The mathematician in the title of this review is of course Newton, but Penrose has a similar claim, though being modest would probably be embarassed at the comparison. Other than their profound knowledge of Mathematics and its closely allied subject Physics, Newton and Penrose were both willing to peer into other areas of science and speculate. Trespassing outside of ones own scientific territory has its perils. Newton wandered into Alchemy and the Occult and Penrose has written on the function of the brain and now in this book has stepped into the great unknown before the Universe exploded in the Big Bang. As a communicator and teacher of advanced Physics Penrose is bettered in my opinion only by Richard Feynman, but where Penrose is extraordinary is in his courage in presenting revolutionary ideas in a world of aggressive and often very unpleasant academics who defend their turf with a vigour verging on mania.<br><br>He expects a lot from his reader but much of what he is trying to explain is not understandable outside of Mathematics and any attempt to explain his arguments using language is not just an oversimplification but may indeed be wrong. Many kinds of mathematical space other than our own three dimensions require some understanding of tensors to follow the reasoning. Quantum physics is based upon complex numbers. Penrose's earlier book The Road to Reality deals with all this in some detail and is understandable with a lot of effort even by non-mathematicians such as me.<br><br>Although marginal to the subject of this book I would have liked to hear more on his views on the cosmological constant that to applied scientists sounds embarassingly like a 'fudge factor' designed to make theory fit facts. Another minor quibble is in the editing. Some sentences sound as if they were taken verbatim from a lecture (perhaps they were). An inevitable problem when a specialist is explaining a subject to a non-specialist editor is that obscurity isn't always because of the nature of the subject but may be because it is badly expressed in language. The spelling of the word discernible and a few other grammatical errors should be corrected in future editions, after all Penrose can't do everything...not quite.