Lectures by a renowned mathematician and physicist on the connections of relativity and quantum theory (the science of the very large and the very small), with an eye to understanding the nature of the mind. Penrose has been over this ground before (in The Emperor's New Mind, not reviewed, and Shadows of the Mind, 1994), and his contention that artificial intelligence is an impossibility has generated a good deal of controversy. Here he reiterates and extends his essential arguments and invites refutation from a trio of critics in related disciplines: Abner Shimony, Nancy Cartwright, and Stephen Hawking. One central point of debate has to do with the relation between mathematics and the ``real universe''—or, as Penrose puts it, between the physical and the platonic worlds. A popular view of how science works is that the scientist, looking to explain a series of observations, finds a mathematical relationship that accounts for the data. Penrose argues that this view has things backwards: The mathematical relationship is the reality, and the data merely an expression of it. Einstein conceived his equations before data were available to verify them; when data became available, his calculations checked out exactly. Penrose goes on to consider the paradoxes of quantum mechanics, but readers without sophisticated mathematics are likely to find this section tough going. Finally, Penrose attempts to apply these issues to the question of whether the activities of mind can ever be duplicated by a computational device, a possibility he denies. His three critics then point to what they feel are weaknesses in his arguments, and finally Penrose counters their rebuttals. Penrose pushes the available analytical tools to the limit, and the result is far from light reading, but those willing to think hard about fundamental questions of mind and matter will find this discussion provocative and rewarding.