The mathematical content of the book may be divided into three main topics. The largest chunk (the first 180 pages) deals with geometry, Euclidean and non-Euclidean. The presentation of Euclid's postulates and first thirty-two propositions is done with full rigor, but due to the relaxed style of the author, the reader is never bored. Davis shows on the way how our geometric understanding developed, from Euclid and the Greeks studying the world we are living in and axiomatizing the facts of life, to later developments in non-Euclidean geometry which made Euclid's geometry one among the many different types of geometries to be studied. A whole chapter on non-Euclidean geometries follows mathematicians like Sacchieri, Gauss, Bolyai, Lobachevsky and Riemann as they stumble across ideas that lead to the new geometries. Abbott's Flatland is brought up as a practice model to understand these different systems. Numerous pictures and diagrams accompany the text and make the visualization of the concepts much easier.
The second topic the book concentrates on is number theory. Basic notions of primality and congruence arithmetic along with most of the standard results like Fermat's little theorem and the Euclidean algorithm are presented. On the way, we are introduced to Fermat and his last theorem, and Turing and the Enigma. We witness the development of the idea of a Turing machine and how this lead to the development of modern computers. We learn about computational complexity. The abstract development is concluded with a clear presentation of public key cryptography, in particular the RSA algorithm.
The third and last part of the book introduces the notion of fractal mathematics. The theory developed here is accompanied with 11 colored plates where the readers can see for themselves the Mandelbrot set and its successive magnifications. Besides the inherent beauty of these pictures, the mathematics involved is very interesting, and many terms like the Feigenbaum constant are introduced as intriguing new notions which still need to be further understood. This chapter provides the reader with computer programs (in BASIC) that produce all sorts of fractal images and the reader is encouraged to work on these and modifications for a full grasp of the concepts involved. Even if the reader may not have access to a computer (or may not be fluent in BASIC), the programs provided are clear enough to be parsed and understood by anyone intent on doing so, and their simplicity adds to the allure of the very complex images that come up.
Davis takes every opportunity to reveal the proof-based nature of mathematics to his readers. Throughout the book, the reader is exposed, with no apologies, to at least a hundred rigorous proofs, many formal axiom systems and all sorts of abstract notions. The mathematician's reluctance to accept as fact any result that has not been rigorously proved is explained as one of the reasons for the interest in Fermat's Last Theorem. The underlying theme is the expectation that mathematical proof is the way to mathematical truth.
A second main theme is, quoting a part of the title of the book, the "power of mathematics", or quoting Wigner, the "unreasonable effectiveness of mathematics". Davis gives us diverse applications of the three topics he studies in detail: Geometry as applied by Kepler to planetary motion, and as applied by Einstein to describe the nature of gravity, number theory as applied to cryptology and computer science, fractals as applied to computer art and data compression. In almost all the cases he presents, the applications have followed the development of the abstract theories and the mathematicians involved had no inkling of what their results could end up being used for. Clearly the mathematician is not really in it for the applications, but if necessary, one could justify mathematics only with its numerous and diverse applications.
However Davis is not satisfied if the reader will use applicability as the only justification for mathematics. His main goal for writing this book, one would guess, is to convince the reader that mathematics is worthy of interest, not because it is useful (although he provides various examples of this resourcefulness), but in fact because it is beautiful and alive. In various parts of the book, he points out the beauty of the ideas involved, and he consistently presents mathematics as a developing subject in progress. He explicitly mentions three problems which were open when he was working on the draft of the book and were proved by the printing time (the infinitude of Carmichael numbers, determination of the Hausdorff dimension of the boundary of the Mandelbrot set, Fermat's last theorem) to emphasize the in-progress nature of mathematics.
Overall this reviewer found this book to be highly entertaining and full of ideas and examples that anyone may use to convince others why one may be interested in studying mathematics after high school. The book covers many exciting topics and explains some rather sophisticated ideas very well. Davis never skips relevant and interesting topics because they might be too complicated, and manages to describe, to a reader who is expected to know only basic high school geometry and algebra, Poincaré's model for hyperbolic geometry, the geometrization conjecture, the RSA algorithm, fractal dimensions, and so on. However, to get the most out of this reading experience, one needs serious dedication, and a clear and persistent mind (paper and pencil may also help) as the details and the numerous exercises contribute a lot to the text.
Gizem Karaali teaches at the University of California in Santa Barbara.