Okay, I think, therefore I am. But who gets to play that game? A newborn? A mosquito? A computer? If my thoughts are elsewhere, am I here or there? When I no longer think as I once did, am I the same person? What composes this "I," molecules or memories?

Questions about the boundaries, location, continuity and constituents of the self stand at the heart of philosophy, but a mathematician and physicist, René Descartes, set the terms of the discussion. Who better to bring us up to date than Douglas Hofstadter? Trained in math and physics, Hofstadter won a 1980 Pulitzer Prize for Gödel, Escher, Bach, a bravura performance linking logic, art and music. He returns now to apply a concept from that book, the strange loop, to the definition of self.

Like consciousness, the strange loop is elusive. When a brilliant author uses one slippery concept to clarify another, the result for the reader can be anxiety. Page after page, we may wonder whether we will reach the limit of our understanding and whether the journey will be worth the effort.

Fortunately, Hofstadter is a gifted raconteur and a master of metaphor. He conjures up a car with a 16-cylinder motor and what the salesman calls Racecar Power®. It's not as if you can get a model that has the engine without the trademark feature. Similarly, Hofstadter writes, "consciousness is not an [added] option" for beings evolved to engage in symbolic thought, recognize patterns, create categories, reason via analogies and wonder about the self. Consciousness is "the upper end of a continuous spectrum of self-perception levels that brains automatically possess as a result of their design."

Hofstadter's strange loop is the feedback loop. Point a video camera at a TV displaying the camera's output, and you will produce a receding corridor of screens. Pixels make up the picture, but our interest is in the image, the tunnel of rectangles. Identity resembles that phenomenon. Never mind the neurons that make up our brain. Our emotions, others' responses and our repeated looks outward to the world and inward to ourselves shape what we call our self. Nor is ours the only loop we contain. We know how our friends see things; our mind houses their perspectives -- it has the formulae for producing their thoughts.

However mechanistic, Hofstadter's account of the self emerges from deep emotion. In 1993, when she was 43, Hofstadter's wife and soul mate, Carol, died suddenly of a brain tumor. Three months into his mourning, Hofstadter initiated a heartfelt correspondence with the philosopher Daniel Dennett. What emerged was Hofstadter's understanding of self as distributed over many minds, a concept that explained how Carol's "personal sense of 'I' " lived on (in "low-resolution fashion") as a "loop" in Hofstadter's consciousness.

I have so far given a superficial account of Hofstadter's position. As his book title indicates, for Hofstadter the self is a strange loop. Strange loops are reflexive and paradoxical, like M.C. Escher's impossible image of right and left hands drawing each other into existence. Hofstadter's example of a real-world strange loop is a key construct in Kurt Gödel's incompleteness theorem, published in 1931, a proof that any seemingly comprehensive mathematical system will contain true statements that cannot be proven. The theorem is notoriously indigestible.

How elusive is this strange loop? I was a young math buff. Last year, when Discover magazine surveyed authors about science writing that had influenced them, atop my list was a popularization of Gödel's proof by Ernest Nagel and James Newman -- the same book that inspired Hofstadter in his teens. If I am, for that reason, an ideal audience for the strange loop theory, there's good news and bad. I found Hofstadter's explication of Gödel revelatory. There were implications of the proof that I had never appreciated. But then (here's evidence for discontinuity), I no longer quite understand the proof. Nor, given what I do grasp, am I convinced that the entities Gödel conceived are apt analogues of the self.

This difficulty does surprisingly little to diminish Hofstadter's achievement. Philosophers of mind are divided between those who see consciousness as a special quality (like Racecar Power®) and those who see it as irredeemably physical (like neural networks). Hofstadter points to another level at which self might exist, up among the symbols and patterns -- or rather, to various levels on which self exists simultaneously. His conclusions mesh well with those of psychotherapy. We are not selves first and social creatures later. It's through empathy that we develop a rich sense of self. Nor is the self neatly demarcated. We contain multitudes.

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