A Mathematician Reads the Newspaper

Praise for A Mathematician Reads the Newspaper

“[A] witty crusade against mathematical illiteracy. . . . Mr. Paulos’s little essay explaining the Banzhaf power index and how it relates to Lani Guinier’s ideas about empowering minorities is itself worth the price of the book. . . . A Mathematician Reads the Newspaper is at bottom a plea for rationality and a repudiation of the illogic and obscurantism that, paradoxically, plague one of history’s best-educated countries.”—New York Times

“It would be great to have John Allen Paulos living next door. Every morning when you read the paper and came across some story that didn’t seem quite right—that had the faint odor of illogic hovering about it—you could just lean out the window and shout, ‘Jack! Get the hell over here!’ . . . [This is] a fun, spunky, wise little book that would be helpful to both the consumers of the news and its purveyors.”—Washington Post

“This book is not about mathematics, and it’s not really about newspapers. It should be called, ‘A Clear Thinker Explains the World.’ . . . This is a wise and thoughtful book, which skewers much of what everyone knows to be true.”—Los Angeles Times

“A truly thought-provoking book.”—USA Today

“A wide-ranging collection of musings on mathematics, the media and life itself.”—Chicago Tribune

“Paulos’ wit and humor—admirably displayed in Innumeracy—are in top form. His irreverent and pointed comments entertain as well as educate. Though Paulos writes about a bewildering number of topics, he has something fresh and interesting to say about each.”—Philadelphia Inquirer

“A strength of the book is Paulos’s self-effacing sense of humor and awareness of the ‘professional myopia’ among mathematicians. . . . This book will help the reader to follow Paulos’s advice to ‘Always be smart, seldom be certain.’”—Christian Science Monitor

“Should be mandatory reading for every journalist—as well as the readers, viewers and former tutors they supposedly serve.”—New Scientist

“Irresistible.”—Scientific American

“If paranoia could be cured by math, Paulos would be the Jonas Salk of the disease. His dissection of conspiracy theories is delicious.”—Molly Ivins

“This is a broad canvas and Paulos uses broad, bold strokes—jumping from baseball salaries to employment equity to the Gulf War to coin-flipping and stock market forecasters. The writing, however, is always clear; the computational demands modest; the rewards potentially great.”—Ottawa Citizen

“[Paulos is] a likable academic with an infectious enthusiasm for his subject.”—Gina Kolata, American Mathematical Society

“There can be no doubt that Paulos writes beautifully, that he loves mathematical ideas, and that he discusses such ideas with zest.”—Technology Review

“Because of Paulos’s quiet good sense and wide-ranging mind, his book . . . is simply nifty, larded with clever and informative tidbits as he strolls his broad, discursive way through typical newspaper reporting.”—Reason

“This book will bring a great deal of pleasure to many—as it did to the reviewer. It is full of fun, full of information, full of insights.”—American Mathematical Monthly

“A Mathematician Reads the Newspaper does as much as one brief book can do to highlight the problem of innumeracy while simultaneously contributing to its solution. . . . Paulos’ book does a valuable public service and should be required reading for all.”—Capital Times

         Preface to the 2013 Paperback Edition

Obviously much has changed since I first wrote A Mathematician Reads the Newspaper. Nevertheless, the same solecisms, both mathematical and otherwise, continue to appear not only in newspapers but also on TV, in blogs, and on social media. I’ve often thought that pointing them out and detoxifying them is somewhat akin to picking up the trash, which must be done continuously or it piles up and begins to smell. Most of the issues discussed in the book—elections, medical tests, coincidences, crime, risk assessment, sports records, philosophical and psychological conundrums, and many others—are subject to solecisms that, indeed, have piled up. Happily, there are more mathematical trash collectors in 2013 than there were when this book first appeared.

A few of the analyses of news stories in A Mathematician Reads the Newspaper were minor and maybe a bit picayune, but I hope they provided more insight than one of my favorite tales about statisticians. Three of them went duck hunting. The first took a shot at a duck and the bullet sailed six inches over it; the second then took a shot that landed six inches in front of the duck. At this the third statistician jumped up and down excitedly, exclaiming, “We got it, we got it.”

Unlike the account of the duck and the statisticians, the stories in this book did benefit from mathematical analysis and many of them had significant policy implications. Let me describe two examples that came along after the book was published and which I first discussed in my Who’s Counting column for ABCnews.com. I choose them because they involve the über-serious issues of war and the environment. The first is relevant to the military strategy that the United States followed during the Iraq War, which, like the war itself, made no sense, and the second deals with the 2010 British Petroleum oil spill in the Gulf of Mexico. Both suggest that journalists of all sorts, not just science journalists, should pay more attention to the mathematical issues that lurk in front-page news stories. Sometimes the reward is just a quirky insight, but often the mathematics casts an undermining light on the whole story.

Squaring Numbers and Military Strategy

The mathematics involved needn’t be esoteric. Squaring the number three, for example, results in nine, and this simple arithmetical operation says something about military strategy in general and the Iraq War in particular. How so?

Early in the war, the failure of the Iraqi regime to collapse immediately surprised many. They blamed Defense Secretary Rumsfeld’s policy of relying on a high-tech “shock and awe” campaign as a substitute for sending enough ground troops and garden-variety equipment to Iraq to dislodge Saddam Hussein. Although I was never in the military, never studied military strategy, and did not support the war, I was struck by public misgivings about the policy expressed by some in the military and vaguely recalled Lanchester’s Square Law and its possible relevance. It had been formulated during World War I, had been taught in military academies ever since, and had been used to model conflicts from ancient times to the battle of Iwo Jima in World War II.

Although usually couched in terms of differential equations (the context in which I first came across it in college), a version of Lanchester’s Law can be paraphrased arithmetically as follows: the strength of a military unit—planes, artillery, tanks, or soldiers with rifles—is proportional not to the size of the unit, but to the square of its size. Derrick Niederman and David Boyum, fellow mathematical trash collectors, came up with the following example. Imagine a conflict between two armies, denoted Quant (for quantity) and Qual (for quality); each has 1,000 pieces of artillery. Further assume that the two sides’ artilleries are more or less equivalent in effectiveness and are capable of destroying each other at a rate of, say, 6 percent per day. That is, after one day each side will have 94 percent of what it had the day before. (Focusing on the loss of artillery pieces obscures the unpleasant fact that people are dying.)

Neither side has an advantage, but consider the differential effects of increasing quantity and improving quality. First, let’s assume that army Quant can increase its artillery to 3,000 pieces, three times as many pieces as army Qual has.

This has two consequences. One is that each of Qual’s artillery pieces will receive three times as much fire from Quant’s artillery as before because Quant now has three times as many guns as Qual. Because of this, Qual will lose artillery at a rate of 18 percent (3 × 6 percent) per day. The other consequence is that each piece of Quant’s artillery will receive one-third as much fire from Qual’s artillery as before because Qual now must spread its fire over three times as many Quant guns as before. Because of this, Quant will lose artillery at a rate of only 2 percent (⅓ × 6 percent) per day.

In this case Lanchester’s Law says that tripling the number of pieces in Quant’s artillery leads to a ninefold advantage (18 percent versus 2 percent) in its relative effectiveness.

In addition to increasing the number of artillery (or planes, tanks, or soldiers with rifles), armies can improve their quality. So let’s alter the balance of power again by assuming that Qual counters Quant’s numerical superiority with better technology. It does so by upgrading its 1,000 pieces of artillery to make them each nine times as accurate as Quant’s 3,000 pieces of artillery.

If Qual can upgrade its artillery in this way, its rate of hitting Quant’s artillery (if Quant had an equal number of pieces as Qual) would no longer be 6 percent per day, but 54 percent (9 × 6 percent). But because we’re still assuming that Quant has three times as many pieces of artillery, Qual destroys only 18 percent (⅓ × 54 percent) of Quant’s artillery each day. Moreover, the rate at which Quant’s artillery takes out Qual’s artillery pieces remains the same as calculated above, at 18 percent per day.

The bottom line is that it takes a ninefold increase in Qual’s quality to equal or make up for a threefold increase in Quant’s quantity. In general, it takes an Nsquaredfold increase in quality to make up for an Nfold increase in quantity.

Now to the Iraq War where, in some limited respects, the Iraqi Republican Guard and irregulars were the Quant army and the American and British forces the Qual army. If the military units under discussion had planes, cruise missiles, and the like, there would be no comparison and Lanchester’s Law would not be relevant. With tanks and enhanced artillery, however, Lanchester’s Law would come into play, and America’s vast qualitative superiority would easily win the day.

It’s only when we get down to the level of individual soldiers with rifles in house-to-house fighting that the balance becomes unclear. Here Lanchester’s Law suggests that American soldiers’ smaller degree of superiority might not always make up for a potential Iraqi numerical advantage (unless weapons more destructive than rifles are unleashed); it takes a big qualitative advantage to overcome a small quantitative one.

Of course this bloodless analysis ignores many confounding factors specific to Iraq, and so any claims made for it remain tentative. Still, Lanchester’s Law does help explain why urban guerrilla warfare appeals to those with a weak conventional military. Only at this level is there less scope for technological superiority. It’s interesting that such a simple (and admittedly simplistic) mathematical analysis involving the heady notion of squaring a number can shed light on military strategy.

Geometry and the BP Oil Spill

Some might argue that the above analysis, simple as it is, lies beyond the ken of journalists and their readers. I don’t think so, but let’s move on to another topic that might have benefited from an early use of mathematics, basic geometry in this case—the British Petroleum oil spill of 2010.

The question at the time was, How much oil has leaked into the Gulf of Mexico? Many commentators, including BP spokesmen, put forth very small estimates ranging from 1,000 to 5,000 barrels a day and stressed the difficulty of determining the answer. True enough, but as Steve Wereley, an engineering professor at Purdue University, showed, and as many others would have shown had BP released underwater pictures of the leak earlier, an approximate estimate was quite easy to come by and indicated a much more voluminous oil spill than BP admitted.

Basically, the method for determining oil spillage depends largely on high school (or even middle school) geometry: the formula for the volume of a cylinder. Let’s first assume that the oil that spilled out through the pipe maintained its cylindrical shape. It didn’t, of course, but the volume of a growing amorphous blob of oil is the same as the volume of a long cylinder of oil extending out of the pipe, and this latter volume is easy to calculate. (Contrast squeezing and emptying a tube of toothpaste into a large blob in the sink with squeezing and emptying the tube into a very long, thin straw—into a cylinder. The volumes would be the same.)

So what was the ever growing volume of this cylinder? Geometry tells us the volume of a cylinder is the circular cross-sectional area of the cylinder (BP certainly knew this because it knew the radius of the leaking pipe was 10.5 inches) multiplied by the height (or because it’s horizontal here, the length), h.

And what was its height (or length), h? Again elementary math comes to the rescue.

The length of the imagined lengthening cylinder of oil equals the rate of flow of the oil multiplied by the time it’s been flowing: h = v*t. We certainly know the time. The break occurred on April 20.

The calculation is then simply a matter of plugging the right values into the formula for the volume of a cylinder: V = (pi*r^2)*v*t, where, once again, V is the volume of a cylinder, pi*r^2, the cross-sectional area of the pipe, t the time the oil has been leaking, and v the flow rate of the oil through the cylinder. The flow rate was not known exactly, so Wereley turned the poor-quality video snippet released by BP into a series of still pictures. Then he calculated the velocity of particles in the flow by analyzing the still pictures frame by frame and pixel by pixel to come up with an estimated flow rate for the oil of two feet per second.

Once one converts all the units and multiplies the numbers, the calculations suggested that about 60,000 barrels of oil had been leaking out of the pipe daily.

A careful scientist, Wereley admitted his estimate was rough for several reasons, but it was much closer to the later official estimate of about 5 million barrels leaked during the three-month debacle than were BP’s absurdly sanguine early estimates. The calculation of 60,000 barrels leaked daily is elementary and could have been obtained as soon as BP had pictures of the leak.

Perhaps the executives of BP, Halliburton, and Transocean were innocent of all technical details and maybe even of elementary geometry. It would have been interesting if, at the Senate hearing where each of them blamed the other two, they were asked the formula for a cylinder and its relevance to the size of the spill. But even if we excuse the executives’ lack of technical knowledge, it’s very hard to believe that the smart engineers who work for them didn’t have a good feel for the extent of the leak early on.

Had the video and pictures been obtained when they first became available, they would have alerted the authorities much earlier to the catastrophic nature of the spill, perhaps induced an even greater urgency to stopping it, and allowed us to plan more realistically for the environmental consequences. Sticking to their estimate of between 1,000 and 5,000 barrels per day was counterproductive at the very least. Did BP and others think they could rely forever on an innumerate public (including too many journalists) to take their word for the size of the spill?

Two Last Points

Just as Nate Silver’s statistical analysis and projections of the 2012 presidential election exposed the emptiness and fatuousness of much conventional political punditry, so, I think, a mathematical approach to other important news stories would reveal traditional coverage to be facile and uninformative. At the very least mathematics should complement standard coverage where appropriate.

One final note. A Mathematician Reads the Newspaper is not as earnest as the two topics discussed above might suggest. In addition to supplying mathematical clarifications and/or debunking news stories, the book also contains a fair amount of whimsy, some personal musings and recollections, and a cautionary message. As I write in the book’s conclusion: Always be smart; seldom be certain. That’s 72.938 percent easier said than done.

            Introduction

       “I read the news today, Oh Boy.”

–JOHN LENNON

My earliest memories, dating from the late 1940s, include hearing a distant train whistle from the back steps of the building we lived in on Chicago’s near north side. I can also see myself crying under the trapezi (Greek for “table”) when my grandmother left to go home to her apartment. I remember watching my mother rub her feet in bed at night, and I remember my father playing baseball and wearing his baseball cap indoors to cover his thinning hair. And, lest you wonder where I’m heading, I can recall watching my grandfather at the kitchen table reading the Chicago Tribune.

The train whistle and the newspaper symbolized the outside world, frighteningly yet appealingly different from the warm family ooze in which I was happily immersed. What was my grandfather reading about? Where was the train going? Were these somehow connected?

When I was five, we moved from a boisterous city block to the sterile environs of suburban Milwaukee, 90 miles and 4 light-years to the north. Better, I suppose, in some conventional 1950s sense, for my siblings and me, but it never felt as nurturing, comfortable, or alive. But this introduction is not intended to be an autobiography, so let me tell you about the Milwaukee Journal’s Green Sheet. This insert, literally green, was full of features that fascinated me. At the top was a saying by Phil Osopher that always contained some wonderfully puerile pun. There was also the “Ask Andy” column: science questions and brief answers. Phil and Andy became friends of mine. And then there was an advice column by a woman with the unlikely name of Ione Quinby Griggs, who gave no-nonsense Midwestern counsel. Of course, I also read the sports pages and occasionally even checked the first section to see what was happening in the larger world.

Every summer my siblings and I left Milwaukee and traveled to Denver, where my grandparents had retired. On long, timeless Saturday afternoons, I’d watch Dizzy Dean narrate the baseball game of the week on television and then listen through the static on my grandmother’s old radio-as-big-as-a-refrigerator as my hero, Eddie Matthews, hit home runs for the distant Milwaukee Braves. The next morning I’d run out to the newspaper box on the corner of Kierney and Colfax, deposit my 5 cents, and eagerly scour the Rocky Mountain News for the box scores. A few years later, I would scour the same paper for news of JFK.

Back home, my affair with the solid Milwaukee Journal deepened (local news, business pages, favorite columnists) until I left for the University of Wisconsin in Madison, at which time the feisty Capitol Times began to alienate my affections. Gradually my attitude toward newspapers matured and, upon moving to Philadelphia after marriage and graduate school, my devotion devolved into a simple adult appreciation of good newspaper reporting and writing. My former fetishism is still apparent, however, in the number of papers I read and in an excessive affection for their look, feel, smell, and peculiarities. I subscribe to the Philadelphia Inquirer and to the paper of record, the New York Times, which arrives in my driveway wrapped in blue plastic. I also regularly skim the Wall Street Journal and the Philadelphia Daily News, occasionally look at USA Today (when I feel a powerful urge to see weather maps in color), the Washington Post, the suburban Ambler Gazette, the Bar Harbor Times, the local paper of any city I happen to be visiting, the tabloids, and innumerable magazines.

At fairly regular intervals and despite the odd credential of a Ph.D. in mathematics, I even cross the line myself to review a book, write an article, or fulminate in an op-ed. But if I concentrate on it, reading the paper can still evoke the romance of distant and uncharted places.

One result of my unnatural attachment to newspapers is this book. Structured like the morning paper, A Mathematician Reads the Newspaper examines the mathematical angles of stories in the news. I consider newspapers not merely out of fondness, however. Despite talk of the ascendancy of multimedia and the decline of print media, I think the rational tendencies that newspapers foster will survive (if we do), and that in some form or other newspapers will remain our primary means of considered public discourse. As such, they should enhance our role as citizens and not reduce it to that of mere consumers and voyeurs (although there’s nothing wrong with a little buying and peeking). In addition to placing increased emphasis on analysis, background, and features, there is another, relatively unappreciated way in which newspapers can better fulfill this responsibility. It is by knowledgeably reflecting the increasing mathematical complexity of our society in its many quantitative, probabilistic, and dynamic facets.

This book provides suggestions on how this can be done. More important, it offers novel perspectives, questions, and recommendations to coffee drinkers, straphangers, policy makers, gossip mongers, bargain hunters, trendsetters, and others who can’t get along without their daily paper. Mathematical naïveté can put such readers at a disadvantage in thinking about many issues in the news that may seem not to involve mathematics at all. Happily, a sounder understanding of these issues can be obtained by reflecting on a few basic mathematical ideas, and even those who despised the subject in school will, I hope, find them fascinating, rewarding, and accessible here.

But perhaps you need a bit more persuasion. Pulitzer, after all, barely fits in the same sentence as Pythagoras. Newspapers are daily periodicals dealing with the changing details of everyday life, whereas mathematics is a timeless discipline concerned with abstract truth. Newspapers deal with mess and contingency and crime, mathematics with symmetry and necessity and the sublime. The newspaper reader is everyman, the mathematician an elitist. Furthermore, because of the mind-numbing way in which mathematics is generally taught, many people have serious misconceptions about the subject and fail to appreciate its wide applicability.

It’s time to let the secret out: Mathematics is not primarily a matter of plugging numbers into formulas and performing rote computations. It is a way of thinking and questioning that may be unfamiliar to many of us, but is available to almost all of us.

As we’ll see, “number stories” complement, deepen, and regularly undermine “people stories.” Probability considerations can enhance articles on crime, health risks, or racial and ethnic bias. Logic and self-reference may help to clarify the hazards of celebrity, media spin control, and reportorial involvement in the news. Business finance, the multiplication principle, and simple arithmetic point up consumer fallacies, electoral tricks, and sports myths. Chaos and nonlinear dynamics suggest how difficult and frequently worthless economic and environmental predictions are. And mathematically pertinent notions from philosophy and psychology provide perspective on a variety of public issues. All these ideas give us a revealing, albeit oblique, slant on the traditional Who, What, Where, When, Why, and How of the journalist’s craft.

The misunderstandings between mathematicians and others run in both directions. Out of professional myopia, the former sometimes fail to grasp the crucial element of a situation, as did the three statisticians who took up duck hunting. The first fired and his shot sailed six inches over the duck. Then the second fired and his shot flew six inches below the duck. At this, the third statistician excitedly exclaimed, “We got it!”

Be warned that, although the intent of this book is serious and its tone largely earnest, a few of the discussions may strike the reader as similarly off the mark. Nonetheless, the duck hunters (and I) will almost always have a point. My emphasis throughout will be on qualitative understanding, pertinence to daily life, and unconventional viewpoints. What new insights does mathematics give us into news stories and popular culture? How does it obscure and intimidate? What mathematical/psychological rules of thumb can guide us in reading the newspaper? Which numbers, relations, and associations are to be trusted, which dismissed as coincidental or nonsensical, which further analyzed, supplemented, or alternatively interpreted? (Don’t worry about the mathematics itself. It is either elementary or else is explained briefly in self-contained portions as needed. If you can find the continuation of a story on page Bl6, column 6, you’ll be okay.)

The format of the book will be loosely modeled after that of a standard newspaper, not that of a more mathematical tome. I’ll proceed through such a generic paper (The Daily E^xponent