I took a lot of math classes in college. I remember Professor Shlomo Sternberg getting up on the first day of his class and telling us we weren’t going to see any numbers other than 0, 1, and 2. I had a great time in that one.
Jordan Ellenberg, the author of How Not to Be Wrong: The Power of Mathematical Thinking, studied under Sternberg, but I think I’d like his book even if we didn’t have that in common. On the surface it’s about math, but it’s really about how much math plays into our daily lives without our even knowing it.
The book starts with a story about Abraham Wald, an Eastern European mathematician who worked for the American government during World War II. One day the military came to him and said, “We have a problem. We send our planes overseas, and when they come back, their engines are fine, but their tails are riddled with bullet holes. If we put more armor on the tails, though, the planes get too heavy to fly. Can you help us figure out how to protect the planes’ tails better?”
And he said, “No.”
They were surprised, but then he explained that they were asking the wrong question. “You need to put more armor where there aren’t bullet holes. Clearly, when the plane gets hit in the tail, it makes it back to you. Your problem is the planes that get hit in the engine, because those are the ones that aren’t coming back.”
Ultimately, that’s really what the book is: a series of stories about how a lot of the apparently non-mathematical systems that underpin our daily lives are actually deeply mathematical, and people couldn’t develop them until they started asking the right questions. Each chapter starts somewhere that seems fairly straightforward—electoral politics, say, or the Massachusetts lottery—and then uses that as a jumping-off point to talk about the math involved.
In some places the math gets quite complicated. Ellenberg deals with cutting-edge thinking about subjects like prime numbers, extra dimensions, and relative infinities. A non-mathematician might get a little lost along the way. But even if you don’t feel like following him all the way to the bottom of things like Fano planes, 24-dimensional spheres, and Condorcet’s paradox, after he goes really deep he always comes back to make sure you’re still with him.
The way he deals with the lottery is a great example. For several years, the Massachusetts lottery ran in a way that allowed three teams—one led by an MIT student, one by a medical researcher, and one by a guy from Michigan—to game the system and win millions of dollars. You might ask, How could the state let them cheat like that for so long? Part of the answer is, the state didn’t care. Massachusetts got 80 cents for every $2 lottery ticket sold, no matter who won. And the second part of the answer is, they weren’t cheating. They were taking advantage of math to give themselves slightly better odds at winning and other people slightly worse odds. They basically turned themselves into the house at a casino.
But Ellenberg extends his analysis even further, because while two of the teams just had the Quic Pic machine choose their numbers randomly, the team of students filled out its tickets by hand. Tens of thousands of tickets, every time they played! Ellenberg has mathematical explanations for the difference—filling out the tickets by hand exposed the students to less risk of losing money in any given week—and then points out that, if you’re on a student’s budget, the thought of losing any money at all is pretty scary.
Toward the end of each chapter, Ellenberg broadens from these specific examples to a series of questions about how else some of the ideas in the chapter might be used, what kinds of mathematical questions are left to answer, and what kinds of real-life problems they might eventually solve.
Given how black-and-white so much of our political dialogue has become, I think it’s great to have somebody advocating for looking at the numbers, explaining the relative costs of things like alternative tax policies or what happens when you implement different voting strategies. Even if you don’t follow the deepest math behind these things, you can still appreciate the argument and the rigor of the thinking, and the world can always use more rigorous thinking.
The writing is funny, smooth, and accessible—not what you might expect from a book about math. What Ellenberg has written is ultimately a love letter to math. If the stories he tells add up to a larger lesson, it’s that “to do mathematics is to be, at once, touched by fire and bound by reason”—and that there are ways in which we’re all doing math, all the time.