How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

Hotelling was totally devoted to research and the generation of knowledge, and in Secrist he may have seen something of a kindred soul. “The labor of compilation and of direct collection of data,” he wrote sympathetically, “must have been gigantic.” Then the hammer drops. The triumph of mediocrity observed by Secrist, Hotelling points out, is more or less automatic whenever we study a variable that’s affected by both stable factors and the influence of chance. Secrist’s hundreds of tables and graphs “prove nothing more than that the ratios in question have a tendency to wander about.” The result of Secrist’s exhaustive investigation is “mathematically obvious from general considerations, and does not need the vast accumulation of data adduced to prove it.” Hotelling drives his point home with a single, decisive observation. Secrist believed the regression to mediocrity resulted from the corrosive effect of competitive forces over time; competition was what caused the top stores in 1916 to be hardly above average in 1922. But what happens if you select the stores with the highest performance in 1922? As in Galton’s analysis, these stores are likely to have been both lucky and good. If you turn back the clock to 1916, whatever intrinsic good management they possess should still be in force, but their luck may be totally different. Those stores will typically be closer to mediocre in 1916 than in 1922. In other words, if regression to the mean were, as Secrist thought, the na...

Biologists are eager to think regression stems from biology, management theorists like Secrist want it to come from competition, literary critics ascribe it to creative exhaustion—but it is none of these. It is mathematics.

Most important, Scared Straight worked. One representative program, in New Orleans, reported that participants were arrested less than half as often after Scared Straight as before. Except it didn’t work. The juvenile offenders are like Secrist’s low-performing stores: selected, not at random, but by virtue of being the worst of their kind. Regression tells you that the very worst-behaved kids this year will likely still be behavior problems next year; but not as much so. The decline in arrest rate is just what you’d expect even if Scared Straight had no effect. Which isn’t to say Scared Straight was completely ineffective. When the program was put through randomized trials, where a randomly selected subgroup of juvenile offenders were put through Scared Straight and then compared to the remaining kids, who didn’t participate, researchers found that the program actually increased antisocial behavior. Maybe it should have been called Scared Stupid.

High eccentricity means that heredity is powerful and regression to the mean is weak; low eccentricity means the opposite, that regression to the mean holds sway. Galton called his measure correlation, the term we still use today. If Galton’s ellipse is almost round, the correlation is near 0; when the ellipse is skinny, lined up along the northeast-southwest axis, the correlation comes close to 1. By means of the eccentricity—a geometric quantity at least as old as the work of Apollonius of Perga in the third century BCE—Galton had found a way to measure the association between two variables, and in so doing had solved a problem at the cutting edge of nineteenth-century biology: the quantification of heredity.

The answer lies in a fundamental property of mathematics—in a sense, the very property that has made mathematics so magnificently useful to scientists. In math there are many, many complicated objects, but only a few simple ones. So if you have a problem whose solution admits a simple mathematical description, there are only a few possibilities for the solution. The simplest mathematical entities are thus ubiquitous, forced into multiple duty as solutions to all kinds of scientific problems. The simplest curves are lines. And it’s clear that lines are everywhere in nature, from the edges of crystals to the paths of moving bodies in the absence of force. The next simplest curves are those cut out by quadratic equations,* in which no more than

two variables are ever multiplied together. So squaring a variable, or multiplying two different variables, is allowed, but cubing a variable, or multiplying one variable by the square of another, is strictly forbidden. Curves in this class, including ellipses, are still called conic sections out of deference to history; but more forward-looking algebraic geometers call them quadrics.* Now there are lots of quadratic equations: any such is of the form A x2 + B xy + C y2 + D x + E y + F= 0 for some values of the six constants A, B, C, D, E, and F. (The reader who feels so inclined can check that no other type of algebraic expression is allowed, subject to our requirement that we are only allowed to multiply two variables together, never three.) That seems like a lot of choices—infinitely many, in fact! But these quadrics turn out to fall into three main classes: ellipses, parabolas, and hyperbolas.*

Bertillon was appalled by the unsystematic and haphazard way in which French police identified criminal suspects. How much better and more modern it would be, Bertillon reasoned, to attach to each miscreant Frenchman a series of numerical measurements: the length and breadth of the head, the length of fingers and feet, and so on. In Bertillon’s system, each arrested suspect was measured and his data filed on cards and stored away for future use. Now, if the same man were nabbed again, identifying him was a simple matter of getting out the calipers, taking his numbers, and comparing them with the cards on file. “Aha, Mr. 15-6-56-42, thought you’d get away, didn’t you?” You can replace your name by an alias, but there’s no alias for the shape of your head. Bertillon’s system, so in keeping with the analytic spirit of the time, was adopted by the Paris Prefecture of Police in 1883, and quickly spread throughout the world. At its height, bertillonage held sway in police departments from Bucharest to Buenos Aires. “The Bertillon cabinet,” Raymond Fosdick wrote in 1915, “became the distinguishing mark of the modern police organization.” In its time, the practice was so common and uncontroversial in the United States that Justice Anthony Kennedy brought it up in his majority opinion in the 2013 case Maryland v. King,

If one pixel is bright green, the next one over is likely to be as well. The actual information contained in the image is much less than 4 million numbers’ worth—and it’s precisely this fact that makes it possible* to have compression, the critical mathematical technology that allows images, videos, music, and text to be stored in much smaller spaces than you’d think. The presence of correlation makes compression possible; actually doing it involves much more modern ideas, like the theory of wavelets developed in the 1970s and ’80s by Jean Morlet, Stéphane Mallat, Yves Meyer, Ingrid Daubechies, and others; and the rapidly developing area of compressed sensing, which started with a 2005 paper by Emmanuel Candès, Justin Romberg, and Terry Tao, and has quickly become its own active subfield of applied math.

And this is Pearson’s formula, in geometric language. The correlation between the two variables is determined by the angle between the two vectors. If you want to get all trigonometric about it, the correlation is the cosine of the angle. It doesn’t matter if you remember what cosine means; you just need to know that the cosine of an angle is 1 when the angle is 0 (i.e., when the two vectors are pointing in the same direction) and −1 when the angle is 180 degrees (vectors pointing in opposite directions). Two variables are positively correlated when the corresponding vectors are separated by an acute angle—that is, an angle smaller than 90 degrees—and negatively correlated when the angle between the vectors is larger than 90 degrees, or obtuse. It makes sense: vectors at an acute angle to each other are, in some loose sense, “pointed in the same direction,” while vectors that form an obtuse angle seem to be working at cross purposes. When the angle is a right angle, neither acute nor obtuse, the two variables have a correlation of zero; they are, at least as far as correlation goes, unrelated to each other. In geometry, we call a pair of vectors that form a right angle perpendicular, or orthogonal. And by extension, it’s common practice among mathematicians and other trig aficionados to use the word “orthogonal” to refer to something unrelated to the issue at hand—“You might expect that mathematical skills are associated with magnificent popularity, but in my experience, t...

If correlation were transitive, medical research would be a lot easier than it actually is. Decades of observation and data collection have given us lots of known correlations to work with. If we had transitivity, doctors could just chain these together into reliable interventions. We know that women’s estrogen levels are correlated with lower risk of heart disease, and we know that hormone replacement therapy can raise those levels, so you might expect hormone replacement therapy to be protective against heart disease. And, indeed, that used to be conventional clinical wisdom. But the truth, as you’ve probably heard, is a lot more complicated.

Doll and Hill’s data showed that lung cancer and smoking were correlated; their relation was not one of strict determination (some heavy smokers don’t get lung cancer, while some nonsmokers do), but neither were the two phenomena independent. Their relation lay in that fuzzy, intermediate zone that Galton and Pearson had been the first to map. The mere assertion of correlation is very different from an explanation. Doll and Hill’s study doesn’t show that smoking causes cancer; as they write, “The association would occur if carcinoma of the lung caused people to smoke or if both attributes were end-effects of a common cause.” That lung cancer causes smoking, as they point out, is not very reasonable; a tumor can’t go back in time and give someone a pack-a-day habit. But the problem of the common cause is more troubling. Our old friend R. A. Fisher, the founding hero of modern statistics, was a vigorous skeptic of the tobacco-cancer link on exactly those grounds. Fisher was the natural intellectual heir to Galton and Pearson; in fact, he succeeded Pearson in 1933 as the Galton Chair of Eugenics at University College, London. (In deference to modern sensibilities, the position is now called the Galton Chair of Genetics.)

Berkson, like Fisher, was more apt to believe the “constitutional hypothesis,” that some preexisting difference between nonsmokers and smokers accounted for the relative healthiness of the abstainers: If 85 to 95 per cent of a population are smokers, then the small minority who are not smokers would appear, on the face of it, to be of some special type of constitution. It is not implausible that they should be on the average relatively longevous, and this implies that death rates generally in this segment of the population will be relatively low. After all, the small group of persons who successfully resist the incessantly applied blandishments and reflex conditioning of the cigaret advertisers are a hardy lot, and, if they can withstand these assaults, they should have relatively little difficulty in fending off tuberculosis or even cancer!

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. When you dig past the front-line numbers of the budget polls, you see that the word problem isn’t so far from the truth. Only 47% of Americans believed balancing the budget would require cutting programs that helped people like them. Just 38% agreed that there were worthwhile programs that would need to be cut. In other words: the infantile “average American,” who wants to cut spending but demands to keep every single program, doesn’t exist. The average American thinks there are plenty of non-worthwhile federal programs that are wasting our money and is ready and willing to put them on the chopping block to make ends meet. The problem is, there’s no consensus on which programs are the worthless ones. In large part, that’s because most Americans think the programs that benefit them personally are the ones that must, at all costs, be preserved. (I didn’t say we weren’t selfish, I just said we weren’t stupid!) The “majority rules” system is simple and elegant and feels fair, but it’s at its best when deciding between just two options. Any more than two, and contradictions start to seep into the majority’s preferences.

52% of respondents said they opposed the law, while only 41% supported it. Bad news for Obama? Not once you break down the numbers. Outright repeal of health care reform was favored by 37%, with another 10% saying the law should be weakened; but 15% preferred to leave it as is, and 36% said the ACA should be expanded to change the current health care system more than it currently does. That suggests that many of the law’s opponents are to Obama’s left, not his right. There are (at least) three choices here: leave the health care law alone, kill it, or make it stronger. And each of the three choices is opposed by most Americans.* The incoherence of the majority creates plentiful opportunities to mislead. Here’s how Fox News might report the poll results above: Majority of Americans oppose Obamacare! And this is how it might look on MSNBC: Majority of Americans want to preserve or strengthen Obamacare! These two headlines tell very different stories about public opinion. Annoyingly enough, both are true.

Livermore’s nightmare came true; we do not now cut people’s ears off, even if they were totally asking for it, and what’s more, we hold that the Constitution forbids us from doing so. Eighth Amendment jurisprudence is now governed by the principle of “evolving standards of decency,” first articulated by the Court in Trop v. Dulles (1958), which holds that contemporary American norms, not the prevailing standards of August 1789, provide the standard of what is cruel and what unusual. That’s where public opinion comes in. In Penry, Justice Sandra Day O’Connor’s opinion held that opinion polls showing overwhelming public opposition to execution of mentally deficient criminals were not to be considered in the computation of “standards of decency.” To be considered by the court, public opinion would need to be codified by state lawmakers into legislation, which represented “the clearest and most reliable objective evidence of contemporary values.” In 1989, only two states, Georgia and Maryland, had made special provisions to prohibit execution of the mentally retarded. By 2002, the situation had changed, with such executions outlawed in many states; even the state legislature of Texas had passed such a law, though it was blocked from enactment by the governor’s veto.

Imagine, they ask, a scenario in which forty-seven state legislatures have outlawed capital punishment, but two of the three nonconforming states allow execution of mentally retarded convicts. In this case, it’s hard to deny that the national standard of decency excludes the death penalty in general, and the death penalty for the mentally retarded even more so. To conclude otherwise concedes an awful lot of moral authority to the three states out of step with the national mood. The right fraction to consider here is 48 out of 50, not 1 out of 3. In real life, though, there is plainly no national consensus against the death penalty itself. This confers a certain appeal to Scalia’s argument. It’s the twelve states that forbid the death penalty* that are out of step with general national opinion in favor of capital punishment; if they don’t think executions should be allowed at all, how can they be said to have an opinion about which executions are permissible?

The mathematical buzzword in play here is “independence of irrelevant alternatives.” That’s a rule that says, whether you’re a slime mold, a human being, or a democratic nation, if you have a choice between two options, A and B, the presence of a third option, C, shouldn’t affect which of A and B you like better. If you’re deciding whether you’d rather have a Prius or a Hummer, it doesn’t matter whether you also have the option of a Ford Pinto. You know you’re not going to choose the Pinto. So what relevance could it have? Or, to keep it closer to politics: in place of an auto dealership, put the state of Florida. In place of the Prius, put Al Gore. In place of the Hummer, put George W. Bush. And in place of the Ford Pinto, put Ralph Nader. In the 2000 presidential election, George Bush got 48.85% of Florida’s votes and Al Gore got 48.84%. The Pinto got 1.6%.

Adam is in the 81st percentile of attractiveness, the 51st percentile of dependability, and the 65th percentile of intelligence, while Bill is in the 61st percentile of attractiveness, 51st of dependability, and 87th of intelligence. The college students, like the slime mold before them, faced a tough choice. And just like the slime mold, they went 50-50, half the group preferring each potential date. But things changed when Chris came into the picture. He was in the 81st percentile of attractiveness and 51st percentile of dependability, just like Adam, but in only the 54th percentile of intelligence. Chris was the irrelevant alternative, an option that was plainly worse than one of the choices already on offer. You can guess what happened. The presence of a slightly dumber version of Adam made the real Adam look better; given the choice between dating Adam, Bill, and Chris, almost two-thirds of the women chose Adam.

So if you’re a single guy looking for love, and you’re deciding which friend to bring out on the town with you, choose the one who’s pretty much exactly like you—only slightly less desirable.

The political virtue Condorcet did possess was a passionate, never-wavering belief in reason, and especially mathematics, as an organizing principle of human affairs. His allegiance to reason was standard stuff for the Enlightenment thinkers, but his further belief that the social and moral world could be analyzed by equations and formulas was novel. He was the first social scientist in the modern sense. (Condorcet’s term was “social mathematics.”) Condorcet, born into the aristocracy, quickly came to the view that universal laws of thought should take precedence over the whims of kings. He agreed with Rousseau’s claim that the “general will” of the people should hold sway on governments but was not, like Rousseau, content to accept this claim as a self-evident principle. For Condorcet, the rule of the majority needed a mathematical justification, and he found one in the theory of probability. Condorcet lays out his theory in his 1785 treatise Essay on the Application of Analysis to the Probability of Majority Decisions. A simple version: suppose a seven-person jury has to decide a defendant’s guilt. Four say the defendant is guilty, and only three believe he’s innocent. Let’s say each of these citizens has a 51% chance of holding the correct view. In that case, you might expect a 4−3 majority in the correct direction to be more likely than a 4−3 majority favoring the incorrect choice.

It’s a little like the World Series. If the Phillies and the Tigers are facing off, and we agree that the Phillies are a bit better than the Tigers—say, they have a 51% chance of winning each game—then the Phillies are more likely to win the Series 4−3 than to lose by the same margin. If the World Series were best of fifteen instead of best of seven, Philadelphia’s advantage would be even greater. Condorcet’s so-called “jury theorem” shows that a sufficiently large jury is very likely to arrive at the right outcome, as long as the jurors have some individual bias toward correctness, no matter how small.* If the majority of people believe something, Condorcet said, that must be taken as strong evidence that it is correct. We are mathematically justified in trusting a sufficiently large majority—even when it contradicts our own preexisting beliefs. “I must act not by what I think reasonable,” Condorcet wrote, “but by what all who, like me, have abstracted from their own opinion must regard as conforming to reason and truth.” The role of the jury is much like the role of the audience on Who Wants to Be a Millionaire? When we have the chance to query a collective, Condorcet thought, even a collective of unknown and unqualified peers, we ought to value their majority opinion above our own.

Condorcet intended to build a mathematical theory of voting from his axiom, just as Euclid had built an entire theory of geometry on his five axioms about the behavior of points, lines, and circles: There is a line joining any two points. Any line segment can be extended to a line segment of any desired length. For every line segment L, there is a circle that has L as a radius. All right angles are congruent to each other. If P is a point and L is a line not passing through P, there is exactly one line through P parallel to L.

Since 2008, umpires have been allowed to consult video replay when they’re unsure of what actually took place on the field. This is good for getting calls right instead of wrong, but many longtime baseball fans feel it’s somehow foreign to the spirit of the sport. I’m one of them. I’ll bet John Roberts is too.

Legal realists, like judge and University of Chicago professor Richard Posner, argue that Supreme Court jurisprudence is never the exercise in formal rule following that Scalia says it is: Most of the cases the Supreme Court agrees to decide are toss-ups, in the sense that they cannot be decided by conventional legal reasoning, with its heavy reliance on constitutional and statutory language and previous decisions. If they could be decided by those essentially semantic methods, they would be resolved uncontroversially at the level of a state supreme court or federal court of appeals and never get reviewed by the Supreme Court.

first theorem I ever proved; I was in college, working on my senior thesis, and I was completely stuck. One night I was at an editorial meeting of the campus literary magazine, drinking red wine and participating fitfully in the discussion of a somewhat boring short story, when all at once something turned over in my mind and I understood how to get past the block. No details, but it didn’t matter; there was no doubt in my mind that the thing was done. That’s the way mathematical creation often presents itself. Here’s the French mathematician Henri Poincaré’s famous account of a geometric breakthrough he made in 1881: Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’s sake I verified the result at my leisure.* But it didn’t really happen in the space of a footstep, Poincaré explains. That moment of inspiration is the product of weeks of work, both conscious and unconscious, which somehow prepare the mind to make the necessary connection of ideas. Sitting around waiting for inspi...

Condorcet, too, held fast to his formalist ideas about politics even when they didn’t conform well to reality. The existence of Condorcet cycles meant that any voting system that obeyed his basic, seemingly inarguable axiom—when the majority prefers A to B, B cannot be the winner—can fall prey to self-contradiction. Condorcet spent much of the last decade of his life grappling with the problem of the cycles, developing more and more intricate voting systems intended to evade the problem of collective inconsistency. He never succeeded.

As F. Scott Fitzgerald said, “The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.” Mathematicians use this ability as a basic tool of thought. It’s essential for the reductio ad absurdum, which requires you to hold in your mind a proposition you believe to be false and reason as if you think it’s true: suppose the square root of 2 is a rational number, even though I’m trying to prove it’s not. . . . It is lucid dreaming of a very systematic kind. And we can do it without short-circuiting ourselves. In fact, it’s a common piece of folk advice—I know I heard it from my PhD adviser, and presumably he from his, etc.—that when you’re working hard on a theorem you should try to prove it by day and disprove it by night. (The precise frequency of the toggle isn’t critical; it’s said of the topologist R. H. Bing that his habit was to split each month between two weeks trying to prove the Poincaré Conjecture and two weeks trying to find a counterexample.*)

If something is true and you try to disprove it, you will fail. We are trained to think of failure as bad, but it’s not all bad. You can learn from failure. You try to disprove the statement one way, and you hit a wall. You try another way, and you hit another wall. Each night you try, each night you fail, each night a new wall, and if you are lucky, those walls start to come together into a structure, and that structure is the structure of the proof of the theorem. For if you have really understood what’s keeping you from disproving the theorem, you very likely understand, in a way inaccessible to you before, why the theorem is true. This is what happened to Bolyai, who bucked his father’s well-meaning advice and tried, like so many before him, to prove that the parallel postulate followed from Euclid’s other axioms. Like all the others, he failed. But unlike the others, he was able to understand the shape of his failure. What was blocking all his attempts to prove that there was no geometry without the parallel postulate was the existence of just such a geometry! And with each failed attempt he learned more about the features of the thing he didn’t think existed, getting to know it more and more intimately, until the moment when he realized it was really there. Proving by day and disproving by night is not just for mathematics. I find it’s a good habit to put pressure on all your beliefs, social, political, scientific, and philosophical. Believe whatever you believe by d...

This salutary mental exercise is not at all what F. Scott Fitzgerald was talking about, by the way. His endorsement of holding contradictory beliefs comes from “The Crack-Up,” his 1936 essay about his own irreparable brokenness. The opposing ideas he has in mind there are “the sense of futility of effort and the sense of the necessity to struggle.” Samuel Beckett later put it more succinctly: “I can’t go on, I’ll go on.”