In "How Not to Be Wrong: The Power of Mathematical Thinking," Jordan Ellenberg writes about when it's a good idea to buy lottery tickets, why tall parents have shorter children, a dead fish in an MRI machine, and overperforming mutual funds. Because he's a mathematician at the University of Wisconsin, these come with handwritten graphs and equations, but his explanations are cultural, his references literary. Ellenberg is a contributor to Slate, and his book debuted on the extended bestseller list at the New York Times last week. He's currently on book tour and answered our questions by email.
It seems like the kind of math you practice at the University of Wisconsin (non-abelian Iwasawa theory! Galois representations!) is very different from the relatively accessible concepts you explain in "How Not to Be Wrong." I've been trying to think of a metaphor – an opera singer teaching nursery rhymes? How do you think of the project of explaining math to non-math-heads?
It would be a different world if there were a commercial market for non-abelian Iwasawa theory! But it's not quite like teaching nursery rhymes; maybe more like a composer teaching the basic idea of the scale and of chords. The ideas I talk about in the book — like linearity, expected value, correlation, formalism — are not "baby" ideas, they're really deep ideas that people worked very hard to create. But at the same time, they're ideas that, once they've been developed and articulated, can be explained quite simply.
Did you have an ideal reader in mind when you were writing, or early readers of the manuscript?
I asked Penguin this question before I even started writing. Who reads this kind of book? Who's going to be across the page from me here? And they gave me an answer which really pleased me, which is — don't worry about that. Your job is to write what you write, it's our job to figure out who's going to buy it. Very freeing!
So what I actually hope is that the book will work for a lot of different audiences. Fourteen-year-olds who are really interested in math (that was me, reading "Godel Escher Bach" over and over again when I was a kid.) College students, retired people with time to read and think, people on the plane, hard-charging businesspeople. Even people who know a lot about math already! There are a lot of stories in the book that I didn't know; that's what made it so fun to write, finding and spinning out those stories.
In writing about statistics, you show that new studies and reports we hear about — and sometimes that get quite a bit of attention — may include statistics that have been misinterpreted. Does the fault for that lie with researchers, reporters, or some combination of the two?
It's systemic, I think. Nobody really has an incentive not to make too much of a new study — not the journalist, not the researchers, not the companies or universities that employ the researcher. It takes a huge amount of willpower to say "No, New York Times, the headline you chose overstates the importance of my latest paper." Not that scientists typically have that level of oversight over science journalism. (Whether they should is not at all an obvious question.)
Is there a way for readers who are better armed with math and statistical insights to see through such reports?
There's no way but to go to the actual paper. The culture of science still has pretty strong norms, and in general by looking at the published paper you can see what was actually done. It's too much to ask people to read the technical guts of a paper in biology or economics — I certainly can't do that for the most part — but even just reading methods and results can be quite enlightening, and I do think learning how to do that is a reachable goal for most people. But it does take a little math.
You connect sometimes complex mathematics to real life situations, like how to think about a mutual fund's performance and when you should get to the airport. What are some of your more counterintuitive demonstrations?
One of my favorites is the example of Abraham Wald, who's working in a top-secret math lab in World War II, and the generals come to him and say, "The planes are coming back from Germany riddled with bullet holes, but there are more bullet holes on the fuselage, less on the engine — how much more armor should we put on the fuselage, where the bullet holes are? And Wald tells them, "No — you have to put the armor where the bullet holes AREN'T." Why? Because it's not that the Germans weren't hitting the planes on the engines; it's that the planes that got hit on the engines weren't coming back from Germany.
Some of your math I could understand, while other times I felt like I was following along with the formulas but would be lost without your explanations. Do you think readers can make sense of your arguments if the math is beyond them? (You can say no, my feelings won't be hurt).
See, now this is a good example of a wrong idea about math that's really popular! You apologize for needing the explanations in order to follow the formulas in some places. But math is about explanations! If you look at a math paper, you'll see it's mostly words. We make arguments, we explain, we attempt to persuade. We do not expect people to follow a long series of naked formulas. When people write papers that look like that we say they are bad at writing math.
That said, I think the book has hard parts and easy parts. I wanted to write a book that really gave people some work to do — if they wanted to — a book where some math actually takes place on the page, and isn't just admired from a distance. But I also tried to write it so that a reader who doesn't want to do that work, which is fine, is quickly brought back to easier waters.
Many of us stopped taking math in high school or college. How can the average person keep their math skills and instincts sharp?
Math is a special form of "thinking carefully." So I think people who keep themselves in a stance of "question assumptions" and "put pressure on arguments to see if they crack" and "when someone makes an assertion, ask what justifies it" are putting themselves in a position where they can learn or relearn mathematics effectively.
You seem to have a lot of fun parsing how mathematicians have evaluated the Torah and Shakespeare, and your book is filled with quotes from poets and philosophers. Do you think we've developed an artificial divide between math and the humanities?
They are different things — math doesn't feel the same as philosophy, or literary study, or literary production — but I do believe they can work together much more than they customarily do. One of the striking things about going back to these thinkers of previous centuries is that the philosophers and writers really didn't think they were doing something weird if they made a mathematical argument, and the mathematicians didn't think they were doing something weird when they thought about political or moral questions. I have a lot of friends who are poets. What they have in common with mathematicians is that the outside world sees them communicating in something other than standard English prose, and concludes they must be doing something really weird and abstract. But most poets and most mathematicians see what we do as very concrete, just looking at the world and pressing on it very hard and trying to figure out how it works. This is why I bring in William Carlos Williams, who was not talking about math when he said "No ideas but in things," but could have been.
Do you ever look at headline-grabbing studies, statistics, or other analyses and think, 'Wow, they got it entirely wrong!' Then what do you do?
Before there were blogs I just complained about it to my wife. Now I usually complain about it on my blog, or, if I think the world will actually care, pitch something to Slate or another magazine. Slate has always been particularly good on this kind of stuff, but I think it's a highly agreeable trend now, this habit of taking the "Newsflash: Here's what everybody's saying right now" article and deepening it to "Here's what everybody's saying and can we figure out why they're saying it, and whether that's actually a good reason?" I thought Jordan Weismann's recent Slate piece about the percentage of "grievously bad" teachers in California was a particularly nice example of the form.