In *Shadows of the Mind*, Roger Penrose explores the relation between consciousness and the current state of our scientific explanation of it. Penrose's aim is clear. He argues that the consciousness experienced by human beings is the kind of thing that *can *be explained by science but, as yet, we do not have the scientific tools to do so. This marks Penrose's position off from those who would claim that consciousness is utterly mysterious and cannot be explained in "physical, computational, or any other scientific terms," as well as those who are convinced that "all thinking is computational" and is thus within reach of our current scientific knowledge, in principle if not in practice (12). To demonstrate this, Penrose employs both a positive and negative approach to the problem.

The positive approach involves bringing the implications of Gödel's incompleteness theorems to bear on what computation can or cannot accomplish. One example is that there are certain mathematical claims that can be known through human reasoning but cannot be reached through brute computation. One classic example is the fact, proved by Lagrange in 1770, that every natural number can be expressed as the sum of four squares (67). This is true but no amount of computation, of "number crunching," could yield such a result.

The practical point of this demonstration is that there are certain things, even when considering only the restricted field of mathematics, that humans can know to be true but which cannot be understood merely by means of computation, even in principle. This is not to say that developing a computer that can prove Lagrange's theorem is necessarily conscious or genuinely intelligent but rather that any claim to achieve artificial intelligence must grapple with this issue. Any would-be intelligence that is incapable, in principle, of proving theorems like Lagrange's are going to fall short of human intelligence.

Part II of the book explores the current state of relativity and quantum physics to see if these revolutionary perspectives provide any alternative ways one might engage in "computation." If it should happen that, through what we learn from these branches of physics, we are able to compute in a new way and solve the kinds of problems that face traditional notions of computation, it might mean that our current science is able to explain consciousness in principle though it has not yet done so in practice.

In a short chapter on the relevant implications of General Relativity, and a much longer discussion on the relevant implications of Quantum Theory, Penrose concludes that, radical as such advances may have been, they do not yet provide us with the tools we need to solve even the restricted case of proving theorems which cannot be solved by pure computation as we know it today.

Penrose's argument performs the admirable task of marking off certain promising research trajectories as being dead-ends and points the way toward new questions that need to be asked.