How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.

A mathematician is always asking, “What assumptions are you making? And are they justified?” This can be annoying.

Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength. Despite the power of mathematics, and despite its sometimes forbidding notation and abstraction, the actual mental work involved is little different from the way we think about more down-to-earth problems.

There’s nothing wrong with the Laffer curve—only with the uses people put it to. Wanniski and the politicians who followed his panpipe fell prey to the oldest false syllogism in the book: It could be the case that lowering taxes will increase government revenue; I want it to be the case that lowering taxes will increase government revenue; Therefore, it is the case that lowering taxes will increase government revenue.

A basic rule of mathematical life: if the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.

It does not occur to a modern mathematician that a collection of mathematical symbols should have a “meaning” until one has been assigned to it by definition.

Niels Henrik Abel, an early fan of Cauchy’s approach, wrote in 1828, “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.”

One of the great joys of mathematics is the incontrovertible feeling that you’ve understood something the right way, all the way down to the bottom; it’s a feeling I haven’t experienced in any other sphere of mental life.

You can do linear regression without thinking about whether the phenomenon you’re modeling is actually close to linear. But you shouldn’t. I said linear regression was like a screwdriver, and that’s true; but in another sense, it’s more like a table saw. If you use it without paying careful attention to what you’re doing, the results can be gruesome.

I think we have to teach a mathematics that values precise answers but also intelligent approximation, that demands the ability to deploy existing algorithms fluently but also the horse sense to work things out on the fly, that mixes rigidity with a sense of play. If we don’t, we’re not really teaching mathematics at all.

Measuring the absolute number of brain cancer deaths is biased toward the big states; but measuring the highest rates—or the lowest ones!—puts the smallest states in the lead. That’s how South Dakota can have one of the highest rates of brain cancer death while North Dakota claims one of the lowest. It’s not because Mount Rushmore or Wall Drug is somehow toxic to the brain; it’s because smaller populations are inherently more variable.

That’s how the Law of Large Numbers works: not by balancing out what’s already happened, but by diluting what’s already happened with new data, until the past is so proportionally negligible that it can safely be forgotten.

Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.

When you sink your savings into the incubated fund with the eye-popping returns, you’re like the newsletter getter who invests his life savings with the Baltimore stockbroker; you’ve been swayed by the impressive results, but you don’t know how many chances the broker had to get those results.

The universe is big, and if you’re sufficiently attuned to amazingly improbable occurrences, you’ll find them. Improbable things happen a lot.

In the British statistician R. A. Fisher’s famous formulation, “the ‘one chance in a million’ will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us.”

The more chances you give yourself to be surprised, the higher your threshold for surprise had better be. If a random Internet stranger who eliminated all North American grains from his food intake reports that he dropped fifteen pounds and his eczema went away, you shouldn’t take that as powerful evidence in favor of the maize-free plan. Somebody’s selling a book about that plan, and thousands of people bought that book and tried it, and the odds are very good that, by chance alone, one among them will experience some weight loss and clear skin the next week. And that’s the guy who’s going to log in as saygoodbye2corn452 and post his excited testimonial, while the people for whom the diet failed stay silent.

Mathematics has a funny relationship with the English language. Mathematical research articles, sometimes to the surprise of outsiders, are not predominantly composed of numerals and symbols; math is made of words. But the objects we refer to are often entities uncontemplated by the editors at Merriam-Webster. New things require new vocabulary. There are two ways to go. You can cut new words from fresh cloth, as we do when we speak of cohomology, syzygies, monodromy, and so on; this has the effect of making our work look forbidding and unapproachable.

I once had an uneasy moment with a colleague in an airport when he made the remark, unexceptional in a mathematical context, that it might be necessary to blow up the plane at one point.

The primes are the atoms of number theory, the basic indivisible entities of which all numbers are made. As such, they’ve been the object of intense study ever since number theory started.

And a lot of twin primes are exactly what number theorists expect to find, no matter how big the numbers get—not because we think there’s a deep, miraculous structure hidden in the primes, but precisely because we don’t think so. We expect the primes to be tossed around at random like dirt. If the twin primes conjecture were false, that would be a miracle, requiring that some hitherto unknown force was pushing the primes apart.

A low-powered study is only going to be able to see a pretty big effect. But sometimes you know that the effect, if it exists, is small. In other words, a study that accurately measures the effect of a gene is likely to be rejected as statistically insignificant, while any result that passes the p < .05 test is either a false positive or a true positive that massively overstates the gene’s effect. Low power is a special danger in fields where small studies are common and effect sizes are typically modest.

A statistically significant finding gives you a clue, suggesting a promising place to focus your research energy. The significance test is the detective, not the judge.

In the Bayesian framework, how much you believe something after you see the evidence depends not just on what the evidence shows, but on how much you believed it to begin with.

Relying purely on null hypothesis significance testing is a deeply non-Bayesian thing to do—strictly speaking, it asks us to treat the cancer drug and the plastic Stonehenge with exactly the same respect.

We don’t allow ourselves to consider every cockamamie theory we can logically devise. Our priors are not flat, but spiky. We assign a lot of mental weight to a few theories, while others, like the RBRRB theory, get assigned a probability almost indistinguishable from zero.

As much as I love numbers, I think people ought to stick to “I don’t believe in God,” or “I do believe in God,” or just “I’m not sure.” And as much as I love Bayesian inference, I think people are probably best off arriving at their faith, or discarding it, in a non-quantitative way. On this matter, math is silent.

“Professor O, I didn’t follow that last step. Why do those two operators commute?” The professor raises his eyebrows and says, “Eet ees obvious.” But the student persists: “I’m sorry, Professor O, I really don’t see it.” So Professor O goes back to the board and adds a few lines of explanation. “What we must do? Well, the two operators are both diagonalized by . . . well, it is not exactly diagonalized but . . . just a moment . . .” Professor O pauses for a little while, peering at what’s on the board and scratching his chin. Then he retreats to his office. About ten minutes go by. The students are about to start leaving when Professor O returns, and again assumes his station in front of the chalkboard. “Yes,” he says, satisfied. “Eet ees obvious.”

as usual, the mathematical approach is a formalized version of our natural mental reckonings, an extension of common sense by other means.

Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite.

Math, like meditation, puts you in direct contact with the universe, which is bigger than you, was here before you, and will be here after you.

In mathematics, we like rules, and we don’t like exceptions.

That is the characteristic of great scientists; they have courage. They will go forward under incredible circumstances; they think and continue to think.

There is real danger that, by strengthening our abilities to analyze some questions mathematically, we acquire a general confidence in our beliefs, which extends unjustifiably to those things we’re still wrong about. We become like those pious people who, over time, accumulate a sense of their own virtuousness so powerful as to make them believe the bad things they do are virtuous too.

“You might expect that mathematical skills are associated with magnificent popularity, but in my experience, the two are orthogonal.”

MR. FRIEDMAN: I think that issue is entirely orthogonal to the issue here because the Commonwealth is acknowledging— CHIEF JUSTICE ROBERTS: I’m sorry. Entirely what? MR. FRIEDMAN: Orthogonal. Right angle. Unrelated. Irrelevant. CHIEF JUSTICE ROBERTS: Oh. JUSTICE SCALIA: What was that adjective? I like that. MR. FRIEDMAN: Orthogonal. JUSTICE SCALIA: Orthogonal? MR. FRIEDMAN: Right, right. JUSTICE SCALIA: Ooh. (Laughter.)

Each voter has a perfectly rational, coherent political stance. But in the aggregate, their position is nonsensical.

For a mathematician, that “Huh, weird” feeling comes as an intellectual challenge. Can you say in some precise way what makes it weird?

I once met a historian of German culture in Columbus, Ohio, who told me that Hilbert’s predilection for wearing sandals with socks is the reason that fashion choice is still noticeably popular among mathematicians today. I could find no evidence this was actually true, but it suits me to believe it, and it gives a correct impression of the length of Hilbert’s shadow.

One of the most painful parts of teaching mathematics is seeing students damaged by the cult of the genius. The genius cult tells students it’s not worth doing mathematics unless you’re the best at mathematics, because those special few are the only ones whose contributions matter. We don’t treat any other subject that way! I’ve never heard a student say, “I like Hamlet, but I don’t really belong in AP English—that kid who sits in the front row knows all the plays, and he started reading Shakespeare when he was nine!” Athletes don’t quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was “ahead” of them.

I think we need more math majors who don’t become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators.

It’s easy to lose sight of the importance of work, because mathematical inspiration, when it finally does come, can feel effortless and instant.

Genius is a thing that happens, not a kind of person.

Göttingen balked at offering a position to the great algebraist Emmy Noether, arguing that students could not possibly be asked to learn mathematics from a woman, Hilbert responded: “I do not see how the sex of the candidate is an argument against her admission. We are a university, not a bathhouse.”

Math gives us a way of being unsure in a principled way: not just throwing up our hands and saying “huh,” but rather making a firm assertion: “I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.”

There is a young man in China named Lu Chao who learned and recited 67,890 digits of pi. That’s an impressive feat of memory. But is it interesting? No, because the digits of pi are not interesting. As far as anyone knows, they’re as good as random. Pi itself is interesting, to be sure. But pi is not its digits; it is merely specified by its digits, in the same way the Eiffel Tower is specified by the longitude and latitude 48.8586° N, 2.2942° E.

To do mathematics is to be, at once, touched by fire and bound by reason.