A version of my review was published in the Huntsville (AL) Times.<br><br>For a book ostensibly about reality, this one talks a lot about "magic", "mystery", occasional "miracles", and especially the Platonic world of mathematics (or more generally, of "truth"), which Penrose forcefully argues has a reality all its own. Those who fear arithmetic may be lulled by Penrose's gentle preface, which goes into some detail on fractions. This introduces the important mathematical principle of equivalence classes; in the case of fractions, pairs of integers are declared equivalent in a simple way that leads to the normal rules of rational numbers.<br><br>But Penrose goes far beyond the simple in his discussion of the mathematical concepts underlying modern physics theories. Penrose's perspective on the mathematics relevant to reality is highly geometrical and visual, and even experts in the field might find new insights in his discussion of hyperbolic geometries, complex numbers (where lies much of the "magic"), and tensors and symmetry groups. Visual insight is difficult when one is talking about higher dimensional spaces, and the lack of visual geometric insight is one of Penrose's complaints about the 10 or more dimensions of typical string theories. Penrose provides startling new ways to look at the four dimensions most relevant to our spacetime, although occasionally his descriptions are marred by poor diagrams. Some of the sections also seem to be missing his unique touch, relying on more pedantic and traditional approaches to the math.<br><br>For the very reasonable list price this is really three books in one, and oddly organized for a book ostensibly directed at a lay audience. First is a textbook on the mathematics used by physical theories, complete with exercises. Penrose is careful to define almost everything so an intelligent lay reader could, in principle, follow, but the definitions are brief and not entirely self-contained, and many readers will lose hope fairly quickly. Second Penrose gives a semi-rigorous introduction to the fundamental theories of modern physics, including the remarkable successes of both General Relativity and the quantum field theor of the standard model of particle physics.  The real payoff is in the final portion of the book, chapters 27-34, which describes the vast gap that still remains between nature and our theories.<br><br>None of the other recent books along these lines - Hawking's<br><br>"Brief History of Time", Brian Greene's books, and other recent works by physicists touting our nearness to a "theory of everything" have approached anything like this level of detail.  In Penrose's view those other books are all at least partially wrong, and he believes the mathematics needs to be taken seriously to see why.<br><br>Penrose dives into this with an attack on inflationary (and other) theories of the big bang, linking the thermodynamic concept of "entropy" to the geometrical structure of the universe, where he is a renowned expert.  The fundamental issue, also discussed in Hawking's book, is why the universe has a definite "arrow of time", given that all our accepted physical theories are symmetrical when time is reversed. Somehow the universe started in a highly unusual low-entropy state; Penrose dismisses all the usual arguments about how this could have happened with a statistical analysis from his understanding of spacetime geometries. Did the universe really require a creator to specially select the astronomically low probability universe we started with, or are there physical processes we don't yet understand that introduce this sort of radical time-asymmetry in a natural way? Penrose favors the latter, but makes clear nobody has a good explanation.<br><br>The most substantive chapter in the book is Penrose's discussion of superstrings and related ideas. Some highly interesting mathematics has come out of these studies, but they may have little or no relation to the real world. Superstring theories have so far predicted essentially nothing that can be confirmed by experiment - Penrose explains why some often-touted cases of prediction (such as black hole entropy) are in fact rather dubious. Another part of Penrose's problem with them, as with the inflationary theories of the big bang, is that they are undergoing continual rapid evolution so that the superstrings of this year are very different from those of ten years back. Penrose gives a reasonably unbiased presentation of the theories and why people are interested in them, but he does not approve of the bandwagon status they have acquired.<br><br>Penrose admits at various points that his views on modern physical theories are somewhat out of the mainstream. His major contributions to physics have ranged widely, but predominantly within the realm of Einstein's "General Relativity", the theory of gravitation. Two of his unique contributions that have not been widely taken up by other physicists are mentioned repeatedly: his view that gravitation is more fundamental than quantum mechanics, and his "twistor" approach to spacetime geometry. The first of these implies that the basic notions of quantum physics may need to be modified with nonlinear terms arising from gravitational interactions within quantum superpositions. For both he identifies new experiments and theoretical work needed to make further progress, including a space-based "Schrodinger's Cat" experiment dubbed FELIX.<br><br>The fundamental thread through this book is this relationship between mathematical theories and the reality of nature. The Platonic world provides us with considerable "magic" to explore and be fascinated by, but do the magic and occasional miracles lead us to the truth about reality, or away from it? The reason mathematical truth is not quite the useless tautology Wittgenstein once argued is the linkage it provides between things we might have thought were distinct, but are in fact the same from a certain perspective, just as 3/6 and 1/2 are equivalent as fractions. Penrose illustrates numerous examples of the way mathematical equivalences can change one's perspective on the nature of the world around us - his "twistors" are yet another way of looking at the world, beautiful but quite different from our usual concepts of space and time.  Sometimes in the past a beautiful mathematical scheme, like that of the quaternions, has turned out to have  little practical use at all, despite much initial promise. Yet Einstein, Dirac, Feynman and others were led by mathematical considerations to theories that had real explanatory and predictive power. Penrose emphasizes the importance of real experiments to guide physical theories.<br><br>It seems we simply don't yet have enough experimental capability or mathematical understanding to find the true "theory of everything". It's possible we're close; if we are this book may help to make that happen. But even if we still have a long and winding road to follow, from Penrose's perspective there will be much that is fascinating and beautiful to observe along the way.