Over the past few decades, electronic computer technology has made enormous strides. Moreover, there can be little doubt that in the decades to follow, there will be further great advances in speed, capacity, and logical design. The computers of today may be made to seem as sluggish and primitive as the mechanical calculators of yesteryear now appear to us. There is something almost frightening about the pace of development. Already computers are able to perform numerous tasks that had previously been the exclusive province of human thinking, with a speed and accuracy which far outstrip anything that a human being can achieve. We have long been accustomed to machinery that easily out-performs us in physical ways. That causes us no distress. On the contrary, we are only too pleased to have devices that regularly propel us at great speeds across the ground - a good five times as fast as the swiftest human athlete - or that can dig holes or demolish unwanted structures at rates which would put teams of dozens of men to shame. We are even more delighted to have machines that can enable us physically to do things we have never been able to do before: they can lift us into the sky and deposit us on the other side of the ocean in a matter of hours. These achievements do not worry our pride. But to be able to think- that has been a very human prerogative. It has, after all, been that ability to think which, when translated to physical terms, has enabled us to transcend our physical limitations and which has seemed to set us above our fellow creatures in achievement. If machines can one day excel us in that one important quality in which we have believed ourselves to be superior, shall we then not have surrendered that unique superiority to our creations? The question of whether a mechanical device could ever be said to think - perhaps even to experience feelings, or to have a mind - is not really a new one. But it has been given a new impetus, even an urgency, by the advent of modern computer technology. The question touches upon deep issues of philosophy. What does it mean to think or to feel? What is a mind? Do minds really exist? Assuming that they do, to what extent are minds functionally dependent upon the physical structures with which they are associated? Might minds be able to exist quite independently of such structures? Or are they simply the functionings of (appropriate kinds of) physical structure? In any case, is it necessary that the relevant structures be biological in nature (brains), or might minds equally well be associated with pieces of electronic equipment? Are minds subject to the laws of physics? What, indeed, are the laws of physics? These are among the issues I shall be attempting to address in this book. To ask for definitive answers to such grandiose questions would, of course, be a tall order. Such answers I cannot provide: nor can anyone else, though some might try to impress us with their guesses. My own guesses will have important roles to play in what follows, but I shall try to be clear in distinguishing such speculation from hard scientific fact, and I shall try also to be clear about the reasons underlying my speculations. My main purpose here, however, is not so much to attempt to guess answers. It is rather to raise certain apparently new issues concerning the relation between the structure of physical law, the nature of mathematics and of conscious thinking, and to present a viewpoint that I have not seen expressed before. It is a viewpoint that I cannot adequately describe in a few words; and this is one reason for my desire to present things in a book of this length. But briefly, and perhaps a little misleadingly, I can at least state that my point of view entails that it is our present lack of understanding of the fundamental laws of physics that prevents us from coming to grips with the concept of 'mind' in physical or biological terms. By this I do not mean that the laws will never be that well known. On the contrary, part of the aim of this work is to attempt to stimulate future research in directions which seem to be promising in this respect, and to try to make certain fairly specific, and apparently new, suggestions about the place that 'mind' might actually occupy within a development of the physics that we know.