that we now have are inadequate to the task. The laws of science that apply to the brain are different from the laws of science that apply to everything else, Penrose said. And these special laws of the brain have yet to be discovered.

Not surprisingly, Penrose’s book touched off a furor of controversy, with many mainstream scientists attacking him for straying from the orthodoxy that all matter is the same and obeys the same laws.

The artificial intelligentsia in particular--those who devote themselves to getting computers to think like people--were livid at Penrose’s assertion that their goal was unattainable and that computers would never have insight or, well, consciousness. (They agree that they have not yet achieved the Holy Grail, but the key word for them is yet. Yet may be a long time coming, they say, but it’s just a matter of time.)

Penrose is no kook. So his views could not simply be dismissed. But he was roundly attacked for them and was accused of the ultimate sin in science: mysticism and its handmaiden, magic thinking.

Now Penrose has written a new book, “Shadows of the Mind,” in which he restates his earlier view more emphatically than before and responds to his critics. “Shadows of the Mind” is an elegant though difficult book, beautifully written and argued, in which Penrose draws on Godel’s Theorem and the paradoxes of quantum theory in support of his claims.

The brain is not a computer, Penrose says. The brain is self-aware. It is capable of belief, understanding, insight and intuition. Computers cannot do these things and never will be able to. “Human intuition and insight cannot be reduced to any set of rules,” he writes.

A computer is a physical object that obeys the known laws of physics. The brain, on the other hand, is a physical object that obeys so-far unknown laws of physics. Penrose surmises that the brain’s physics is like quantum physics, in which subatomic particles can seemingly exist and not exist at the same time, and you can’t be sure which until you observe them.

(Never mind about how this can be. No one knows. Quantum theory is an extremely accurate predictor of experimental results. So what if its philosophical implications don’t make sense?)

Penrose speculates that the interaction between quantum mechanics and the brain occurs in cellular structures called microtubules. This is, to be generous, something of a stretch, as the evidence in support of this theory is scant at best.

Let’s face it: Penrose is not going to persuade his critics, and they are not going to persuade him. Descartes’s question remains unanswered.

But Penrose has taken a mighty swing at it, using all of the tools of 20th-Century thought at his command. If he hasn’t quite solved the problem, he has nonetheless plumbed the depths of physics, philosophy and brain research in a powerful attempt to come up with an answer. Where is the computer that could do half as much?