Ian Stewart has written so many books, and is so widely read, that his reputation carries more weight than could any brief review. Nevertheless, here goes.
It is critical for mathematics educators to keep up with popular mathematics. It is important to be able to provide students with concrete glimpses of the usefulness of mathematics, and the diversity of this usefulness. Now as much as ever, the attitude of the educated public toward mathematics is that it is esoteric and strange, and practiced by people out of touch with the "real world." This is where books like this one come in.
The Magical Maze grew out of an invitation Stewart received to present the Christmas Lectures, a series "for young people" sponsored by the Royal Institution of Great Britain. His intent is to introduce a non-mathematical audience (but one interested in mathematics) to what mathematicians do, the relationships that exist between mathematics and the natural world, and to showcase some of today's mathematics.
He patterns his book after a maze, his metaphor for the body of knowledge that is mathematics. At a given junction there are many possible ways to proceed. A wise choice leads to advances in mathematical knowledge. Poor choices (or random exploration) lead to time wasted or dead ends, but sometimes also to surprising new ideas. To Stewart then, a mathematician is someone with the knowledge and skills to be able to navigate small pieces of the maze. I suppose he comes down firmly on the side of the discussion that says we discover, not create, mathematics. Are there readers who still believe, as I do, that mathematics is a creative endeavor?
As books on popular mathematics go this one is a mixed bag. Stewart has managed to sneak in quite a large number of topics. Some are tried and true, and anyone who teaches at the college or university level is certainly familiar with them, at least at a superficial level. A few are the Fibonacci numbers, the golden section, and phyllotaxis; the Monty Hall problem; symmetry of plane and three-dimensional figures; Turing machines; chaos and fractals. Stewart also weaves in more recent material related to these ideas. Sections on the Interrogator's Fallacy, animals' gaits as coupled oscillators, and the computational abilities of train sets (yes, train sets!) both extend and add a contemporary flavor to the above-listed topics. (Read about the "Prosecutor's Fallacy;" it could open as big a can of worms as Marilyn vos Savant did when she wrote about the Monty Hall problem several years ago.) Though he sometimes stretches the maze metaphor a bit thin, it should be apparent to the most casual reader of his book that mathematics is alive and growing.
Stewart succeeds in providing a window through which the general public can glimpse mathematicians at work. He writes about the power of recognizing patterns, both subtle and not so subtle, the importance of being able to relate different kinds of mathematics to each other, and the occasional necessity of being able to admit one's mistakes. He discusses how mathematicians proceed toward proofs of their conjectures, and touches upon the role serendipity plays in the discovery of new mathematics. He does all this in a manner accessible to a popular audience.
Since the book requires almost no mathematical background, an interested high school student can read it. Let me emphasize though that they must be interested. Effort will be necessary even for typical undergraduate mathematics majors to wade through some of the material. There is no harm in that, though. Stewart writes as a journalist reporting on the mathematics he knows. His conversational style ("When you're hot you're hot!" and "Convinced? Good.") may be attractive to readers who do not know or read much mathematics.
Several times (most notably in the section on probability) Stewart bites off more than his intended audience will be able to chew. In trying to make complicated ideas transparent, perhaps an impossible task, he resorts to confusing explanations, or glosses over important ideas. For example, after stating that "probabilities are not especially intuitive" he goes on to say that to give the reader a feel for probability he "will take a more freewheeling attitude -- without trying to convey the content of the mathematician's definition." It is precisely the lack of intuition when it comes to probabilities that makes the mathematician's formal definitions necessary. I am not arguing that he should discuss measure theory in this book, but that he has missed an opportunity to make an important point. Based on my experiences trying to teach probability to non-math folks, I think the rest of the discussion falls short of being convincing for the intended audience. He is not the first person to think he can explain somewhat deep ideas about probability to a general audience in less than an hour.
The method used for footnotes (a hand icon, pointing to the back of the book) is awkward. Since the notes are not numbered I had to continually count how many hands had appeared in the current chapter to know which one was being referenced. There is one more significant problem: the book contains several careless errors. These are forgivable if only misspelled words, but not when incorrect words or symbols hinder the reader trying to follow a logical argument.
If you keep up with popular mathematics, you will have read almost all this book contains elsewhere. If you are an educator looking for a shortcut to interesting ideas you can share with students, Stewart's book will do. Overall, despite occasionally bewildering a non-mathematical audience, Stewart succeeds in writing a book that gives the uninitiated a feel for what mathematics is and what mathematicians do. I hope that every non-mathematician, at least once, will read a book like The Magical Maze.
Sean Bradley ( firstname.lastname@example.org) is assistant professor of mathematics at Clarke College in Dubuque, Iowa.