Brown Walker Press, Boca Raton, Florida, 2014133 pp. Paper, $25.95 ISBN: 9781627345064. Reviewed by Phil DykePlymouth University p.dyke@plymouth.ac.uk The first thing to say about this book is that it is not an easy read; it is really an extended essay rather than a book. It is very scholarly and is full of interesting facts. The layout of the book is almost chronological, but not quite. The first few chapters concentrate on religious notions of infinity, not a great surprise as the Bible and the Qu'ran are rich sources in a field where there are few rivals. The Greeks' intolerance of Zeno's paradox and the infinite decimal representation of square root of two are discussed. There follows an account that includes the geometry of tassellations and the attempts to represent God and link this representation to the infinite. His thesis also spills into architecture; the Alhambra Palace with its geometrical designs, gothic fan vaulting in Wells cathedral, and then there's the art of M. C. Escher. It is at this point one is pondering where the book is going; is it actually going to talk about the meaning of infinity or just tell tales? We, then, land on firmer ground with the renaissance covering perspective in art as well as the scientific revolution. Copernicus, Galileo, and Newton get a mention. Calculus is explained, rather naï vely, and Newton's ideas on gravitation is contrasted with Einstein's 20^{th} century space-time notions of gravity. This leads to a discussion of modern astronomy and theories of how the universe began. The idea of an infinite but bounded universe, the steady state, and big bang theories all get more than a mention. There's no deep philosophical discussion, no complex mathematics. We are informed about a lot of stuff, but rather like a tourist, the reader is shown around then led on. Finally after 100 pages, transfinite numbers are covered. This reviewer had been waiting for this information as it is a central idea in the meaning of infinity, and the treatment is accurate but limited. It is this part of the book that best demonstrates the difficulty presenting a coffee table book on a subject that does get rather technical. To go through Georg Cantor's rigorous definitions of transfinite numbers is outside the scope of the book (as Topper confesses on page 106) but not to do so makes the explanation incomplete and frustrates the reader, maybe I should say frustrates this reader. The book finishes with a short history of Cantor and his attempts to relate transfinite numbers to the theories of the "size" of the universe, to theology and to God. Finally the lack of transfinite numbers when describing the physical world, despite their use in the descriptions of computability, is covered. Altogether it is a very worthy serious book, and a book that has my admiration. |

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