Book Reviews - Spring 2015

A Taste for Higher Math

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The numbers that count

Dennis Wilkinson/Flickr

By Natalie Angier

March 4, 2015


 

How to Bake π: An Edible Exploration of the Mathematics of Mathematics, by Eugenia Cheng, Basic Books, 304 pp., $27.50

A few years ago, I wrote a big feature for The New York Times on the evolution of cursing. Of course, the Times being a proudly fastidious family newspaper, I couldn’t quote any foul language directly, but at least I had Shakespeare and his love of a good, archaic “fack.”

Covering mathematics presents a similar challenge. Sure, publishers say, go ahead and write a popular book about math—so long as you can do it without using math. Most readers are thought to be liberal-arts innumerates who never recovered from high school algebra, and the inclusion of even a single equation is said to cut a book’s potential market in half. Math is a four-letter word that must be euphemized, disguised, or sugar-frosted with fack.

Or must it? In her slyly illuminating dispatch on the deep meaning of mathematics, Eugenia Cheng certainly dons a pastry chef’s toque, and I’m not being metaphorical. Every chapter of How to Bake π—that is, pi—begins with one of the author’s recipes, which all sound delicious and bakeable. Yet they are not mere cheesecake camouflage for the book’s meatier mathematical fare. Instead, each one elegantly encapsulates a key concept of the chapter it prefaces, underscoring how readily life and math intersect.  For example, to emphasize the importance of structure in mathematics, the author leads with a recipe for Baked Alaska, a confection that succeeds or flops by the integrity of its form. If the stiff meringue on top and the sponge cake on the bottom aren’t joined into a perfect insulating seal around the ice cream in the middle, you know what will happen once the dish hits the oven. Similarly, Cheng writes, “A lot of math is about structures. … What is holding something up? Which parts could you remove without making the whole thing fall down?”

A lecturer in mathematics at the universities of Sheffield and Chicago, Cheng asks us to think about prime numbers, which mathematicians love to do, and after reading this book, so do I. A prime is any whole number that can be divided only by 1 and itself. The number 1 doesn’t count, but 2 does—followed by 3, 5, 7, 11, and onward to infinity. Prime numbers have many admirable properties, Cheng says, particularly their essential role as the “building blocks” of numberhood. It turns out that every whole number is the product of a unique set of prime numbers. To get 6, you multiply 2 times 3. For 16, the singular prime recipe is 2 times 2 times 2 times 2. To bake 62, you multiply 31 and 2.

Coming back to the question of structure, Cheng asks us to imagine a world where the number 2 doesn’t exist. What, then, becomes of the prime numbers? No more 2 to start the game, 3 remains prime, but what about 4? It can’t be divided by 2 because 2 doesn’t exist, so it’s elevated to a prime position. The same goes for 6, 8, and 10—they’re all primes now. What’s more, you can scratch the old rule that every number can be expressed as a product of primes in just one way: 24, for example, which before was primally, exclusively defined as 3×2×2×2, now can be framed as the product of two sets of nouvelle primes: 3 and 8, and 6 and 4. “So by throwing away the number 2 we have destroyed one of the fundamental principles of numbers,” Cheng writes. And our poor Baked Alaska runneth all over the oven floor.

Strictly speaking, How to Bake π is about a relatively new branch of mathematics called category theory, which the author describes as “the mathematics of mathematics,” and the relationships between different types of math. But because category theory is so encompassing, so meta, the book ends up being about the nature of math itself. Many people believe that math is “hard,” Cheng says, and that she, as a mathematician, must be some sort of genius. “This is one of the great myths of mathematics,” she writes, and she’s devoted to “exploding it.” Life is hard: it’s full of ambiguities and subtleties and “things that are not attainable by logical thought processes.” By contrast, she insists, “mathematics is easy.” In fact, it is designed to make life much easier than it is, a method for making “precise statements about concepts” and idealizing those concepts so that they can be applied to a wide range of problems, and so that diverse problems can be compared with each other. People get unnerved by the constant abstractions of math. They’re fine with a tincture of generalization, statements like, “If A is divisible by 6, and 6 is divisible by 2, then A is divisible by 2.” But put it into a form of “If A is divisible by B, and B is divisible by C, then A is divisible by C,” and it becomes “a step of abstraction too far for some people.” Yet the abstraction is crucial to math’s power. With A, B, and C standing in for any numbers, “we can now understand something more broadly true about numbers, not just about the particular numbers 6 and 2.”

We can also have fun testing the veracity of our so-called transitivity construct in a wide array of circumstances.  “If A is older than B, and B is older than C, does that mean A is older than C?” Yes, that works. “If A is the mother of B, and B is the mother of C, does that mean A is the mother of C?” No, she’s the grandmother of C, or maybe step-grandmother.

Cheng also explains that although math is a vital tool of scientific research, doing math is very different from doing science. Scientists form hypotheses and gather evidence that either supports or disproves the hypothesis. Mathematicians make their argument using logic. “Not experiments, not physical evidence, not blind faith,” she writes. “Just logic.”  She takes the reader step by step through examples of abstraction in action. Here’s a good one: My father is three times as old as I am now but in 10 years’ time he will be twice as old as me. How old am I? Well, you start by abstracting Dad’s and child’s current ages as y=3x. In 10 years, you’ll have y+10=2(x+10). With two equations like that, you can miraculously get rid of the y in the second equation by substituting 3x, write it out as 3x+10=2(x +10), and then solve for x. Using some multiplication and subtraction, which Cheng unobtrusively lays out, you find the child is 10 years old. In sum, she says, we began with a “real-life” problem expressed in words, abstracted the problem into elements that could be manipulated with logic, and then retranslated the abstraction back to real life.

Cheng manages to do for us what the mathematician Keith Devlin has said mathematicians do for themselves: she compels us to see numbers and symbols as vivid characters in an ongoing drama, a narrative in which we are alternately observers and participants. Sure, life is hard, but stories are easy, especially when they end with the perfect dessert.


Natalie Angier is a science columnist for The New York Times and the author of Woman, The Canon, and other books.


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