How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

When you read a report that one thing is correlated with another, you’re implicitly being told that the correlation is “strong enough” to be worth reporting—usually because it passed a test of statistical significance.

Here we come to the really sticky part. Marriage is negatively correlated with smoking; that’s a fact. A typical way to express that fact is to say “If you’re a smoker, you’re less likely to be married.” But one small change makes the meaning very different: “If you were a smoker, you’d be less likely to be married.” It seems strange that changing the sentence from the indicative to the subjunctive mood can change what it says so drastically. But the first sentence is merely a statement about what is the case. The second concerns a much more delicate question: What would be the case if we changed something about the world? The first sentence expresses a correlation; the second suggests a causation.

Clinical researchers call this the surrogate endpoint problem. It’s time consuming and expensive to check whether a drug improves average life span, because in order to record someone’s life span you have to wait for them to die. HDL level is the surrogate endpoint, the easy-to-check biomarker that’s supposed to stand in for “long life with no heart attack.” But the correlation between HDL and absence of heart attack might not indicate any causal link.

What had changed? By 1964, the association between smoking and cancer had appeared consistently across study after study. Heavier smokers suffered more cancer than lighter smokers, and cancer was most likely at the point of contact between tobacco and human tissue; cigarette smokers got more lung cancer, pipe smokers more lip cancer. Ex-smokers were less prone to cancer than smokers who kept up the habit. All these factors combined to lead the surgeon general’s committee to the conclusion that smoking was not just correlated with lung cancer, but caused lung cancer, and that efforts to reduce tobacco consumption would be likely to lengthen American lives.

Makers of public policy don’t have the luxury of uncertainty that scientists do. They have to form their best guesses and make decisions on the basis thereof. When the system works—as it unquestionably did in the case of tobacco—the scientist and the policy maker work in concert, the scientist reckoning how uncertain we ought to be and the policy maker deciding how to act under the uncertainty thus specified.

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.

Perhaps we are 75% sure that our conclusion is correct and that a campaign against eggplant would save a thousand American lives per year. But there’s also a 25% chance our conclusion is wrong; and if it’s wrong, we’ve induced many people to give up what might be a favorite vegetable, leading them to eat a less healthy diet overall, and causing, let’s say, two hundred excess deaths annually.* As always, we obtain the expected value by multiplying the result of each possible outcome by the corresponding probability, and then adding everything up. In this case, we find that 75% × 1000 + 25% × (−200)= 750 − 50= 700. So our recommendation has an expected value of seven hundred lives saved per year. Over the loud and well-financed complaints of the Eggplant Council, and despite our very real uncertainty, we go public.

And if we held ourselves to a stricter evidentiary standard, declining to issue any of these recommendations because we weren’t sure we were right? Then the lives we would have saved would be lost instead.

The very nature of uncertainty is that we don’t know which of our choices will help, like attacking tobacco, and which will hurt, like recommending hormone replacement therapy. But one thing’s for certain: refraining from making recommendations at all, on the grounds that they might be wrong, is a losing strategy. It’s a lot like George Stigler’s advice about missing planes. If you never give advice until you’re sure it’s right, you’re not giving enough advice.

I think the right answer is that there are no answers. Public opinion doesn’t exist. More precisely, it exists sometimes, concerning matters about which there’s a clear majority view. Safe to say it’s the public’s opinion that terrorism is bad and The Big Bang Theory is a great show. But cutting the deficit is a different story. The majority preferences don’t meld into a definitive stance.

Scalia—much like Samuel Livermore two hundred years earlier—foresees and deplores a world in which the populace loses by inches its ability to impose effective punishments on wrongdoers. I can’t manage to share their worry. The immense ingenuity of the human species in devising ways to punish people rivals our abilities in art, philosophy, and science. Punishment is a renewable resource; there is no danger we’ll run out.

The mathematical buzzword in play here is “independence of irrelevant alternatives.”

votes produce paradoxical outcomes, in which majorities don’t always get their way and irrelevant alternatives control the outcome.

This phenomenon is called the “asymmetric domination effect,” and slime molds are not the only creatures subject to it. Biologists have found jays, honeybees, and hummingbirds acting in the same seemingly irrational way.

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.

Here you see one of IRV’s weaknesses. A centrist candidate who’s liked pretty well by everyone, but is nobody’s first choice, has a very hard time winning. To sum up: Traditional American voting method: Wright wins Instant-runoff method: Kiss wins Head-to-head matchups: Montroll wins

On the positive side, Condorcet’s practice of following ideas to their logical conclusions led him to insist, almost alone among his peers, that the much-discussed Rights of Man belonged to women, too.

Condorcet thought he could. He wrote down an axiom—that is, a statement he took to be so self-evident as to require no justification. Here it is: If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people’s choice.

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.

This is not just an argument about politics—it’s a fundamental question that applies to every field of mental endeavor. Are we trying to figure out what’s true, or are we trying to figure out what conclusions are licensed by our rules and procedures? Hopefully the two notions frequently agree; but all the difficulty, and thus all the conceptually interesting stuff, happens at the points where they diverge.

The younger Bolyai kept working on the parallels, and by 1823 he had formed the outline of a solution to the ancient problem. He wrote back to his father, saying: I have discovered such wonderful things that I was amazed, and it would be an everlasting piece of bad fortune if they were lost. When you, my dear Father, see them, you will understand; at present I can say nothing except this: that out of nothing I have created a strange new universe. János Bolyai’s insight was to come at the problem from behind. Rather than try to prove the parallel postulate from the other axioms, he allowed his mind the freedom to wonder: What if the parallel axiom were false? Does a contradiction follow? And he found that the answer was no—that there was another geometry, not Euclid’s but something else, in which the first four axioms were correct but the parallel postulate was not.

Here is what they mean in Riemann’s spherical geometry. A Point is a pair of points on the sphere which are antipodal, or diametrically opposite each other. A Line is a “great circle”—that is, a circle on the sphere’s surface—and a Line segment is a segment of such a circle. A Circle is a circle, now allowed to be of any size.

If that strange condition, where no two lines are ever parallel, sounds familiar, it’s because we’ve been here before. It’s just the same phenomenon we saw in the projective plane, which Brunelleschi and his fellow painters used to develop the theory of perspective.* There, too, every pair of lines met. And this is no coincidence—one can prove that Riemann’s geometry of Points and Lines on a sphere is the same as that of the projective plane.

Here’s the thing; once you understand that the first four axioms apply to many different geometries, then any theorem Euclid proves from only those axioms must be true, not only in Euclid’s geometry, but in all the geometries where those axioms hold. It’s a kind of mathematical force multiplier; from one proof, you get many theorems.

And these theorems are not just about abstract geometries made up to prove a point. Post-Einstein, we understand that non-Euclidean geometry is not just a game; like it or not, it’s the way space-time actually looks.

This is a story told in mathematics again and again: we develop a method that works for one problem, and if it is a good method, one that really contains a new idea, we typically find that the same proof works in many different contexts, which may be as ...

The tradition is called “formalism.” It’s what G. H. Hardy was talking about when he remarked, admiringly, that mathematicians of the nineteenth century finally began to ask what things like 1 − 1 + 1 − 1 + . . . should be defined to be, rather than what they were.

Hardy would certainly have recognized Condorcet’s anguish as perplexity of the most unnecessary kind. He would have advised Condorcet not to ask who the best candidate really was, or even who the public really intended to install in office, but rather which candidate we should define to be the public choice. And this formalist take on democracy is more or less general in the free world today. In the contested 2000 presidential election in Florida, thousands of Palm Beach County voters who believed they were voting for Al Gore in fact recorded votes for the paleoconservative Reform Party candidate Pat Buchanan instead, thanks to the confusingly designed “butterfly ballot.”

But Gore doesn’t get those votes; he never even seriously argued for them. Our electoral system is formalist: what counts is the mark made on the ballot, not whatever feature of the voter’s mind we may take it to indicate.

We define the public will to be that mark that appears most frequently on the pieces of paper collected at the voting booth.

More counting would presumably have resulted in a more accurate enumeration of the votes; but this, the court said, is not the overriding goal of an election. Recounting some counties but not others, they said, would be unfair to the voters whose ballots were not revisited. The proper business of the state is not to count the votes as accurately as possible—to know what actually happened—but to obey the formal protocol that tells us, in Hardy’s terms, who the winner should be defined to be.

More generally, formalism in the law manifests itself as an adherence to procedure and the words of the statutes, even when—or especially when—they cut against what common sense prescribes. Justice Antonin Scalia, the fiercest advocate of legal formalism there is, puts it very directly: “Long live formalism. It is what makes a government a government of laws and not of men.”

In Scalia’s view, when judges try to understand what the law intends—its spirit—they’re inevitably bamboozled by their own prejudices and desires. Better to stick to the words of the Constitution and the statutes, treating them as axioms from wh...

Chief Justice John Roberts isn’t a fervent advocate of formalism like Scalia, but he’s broadly in sympathy with Scalia’s philosophy. In his confirmation hearing in 2005, he famously described his job in baseball terms: Judges and justices are servants of the law, not the other way around. Judges are like umpires. Umpires don’t make the rules; they apply them. The role of an umpire and a judge is critical. They make sure everybody plays by the rules. But it is a limited role. Nobody ever went to a ball game to see the umpire.

That’s because baseball is a formalist sport. What a thing is is what an umpire declares it to be, and nothing else. Or, as Klem put it, in what must be the bluntest assertion of an ontological stance ever made by a professional sports official: “It ain’t nothin’ till I call it.”

In this view, the hard questions about law, the ones that make it all the way to the Supremes, are left indeterminate by the axioms. The justices are thus in the same position Pascal was when he found he couldn’t reason his way to any conclusion about God’s existence. And yet, as Pascal wrote, we don’t have the choice not to play the game. The court must decide, whether it can do so by conventional legal reasoning or not. Sometimes it takes Pascal’s route: if reason does not determine the judgment, make the judgment that seems to have the best consequences.

Formalism’s greatest champion in mathematics was David Hilbert, the German mathematician whose list of twenty-three problems, prepared for a lecture in Paris at the 1900 International Congress of Mathematics, set the course for much of twentieth-century math. Hilbert is so revered that any work that touches even tangentially on one of his problems takes on a little extra shine, even a hundred years later.

Hilbert’s second problem was different from the others, because it was not so much a mathematical question as a question about mathematics itself. He began with a full-throated endorsement of the formalist approach to mathematics: When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps.

And then, one day, a mathematician in Amsterdam proves that a certain mathematical assertion is the case, while another mathematician in Kyoto proves that it is not. Now what? Starting from assertions one cannot possibly doubt, one has arrived at a contradiction. Reductio ad absurdum. Do you conclude that the axioms were wrong? Or that there’s something wrong with the structure of logical deduction itself? And what do you do with the decades of work based on those axioms?*

Thus, the second problem among those Hilbert presented to the mathematicians gathered in Paris: But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.

Mathematics has a nasty habit of showing that, sometimes, what’s obviously true is absolutely wrong.

A set can be a collection of pigs, or real numbers, or ideas, possible universes, or other sets. And it’s that last one that causes all the problems. Is there a set of all sets? Sure. A set of all infinite sets? Why not? In fact, both of these sets have a curious property: they are elements of themselves.

Now here comes the punch line. Let NO be the set of all non-ouroboric sets. NO seems like a weird thing to think about, but if Frege’s definition allows it into the world of sets, so must we. Is NO ouroboric or not? That is, is NO an element of NO? By definition, if NO is ouroboric, then NO cannot be in NO, which consists only of non-ouroboric sets. But to say NO is not an element of NO is precisely to say NO is non-ouroboric; it does not contain itself. But wait a minute—if NO is non-ouroboric, then it is an element of NO, which is the set of all non-ouroboric sets. Now NO is an element of NO after all, which is to say that NO is ouroboric. If NO is ouroboric, it isn’t, and if it isn’t, it is.

But then: “I have encountered a difficulty only on one point.” And Russell explains the quandary of NO, now known as Russell’s paradox.

Frege’s explanation is perhaps the saddest sentence ever written in a technical work of mathematics: “Einem wissenschaftlichen Schriftsteller kann kaum etwas Unerwünschteres begegnen, als dass ihm nach Vollendung einer Arbeit eine der Grundlagen seines Baues erschüttert wird.” Or: “A scientist can hardly encounter anything more undesirable than, just as a work is completed, to have its foundation give way.”

Hilbert knew very well what Russell had done to Frege and was keenly aware of the dangers posed by casual reasoning about infinite sets. “A careful reader,” he wrote in 1926, “will find that the literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite.”

But Hilbert was to be disappointed. In 1931, Kurt Gödel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert’s program with a single stroke.

Even if geometry can be recast as an exercise in manipulating meaningless strings of symbols, no human being can generate geometric ideas without drawing pictures, without imagining figures, without thinking of the objects of geometry as real things. My philosopher friends typically find this point of view, usually called Platonism, fairly disreputable; how can a fifteen-dimensional hypercube be a real thing? I can only reply that they seem as real to me as, say, mountains. After all, I can define a fifteen-dimensional hypercube. Can you do the same for the mountain?

In the famous formulation of Philip Davis and Reuben Hersh, “The typical working mathematician is a Platonist on weekdays and a formalist on Sundays.”

The cult of the genius also tends to undervalue hard work. When I was starting out, I thought “hardworking” was a kind of veiled insult—something to say about a student when you can’t honestly say they’re smart. But the ability to work hard—to keep one’s whole attention and energy focused on a problem, systematically turning it over and over and pushing at everything that looks like a crack, despite the lack of outward signs of progress—is not a skill everybody has.