*In Pursuit of the Unknown: 17 Equations That Changed the World*

**Ian Stewart**

Basic Books, 2013

360 pp., 19.99

*The Universe in Zero Words: The Story of Mathematics as Told through Equations*

**Dana Mackenzie**

Princeton University Press, 2013

224 pp., 19.95

## Karl Crisman

### Playing the Numbers

If you're a member of group that gets club oufits, it's typical to personalize the shirts or jackets with a nickname or in-joke. My students often use versions of their names (e.g., "Jules" for Julie), but the names can be more cryptic: a swimmer I knew in high school used "Plecostomus."[1] When I was on the student council, I chose "E=MC2" for the back of my sweatshirt, sure that this was the clearest way to show my identification as a future scientist.

To my chagrin, my choice came off as more puzzling than anything else. It's one thing to be nerdy, but for your nerdiness to not even be intelligible … that's the worst. I got so many questions that, finally, I just stopped wearing it. I had naively figured that most of my contemporaries would have at least *seen* Einstein's mass-energy equivalence; what about the Cold War, hadn't they heard of nuclear bombs? But this most famous equation was a mystery even to otherwise well-read friends.

The books under review would like to rectify this. The Universe in Zero Words and In Pursuit of the Unknown are books by mathematicians who promise to explain some of the secrets behind the most important equations. The first book, by science writer[2] Dana Mackenzie, leans more toward using equations as a way to tell interesting stories about mathematics, while veteran math popularizer Ian Stewart focuses a lot of time and attention on the background behind, and applications of, these formulas.

Naturally, although each book has around 20 equations as the starting point, the individual chapters range beyond that strictness, with lots of interesting anecdotes about the humanity of those involved as well. Both books have good morsels to offer to the educated "layman"; just who that ideal reader might be, we'll get to in a bit.

In Pursuit of the Unknown has particularly good bites to recommend it. Each chapter is really a self-standing essay, and I believe this is the best way to read this book. Want to learn about math connected to probability, astronomy, and eugenics? Read Chapter 7 on the normal distribution.[3] Want to know how your camera saves and stores its pictures and what this has to do with fingerprints? Read Chapter 9 on the Fourier Transform. On the other hand, if you want a (condensed) explanation of what Fermat's Last Theorem is about, or one of several possible approaches to chaos theory (lushly illustrated), then The Universe in Zero Words has it for you.

Big-name mathematical topics such as calculus and the Pythagorean Theorem get full chapters in both books; the most celebrated physics equations, like those of electromagnetism, relativity, and quantum theory, are also shared, though sometimes from quite different viewpoints. Most readers of this journal will be more comfortable with the level Stewart starts out with, and I recommend this book more when it comes to applications and a broader sense of how each topic fits into a cultural context. He doesn't just tell you that relativity makes your GPS work, he tells you *exactly* how useless a Newtonian GPS would be![4] Mackenzie does a very nice job explaining a variety of "pure" topics, like the quaternions, the prime number theorem, and the (Chern-)Gauss-Bonnet formula, for those whose background includes more than a semester or two of college mathematics; his "whale geometry" example is a wonderful way for someone of any background to think about why the shortest distance between two points *isn't* always a straight line. The books aren't perfect in editing—for example, Stewart repeats the long-debunked quote about IBM chair Thomas J. Watson thinking there would be a world demand for five computers, and Mackenzie's book has quite a few just-barely-relevant illustrations. But rather than desiring more such details, the reader at this point is more likely wondering whether he or she is in the target audience.

Indeed! Who *is* this semi-mythical reader who is neither mathematician nor physicist but still wants to see, as Mackenzie puts it, the masterpieces of figures like Einstein or Newton? This question is important to the authors; Mackenzie speaks of a "vast cultural gap," and Stewart grants an entire page to C. P. Snow's "Two Cultures" essay. And it *is* a shame that, in some circles, one can profess to not knowing "E=MC2" but not to being ignorant of Shakespeare.[5]

But I don't see these books as contributing to bridging that gap. Even though, particularly in Stewart's book, there is not a huge amount of mathematical background needed, I feel that the intended reader probably is already interested in learning more—perhaps like students I occasionally encounter who always loved math but didn't find room for it in their schedule, or the fellow I see on the train who sometimes asks for a tip for a math history book. In which case, the reader already has a multitude of general math and science books to choose from at his or her favorite (bricks-and-mortar or online) retailer, and ones which have the space to really tell a story in engrossing depth. As another reviewer put it, "I don't think we now have a surfeit of 'great equations' books, but we do have a sufficiency."

One "bridge" that might contribute to bringing readers a little closer to the point where they might want to take up Stewart or Mackenzie is Clifford Pickover's The Math Book. With just one glossy page (and opposing illustration) per fact, person, pretty picture, or formula, some topics are only brushed on; on the other hand, with 250 mathematical milestones, it's easy to pick another one if your first isn't intriguing. With well-chosen pictures[6] that invite discussion, Pickover's volume is a coffee-table book which (so my personal experience attests) people actually want to talk about, but it also offers some real depth.

Nine Algorithms that Changed the Future, by computer scientist John MacCormick, could contribute in a different direction. Readers averse to equations may nonetheless be interested in learning about the step-by-step instructions that have changed our society so radically in the last half-century. (Some commentators would argue that these algorithms are *more* important than the equations!) Want to know how Google works its magic, or how handwriting-recognition software does its job? It's here, and in a surprisingly accessible treatment—I really liked the color-mixing metaphor for public key exchange, for instance. A few chapters of this could provide a bridge to the bridge, so to speak.

Why should I care about people getting to the place where they might want to explore these or similar books? After all, my fellow mathematicians often deprecate such efforts. (As MIT's Gian-Carlo Rota put it, "Attempts have been made to string together beautiful mathematical results and to present them in books bearing … attractive titles …. Such anthologies are seldom found on a mathematician's bookshelf.")

A big answer is exemplified by one formula both books under review spend quite a bit of time[7] on—one you haven't heard of, but should have. The Black-Scholes(-Merton) equation is a second-order partial differential equation, possibly the most forbidding-looking one in either book—and well it should be. For both authors (correctly, in my view) lay the blame for the recent Great Recession largely on misuse of solutions to this Nobel-winning formula and those derived from it. These formulas give the correct price of financial derivatives … in a perfect market … with a certain statistically defined type of volatility … which doesn't obtain in a panic.

More books about equations can't prevent such abuses by themselves; there was a *lot* of money to be made, and some will always take the risk. But what if some of the higher echelons of AIG, Bear Stearns, *et al*. (or those crafting regulations) had asked themselves whether the quants really even knew what their equations were saying? As Stewart says, the financial system "desperately needs *more* mathematics, not less. But it also needs to learn how to use mathematics intelligently, rather than as a kind of magical talisman." Understanding how mathematics, science, and their equations lead to applications—and when they can go wrong—is something anyone in authority in our technology-saturated culture ought to seek out. These books, and many others like them, can help bring us closer to that point.

Karl-Dieter Crisman is associate professor of mathematics at Gordon College.

1. The generic name for a bottom-dwelling "sucker fish," often used in aquariums to remove algae. I'm not sure what he was trying to convey with that name.

2. Mackenzie's story, available online, of how he ended up in science writing after not receiving tenure is itself interesting (and sobering) reading.

3. To the experts: in what is surely an editorial slip, the distribution is at one point called a probability, which it isn't.

4. For those who can't wait to find out: the error grows by about 10 kilometers a day. See also http://xkcd.com/808/.

5. In many circles it is considered a badge of honor to be ignorant of both, and readers of this journal would do well to continue to productively engage those circles as well, but these books do not address that audience.

6. Often of beautiful fractals or somber mathematicians.

7. And which, in this case, Mackenzie has the more elementary treatment of.

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