How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

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. Revenue from personal income taxes (per person, adjusted for inflation) fell by 9 percent from 1980 to 1984, even though average income (per person, adjusted for inflation) grew by 4 percent over this period. Yet once the policy was in place, it was hard to reverse.

Lower Mississippi River was upward of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

Relationships are often only linear in the middle

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.

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.”

But the Obesity paper conceals a worse crime against mathematics and common sense. Linear regression is easy to do—and once you’ve done one, it’s cake to do more. So Wang and company broke down their data by ethnic group and sex. Black men, for instance, were less likely to be overweight than the average American; and, more important, their rate of overweight was growing only half as quickly. If we superimpose the proportion of overweight black men on the proportion of overweight Americans overall, together with the linear regressions Wang et al. worked out, we get a picture that looks like this. 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. It’s the epidemiological equivalent of saying there are −4 grams of water left in the bucket. Zero credit.

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.

The slogan to live by here is: Don’t talk about percentages of numbers when the numbers might be negative.

Once you’ve truly absorbed this fundamental truth, the Baltimore stockbroker has no power over you. That the stockbroker handed you ten straight good stock picks is very unlikely; that he handed somebody such a good run of picks, given ten thousand chances, is not even remotely surprising. 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.”

But a conventional boundary, obeyed long enough, can be easily mistaken for an actual thing in the world. Imagine if we talked about the state of the economy this way! Economists have a formal definition of a “recession,” which depends on arbitrary thresholds just as “statistical significance” does. One doesn’t say, “I don’t care about the unemployment rate, or housing starts, or the aggregate burden of student loans, or the federal deficit; if it’s not a recession, we’re not going to talk about it.” One would be nuts to say so.

If you do happen to find yourself partially believing a crazy theory, don’t worry—probably the evidence you encounter will be inconsistent with it, driving down your degree of belief in the craziness until your beliefs come into line with everyone else’s. Unless, that is, the crazy theory is designed to survive this winnowing process. That’s how conspiracy theories work.

This reminds me of an old story from the Harvard math department, concerning one of the grand old Russian professors, whom we shall call O. Professor O is midway through an intricate algebraic derivation when a student in the back row raises his hand. “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.”

Barbier’s proof reminds me of what the algebraic geometer Pierre Deligne wrote of his teacher, Alexander Grothendieck: “Nothing seems to happen, and yet at the end a highly nontrivial theorem is there.”

$31 million represents .004% of the benefits disbursed annually by the SSA. In other words, the agency is already extremely good at knowing who’s alive and who’s no more. Getting even better at that distinction, in order to eliminate those last few mistakes, might be expensive. If we’re going to count utils, we shouldn’t be asking, “Why are we wasting the taxpayers’ money?” but “What’s the right amount of the taxpayers’ money to be wasting?” To paraphrase Stigler: if your government isn’t wasteful, you’re spending too much time fighting government waste.

But the simplest explanation doesn’t require much theoretical heavy lifting at all. Simply: buying a lottery ticket, whether you win or not, is, in some small way, fun. Not Caribbean vacation fun, not all-night dance party fun, but one or two dollars’ worth of fun? Quite possibly so. There are reasons to doubt this explanation (for instance, lottery players themselves tend to cite the prospect of winning as their primary reason for playing), but it does an admirable job of explaining the behavior we see.

In 1976 and again in 2009, the U.S. government embarked on massive and expensive vaccination campaigns against the swine flu, having received warnings from epidemiologists each time that the currently prevailing strain was particularly likely to go catastrophically pandemic. In fact, both flus, while severe, fell well short of disastrous. It’s easy to criticize the policy makers in these scenarios for letting their decision making get ahead of the science. But it’s not that simple. It’s not always wrong to be wrong.

And look at it now in Sept 2020

That’s the familiar self-contradicting position we see in polls: We want to cut! But we also want each program to keep all its funding! How did we get to this impasse? Not because the voters are stupid or delusional. Each voter has a perfectly rational, coherent political stance. But in the aggregate, their position is nonsensical.

“I pledge not to raise taxes a single cent. I will give municipalities the tools they need to deliver top-quality public services at less cost to the taxpayers.” Now each locality, supplied with less revenue by the state, has to decide on its own between the remaining two options: cut roads or cut schools. See the genius here? The governor has specifically excluded raising taxes, the most popular of the three options, yet his firm stand has majority support: 59% of voters agree with the governor that taxes shouldn’t rise. Pity the mayor or county executive who has to swing the axe. That poor sap has no choice but to execute a policy most voters won’t like, and suffers the consequence while the governor sits pretty. In the budget game, as in so many others, playing first can be a big advantage.

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.

The popular image of the lone (and possibly slightly mad) genius—who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts—is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics.

For example, Gödel, whose theorem ruled out the possibility of definitively banishing contradiction from arithmetic, was also worried about the Constitution, which he was studying in preparation for his 1948 U.S. citizenship test. In his view, the document contained a contradiction that could allow a Fascist dictatorship to take over the country in a perfectly constitutional manner. Gödel’s friends Albert Einstein and Oskar Morgenstern begged him to avoid this matter in his exam, but, as Morgenstern recalls it, the conversation ended up going like this: The examiner: Now, Mr. Gödel, where do you come from? Gödel: Where I come from? Austria. The examiner: What kind of government did you have in Austria? Gödel: It was a republic, but the constitution was such that it finally was changed into a dictatorship. The examiner: Oh! This is very bad. This could not happen in this country. Gödel: Oh, yes, I can prove it.

Hilbert’s commitment to logical principle and deduction often led him, like Condorcet, to adopt a surprisingly modern outlook in non-mathematical matters.* At some political cost to himself, he refused to sign the 1914 Declaration to the Cultural World, which defended the kaiser’s war in Europe with a long list of denials, each one starting “It is not true”: “It is not true that Germany violated the neutrality of Belgium,” and so on. Many of the greatest German scientists, like Felix Klein, Wilhelm Roentgen, and Max Planck, signed the declaration. Hilbert said, quite simply, that he was unable to verify to his exacting standards that the assertions in question were not true. A year later, when the faculty at 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.”

Logic can make us aware of our inner biases

In 1785 he wrote, rather forlornly, “We cannot usually avoid being presented with decisions of this kind, which we might call equivocal, except by requiring a large plurality or allowing only very enlightened men to vote. . . . If we cannot find voters who are sufficiently enlightened, we must avoid making a bad choice by accepting as candidates only those men in whose competence we can trust.”

Each one of these little local uncertainties percolated through the simulation, and the uncertainties about different parameters of the model fed back into each other, and by 2050, the noise had engulfed the signal. I could make the simulation come out any which way. Maybe there was going to be no such thing as tuberculosis in 2050, or maybe most of the world’s population would be infected. I had no principled way to choose. This was not what the researcher wanted to hear. It was not what he was paying me for. He was paying me for a number, and he patiently repeated his request for one. I know there’s uncertainty, he said, that’s how medical research is, I get that, just give me your best guess. It didn’t matter how much I protested that any single guess would be worse than no guess at all. He insisted. And he was my boss, and eventually I gave in. I have no doubt he told many people, afterward, that X million people were going to have tuberculosis in the year 2050. And I’ll bet if anyone asked him how he knew this, he would say, I hired a guy who did the math.

And yet—when Roosevelt says, “The closet philosopher, the refined and cultured individual who from his library tells how men ought to be governed under ideal conditions, is of no use in actual governmental work,” I think of Condorcet, who spent his time in the library doing just that, and who contributed more to the French state than most of his time’s more practical men. And when Roosevelt sneers at the cold and timid souls who sit on the sidelines and second-guess the warriors, I come back to Abraham Wald, who as far as I know went his whole life without lifting a weapon in anger, but who nonetheless played a serious part in the American war effort, precisely by counseling the doers of deeds how to do them better. He was unsweaty, undusty, and unbloody, but he was right. He was a critic who counted.