How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843)

by Ellenberg, Jordan

  Wald’s other advantage was his tendency toward abstraction. Wolfowitz, who had studied under Wald at Columbia, wrote that the problems he favored were “all of the most abstract sort,” and that he was “always ready to talk about mathematics, but uninterested in popularization and special applications.”

  Wald’s personality made it hard for him to focus his attention on applied problems, it’s true. The details of planes and guns were, to his eye, so much upholstery—he peered right through to the mathematical struts and nails holding the story together. Sometimes that approach can lead you to ignore features of the problem that really matter. But it also lets you see the common skeleton shared by problems that look very different on the surface. Thus you have meaningful experience even in areas where you appear to have none.

  To a mathematician, the structure underlying the bullet hole problem is a phenomenon called survivorship bias. It arises again and again, in all kinds of contexts. And once you’re familiar with it, as Wald was, you’re primed to notice it wherever it’s hiding.

  Like mutual funds. Judging the performance of funds is an area where you don’t want to be wrong, even by a little bit. A shift of 1% in annual growth might be the difference between a valuable financial asset and a dog. The funds in Morningstar’s Large Blend category, whose mutual funds invest in big companies that roughly represent the S&P 500, look like the former kind. The funds in this class grew an average of 178.4% between 1995 and 2004: a healthy 10.8% per year.* Sounds like you’d do well, if you had cash on hand, to invest in those funds, no?

  Well, no. A 2006 study by Savant Capital shone a somewhat colder light on those numbers. Think again about how Morningstar generates its number. It’s 2004, you take all the funds classified as Large Blend, and you see how much they grew over the last ten years.

  But something’s missing: the funds that aren’t there. Mutual funds don’t live forever. Some flourish, some die. The ones that die are, by and large, the ones that don’t make money. So judging a decade’s worth of mutual funds by the ones that still exist at the end of the ten years is like judging our pilots’ evasive maneuvers by counting the bullet holes in the planes that come back. What would it mean if we never found more than one bullet hole per plane? Not that our pilots are brilliant at dodging enemy fire, but that the planes that got hit twice went down in flames.

  The Savant study found that if you included the performance of the dead funds together with the surviving ones, the rate of return dropped down to 134.5%, a much more ordinary 8.9% per year. More recent research backed that up: a comprehensive 2011 study in the Review of Finance covering nearly 5,000 funds found that the excess return rate of the 2,641 survivors is about 20% higher than the same figure recomputed to include the funds that didn’t make it. The size of the survivorship effect might have surprised investors, but it probably wouldn’t have surprised Abraham Wald.

  MATHEMATICS IS THE EXTENSION OF COMMON SENSE BY OTHER MEANS

  At this point my teenaged interlocutor is going to stop me and ask, quite reasonably: Where’s the math? Wald was a mathematician, that’s true, and it can’t be denied that his solution to the problem of the bullet holes was ingenious, but what’s mathematical about it? There was no trig identity to be seen, no integral or inequality or formula.

  First of all: Wald did use formulas. I told the story without them, because this is just the introduction. When you write a book explaining human reproduction to preteens, the introduction stops short of the really hydraulic stuff about how babies get inside Mommy’s tummy. Instead, you start with something more like “Everything in nature changes; trees lose their leaves in winter only to bloom again in spring; the humble caterpillar enters its chrysalis and emerges as a magnificent butterfly. You are part of nature too, and . . .”

  That’s the part of the book we’re in now.

  But we’re all adults here. Turning off the soft focus for a second, here’s what a sample page of Wald’s actual report looks like:

  I hope that wasn’t too shocking.

  Still, the real idea behind Wald’s insight doesn’t require any of the formalism above. We’ve already explained it, using no mathematical notation of any kind. So my student’s question stands. What makes that math? Isn’t it just common sense?

  Yes. Mathematics is common sense. On some basic level, this is clear. How can you explain to someone why adding seven things to five things yields the same result as adding five things to seven? You can’t: that fact is baked into our way of thinking about combining things together. Mathematicians like to give names to the phenomena our common sense describes: instead of saying, “This thing added to that thing is the same thing as that thing added to this thing,” we say, “Addition is commutative.” Or, because we like our symbols, we write:

  For any choice of a and b, a + b = b + a.

  Despite the official-looking formula, we are talking about a fact instinctively understood by every child.

  Multiplication is a slightly different story. The formula looks pretty similar:

  For any choice of a and b, a × b = b × a.

  The mind, presented with this statement, does not say “no duh” quite as instantly as it does for addition. Is it “common sense” that two sets of six things amount to the same as six sets of two?

  Maybe not; but it can become common sense. Here’s my earliest mathematical memory. I’m lying on the floor in my parents’ house, my cheek pressed against the shag rug, looking at the stereo. Very probably I am listening to side two of the Beatles’ Blue Album. Maybe I’m six. This is the seventies, and therefore the stereo is encased in a pressed wood panel, which has a rectangular array of airholes punched into the side. Eight holes across, six holes up and down. So I’m lying there, looking at the airholes. The six rows of holes. The eight columns of holes. By focusing my gaze in and out I could make my mind flip back and forth between seeing the rows and seeing the columns. Six rows with eight holes each. Eight columns with six holes each.

  And then I had it—eight groups of six were the same as six groups of eight. Not because it was a rule I’d been told, but because it could not be any other way. The number of holes in the panel was the number of holes in the panel, no matter which way you counted them.

  We tend to teach mathematics as a long list of rules. You learn them in order and you have to obey them, because if you don’t obey them you get a C-. This is not mathematics. Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.

  Now let’s be fair: not everything in mathematics can be made as perfectly transparent to our intuition as addition and multiplication. You can’t do calculus by common sense. But calculus is still derived from our common sense—Newton took our physical intuition about objects moving in straight lines, formalized it, and then built on top of that formal structure a universal mathematical description of motion. Once you have Newton’s theory in hand, you can apply it to problems that would make your head spin if you had no equations to help you. In the same way, we have built-in mental systems for assessing the likelihood of an uncertain outcome. But those systems are pretty weak and unreliable, especially when it comes to events of extreme rarity. That’s when we shore up our intuition with a few sturdy, well-placed theorems and techniques, and make out of it a mathematical theory of probability.

  The specialized language in which mathematicians converse with each other is a magnificent tool for conveying complex ideas precisely and swiftly. But its foreignness can create among outsiders the impression of a sphere of thought totally alien to ordinary thinking. That’s exactly wrong.

  Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength. Despite the power of mathematics, and despite its sometimes forbidding notation and abstraction, the actual mental work involved is little different from the way we think about more down-to-earth prob
lems. I find it helpful to keep in mind an image of Iron Man punching a hole through a brick wall. On the one hand, the actual wall-breaking force is being supplied, not by Tony Stark’s muscles, but by a series of exquisitely synchronized servomechanisms powered by a compact beta particle generator. On the other hand, from Tony Stark’s point of view, what he is doing is punching a wall, exactly as he would without the armor. Only much, much harder.

  To paraphrase Clausewitz: Mathematics is the extension of common sense by other means.

  Without the rigorous structure that math provides, common sense can lead you astray. That’s what happened to the officers who wanted to armor the parts of the planes that were already strong enough. But formal mathematics without common sense—without the constant interplay between abstract reasoning and our intuitions about quantity, time, space, motion, behavior, and uncertainty—would just be a sterile exercise in rule-following and bookkeeping. In other words, math would actually be what the peevish calculus student believes it to be.

  That’s a real danger. John von Neumann, in his 1947 essay “The Mathematician,” warned:

  As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration.*

  WHAT KINDS OF MATHEMATICS WILL APPEAR IN THIS BOOK?

  If your acquaintance with mathematics comes entirely from school, you have been told a story that is very limited, and in some important ways false. School mathematics is largely made up of a sequence of facts and rules, facts which are certain, rules which come from a higher authority and cannot be questioned. It treats mathematical matters as completely settled.

  Mathematics is not settled. Even concerning the basic objects of study, like numbers and geometric figures, our ignorance is much greater than our knowledge. And the things we do know were arrived at only after massive effort, contention, and confusion. All this sweat and tumult is carefully screened off in your textbook.

  There are facts and there are facts, of course. There has never been much controversy about whether 1 + 2 = 3. The question of how and whether we can truly prove that 1 + 2 = 3, which wobbles uneasily between mathematics and philosophy, is another story—we return to that at the end of the book. But that the computation is correct is a plain truth. The tumult lies elsewhere. We’ll come within sight of it several times.

  Mathematical facts can be simple or complicated, and they can be shallow or profound. This divides the mathematical universe into four quadrants:

  Basic arithmetic facts, like 1 + 2 = 3, are simple and shallow. So are basic identities like sin(2x) = 2 sin x cos x or the quadratic formula: they might be slightly harder to convince yourself of than 1 + 2 = 3, but in the end they don’t have much conceptual heft.

  Moving over to complicated/shallow, you have the problem of multiplying two ten-digit numbers, or the computation of an intricate definite integral, or, given a couple of years of graduate school, the trace of Frobenius on a modular form of conductor 2377. It’s conceivable you might, for some reason, need to know the answer to such a problem, and it’s undeniable that it would be somewhere between annoying and impossible to work it out by hand; or, as in the case of the modular form, it might take some serious schooling even to understand what’s being asked for. But knowing those answers doesn’t really enrich your knowledge about the world.

  The complicated/profound quadrant is where professional mathematicians like me try to spend most of our time. That’s where the celebrity theorems and conjectures live: the Riemann Hypothesis, Fermat’s Last Theorem,* the Poincaré Conjecture, P vs. NP, Gödel’s Theorem . . . Each one of these theorems involves ideas of deep meaning, fundamental importance, mind-blowing beauty, and brutal technicality, and each of them is the protagonist of books of its own.

  But not this book. This book is going to hang out in the upper left quadrant: simple and profound. The mathematical ideas we want to address are ones that can be engaged with directly and profitably, whether your mathematical training stops at pre-algebra or extends much further. And they are not “mere facts,” like a simple statement of arithmetic—they are principles, whose application extends far beyond the things you’re used to thinking of as mathematical. They are the go-to tools on the utility belt, and used properly they will help you not be wrong.

  Pure mathematics can be a kind of convent, a quiet place safely cut off from the pernicious influences of the world’s messiness and inconsistency. I grew up inside those walls. Other math kids I knew were tempted by applications to physics, or genomics, or the black art of hedge fund management, but I wanted no such rumspringa.* As a graduate student, I dedicated myself to number theory, what Gauss called “the queen of mathematics,” the purest of the pure subjects, the sealed garden at the center of the convent, where we contemplated the same questions about numbers and equations that troubled the Greeks and have gotten hardly less vexing in the twenty-five hundred years since.

  At first I worked on number theory with a classical flavor, proving facts about sums of fourth powers of whole numbers that I could, if pressed, explain to my family at Thanksgiving, even if I couldn’t explain how I proved what I proved. But before long I got enticed into even more abstract realms, investigating problems where the basic actors—“residually modular Galois representations,” “cohomology of moduli schemes,” “dynamical systems on homogeneous spaces,” things like that—were impossible to talk about outside the archipelago of seminar halls and faculty lounges that stretches from Oxford to Princeton to Kyoto to Paris to Madison, Wisconsin, where I’m a professor now. When I tell you this stuff is thrilling, and meaningful, and beautiful, and that I’ll never get tired of thinking about it, you may just have to believe me, because it takes a long education just to get to the point where the objects of study rear into view.

  But something funny happened. The more abstract and distant from lived experience my research got, the more I started to notice how much math was going on in the world outside the walls. Not Galois representations or cohomology, but ideas that were simpler, older, and just as deep—the northwest quadrant of the conceptual foursquare. I started writing articles for magazines and newspapers about the way the world looked through a mathematical lens, and I found, to my surprise, that even people who said they hated math were willing to read them. It was a kind of math teaching, but very different from what we do in a classroom.

  What it has in common with the classroom is that the reader gets asked to do some work. Back to von Neumann on “The Mathematician”:

  “It is harder to understand the mechanism of an airplane, and the theories of the forces which lift and which propel it, than merely to ride in it, to be elevated and transported by it—or even to steer it. It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way.”

  In other words: it is pretty hard to understand mathematics without doing some mathematics. There’s no royal road to geometry, as Euclid told Ptolemy, or maybe, depending on your source, as Menaechmus told Alexander the Great. (Let’s face it, famous old maxims attributed to ancient scie
ntists are probably made up, but they’re no less instructive for that.)

  This will not be the kind of book where I make grand, vague gestures at great monuments of mathematics, and instruct you in the proper manner of admiring them from a great distance. We are here to get our hands a little dirty. We’ll compute some things. There will be a few formulas and equations, when I need them to make a point. No formal math beyond arithmetic will be required, though lots of math way beyond arithmetic will be explained. I’ll draw some crude graphs and charts. We’ll encounter some topics from school math, outside their usual habitat; we’ll see how trigonometric functions describe the extent to which two variables are related to each other, what calculus has to say about the relationship between linear and nonlinear phenomena, and how the quadratic formula serves as a cognitive model for scientific inquiry. And we’ll also run into some of the mathematics that usually gets put off to college or beyond, like the crisis in set theory, which appears here as a kind of metaphor for Supreme Court jurisprudence and baseball umpiring; recent developments in analytic number theory, which demonstrate the interplay between structure and randomness; and information theory and combinatorial designs, which help explain how a group of MIT undergrads won millions of dollars by understanding the guts of the Massachusetts state lottery.