Mathematics Is Not a Spectator Sport
by George M. Phillips

In the Preface to the book the author wrote

... there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are all doers, not spectators. Doing is much more fun than merely watching or listening, and I celebrate this in the title of the book.

Unfortunately, the celebration does not seem to propagate into the body of the book leaving the title quite detached from the contents and the manner in which the book has been written.

Some false notes can be detected already in the Preface:

Some very fine books have been written about mathematics. These give us the flavor of certain areas of mathematics, but they don't make us mathematicians. This book will introduce you to a range of topics in mathematics and will help you on your way to becoming a mathematician, even if you have only begun this journey.

I am not sure whether the above is an intentional attempt at misleading the audience. The author is Professor of Mathematics at St. Andrews University, Scotland. Even assuming strained relations between The Highlands and the rest of Great Britain, there are quite a few books by English authors that could be thought as written for about the same purpose and about the same audience and of which the author might be expected to be aware. The three that most readily come to mind are You Are a Mathematician by D. Wells, The Mathematical Olympiad Handbook: An Introduction to Problem Solving by A. Gardiner, and Thinking Mathematically by J. Mason.

Curiously, in the Introduction to his book, D. Wells quotes the same dictum:

Mathematics, as teachers are fond of repeating, is not a spectator sport.

Along the same lines, A. Gardiner writes that the potential to do mathematics has to be developed - "and that requires effort and commitment." J. Mason's aim "is to show how to make a start on any question, how to attack it effectively and how to learn from the experience." I am bewildered as to why the author of the book under review chose to juxtapose his book to those written "about mathematics".

Going through the book made me also wonder what pedagogical stratagem has been employed by the author in order to lure the reader from the audience and onto the stage. D. Wells weaves a story connecting diverse mathematical items which he interrupts with boxed questions. The purposeful topic selection allows him to constantly switch between graphical visualization and logical reasoning. A. Gardiner employs his trademark method (that he developed and polished in several of his books) of outline proofs whereby the reader is literally forced to complement the author's exposition by his or her own efforts. J. Mason virtually walks the reader by hand through a sequence of problem solving steps, making the book as interactive as a printed matter can possibly be.

In contrast, the book under review has been written in a relatively formal style of numbered definitions, theorems, examples, remarks and problems. All five categories have independent numberings, so that, for instance, one has in sequence Theorem 4.3.1, Definition 4.3.1, Remark 4.3.1, Theorem 4.3.2, Example 4.3.1, which I found awkward.

The book is primarily devoted to the topics in number theory, with chapters named Squares (1), Numbers, Numbers Everywhere (2), Fibonacci Numbers (3), Prime Numbers (4). Chapter 5, Choice and Chance, deals with the counting combinatorics. Chapter 6, Geometrical Constructions, besides the traditional Ruler and Compass geometry and Unsolvable Problems, offers sections on Properties of Triangle (existence of some remarkable triangle centers), Coordinate Geometry and Regular Polyhedra. Chapter 7 introduces the Algebra of Groups.

The topic selection does not look very original, although their grouping in chapters is unusual at times. Just to give you an idea, the first chapter combines a nice discourse on square and triangular numbers and on those that are both, the Pythagorean theorem with several proofs, a derivation of a formula for Pythagorean Triples, a discussion on representation of integers as a Sum of Two Squares, and an introduction into Complex Numbers.

The second chapter introduces Different Bases, Congruences, Continued Fractions, The Euclidean Algorithm, and Infinity. The latter includes introduction into the cardinal numbers and their infinitude and a story of the mythical Hilbert's Hotel.

The exposition on the whole is competent but unexciting, perhaps uneven. The author often, but not systematically so, provides historic attributions. For example, he mentions that the Pythagorean theorem appears as Proposition 47 in Book I of Euclid's Elements, but does not relate a proof by "similar triangles" (p. 19) to Euclid VI.31. On a positive side, a well known Chinese diagram is not presented, as is often done, as a proof of the Pythagorean theorem, but only as a proof that the 3-4-5 triangle is right.

Sometimes the book creates an impression of not being thought through. E.g., the diagram for the addition of complex numbers x1 + y1i and x2 + y2i appears in Chapter 1, while a discussion of Coordinate Geometry is postponed till Chapter 6. A basic fact that the conjugate of the product of two complex numbers is the product of the conjugates is used but not stated. The addition formulas for sine and cosine (p. 40) are mentioned without proof but said to be amenable for a verification by "elementary geometrical arguments". I wonder. In Chapter 7, matrices and quaternions are used to motivate the introduction of more abstract group theoretic concepts. The product of 2×2 matrices is defined explicitly (p. 196) as is the product of a matrix by a scalar (p. 197), but the definition of matrix addition may only be guessed from a formula for a matrix representation of quaternions (which is a linear combination of 4 matrices.) In a discussion on the distribution of primes, the author does not shy (p. 108) from introducing the logarithmic integral and "asymptotically equal" functions. In Chapter 1 (p. 13), however, he finds it necessary to mention in parentheses that "ak means a multiplied by itself k times and a-k = 1/ak."

The book "is intended for the senior students in high school and those who are beginning their study of mathematics at university level ... (and for) the large set of people who are not in school or university but are intellectually curious and active." At this time and day, there is a large body of mathematical literature that targets the same audience. It is my impression that the book under review may have a hard time competing in the intended market segment.

Mathematics Is Not a Spectator Sport, by George M. Phillips. Springer (2005), 240 pages, $39.95. ISBN 0-387-25528-1.

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