How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

replication; if an effect can’t be replicated, despite repeated trials, science backs apologetically away.

Fisher, at any rate, didn’t believe in a hard and fast rule that tells us what to do. He was a distruster of pure mathematical formalism.

It was scary when a statistical model deployed by the Guest Marketing Analytics team at Target correctly inferred based on purchasing data that one of its customers—sorry, guests—a teenage girl in Minnesota, was pregnant, based on an arcane formula involving elevated rates of buying unscented lotion, mineral supplements, and cotton balls.

Target started sending her coupons for baby gear, much to the consternation of her father, who, with his puny human inferential power, was still in the dark.

the data fed to them more voluminous and nutritious.

Great adjective for data

In fact, there’s a 99.99% chance that your neighbor is innocent. In a way, this is the birth control scare revisited.

Human beings are always inferring, always using observations to refine our judgments about the various competing theories that jostle around inside our mental representation of the world. We are very confident, almost unshakably confident, about some of our theories (“The sun will rise tomorrow,” “When you drop things, they fall”) and less sure about others (“If I exercise today, I’ll sleep well tonight,” “There’s no such thing as telepathy”). We have theories about big things and little things, things we encounter every day and things we’ve run into only once. As we encounter evidence for and against those theories, our confidence in them bobs up and down.

When Fisher says that “no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas,” he is saying exactly that scientific inference can’t, or at least shouldn’t, be carried out purely mechanically; our preexisting ideas and beliefs

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.

The theory U acts as a kind of Bayesian coating to T, keeping new evidence from getting to it and dissolving it. This is a property most successful crackpot theories have in common; they’re encased in just enough protective stuff that they’re equally consistent with many possible observations, making them hard to dislodge. They’re like the multi-drug-resistant E. coli of the information ecosystem. In a weird way you have to admire them.

David Foster Wallace, the most mathematical of modern novelists (he once took a break from fiction to write a whole book about transfinite set theory!) described the myth as the “Math Melodrama,” and described its protagonist as “a kind of Prometheus-Icarus figure whose high-altitude genius is also hubris and Fatal Flaw.”

Mathematics tends to strengthen the mind rather than strain it to its breaking point. If anything, I’ve found that in moments of emotional extremity there is nothing like a math problem to quiet the complaints the rest of the psyche serves up. 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. It might drive me crazy not to do it.

Some people even like to measure utility in standard units, called utils.*

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.

Christians and obey the forms of piety, until, just by force of propinquity, you start to truly believe. Can I put Pascal’s argument in modern terms better than David Foster Wallace did in Infinite Jest? I cannot. The desperate, newly sober White Flaggers are always encouraged to invoke and pay empty lip-service to slogans they don’t yet understand or believe—e.g. “Easy Does It!” and “Turn It Over!” and “One Day at a Time!” It’s called “Fake It Till You Make It,” itself an oft-invoked slogan. Everyone on a Commitment who gets up publicly to speak starts out saying he’s an alcoholic, says it whether he believes he is yet or not; then everybody up there says how Grateful he is to be sober today and how great it is to be Active and out on a Commitment with his Group, even if he’s not grateful or pleased about it at all. You’re encouraged to keep saying stuff like this until you start to believe it, just like if you ask somebody with serious sober time how long you’ll have to keep schlepping to all these goddamn meetings he’ll smile that infuriating smile and tell you just until you start to want to go to all these goddamn meetings.

a man that discovered a mountain of gold would not be richer than the one that found only one cubic fathom.

The doctrine of expected utility is appealingly straightforward and simple: presented with a set of choices, pick the one with the highest expected utility.

“We see, in this Essay, that the theory of probabilities is, in the end, only common sense boiled down to ‘calculus’; it points out in a precise way what rational minds understand by means of a sort of instinct, without necessarily being aware of it. It leaves nothing to doubt, in the choice of opinions and decisions; by its use one can always determine the most advantageous choice.”

In 1961, a decade before his explosion into public view, Ellsberg was a brilliant young analyst at the RAND Corporation, consulting with the U.S. government on strategic matters surrounding nuclear war—how it could be prevented, or, barring that, effectively conducted. At the same time, he was working toward a Harvard PhD in economics. On both tracks, he was thinking deeply about the process by which human beings made decisions in the face of the unknown.

This reasoning is well captured by utility theory. If I’m a corporation with limitless funds, losing $100,000 might not be so bad—let’s say it’s worth −100 utils—while winning $200,000 brings me 200 utils. In that case, dollars and utils might match up to be nicely linear; a util is just another name for a grand.

This is a mathematical way of formalizing a principle you already know: the richer you are, the more risks you can afford to take.

First try: just ask your computer to do it. Harvey and his team were MIT students, presumably able to knock off a few dozen lines of code before their morning coffee.

Why not just write a program to run through all combinations of 300,000 WinFall tickets to see which one provided the lowest-variance strategy? That wouldn’t be a hard program to write. The one small problem would be the way all matter and energy in the universe decayed into heat death by the time your program had handled the first tiny fragment of a microsliver of the data it was trying to analyze.

This view goes at least as far back as Alcmaeon of Croton, one of the Pythagorean weirdos we met in chapter 2. The eye must generate light, Alcmaeon argued: what other source could there be for the phosphene, the stars you see when you shut your eyes and press down on your eyeball? The theory of vision by reflected rays was worked out in great detail by the eleventh-century Cairene mathematician Abu ‘Ali al-Hasan ibn al-Haytham (but let’s call him Alhazen, as most Western writers do). His treatise on optics, the Kitab al-Manazir, was translated into Latin and taken up eagerly by philosophers and artists seeking a more systematic understanding of the relation between sight and the thing seen. The main point is this: a point P on your canvas represents a line in three-dimensional space. Thanks to Euclid, we know there’s a unique line containing any two specified points. In this case, the line is the one containing P and your eye. Any object in the world that lies on that line gets painted at point P.

In fact, point R is not just the endpoint of the horizontal axis, but of any horizontal line. If two different lines are both horizontal, they are parallel; and yet, in projective geometry, they meet, at the point at infinity. David Foster Wallace was asked in a 1996 interview about the ending of Infinite Jest, which many people found abrupt: Did he, the interviewer asked, avoid writing an ending because he “just got tired of writing it”? Wallace replied, rather testily: “There is an ending as far as I’m concerned. Certain kinds of parallel lines are supposed to start converging in such a way that an ‘end’ can be projected by the reader somewhere beyond the right frame. If no such convergence or projection occurred to you, then the book’s failed for you.”

Shannon, in the paper that launched the theory of information, identified the basic tradeoff that engineers still grapple with today: the more resistant to noise you want your signal to be, the slower your bits are transmitted. The presence of noise places a cap on the length of a message your channel can reliably convey in a given amount of time; this limit was what Shannon called the capacity of the channel. Just as a pipe can handle only so much water, a channel can handle only so much information.

Good political discourse rule too

We’ve managed to stay just ahead of the ever-increasing sphere of computer dominance, like action heroes outracing a fireball.

Then there’s the rise and fall of Scared Straight.

Gotta use this passage for book club + consider mentioning with Cornel West’s and TaNheisi Coates preemptive violence on children by black parents; need to protect black bodies

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.

Maybe it should have been called Scared Stupid.

It is not so. I think that stern compulsion ought to be exerted to prevent the free propagation of the stock of those who are seriously afflicted by lunacy, feeble-mindedness, habitual criminality, and pauperism, but that is quite different from compulsory marriage.

How to restrain ill-omened marriages is a question by itself, whether it should be effected by seclusion, or in other ways yet to be devised that are consistent with a humane and well-informed public opinion.

Wrongness is like original sin; we are born to it and it remains always with us, and constant vigilance is necessary if we mean to restrict its sphere of influence over our actions. 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.

It is difficult to overstate the impact of Galton’s creation of correlation on the conceptual world we now inhabit—not only in statistics, but in every precinct of the scientific enterprise.

Great sentence

remaining completely agnostic about the existence of any particular causal relationship,

I seldom feel I really understand a piece of mathematics until I know what it’s all about in geometric language.

And what about when two variables are correlated? What does that really mean? To make this simple, let’s start with the simplest kind of variable, a binary variable with only two possible values. Oftentimes a binary variable is the answer to a yes-or-no question: “Are you married?” “Do you smoke?” “Are you now, or have you ever been, a member of the Communist Party?” When you’re comparing two binary variables, correlation takes on a particularly simple form.

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.

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.

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.

If there’s no such thing as public opinion, what’s an elected official to do? The simplest answer: when there’s no coherent message from the people, do whatever you want. As we’ve seen, simple logic demands that you’ll sometimes be acting contrary to the will of the majority. If you’re a mediocre politician, this is where you point out that the polling data contradicts itself. If you’re a good politician, this is where you say, “I was elected to lead—not to watch the polls.”

constitutes cruelty and unusualness has been the subject of energetic legal dispute.

contemporary American norms, not the prevailing standards of August 1789, provide the standard of what is cruel and what unusual.

I can’t manage to share their worry.

Punishment is a renewable resource; there is no danger we’ll run out.

Remember, the slime mold doesn’t have a brain to coordinate its decision making, just thousands of nuclei enclosed in the plasmodium, each pushing the collective in one direction or another. Somehow the slime mold has to aggregate the information available to it into a decision.

We’ve seen already that the apparent irrationality of popular opinion can arise from the collective behavior of perfectly rational individual people. But individual people, as we know from experience, are not perfectly rational.

The story of the slime mold suggests that the paradoxes and incoherencies of our everyday behavior might themselves be explainable in a more systematic way. Maybe individual people seem irrational because they aren’t really individuals! Each one of us is a little nation-state, doing our best to settle disputes and broker compromises between the squabbling voices that drive us. The results don’t always make sense. But they somehow allow us, like the slime molds, to shamble along without making too many terrible mistakes. Democracy is a mess—but it kind of works.

This combination of timidity and temper led his mentor Jacques Turgot to nickname him “le mouton enragé,” or “the rabid sheep.”

Condorcet did possess was a passionate, never-wavering belief in reason, and especially mathematics, as an organizing principle of human affairs.