How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

So let’s posit a lottery draw with just seven balls, of which the state picks three as the jackpot combination. There are thirty-five possible jackpot combos, corresponding to the thirty-five different ways that three numbers can be chosen from the set 1, 2, 3, 4, 5, 6, 7. (Mathematicians like to say, for short, “7 choose 3 is 35.”)

What we’re up against here is the dreaded phenomenon known by computer-science types as “the combinatorial explosion.” Put simply: very simple operations can change manageably large numbers into absolutely impossible ones. If you want to know which of the fifty states is the most advantageous place to site your business, that’s easy; you just have to compare fifty different things. But if you want to know which route through the fifty states is the most efficient—the so-called traveling salesman problem—the combinatorial explosion goes off, and you face difficulty on a totally different scale. There are about 30 vigintillion routes to choose from. In more familiar terms, that’s 30 thousand trillion trillion trillion trillion trillion.

The moment, early in the Italian Renaissance, at which painters understood perspective was the moment visual representation changed forever, the moment when European paintings stopped looking like your kid’s drawings on the refrigerator door (if your kid mostly drew Jesus dead on the cross) and started looking like the things they were paintings of.*

The two rails are parallel. But the two planes are not. How could they be? They meet at your eye, and parallel planes do not meet anywhere. But planes that aren’t parallel have to intersect in a line. In this case, the line is horizontal, emanating from your eye and proceeding parallel to the train tracks.

If you travel infinitely far to the right, until you arrive at R, and then keep on going, you find yourself still traveling rightward but now heading back toward the center from the left edge of the picture. This kind of leaving-one-way-and-coming-back-the-other enthralled the young Winston Churchill, who recalled vividly the one mathematical epiphany of his life: I had a feeling once about Mathematics, that I saw it all—Depth beyond depth was revealed to me—the Byss and the Abyss. I saw, as one might see the transit of Venus—or even the Lord Mayor’s Show, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!

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.”

In Euclid’s plane, two different points determine a single line, and two different lines determine a single intersection point—unless they’re parallel, in which case they don’t meet at all. In mathematics, we like rules, and we don’t like exceptions. In the projective plane, you don’t have to make any exceptions to the rule that two lines meet at a point, because parallel lines meet too.

The geometry of the projective plane is governed by two axioms: Every pair of points is contained in exactly one common line. Every pair of lines contains exactly one common point.

For Fano and his intellectual heirs, it doesn’t matter whether a line “looks like” a line, a circle, a mallard duck, or anything else—all that matters is that lines obey the laws of lines, set down by Euclid and his successors. If it walks like geometry, and it quacks like geometry, we call it geometry. To one way of thinking, this move constitutes a rupture between mathematics and reality, and is to be resisted. But that view is too conservative. The bold idea that we can think geometrically about systems that don’t look like Euclidean space,* and even call these systems “geometries” with head held high, turned out to be critical to understanding the geometry of the relativistic space-time we live in; and nowadays we use generalized geometric ideas to map Internet landscapes, which are even further removed from anything Euclid would recognize.

That’s part of the glory of math; we develop a body of ideas, and once they’re correct, they’re correct, even when applied far, far outside the context in which they were first conceived.

Understand this: I warmly endorse, in fact highly recommend, a bristly skepticism in the face of all claims that such-and-such an entity can be explained, or tamed, or fully understood, by mathematical means.

And yet the history of mathematics is a history of aggressive territorial expansion, as mathematical techniques get broader and richer, and mathematicians find ways to address questions previously thought of as outside their domain. “A mathematical theory of probability” sounds unexceptional now, but once it would have seemed a massive overreach; math was about the certain and the true, not the random and the maybe-so! All that changed when Pascal, Bernoulli, and others found mathematical laws that governed the workings of chance.* A mathematical theory of infinity? Before the work of Georg Cantor in the nineteenth century, the study of the infinite was as much theology as science; now we understand Cantor’s theory of multiple infinities, each one infinitely larger than the last, well enough to teach it to first-year math majors.

Will there be a mathematical theory of consciousness? Of society? Of aesthetics? People are trying, that’s for sure, with only limited success so far. You should distrust all such claims on instinct. But you should also keep in mind that they might end up getting some important things right.

In other words, thanks to the axioms of geometry, the Hamming code has the same magical error-correcting property as “repeat three times”; if a message gets modified by a single bit en route, the receiver can always figure out what message the transmitter meant to send. But instead of multiplying your transmission time by three, your new improved code sends just seven bits for every three bits of your original message, a more efficient ratio of 2.33.

This same principle underlies all manner of communications that are robust to noise. Natural language works this way: if I write lanvuage instead of language, you can figure out what it was I meant to say, because there’s no other word in English that’s one letter substitution away from lanvuage.

And if machine intelligences of the future can take over from us much of the work we know as research now? We’ll reclassify that research as “computation.” And whatever we quantitatively minded humans are doing with our newly freed-up time, that’s what we’ll call “mathematics.”

What Shannon proved—and once he understood what to prove, it was really not so hard—was that almost all sets of code words exhibited the error-correcting property; in other words, a completely random code, with no design at all, was extremely likely to be an error-correcting code.

Milton Friedman and Leonard Savage, who proposed that lottery players follow a squiggly utility curve, reflecting that people think about wealth in terms of classes, not numerical amounts. If you’re a middle-class worker who spends five bucks a week on the lottery, and you lose, that choice costs you a little money but doesn’t change your class position; despite the loss of money, the negative utility is pretty close to zero. But if you win, well, that moves you into a different stratum of society. You can think of this as the “deathbed” model—on your deathbed, will you care that you died with a little less money because you played the lottery? Probably not at all. Will you care that you retired at thirty-five and spent the rest of your life snorkeling off Cabo because you hit the Powerball? Yes. Yes you will.

In a bigger departure from classical theory, Daniel Kahneman and Amos Tversky suggested that people in general tend to follow a different path from the one the utility curve demands, not just when Daniel Ellsberg sticks an urn in front of them, but in the general course of life. Their “prospect theory,” for which Kahneman later won the Nobel Prize, is now seen as the founding document of behavioral economics, which aims to model with the greatest possible fidelity the way people do act, not the way that, according to an abstract notion of rationality, they should. In the Kahneman-Tversky theory, people tend to place more weight on low-probability events than a person obedient to the von Neumann-Morgenstern axioms would; so the allure of the jackpot exceeds what a strict expected utility calculation would license.

Economics isn’t like physics and utility isn’t like energy. It is not conserved, and an interaction between two beings can leave both with more utility than they started with.

rich states vote for Democrats but rich people vote for Republicans,

But now here is Galton’s remarkable discovery: those children are not likely to be as tall as their parents. The same goes for short parents, in the opposite direction; their kids will tend to be short, but not as short as they themselves are. Galton had discovered the phenomenon now called regression to the mean. His data left no doubt that it was real.

Tall people are tall because their heredity predisposes them to be tall, or because external forces encourage them to be tall, or both. And the taller a person is, the likelier it is that both factors are pointing in the upward direction.

That’s what causes regression to the mean: not a mysterious mediocrity-loving force, but the simple workings of heredity intermingled with chance. That’s why Galton writes that regression to the mean is “theoretically a necessary fact.” At first, it came to him as a surprising feature of his data, but once he understood what was going on, he saw it couldn’t possibly have come out any other way.

Why is the second novel by a breakout debut writer, or the second album by an explosively popular band, so seldom as good as the first? It’s not, or not entirely, because most artists only have one thing to say. It’s because artistic success is an amalgam of talent and fortune, like everything else in life, and thus subject to regression to the mean.*

But it’s still wrong. If there’s regression to the mean in April, there’s regression to the mean in July. Ballplayers get this. Derek Jeter, when bugged about being on pace to break Pete Rose’s career hit record, told the New York Times, “One of the worst phrases in sports is ‘on pace for.’” Wise words!

how many home runs should I expect to hit the rest of the way? The All-Star break divides the baseball season into a “first half” and a “second half,” but the second half is actually a bit shorter: in recent years, between 80% and 90% as long as the first half. So you might expect me to hit about 85% as many home runs in the second half as I did in the first.* But history says this is the wrong thing to expect.

There’s something the mind resists about regression to the mean. We want to believe in a force that brings down the mighty.

To ‘prove’ such a mathematical result by a costly and prolonged numerical study of many kinds of business profit and expense ratios is analogous to proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals. The performance, though perhaps entertaining, and having a certain pedagogical value, is not an important contribution either to zoölogy or mathematics.”

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.

My colleague Michael Harris, a distinguished number theorist at the Institut de Mathématiques de Jussieu in Paris, has a theory that three of Thomas Pynchon’s major novels are governed by the three conic sections: Gravity’s Rainbow is about parabolas (all those rockets, launching and dropping!), Mason & Dixon about ellipses, and Against the Day about hyperbolas. This seems as good to me as any other organizing theory of these novels I’ve encountered; certainly Pynchon, a former physics major who likes to drop references to Möbius strips and the quaternions in his novels, knows very well what the conic sections are.

If, for instance, you have the “Ode to Joy” in your head but can’t remember what it’s called, you can go to a website like Musipedia and type in *ruurdddd. That short string is enough to cut the possibilities down to “Ode to Joy” or Mozart’s Piano Concerto No. 12.

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.

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. If you know one thing about the word “correlation” it’s that “correlation does not imply causation”—two phenomena can be correlated, in Galton’s sense, even if one doesn’t cause the other.

In the post-Galton world, you could talk about an association between two variables while remaining completely agnostic about the existence of any particular causal relationship, direct or indirect. In its way, the conceptual revolution Galton engendered has something in common with the insight of his more famous cousin, Charles Darwin. Darwin showed that one could meaningfully talk about progress without any need to invoke purpose. Galton showed that one could meaningfully talk about association without any need to invoke underlying cause.

Mathematicians ever since Descartes have enjoyed the wonderful freedom to flip back and forth between algebraic and geometric descriptions of the world. The advantage of algebra is that it’s easier to formalize and to type into a computer. The advantage of geometry is that it allows us to bring our physical intuition to bear on the situation, particularly when you can draw a picture.

High-dimensional geometry can seem a little arcane, especially since the world we live in is three-dimensional (or four-dimensional, if you count time, or maybe twenty-six-dimensional, if you’re a certain kind of string theorist, but even then, you think the universe doesn’t extend very far along most of those dimensions).

Remember the digital photo from the four-megapixel camera: it’s described by 4 million numbers, one for each pixel. (And that’s before we take color into account!) So that image is a 4-million-dimensional vector; or, if you like, a point in 4-million-dimensional space. And an image that changes with time is represented by a point that’s moving around in a 4-million-dimensional space, which traces out a curve in 4-million-dimensional space, and before you know it you’re doing 4-million-dimensional calculus, and then the fun can really start.

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.

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, the two are orthogonal.”

I’m rooting for orthogonal to catch on. It’s been a while since a mathy word really broke out into demotic English. Lowest common denominator has by now lost its mathematical flavor almost entirely, and exponentially—just don’t get me started on exponentially.*

The application of trigonometry to high-dimensional vectors in order to quantify correlation is not, to put it mildly, what the developers of the cosine had in mind. The Nicaean astronomer Hipparchus, who wrote down the first trigonometric tables in the second century BCE, was trying to compute the time lapse between eclipses; the vectors he dealt with described objects in the sky, and were solidly three-dimensional. But a mathematical tool that’s just right for one purpose tends to make itself useful again and again.

In some states, like Texas and Wisconsin, richer counties tend to vote more Republican. In others, like Maryland, California, and New York, the richer counties are more Democratic. Those last states happen to be the ones where many political pundits live.

Some relationships, like “bigger than,” are transitive; if I weigh more than my son and my son weighs more than my daughter, it’s an absolute certainty that I weigh more than my daughter. “Lives in the same city as” is transitive, too—if I live in the same city as Bill, who lives in the same city as Bob, then I live in the same city as Bob. Correlation is not transitive. It’s more like “blood relation”—I’m related to my son, who’s related to my wife, but my wife and I aren’t blood relatives to each other. In fact, it’s not a terrible idea to think of correlated variables as “sharing part of their DNA.”

correlation isn’t transitive. Niacin is correlated with high HDL, and high HDL is correlated with low risk of heart attack, but that doesn’t mean that niacin prevents heart attacks.

As the voters get more informed, they don’t get more Democratic or more Republican, but they do get more polarized: lefties go farther left, right-wingers get farther right, and the sparsely populated space in the middle gets even sparser.

Undecided voters, by and large, aren’t undecided because they’re carefully weighing the merits of each candidate, unprejudiced by political dogma. They’re undecided because they’re barely paying attention.

a correlation computation can’t see the heart-shapedness (cardiomorphism?) of this scatterplot any more than your camera can detect gamma rays. Keep this in mind when you’re told that two phenomena in nature or society were found to be uncorrelated. It doesn’t mean there’s no relationship, only that there’s no relationship of the sort that correlation is designed to detect.

To say that marital status and smoking status are negatively correlated, for example, is simply to say that married people are less likely than the average person to smoke. Or, to put it another way, smokers are less likely than the average person to be married. It’s worth taking a moment to persuade yourself that those two things are indeed the same! The first statement can be written as an inequality married smokers / all married people < all smokers / all people and the second as married smokers / all smokers < all married people / all people

Surely the chance is very small that the proportion of smokers among married people is exactly the same as the proportion of smokers in the whole population. So, absent a crazy coincidence, marriage and smoking will be correlated, either positively or negatively. And so will sexual orientation and smoking, U.S. citizenship and smoking, first-initial-in-the-last-half-of-the-alphabet and smoking, and so on. Everything will be correlated with smoking, in one direction or the other. It’s the same issue we encountered in chapter 7; the null hypothesis, strictly speaking, is just about always false.