The past decade has seen an explosion in the number of books about mathematics written for the general public — one measure of this phenomenon is that there have been over 110 books "for the general reader" reviewed in "Read This!" alone. These books vary greatly in a number of ways. Some of them are actually quite technical, while some assume essentially no mathematical background. Some of them focus more on the history than the mathematics, while others are not afraid to include some (or lots of) equations. Some cover a single topic in depth and some meander all around the world of mathematics. Some are written by professional mathematicians and some by professional expositors. (Before I continue, I should mention that I have read both good and bad books in all of these categories).
Another difference between these books concerns their target audiences: some authors choose to write a book that, while designed for general audiences could also be engaging to professional mathematicians — I would include Keith Devlin's The Millenium Problems and most of Martin Gardner's writings in this category — while other authors aim their book squarely at general audiences, and professional mathematicians would have little to gain from their book. It is in this latter category that Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, the new book by Edward Burger and Michael Starbird, fits. Therefore, I feel that I must begin my review of the book with the following disclaimer:
I am not in the target audience for this book, and most likely you are not either.
This book will contain lots of ideas new to high school students — or to people who have not taken any math since high school. It even includes some topics that would appeal to other kinds of academics, though I think there are more appropriate books for you to recommend to convince your colleagues across campus of the beauty of mathematics. But I would be surprised if any mathematician or math teacher would find many new topics contained within, and if they do they will find the treatment lacking for their own interests.
With that disclaimer disposed of, I can get on to the business at hand and discuss how the book would be received by its actual target audience. For anyone who is familiar with Burger and Starbird's previous book The Heart of Mathematics, it will not come as a surprise that this book is full of very lively and engaging explanations of a wide range of mathematics. That book (which was reviewed in "Read This!" several years ago) is a wonderful textbook designed for the kind of "Math For Liberal Arts Majors" courses which are popping up on campuses across the country. Chaos, Coincidences, and All That Jazz covers much of the same ground as that book, but it is written in an even more informal style (which means that, yes, there are fewer equations) and is in a format which will likely be less daunting for a civilian reader to just pick up and read than a textbook would be.
The book consists of four parts, each of which is subdivided into three chapters. The first part is on "Understanding Uncertainty" and covers topics related to chaos, coincidences, and statistics. (It is interesting to note that the Booklist review of this book says that the authors "home in on include the amazing similarities between John Kennedy and Abraham Lincoln", but I found the moral of that discussion to be that the "similarities" are not so amazing after all)
The second part, "Embracing Figures", deals with cryptography and patterns and has an especially nice section on "sizing up numbers" which deals with orders of magnitude and topics which should be a part of anybody's quantitative literacy. "Exploring Aesthetics" is the subject of the third part, which includes discussions of fractals and chaos and a nice introduction to the coffee cups and doughnuts of topology. There is also a nice section on Turing machines, and how they can be used to create certain kind of origami designs. They also discuss Möbius Bands and Klein Bottles, which lead nicely into the final section, which is entitled "Transcending Reality", and deals with the fourth dimension and various notions of infinity. This chapter also includes an extended discussion of what happens when the infinite baseball team (the St Louis Cardinals, naturally) check in to The Infinite Inn.
That is a large number of topics to cover in 288 pages, and doing a little division will tell you that many topics are treated extremely briefly. And that would probably be many readers' main criticism of the book: while it certainly gives a sampling platter of a large number of ideas throughout mathematics, it does not give you an entire meal of any of them, and before you are even done chewing one bite, the authors bring out the next topic. While I certainly understand, and to some extent agree with, this criticism, I do think that many readers will prefer their mathematics served this way, and it certainly will open the door for many of them to explore these ideas further. To this end, I wish that the authors had done a better job of pointing the interested reader to other books and sources; instead the bibliography only lists the authors' other collaborations.
Burger and Starbird take the subtitle of their book — "Making Light Out Of Weighty Matters" — quite seriously, and their exposition is filled with jokes and asides ranging from the corny to the extremely corny. I found the writing style to be fun, and I think that it would help bring in many readers who would be turned off by a more serious approach to exposition. On the other hand, I can imagine that many readers might find it annoying, or that some people who are reading the book to try to learn mathematics (as opposed to most MAA members who will be reading it to sample the exposition) might find it frustrating when the joke-to-math ratio approaches (and occassionaly even passes) 1:1. It is also worth noting that the book is filled with illustrations which help make the material come to life.
I do have a few other minor quibbles with the book, most of which stem from the brevity of the sections on the topics that I would like to last longer and the shortcuts that this brevity forces the authors to take. One example is in the section on public key cryptography: it seems to me that the authors conflate public key cryptography and the RSA encryption method (which is, to be fair, the most popularized method of public key cryptography) in a way that I think would leave many readers not realizing that there was a difference between the two or what public key cryptography is all about. Another issue is their discussion of the golden ratio, which is mathematically interesting, but perpetuates some of the myths about places that the golden ratio shows up in art and architecture which are debunked in (among other places) Mario Livio's book The Golden Ratio and George Markowsky's paper "Misconceptions About The Golden Ratio."
But, as I said above, these are just minor quibbles, and on the whole I think the authors succeed in their goal remarkably well: readers who are in the target audience will walk away from the book having learned a little bit about a lot of different mathematical topics. Hopefully, they also will have their appetites whetted for further — and deeper — learning and they will find some of the other popular math books populating their bookstore's shelves to satisfy this hunger. Most importantly, any reader of Burger and Starbird's book will realize that mathematics is a far more creative and exciting field than they may have gathered from their prior courses and experiences. On the flipside, they will probably also walk away thinking that mathematicians all have corny sense of humor, but even if that isn't true (and I'm not sure that it isn't), I think it is a small price to pay.
Darren Glass teaches at Gettysburg College.