How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

One thing the American defense establishment has traditionally understood very well is that countries don’t win wars just by being braver than the other side, or freer, or slightly preferred by God. The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost. That’s not the stuff war movies are made of, but it’s the stuff wars are made of. And there’s math every step of the way.

A mathematician is always asking, “What assumptions are you making? And are they justified?”

Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.

Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.

To paraphrase Clausewitz: Mathematics is the extension of common sense by other means.

This divides the mathematical universe into four quadrants:

The difference between the two pictures is the difference between linearity and nonlinearity, one of the central distinctions in mathematics.

A line is one kind of curve, but not the only kind, and lines enjoy all kinds of special properties that curves in general may not.

The highest point on a line segment—the maximum prosperity, in this example—has to be on one end or the other. That’s just how lines are. If lowering taxes is good for prosperity, then lowering taxes even more is even better.

A moment of reflection will tell you that the real curves of economics look like the second picture, not the first. They’re nonlinear. Mitchell’s reasoning is an example of false linearity—he’s assuming, without coming right out and saying so, that the course of prosperity is described by the line segment in the first picture, in which case Sweden stripping down its social infrastructure means we should do the same.

Some principle more complicated than “More government bad, less government good” is in effect.

Nonlinear thinking means which way you should go depends on where you already are.

Already in Roman times we find Horace’s famous remark “Est modus in rebus, sunt certi denique fines, quos ultra citraque nequit consistere rectum” (“There is a proper measure in things. There are, finally, certain boundaries short of and beyond which what is right cannot exist”).

in the Nicomachean Ethics, Aristotle observes that eating either too much or too little is troubling to the constitution. The optimum is somewhere in between, because the relation between eating and health i...

It’s called the Laffer curve, and it’s played a central role in Republican economics for almost forty years.

The legend of the Laffer curve goes like this: Arthur Laffer, then an economics professor at the University of Chicago, had dinner one night in 1974 with Dick Cheney, Donald Rumsfeld, and Wall Street Journal editor Jude Wanniski at an upscale hotel restaurant in Washington, DC. They were tussling over President Ford’s tax plan, and eventually, as intellectuals do when the tussling gets heavy, Laffer commandeered a napkin* and drew a picture.

(Aside: it’s important to point out here that people with out-of-the-mainstream ideas who compare themselves to Edison and Galileo are never actually right.

Greg Mankiw, an economist at Harvard and a Republican who chaired the Council of Economic Advisors under the second President Bush, writes in his microeconomics textbook: Subsequent history failed to confirm Laffer’s conjecture that lower tax rates would raise tax revenue. When Reagan cut taxes after he was elected, the result was less tax revenue, not more.

Friedman’s famous slogan on taxation is “I am in favor of cutting taxes under any circumstances and for any excuse, for any reason, whenever it’s possible.”

More moderate supply-side thinkers, like Mankiw, argue that lower taxes can increase the motivation to work hard and launch businesses, leading eventually to a bigger, stronger economy, even if the immediate effect of the tax cut is decreased government revenue and bigger deficits. An economist with more redistributionist sympathies would observe that this cuts both ways; maybe the government’s diminished ability to spend means it constructs less infrastructure, regulates fraud less stringently, and generally does less of the work that enables free enterprise to thrive.

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.

The Pythagoreans, you have to remember, were extremely weird. Their philosophy was a chunky stew of things we’d now call mathematics, things we’d now call religion, and things we’d now call mental illness. They believed that odd numbers were good and even numbers evil; that a planet identical to our own, the Antichthon, lay on the other side of the sun; and that it was wrong to eat beans, by some accounts because they were the repository of dead people’s souls. Pythagoras himself was said to have had the ability to talk to cattle (he told them not to eat beans) and to have been one of the very few ancient Greeks to wear pants.

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.

I will now teach you calculus. Ready? The idea, for which we have Isaac Newton to thank, is that there’s nothing special about a perfect circle. Every smooth curve, when you zoom in enough, looks just like a line.

That’s yet another Newtonian insight; objects in motion tend to proceed in a straight-line path, unless some other force intercedes to nudge the object one way or the other. That’s one reason linear thinking comes so naturally to us: our intuition about time and motion is formed by the phenomena we observe in the world.

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;

The stand-up comic Eugene Mirman tells this joke about statistics. He says he likes to tell people, “I read that 100% of Americans were Asian.” “But Eugene,” his confused companion protests, “you’re not Asian.” And the punch line, delivered with magnificent self-assurance: “I read that I was!”

Because not every curve is a line. But every curve, as we just learned from Newton, is pretty close to a line. That’s the idea that drives linear regression, the statistical technique that is to social science as the screwdriver is to home repair. It’s the one tool you’re pretty much definitely going to use, whatever the task.

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.

Not every curve is a line. And the curve of a missile’s flight is most emphatically not a line; it’s a parabola.

There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.

When doing any serious mathematical thinking, you’re going to have to multiply 6 by 8 sometimes, and if you have to reach for your calculator each time you do that, you’ll never achieve the kind of mental flow that actual thinking requires. You can’t write a sonnet if you have to look up the spelling of each word as you go.

Calculators are also useful tools that people worked hard to make—we should use them, too, when the situation demands! I don’t even care whether my students can divide 430 by 12 using long division—though I do care that their number sense is sufficiently developed to reckon mentally that the answer’s a little more than 35.

If we settle on a vision of mathematics that consists of “getting the answer right” and no more, and test for that, we run the risk of creating students who test very well but know no mathematics at all.

Of course it’s no better (in fact, it’s substantially worse) to pass along a population of students who’ve developed some wispy sense of mathematical meaning but can’t work examples swiftly and correctly. A math teacher’s least favorite thing to hear from a student is “I get the concept, but I couldn’t do the problems.” Though the student doesn’t know it, this is shorthand for “I don’t get the concept.”

William Carlos Williams put it crisply: no ideas but in things.

Nice work, black men! Not until 2095 will all of you be overweight. In 2048, only 80% of you will be. See the problem? If all Americans are supposed to be overweight in 2048, where are those one in five future black men without a weight problem supposed to be? Offshore? That basic contradiction goes unmentioned in the paper.

Anytime a lot of people in a small country come to a bad end, editorialists get out their slide rules and start figuring: how much is that in dead Americans?

Eventually (or perhaps immediately?) this reasoning starts to break down. When there are two men left in the bar at closing time, and one of them coldcocks the other, it is not equivalent in context to 150 million Americans getting simultaneously punched in the face.

Or: when 11% of the population of Rwanda was wiped out in 1994, all agree that it was among the worst crimes of the century. But we don’t describe the bloodshed there by saying, “In the context of 1940s Europe, it was nine times as bad as the Holocaust.” And to do so would set teeth rightly on edge.

An important rule of mathematical hygiene: when you’re field-testing a mathematical method, try computing the same thing several different ways. If you get several differe...

One can’t, of course, reject proportions entirely. Proportions matter! If you want to know which parts of America have the biggest brain cancer problem, it doesn’t make much sense to look at the states with the most deaths from brain cancer: those are California, Texas, New York, and Florida, which have the most brain cancer because they have the most people.

These days we don’t spare much grief for antique bloodlettings like the Thirty Years’ War. But that war took place in a smaller world, and by Pinker’s estimate killed one out of every hundred people on earth. To do that now would mean wiping out 70 million people, nearly as many as died in both world wars combined.

The five states at the top have something in common, and the five states at the bottom do, too. And it’s the same thing: hardly anyone lives there.

Something is pushing those numbers closer and closer to 50%. That something is the cold, strong hand of the Law of Large Numbers.

the more coins you flip, the more and more extravagantly unlikely it is that you’ll get 80% heads. In fact, if you flip enough coins, there’s only the barest chance of getting as many as 51%!

an informal form of the principle was asserted in the sixteenth century by Girolamo Cardano, though it was not until the early 1800s that Siméon-Denis Poisson came up with the pithy name “la loi des grands nombres” to describe it.

De Moivre’s observation is the same one that underlies the computation of the standard error in a political poll. If you want to make the error bar half as big, you need to survey four times as many people.

But de Moivre wasn’t done. He found that the discrepancies from 50-50, in the long run, always tend to form themselves into a perfect bell curve, or, as we call it in the biz, the normal distribution. (Statistics pioneer Francis Ysidro Edgeworth proposed that the curve be called the gendarme’s hat, and I have to say I’m sorry this didn’t catch on.)