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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 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
“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 U.S. ‘obesity apocalypse,’ or unpicking an attempt by psychologist B. F. Skinner to prove statistically that Shakespeare was a dud at alliteration.”
“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.”
“Refreshingly lucid while still remaining conceptually rigorous, this book lends insight into how mathematicians think—and 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 day—problems of politics, of medicine, of commerce, of theology—are 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.”
“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.”
“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.”
“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 (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 person—of 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 savviness—and 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
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 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 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 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 t...
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