I first read this book as a sixth former, shortly after I'd read A Brief History of Time. Having re-read it, I wish I had paid it more attention whist studying relativity in the first year of my physics degree, because this book contains the best explanation I've come across. It also sheds some light on Hamiltonian mechanics - a subject not present in my course yet which is the starting point for Schrödinger's equation in quantum mechanics.
Penrose chooses his agenda carefully, perhaps not only because his investigation into minds and consciousness is interesting, but also because his argument allows him to proceed through relativity, quantum mechanics, black holes and other popular scientific phenomena. It's not a journey for the faint-hearted, though: sparing use is made of equations, but Penrose writes intelligently and for the intelligent reader. These days I wonder not so much about whether a computer might pass the Turing test, as whether a human might fail it.
Indeed, the Turing test features on page 5 of the book, as Penrose begins with a couple of chapters of history and philosophy on artificial intelligence, Hilbert, Gödel, Turing and Church. Apparently switching tack, he then presents the Mandelbrot set - but its relationship to the previous topic is surely relevant, the fractal demonstrating infinite detail arising from a very simple rule, the nature of truth and proof, the problem of determinacy, and a finite region surrounded by an infinite boundary.
Moving on once again, the next chapters describe first classical physics (including relativity), and then quantum physics. The equivalence of gravitational mass and inertial mass is hinted at, but perhaps could have been explored more, as it's otherwise difficult to grasp why gravity is so special (except in the case of relativity, where it is general). On quantum mechanics, he is careful to explain the determinism of much of the theory, something which is often misunderstood or misrepresented. Switching then to thermodynamics, Penrose really begins to construct his argument. The second law of thermodynamics is one of very few physical laws which are time-asymmetric, and we need an explanation of why time flows only in one direction, if it is to "flow" at all. Penrose goes on to speculate about quantum gravity: his hypothesis that it is the non-algorithmic action of quantum gravity which may give the appearance of the probabilistic behaviour in quantum physics is intriguing and appealing, but he makes it clear that it is speculative and that he has no firm theory to put either before the layman or the specialist. It is an intuition, perhaps firmly described as such in order to emphasise the nature of the human thought process.
Finally, Penrose describes the structure and function of the brain, and speculates on the nature of consciousness. Reductionist theories cannot explain this, because at the atomic and sub-atomic levels, all particles of the same type are identical: there is no "special stuff" from which life or consciousness forms. Again his argument, perhaps less clear here than in earlier chapters, is that quantum gravity must play a role, that quantum superpositions are the closest thing we have to "free will". His views on the non-algorithmic nature of quantum gravity close the door on "strong AI" and thinking computers (at least, in the form we presently understand computers). There are hints that Penrose is looking for some sort of spiritual dimension, dissatisfied in some way with a mechanical approach. But this book dates from 1989, and I re-read it in part because I have his newer work The Road to Reality also on my shelves, awaiting my attention.