Veuillez patienter - l'impression de la page est en cours de préparation.
La boîte de dialogue d'impression s'ouvre dès que la page a été entièrement chargée.
Si l'aperçu avant impression est incomplet, veuillez le fermer et sélectionner "Imprimer à nouveau".

How Not to Be Wrong

The Power of Mathematical Thinking
Einband grossHow Not to Be Wrong
ISBN/GTIN
CHF17.90
inkl. 2.6 % MwSt.

Produkt

KlappentextThe Freakonomics of math-a math-world superstar unveils the hidden beauty and logic of the world and puts its power in our hands

The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn't confined to abstract incidents that never occur in real life, but rather touches everything we do-the whole world is shot through with it.

Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It's a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does "public opinion" really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer?

How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician's method of analyzing life and exposing the hard-won insights of the academic community to the layman-minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia's views on crime and punishment, the psychology of slime molds, what Facebook can and can't figure out about you, and the existence of God.

Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is "an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength." With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how.
Zusatztext"Brilliantly engaging . . . Ellenberg's talent for finding real-life situations that enshrine mathematical principles would be the envy of any math teacher. He presents these in fluid succession, like courses in a fine restaurant, taking care to make each insight shine through, unencumbered by jargon or notation. Part of the sheer intellectual joy of the book is watching the author leap nimbly from topic to topic, comparing slime molds to the Bush-Gore Florida vote, criminology to Beethoven's Ninth Symphony. The final effect is of one enormous mosaic unified by mathematics." -Manil Suri, The Washington Post

"Easy-to-follow, humorously presented . . . This book will help you to avoid the pitfalls that result from not having the right tools. It will help you realize that mathematical reasoning permeates our lives-that it can be, as Mr. Ellenberg writes, a kind of 'X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.'" -Mario Livio, The Wall Street Journal

"Witty, compelling, and just plain fun to read . . . How Not to Be Wrong can help you explore your mathematical superpowers." -Evelyn Lamb, Scientific American

"A poet-mathematician offers an empowering and entertaining primer for the age of Big Data . . . A rewarding popular math book for just about anyone." -Laura Miller, Salon

"Mathematicians from Charles Lutwidge Dodgson to Steven Strogatz have celebrated the power of mathematics in life and the imagination. In this hugely enjoyable exploration of everyday maths as 'an atomic-powered prosthesis that you attach to your common sense', Jordan Ellenberg joins their ranks. Ellenberg, an academic and Slate's 'Do the Math' columnist, explains key principles with erudite gusto-whether poking holes in predictions of a US 'obesity apocalypse', or unpicking an attempt by psychologist B. F. Skinner to prove statistically that Shakespeare was a dud at alliteration." -Nature

"A fresh application of complex mathematical thinking to commonplace events . . . How Not to Be Wrong is beautifully written, holding the reader's attention throughout with well-chosen material, illuminating exposition, wit and helpful examples. I am reminded of the great writer of recreational mathematics, Martin Gardner: Ellenberg shares Gardner's remarkable ability to write clearly and entertainingly, bringing in deep mathematical ideas without the reader registering their difficulty." -Times Higher Education

"The author avoids heavy jargon and relies on real-world anecdotes and basic equations and illustrations to communicate how even simple math is a powerful tool . . . [Ellenberg] writes that, at its core, math is a special thing and produces a feeling of understanding unattainable elsewhere: 'You feel you've reached into the universe's guts and put your hand on the wire.' Math is profound, and profoundly awesome, so we should use it well-or risk being wrong . . . Witty and expansive, Ellenberg's math will leave readers informed, intrigued and armed with plenty of impressive conversation starters." -Kirkus Reviews

"Readers will indeed marvel at how often mathematics sheds unexpected light on economics (assessing the performance of investment advisors), public health (predicting the likely prevalence of obesity in 30 years), and politics (explaining why wealthy individuals vote Republican but affluent states go for Democrats). Relying on remarkably few technical formulas, Ellenberg writes with humor and verve as he repeatedly demonstrates that mathematics simply extends common sense. He manages to translate even the work of theoretical pioneers such as Cantor and Gödel into the language of intelligent amateurs. The surprises that await readers include not only a discovery of the astonishing versatility of mathematical thinking but also a realization of its very real limits. Mathematics, as it turns out, simply cannot resolve the real-world ambiguities surrounding the Bush-Gore cliff-hanger of 2000, nor can it resolve the much larger question of God's existence. A bracing encounter with mathematics that matters." -Booklist

"The title of this wonderful book explains what it adds to the honorable genre of popular writing on mathematics. Like Lewis Carroll, George Gamow, and Martin Gardner before him, Jordan Ellenberg shows how mathematics can delight and stimulate the mind. But he also shows that mathematical thinking should be in the toolkit of every thoughtful person-of everyone who wants to avoid fallacies, superstitions, and other ways of being wrong." -Steven Pinker, Johnstone Family Professor of Psychology, Harvard University; author of How the Mind Works

"With math as with anything else, there's smart, and then there's street smart. This book will help you be both. Fans of Freakonomics and The Signal and the Noise will love Ellenberg's surprising stories, snappy writing, and brilliant lessons in numerical savvy. How Not to Be Wrong is sharp, funny, and right." -Steven Strogatz, Schurman Professor of Applied Mathematics, Cornell University, and author, The Joy of x

"Through a powerful mathematical lens Jordan Ellenberg engagingly examines real-world issues ranging from the fetishizing of straight lines in the reporting of obesity to the game theory of missing flights, from the relevance to digestion of regression to the mean to the counter-intuitive Berkson's paradox, which may explain why handsome men don't seem to be as nice as not so handsome ones. The coverage is broad, but not shallow and the exposition is non-technical and sprightly." -John Allen Paulos, author of Innumeracy and A Mathematician Reads the Newspaper

"Jordan Ellenberg is a top mathematician and a wonderful expositor, and the theme of his book is important and timely. How Not to Be Wrong is destined to be a classic." -Timothy Gowers

"Jordan Ellenberg promises to share ways of thinking that are both simple to grasp and profound in their implications, and he delivers in spades. These beautifully readable pages delight and enlighten in equal parts. Those who already love math will eat it up, and those who don't yet know how lovable math is are in for a most pleasurable surprise." -Rebecca Newberger Goldstein, author of Plato at the Googleplex

"Brilliant and fascinating! Ellenberg shows his readers how to magnify common sense using the tools usually only accessible to those who have studied higher mathematics. I highly recommend it to anyone interested in expanding their worldly savviness-and math IQ!" -Danica McKellar, actress and bestselling author of Math Doesn't Suck and Kiss My Math

"How Not to Be Wrong is a cheery manifesto for the utility of mathematical thinking. Ellenberg's prose is a delight-informal and robust, irreverent yet serious. Maths is 'an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength,' he writes. Doing maths 'is to be, at once, touched by fire and bound by reason. Logic forms a narrow channel through which intuition flows with vastly augmented force.'" -The Guardian
Details
ISBN/GTIN978-0-14-312753-6
ProduktartBuch
EinbandartKartonierter Einband
ProduktionslandUSA
Erscheinungsjahr2015
Erscheinungsdatum26.05.2015
Erstverkaufstag26.05.2015
Seiten480 Seiten
SpracheEnglisch
MasseBreite 141 mm, Höhe 213 mm, Dicke 25 mm
Gewicht376 g
BZ-Nr.16768878

Inhalt/Kritik

Leseprobe
Praise for How Not to Be Wrong

 

Brilliantly engaging . . . Ellenberg's talent for finding real-life situations that enshrine mathematical principles would be the envy of any math teacher. He presents these in fluid succession, like courses in a fine restaurant, taking care to make each insight shine through, unencumbered by jargon or notation. Part of the sheer intellectual joy of the book is watching the author leap nimbly from topic to topic, comparing slime molds to the Bush-Gore Florida vote, criminology to Beethoven's Ninth Symphony. The final effect is of one enormous mosaic unified by mathematics.

Manil Suri, The Washington Post

Easy to follow, humorously presented . . . This book will help you to avoid the pitfalls that result from not having the right tools. It will help you realize that mathematical reasoning permeates our livesthat it can be, as Mr. Ellenberg writes, a kind of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.'

Mario Livio, The Wall Street Journal

Witty, compelling, and just plain fun to read . . . How Not to Be Wrong can help you explore your mathematical superpowers.

Evelyn Lamb, Scientific American

Mathematicians from Charles Lutwidge Dodgson to Steven Strogatz have celebrated the power of mathematics in life and the imagination. In this hugely enjoyable exploration of everyday maths as an atomic-powered prosthesis that you attach to your common sense,' Jordan Ellenberg joins their ranks. Ellenberg, an academic and Slate 'sDo the Math' columnist, explains key principles with erudite gustowhether poking holes in predictions of a U.S. obesity apocalypse,' or unpicking an attempt by psychologist B. F. Skinner to prove statistically that Shakespeare was a dud at alliteration.

Nature

The book is filled to the rim with anecdotes and good-to-know' facts. And Ellenberg does not shy away from delving deeply into most topics, both in terms of the underlying mathematical concepts and the background material, which he has researched meticulously. . . . Whereas the book may be aimed at a general audience, who wonder how the mathematics they learned at school might ever be useful, there is much on offer for those who have chosen a professional career in the sciences even when the fundamental ideas discussed are not new. It's a bit like walking through a well-curated exhibition of a favored painter. Many works you know inside out, but the context and the logic of the presentation may offer refreshing new perspectives and insights.

Nature Physics

Refreshingly lucid while still remaining conceptually rigorous, this book lends insight into how mathematicians thinkand shows us how we can start to think like mathematicians as well.

The New York Times Book Review

A poet-mathematician offers an empowering and entertaining primer for the age of Big Data. . . . A rewarding popular math book for just about anyone.

Laura Miller, Salon

A fresh application of complex mathematical thinking to commonplace events . . . How Not to Be Wrong is beautifully written, holding the reader's attention throughout with well-chosen material, illuminating exposition, wit, and helpful examples. I am reminded of the great writer of recreational mathematics, Martin Gardner: Ellenberg shares Gardner's remarkable ability to write clearly and entertainingly, bringing in deep mathematical ideas without the reader registering their difficulty.

Times Higher Education (London)

Ellenberg tells engaging, even exciting stories about how the problems we think about every dayproblems of politics, of medicine, of commerce, of theologyare shot through with mathematics.'

The Washington Post (blog)

A collection of fascinating examples of math and its surprising applications . . . How Not to Be Wrong is full of interesting and weird mathematical tools and observations.

Business Insider

Wry, accessible, and entertaining . . . Ellenberg finds the commonsense math at work in the everyday world, and his vivid examples and clear descriptions show how math is woven into the way we reason.'

Publishers Weekly (starred review)

Witty and expansive, Ellenberg's math will leave readers informed, intrigued, and armed with plenty of impressive conversation starters.

Kirkus Reviews

Readers will indeed marvel at how often mathematics shed unexpected light on economics (assessing the performance of investment advisors), public health (predicting the likely prevalence of obesity in thirty years), and politics (explaining why wealthy individuals vote Republican but affluent states go for Democrats). Relying on remarkably few technical formulas, Ellenberg writes with humor and verve as he repeatedly demonstrates that mathematics simply extends common sense.

Booklist

How Not to Be Wrong is a cheery manifesto for the utility of mathematical thinking. Ellenberg's prose is a delightinformal and robust, irreverent yet serious. Maths is an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength,' he writes. Doing maths is to be, at once, touched by fire and bound by reason. Logic forms a narrow channel through which intuition flows with vastly augmented force.'

The Guardian (London)

The title of this wonderful book explains what it adds to the honorable genre of popular writing on mathematics. Like Lewis Carroll, George Gamow, and Martin Gardner before him, Jordan Ellenberg shows how mathematics can delight and stimulate the mind. But he also shows that mathematical thinking should be in the toolkit of every thoughtful ­personof everyone who wants to avoid fallacies, superstitions, and other ways of being wrong.

Steven Pinker, Johnstone Family Professor of Psychology, Harvard University, and author of How the Mind Works

Brilliant and fascinating! Ellenberg shows his readers how to magnify common sense using the tools usually only accessible to those who have studied higher mathematics. I highly recommend it to anyone interested in expanding their worldly savvinessand math IQ!

Danica McKellar, actress and bestselling author of Math Doesn't Suck and Kiss My Math

Jordan Ellenberg promises to share ways of thinking that are both simple to grasp and profound in their implications, and he delivers in spades. These beautifully readable pages delight and enlighten in equal parts. Those who already love math will eat it up, and those who don't yet know how lovable math is are in for a most pleasurable surprise.

Rebecca Newberger Goldstein, author of Plato at the Googleplex

With math as with anything else, there's smart, and then there's street smart. This book will help you be both. Fans of Freakonomics and The Signal and the Noise will love Ellenberg's surprising stories, snappy writing, and brilliant lessons in numerical savvy. How Not to Be Wrong is sharp, funny, and right.

Steven Strogatz, Jacob Gould Schurman Professor of Applied Mathematics, Cornell University, and author of The Joy of x

Every page is a stand-alone, positive, and ontological examination of the beauty and surprise of mathematical discovery.

Cathy O'Neil, Mathbabe.com



PENGUIN BOOKS

HOW NOT TO BE WRONG

Jordan Ellenberg is the Vilas Distinguished Achievement Professor of Mathematics at the University of Wisconsin-Madison. His writing has appeared in Slate, The Wall Street Journal, The New York Times, The Washington Post, The Boston Globe, and The Believer.





 

WHEN AM I GOING TO USE THIS?

Right 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 differentbut not very differentlist 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 chooseyou 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 3, 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 outalthough 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 soccerI mean, really play, at a competitive levelyou'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 finemost 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 dayproblems of politics, of medicine, of commerce, of theologyare 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 mightif she is really sharpremain 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 kinds 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 Austro-Hungarian 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 effortsomething 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 GroupColumbia, 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.

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 to 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 nosebut 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 data-slinging 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 takes only 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 upholsteryhe 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 teenage 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....mehr

Autor

Jordan Ellenberg is the John D. MacArthur Professor of Mathematics at the University of Wisconsin-Madison and a 2015 Guggenheim fellow. He has lectured around the world on his research in number theory and delivered one of the invited addresses at the 2015 Joint Mathematics Meetings, the largest math conference in the world. His writing has appeared in The New York Times, The Washington Post, The Wall Street Journal, The Boston Globe, Wired, and The Believer, and he has been featured on the Today show and NPR's All Things Considered. He writes a popular column called "Do the Math" for Slate.
Verwendung von Cookies
Cookies helfen uns, das Benutzererlebnis zu verbessern. In den Einstellungen können einzelne Cookie-Arten ausgewählt werden. Weitere Informationen finden sich hier:
Datenschutzvereinbarung
Alle Cookies akzeptierenEinstellungen