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First published by The Penguin Press, a member of Penguin Group (USA) LLC, 2014
Copyright © 2014 by Jordan Ellenberg
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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Ellenberg, Jordan, 1971- author.
How not to be wrong : the power of mathematical thinking / Jordan Ellenberg.
pages cm
Includes bibliographical references and index.
ISBN 978-0-698-16384-3
1. Mathematics—Miscellanea. 2. Mathematical analysis—Miscellanea. I. Title.
QA99.E45 2014
510—dc23 2014005394
Version_1 for Tanya “What is best in mathematics deserves not merely to be learnt as
a task, but to be assimilated as a part of daily thought, and
brought again and again before the mind with ever-renewed
encouragement.”
BERTRAND RUSSELL, “The Study of Mathematics” (1902) CONTENTS Title Page
Copyright
Dedication
Epigraph WHEN AM I GOING TO USE THIS? PART I Linearity
One. LESS LIKE SWEDEN
Two. STRAIGHT LOCALLY, CURVED GLOBALLY
Three. EVERYONE IS OBESE
Four. HOW MUCH IS THAT IN DEAD AMERICANS?
Five. MORE PIE THAN PLATE PART II Inference
Six. THE BALTIMORE STOCKBROKER AND THE BIBLE CODE
Seven. DEAD FISH DON’T READ MINDS
Eight. REDUCTIO AD UNLIKELY
Nine. THE INTERNATIONAL JOURNAL OF HARUSPICY
Ten. ARE YOU THERE, GOD? IT’S ME, BAYESIAN INFERENCE PART III Expectation
Eleven. WHAT TO EXPECT WHEN YOU’RE EXPECTING TO WIN THE
LOTTERY
Twelve. MISS MORE PLANES!
Thirteen. WHERE THE TRAIN TRACKS MEET PART IV Regression
Fourteen. THE TRIUMPH OF MEDIOCRITY
Fifteen. GALTON’S ELLIPSE Sixteen. DOES LUNG CANCER MAKE YOU SMOKE CIGARETTES? PART V Existence
Seventeen. THERE IS NO SUCH THING AS PUBLIC OPINION
Eighteen. “OUT OF NOTHING I HAVE CREATED A STRANGE NEW
UNIVERSE”
HOW TO BE RIGHT Acknowledgments
Notes
Index WHEN AM I GOING TO USE THIS? R ight now, in a classroom somewhere in the world, a student is mouthing
off to her math teacher. The teacher has just asked her to spend a
substantial portion of her weekend computing a list of thirty definite integrals.
There are other things the student would rather do. There is, in fact, hardly
anything she would not rather do. She knows this quite clearly, because she
spent a substantial portion of the previous weekend computing a different—but
not very different—list of thirty definite integrals. She doesn’t see the point, and
she tells her teacher so. And at some point in this conversation, the student is
going to ask the question the teacher fears most:
“When am I going to use this?”
Now the math teacher is probably going to say something like:
“I know this seems dull to you, but remember, you don’t know what career
you’ll choose—you may not see the relevance now, but you might go into a field
where it’ll be really important that you know how to compute definite integrals
quickly and correctly by hand.”
This answer is seldom satisfying to the student. That’s because it’s a lie. And
the teacher and the student both know it’s a lie. The number of adults who will
ever make use of the integral of (1 − 3x + 4x2)−2 dx, or the formula for the
cosine of 3θ, or synthetic division of polynomials, can be counted on a few
thousand hands.
The lie is not very satisfying to the teacher, either. I should know: in my
many years as a math professor I’ve asked many hundreds of college students to
compute lists of definite integrals.
Fortunately, there’s a better answer. It goes something like this:
“Mathematics is not just a sequence of computations to be carried out by rote
until your patience or stamina runs out—although it might seem that way from what you’ve been taught in courses called mathematics. Those integrals are to
mathematics as weight training and calisthenics are to soccer. If you want to play
soccer—I mean, really play, at a competitive level—you’ve got to do a lot of
boring, repetitive, apparently pointless drills. Do professional players ever use
those drills? Well, you won’t see anybody on the field curling a weight or
zigzagging between traffic cones. But you do see players using the strength,
speed, insight, and flexibility they built up by doing those drills, week after
tedious week. Learning those drills is part of learning soccer.
“If you want to play soccer for a living, or even make the varsity team,
you’re going to be spending lots of boring weekends on the practice field.
There’s no other way. But now here’s the good news. If the drills are too much
for you to take, you can still play for fun, with friends. You can enjoy the thrill of
making a slick pass between defenders or scoring from distance just as much as
a pro athlete does. You’ll be healthier and happier than you would be if you sat
home watching the professionals on TV.
“Mathematics is pretty much the same. You may not be aiming for a
mathematically oriented career. That’s fine—most people aren’t. But you can
still do math. You probably already are doing math, even if you don’t call it that.
Math is woven into the way we reason. And math makes you better at things.
Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden
structures underneath the messy and chaotic surface of the world. Math is a
science of not being wrong about things, its techniques and habits hammered out
by centuries of hard work and argument. With the tools of mathematics in hand,
you can understand the world in a deeper, sounder, and more meaningful way.
All you need is a coach, or even just a book, to teach you the rules and some
basic tactics. I will be your coach. I will show you how.”
For reasons of time, this is seldom what I actually say in the classroom. But
in a book, there’s room to stretch out a little more. I hope to back up the grand
claims I just made by showing you that the problems we think about every day—
problems of politics, of medicine, of commerce, of theology—are shot through
with mathematics. Understanding this gives you access to insights accessible by
no other means.
Even if I did give my student the full inspirational speech, she might—if she
is really sharp—remain unconvinced.
“That sounds good, Professor,” she’ll say. “But it’s pretty abstract. You say
that with mathematics at your disposal you can get things right you’d otherwise
get wrong. But what kind of things? Give me an actual example.” And at that point I would tell her the story of Abraham Wald and the missing
bullet holes. ABRAHAM WALD AND THE MISSING BULLET
HOLES
This story, like many World War II stories, starts with the Nazis hounding a Jew
out of Europe and ends with the Nazis regretting it. Abraham Wald was born in
1902 in what was then the city of Klausenburg in what was then the AustroHungarian Empire. By the time Wald was a teenager, one World War was in the
books and his hometown had become Cluj, Romania. He was the grandson of a
rabbi and the son of a kosher baker, but the younger Wald was a mathematician
almost from the start. His talent for the subject was quickly recognized, and he
was admitted to study mathematics at the University of Vienna, where he was
drawn to subjects abstract and recondite even by the standards of pure
mathematics: set theory and metric spaces.
But when Wald’s studies were completed, it was the mid-1930s, Austria was
deep in economic distress, and there was no possibility that a foreigner could be
hired as a professor in Vienna. Wald was rescued by a job offer from Oskar
Morgenstern. Morgenstern would later immigrate to the United States and help
invent game theory, but in 1933 he was the director of the Austrian Institute for
Economic Research, and he hired Wald at a small salary to do mathematical odd
jobs. That turned out to be a good move for Wald: his experience in economics
got him a fellowship offer at the Cowles Commission, an economic institute then
located in Colorado Springs. Despite the ever-worsening political situation, Wald
was reluctant to take a step that would lead him away from pure mathematics for
good. But then the Nazis conquered Austria, making Wald’s decision
substantially easier. After just a few months in Colorado, he was offered a
professorship of statistics at Columbia; he packed up once again and moved to
New York.
And that was where he fought the war.
The Statistical Research Group (SRG), where Wald spent much of World
War II, was a classified program that yoked the assembled might of American statisticians to the war effort—something like the Manhattan Project, except the
weapons being developed were equations, not explosives. And the SRG was
actually in Manhattan, at 401 West 118th Street in Morningside Heights, just a
block away from Columbia University. The building now houses Columbia
faculty apartments and some doctor’s offices, but in 1943 it was the buzzing,
sparking nerve center of wartime math. At the Applied Mathematics Group
−Columbia, dozens of young women bent over Marchant desktop calculators
were calculating formulas for the optimal curve a fighter should trace out
through the air in order to keep an enemy plane in its gunsights. In another
apartment, a team of researchers from Princeton was developing protocols for
strategic bombing. And Columbia’s wing of the atom bomb project was right
next door.
But the SRG was the most high-powered, and ultimately the most influential,
of any of these groups. The atmosphere combined the intellectual openness and
intensity of an academic department with the shared sense of purpose that comes
only with high stakes. “When we made recommendations,” W. Allen Wallis, the
director, wrote, “frequently things happened. Fighter planes entered combat with
their machine guns loaded according to Jack Wolfowitz’s* recommendations
about mixing types of ammunition, and maybe the pilots came back or maybe
they didn’t. Navy planes launched rockets whose propellants had been accepted
by Abe Girshick’s sampling-inspection plans, and maybe the rockets exploded
and destroyed our own planes and pilots or maybe they destroyed the target.”
The mathematical talent at hand was equal to the gravity of the task. In
Wallis’s words, the SRG was “the most extraordinary group of statisticians ever
organized, taking into account both number and quality.” Frederick Mosteller,
who would later found Harvard’s statistics department, was there. So was
Leonard Jimmie Savage, the pioneer of decision theory and great advocate of the
field that came to be called Bayesian statistics.* Norbert Wiener, the MIT
mathematician and the creator of cybernetics, dropped by from time to time.
This was a group where Milton Friedman, the future Nobelist in economics, was
often the fourth-smartest person in the room.
The smartest person in the room was usually Abraham Wald. Wald had been
Allen Wallis’s teacher at Columbia, and functioned as a kind of mathematical
eminence to the group. Still an “enemy alien,” he was not technically allowed to
see the classified reports he was producing; the joke around SRG was that the
secretaries were required to pull each sheet of notepaper out of his hands as soon
as he was finished writing on it. Wald was, in some ways, an unlikely participant. His inclination, as it always had been, was toward abstraction, and
away from direct applications. But his motivation to use his talents against the
Axis was obvious. And when you needed to turn a vague idea into solid
mathematics, Wald was the person you wanted at your side. —
So here’s the question. You don’t want your planes to get shot down by enemy
fighters, so you armor them. But armor makes the plane heavier, and heavier
planes are less maneuverable and use more fuel. Armoring the planes too much
is a problem; armoring the planes too little is a problem. Somewhere in between
there’s an optimum. The reason you have a team of mathematicians socked away
in an apartment in New York City is to figure out where that optimum is.
The military came to the SRG with some data they thought might be useful.
When American planes came back from engagements over Europe, they were
covered in bullet holes. But the damage wasn’t uniformly distributed across the
aircraft. There were more bullet holes in the fuselage, not so many in the
engines.
Section of plane Bullet holes per square foot Engine 1.11 Fuselage 1.73 Fuel system 1.55 Rest of the plane 1.8 The officers saw an opportunity for efficiency; you can get the same
protection with less armor if you concentrate the armor on the places with the
greatest need, where the planes are getting hit the most. But exactly how much
more armor belonged on those parts of the plane? That was the answer they
came to Wald for. It wasn’t the answer they got.
The armor, said Wald, doesn’t go where the bullet holes are. It goes where
the bullet holes aren’t: on the engines.
Wald’s insight was simply to ask: where are the missing holes? The ones that
would have been all over the engine casing, if the damage had been spread
equally all over the plane? Wald was pretty sure he knew. The missing bullet
holes were on the missing planes. The reason planes were coming back with
fewer hits to the engine is that planes that got hit in the engine weren’t coming
back. Whereas the large number of planes returning to base with a thoroughly
Swiss-cheesed fuselage is pretty strong evidence that hits to the fuselage can (and therefore should) be tolerated. If you go the recovery room at the hospital,
you’ll see a lot more people with bullet holes in their legs than people with bullet
holes in their chests. But that’s not because people don’t get shot in the chest; it’s
because the people who get shot in the chest don’t recover.
Here’s an old mathematician’s trick that makes the picture perfectly clear: set
some variables to zero. In this case, the variable to tweak is the probability that a
plane that takes a hit to the engine manages to stay in the air. Setting that
probability to zero means a single shot to the engine is guaranteed to bring the
plane down. What would the data look like then? You’d have planes coming
back with bullet holes all over the wings, the fuselage, the nose—but none at all
on the engine. The military analyst has two options for explaining this: either the
German bullets just happen to hit every part of the plane but one, or the engine is
a point of total vulnerability. Both stories explain the data, but the latter makes a
lot more sense. The armor goes where the bullet holes aren’t.
Wald’s recommendations were quickly put into effect, and were still being
used by the navy and the air force through the wars in Korea and Vietnam. I
can’t tell you exactly how many American planes they saved, though the dataslinging descendants of the SRG inside today’s military no doubt have a pretty
good idea. One thing the American defense establishment has traditionally
understood very well is that countries don’t win wars just by being braver than
the other side, or freer, or slightly preferred by God. The winners are usually the
guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5%
more nutrition into their infantry at 95% of the cost. That’s not the stuff war
movies are made of, but it’s the stuff wars are made of. And there’s math every
step of the way. —
Why did Wald see what the officers, who had vastly more knowledge and
understanding of aerial combat, couldn’t? It comes back to his math-trained
habits of thought. A mathematician is always asking, “What assumptions are you
making? And are they justified?” This can be annoying. But it can also be very
productive. In this case, the officers were making an assumption unwittingly:
that the planes that came back were a random sample of all the planes. If that
were true, you could draw conclusions about the distribution of bullet holes on
all the planes by examining the distribution of bullet holes on only the surviving
planes. Once you recognize that you’ve been making that hypothesis, it only
takes a moment to realize it’s dead wrong; there’s no reason at all to expect the planes to have an equal likelihood of survival no matter where they get hit. In a
piece of mathematical lingo we’ll come back to in chapter 15, the rate of
survival and the location of the bullet holes are correlated.
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 r...

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