# How Not to Be Wrong: The Power of Mathematical Thinking

## Excerpts

### WHEN AM I GOING TO USE THIS?

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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

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mathematician is always asking, “What assumptions are you making? And are they justified

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Mathematics is the study of things that come out a certain way because there is no other way they could possibly

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You can’t do calculus by common sense. But calculus is still derived from our common sense

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

### PART I: Linearity

#### One. LESS LIKE SWEDEN

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Mitchell’s reasoning is an example of false linearity — he’s assuming, without coming right out and saying so, that the course of prosperity is described by the line segment in the first picture, in which case Sweden stripping down its social infrastructure means we should do the same

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Nonlinear thinking means which way you should go depends on where you already are

#### Three. EVERYONE IS OBESE

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trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upward of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen

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Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense — or deciding whether the method is the right one to use in the first place — requires a guiding human hand

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who think math teaching should be about

#### Four. HOW MUCH IS THAT IN DEAD AMERICANS?

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An important rule of mathematical hygiene: when you’re field-testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something’s wrong with your method

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The understanding that the results of an experiment tend to settle down to a fixed average when the experiment is repeated again and again is not new. In fact, it’s almost as old as the mathematical study of chance itself; an informal form of the principle was asserted in the sixteenth century by Girolamo Cardano, though it was not until the early 1800s that Siméon-Denis Poisson came up with the pithy name “la loi des grands nombres” to describe it

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He did, and they are. But if you’ve already had three sons, a fourth son is not so unlikely at all. In fact, you’re just as likely to have a son as a first-time parent. This seems at first to be in conflict

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with the Law of Large Numbers, which ought to be pushing your brood to be split half and half between boys and girls

### PART II: Inference

#### Six. THE BALTIMORE STOCKBROKER AND THE BIBLE CODE

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company launches a mutual fund, they often maintain the fund in-house for some time before opening it to the public, a practice called incubation.

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It really is improbable that ten stock picks in a row would come out the right way, or that a magician who bet on six horse races would get the winner right every time, or that a mutual fund would beat the market by 10 %. The mistake is in being surprised by this encounter with the improbable. The universe is big, and if you’re sufficiently attuned to amazingly improbable occurrences, you’ll find them.

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Improbable things happen a lot.

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Aristotle, as usual, was here first: despite lacking any formal notion of probability, he was able to understand that “it is probable that improbable things will happen. Granted this, one might argue that what is improbable is probable.”

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McKay and Bar-Natan are making a potent point about the power of wiggle room. Wiggle room is what the Baltimore stockbroker has when he gives himself plenty of chances to win; wiggle room is what the mutual fund company has when it decides which of its secretly incubating funds are winners and which are trash. Wiggle room is what McKay and Bar-Natan used to work up a list of rabbinical names that jibed well with War and Peace.

#### Seven. DEAD FISH DON’T READ MINDS

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(And a well-executed one: I especially like the “Methods” section, which starts “One mature Atlantic Salmon (Salmo salar) participated in the fMRI study. The salmon was approximately 18 inches long, weighed 3.8 lbs, and was not alive at the time of scanning. .. . Foam padding was placed within the head coil as a method of limiting salmon movement during the scan, but proved to be largely unnecessary as subject motion was exceptionally low.”)

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Improbability, as described here, is a relative notion, not an absolute one; when we say an outcome is improbable, we are always saying, explicitly or not, that it is improbable under some set of hypotheses we’ve made about the underlying mechanisms of the world.

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If you flip a coin 82 times and get 82 heads, you ought to be thinking, “Something is biased about this coin,” not “God loves heads.”

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Back in 1966, the psychologist David Bakan wrote about the “crisis of psychology,” which in his view was a “crisis in statistical theory”:

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the dominant form of literary criticism at the time, “close reading,” bore the mark of Watson’s philosophy (though not as directly as Skinner did), displaying a very behaviorist preference for the words on the page over the unobservable intentions of the author.

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Stuff your sonnets with one or two extra alliterations each and you become one of the stone-footed poets mocked by Shakespeare’s fellow Elizabethan George Gascoigne: “Many writers indulge in repeticion of sundrie wordes all beginning with one letter, the whiche (beyng modestly used) lendeth good grace to a verse; but they do so hunt a letter to death, that they make it Crambe, and Crambe bis positum mors est.”

#### Eight. REDUCTIO AD UNLIKELY

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Put this way, the reductio sounds almost trivial, and in a sense, it is; but maybe it’s more accurate to say it’s a mental tool we’ve grown so used to handling that we forget how powerful it is. In fact, it’s a simple reductio that drives the Pythagoreans’ proof of the irrationality of the square root of 2; the one so awesomely paradigm-busting they had to kill its author; a proof so simple, refined, and compact that I can write it out whole in a page. Suppose H: the square root of 2 is a rational number

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Not a reductio ad absurdum, in other words, but a reductio ad unlikely.

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We are inexorably led to conclude, with a high degree of statistical confidence, that H is incorrect: the subjects in the sample are not human beings.

#### Nine. THE INTERNATIONAL JOURNAL OF HARUSPICY

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The loudest drummer is John Ioannidis, a Greek high school math star turned biomedical researcher whose 2005 paper “Why Most Published Research Findings Are False” touched off a fierce bout of self-criticism (and a second wave of self-defense) in the clinical sciences.

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problem — a scientific field has a drastically distorted view of the evidence for a hypothesis when public dissemination is cut off by a statistical significance threshold. But we’ve already given the problem another name. It’s the Baltimore stockbroker.

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The significance test is the detective, not the judge.

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oh-so-statistically-significant finding isn’t the conclusion of the scientific process, but the bare beginning.

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“in fact no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas.” In the next chapter we will see one way in which “the light of the evidence” might be made more specific.

#### Ten. ARE YOU THERE, GOD? IT’S ME, BAYESIAN INFERENCE

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chaotic. In fact, it was in the numerical study of weather that Edward Lorenz discovered the mathematical notion of chaos in the first place. He wrote, “One meteorologist remarked that if the theory were correct, one flap of a sea gull’s wing would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.”

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Try worrying about this. Suppose a team at Facebook decides to develop a method for guessing which of its users are likely to be involved in terrorism against the United States. Mathematically, it’s not so different from the problem of figuring out whether a Netflix user is likely to enjoy Ocean’s Thirteen. Facebook generally knows its users’ real names and locations, so it can use public records to generate a list of Facebook profiles belonging to people who have already been convicted of terroristic crimes or support of terrorist groups. Then the math starts. Do the terrorists tend to make more status updates per day than the general population, or fewer, or on this metric do they look basically the same?

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That’s not how the Iranian vote counts looked. There were too many 7s, almost twice as many as their fair share; not like digits derived from a random process, but very much like digits written down by humans trying to make them look random. This, by itself, isn’t proof that the election was fixed, but it’s evidence in that direction.

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putting patients inside a plastic replica of Stonehenge, would you grudgingly accept that the ancient formations were actually focusing vibrational earth energy on the body and stunning the tumors? You would not, because that’s nutty. You’d think Stonehenge probably got lucky.

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Relying purely on null hypothesis significance testing is a deeply non-Bayesian thing to do — strictly speaking, it asks us to treat the cancer drug and the plastic Stonehenge with exactly the same respect.

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there’s a 5 % chance that RED is true, we are making a statement not about the global distribution of biased roulette wheels (how could we know?) but rather about our own mental state. Five percent is the degree to which we believe that a roulette wheel we encounter is weighted toward the red. This is the point at which Fisher totally got off the bus, by the way. He wrote an unsparing pan of John Maynard Keynes’s Treatise on Probability, in which probability “measures the ‘degree of rational belief’ to which a proposition is entitled in the light of given evidence.” Fisher’s opinion of this viewpoint is well summarized by his closing lines: “If the views of the last section of Mr. Keynes’s book were accepted as authoritative by mathematical students in this country, they would be turned away, some in disgust, and most in ignorance, from one of the most promising branches of applied mathematics.”

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The Bayesian outlook is already enough to explain why RBRRB looks random while RRRRR doesn’t, even though both are equally improbable. When we see RRRRR, it strengthens a theory — the theory that the wheel is rigged to land red — to which we’ve already assigned some prior probability.

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We tend to like simpler theories better than more complicated ones, theories that rest on analogies to things we already know about better than theories that posit totally novel phenomena.

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“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth.” Doesn’t that sound cool, reasonable, indisputable? But it doesn’t tell the whole story. What Sherlock Holmes should have said was: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth, unless the truth is a hypothesis it didn’t occur to you to consider.” Less pithy, more correct.

### PART III: Expectation

#### Eleven. WHAT TO EXPECT WHEN YOU’RE EXPECTING TO WIN THE LOTTERY

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Harvard, back in the days before it enjoyed a nine-figure endowment, ran lotteries in 1794 and 1810 to fund two new college buildings. (They’re still used as dorms for first-year students today.)

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Adam Smith, too, was a lottery naysayer. In The Wealth of Nations, he wrote: That the chance of gain is naturally overvalued, we may learn from the universal success of lotteries. The world neither ever saw, nor ever will see, a perfectly fair lottery, or one in which the whole gain compensated the whole loss; because the undertaker could make nothing by it. .. . In a lottery in which no prize exceeded twenty pounds, though in other respects it approached much nearer to a perfectly fair one than the common state lotteries, there would not be the same demand for tickets. In order to have a better chance for some of the great prizes, some people purchase several tickets; and others, small shares in a still greater number. There is not, however, a more certain proposition in mathematics, than that the more tickets you adventure upon, the more likely you are to be a loser. Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty.

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EXPECTED VALUE IS NOT THE VALUE YOU EXPECT

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“But it’s obvious that you should charge younger people more!” It is not obvious. Rather, it is obvious if you already know it, as modern people do. But the fact that people who administered annuities failed to make this observation, again and again, is proof that it’s not actually obvious.

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Mathematics is filled with ideas that seem obvious now — that negative quantities can be added and subtracted, that you can usefully represent points in a plane by pairs of numbers, that probabilities of uncertain events can be mathematically described and manipulated — but are in fact not obvious at all. If they were, they would not have arrived so late in the history of human thought.

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This reminds me of an old story from the Harvard math department, concerning one of the grand old Russian professors, whom we shall call O. Professor O is midway through an intricate algebraic derivation when a student in the back row raises his hand. “Professor O, I didn’t follow that last step. Why do those two operators commute?” The professor raises his eyebrows and says, “Eet ees obvious.” But the student persists: “I’m sorry, Professor O, I really don’t see it.” So Professor O goes back to the board and adds a few lines of explanation. “What we must do? Well, the two operators are both diagonalized by. .. well, it is not exactly diagonalized but. .. just a moment. ..” Professor O pauses for a little while, peering at what’s on the board and scratching his chin. Then he retreats to his office. About ten minutes go by. The students are about to start leaving when Professor O returns, and again assumes his station in front of the chalkboard. “Yes,” he says, satisfied. “Eet ees obvious.”

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It starts with the game of franc-carreau, which, like the Genoese lottery, reminds you that people in olden times would gamble on just about anything. All you need for franc-carreau is a coin and a floor with square tiles. You throw the coin on the floor and make a bet: will it land wholly within one tile, or end up touching one of the cracks? (“Franc-carreau” translates roughly as “squarely within the square” — the coin used for this game was not a franc, which wasn’t in circulation at the time, but the écu.)

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Instead, we turn to the other strategy, which is the one Barbier used: make the problem harder. That doesn’t sound promising. But when it works, it works like a charm.

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It doesn’t matter where the circle falls — it crosses the lines in the floor exactly twice. So the expected number of crossings is 2; and it is also πp; and so we have discovered that p = 2 / π, just as Buffon said. In fact, the argument above applies to any needle, however polygonal and curvy it might be; the expected number of crossings is Lp, where L is the length of the needle in slat-width units. Throw a mass of spaghetti on the tile floor and I can tell you exactly how many times to expect a strand to cross a line. This generalized version of the problem is called, by mathematical wags, Buffon’s noodle problem.

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Barbier’s proof reminds me of what the algebraic geometer Pierre Deligne wrote of his teacher, Alexander Grothendieck: “Nothing seems to happen, and yet at the end a highly nontrivial theorem is there

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Grothendieck, who remade much of pure mathematics in his own image in the 1960s and’ 70s, had a different view: “The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration. .. the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it. .. yet it ﬁnally surrounds the resistant substance

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The unknown is a stone in the sea, which obstructs our progress. We can try to pack dynamite in the crevices of rock, detonate it, and repeat until the rock breaks apart, as Buffon did with his complicated computations in calculus. Or you can take a more contemplative approach, allowing your level of understanding gradually and gently to rise, until after a time what appeared as an obstacle is overtopped by the calm water, and is gone

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One person who figured this out was the mathematician and explorer Charles-Marie de La Condamine; just as Harvey would do almost three centuries later, he gathered his friends into a ticket-buying cartel. One of these was the young writer François-Marie Arouet, better known as Voltaire. While he might not have contributed to the mathematics of the

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scheme, Voltaire placed his stamp on it. Lottery players were to write a motto on their ticket, to be read aloud when a ticket won the jackpot; Voltaire, characteristically, saw this as a perfect opportunity to epigrammatize, writing cheeky slogans like “All men are equal!” and “Long live M. Peletier des Forts!” on his tickets for public consumption when the cartel won the prize

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Eventually, the state caught on and canceled the program, but not before La Condamine and Voltaire had taken the government for enough money to be rich men for the rest of their lives. What — you thought Voltaire made a living writing perfectly realized essays and sketches? Then, as now, that’s no way to get rich

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Their approach was governed by a simple maxim: if gambling is exciting, you’re doing it wrong.

#### Twelve. MISS MORE PLANES!

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George Stigler, the 1982 Nobelist in economics, used to say, “If you never miss the plane, you’re spending too much time in airports

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Some people even like to measure utility in standard units, called utils

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But what the Pensées are most famous for is thought 233, which Pascal titled “Infinite-rien” (“Infinity-nothing”) but which is universally known as “Pascal’s wager.”

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Again we see it: mathematics is the extension of common sense by other means

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he devised a famous experiment now known as Ellsberg’s paradox.

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We see here the characteristic push and pull of the mathematical approach to science. Mathematicians like Bernoulli and von Neumann construct formalisms that apply a penetrating light to a sphere of inquiry only dimly understood before; mathematically fluent scientists like Ellsberg work to understand the limits of those formalisms, to refine and improve them where it’s possible to do so, and to post strongly worded warning signs where it’s not.

#### Thirteen. WHERE THE TRAIN TRACKS MEET

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it’s dull, but it works. If retirement planning is exciting

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You don’t see people swagger up to the roulette wheel and lay one chip on every number; that’s just an unnecessarily elaborate way of handing chips to the dealer.

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Mathematical elegance and practical utility are close companions, as the history of science has shown again and again.

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In fact, the seven nonzero code words in the Hamming code match up exactly to the seven lines in the Fano plane.

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The Mars orbiter Mariner 9 sent pictures of the Martian surface back to Earth using one such code, the Hadamard code. Compact discs are encoded with the Reed-Solomon code, which is why you can scratch them and they still sound perfect.

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Hamming’s notion of “distance” follows Fano’s philosophy — a quantity that quacks like distance has the right to be called distance.

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The sphere-packing problem is a lot older than error-correcting codes; it goes back to the astronomer Johannes Kepler, who wrote a short booklet in 1611 called Strena Seu De Nive Sexangula, or “The Six-Cornered Snowflake.” Despite the rather specific title, Kepler’s book contemplates the general question of the origin of natural form.

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Here’s Kepler’s explanation. The pomegranate wants to fit as many seeds as possible inside its skin; in other words, it is carrying out a sphere-packing problem.

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almost all sets of code words exhibited the error-correcting property; in other words, a completely random code, with no design at all, was extremely likely to be an error-correcting code.

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Hamming, still impressed forty years later, said of Shannon’s proof in 1986: Courage is one of the things that Shannon had supremely. You have only to think of his major theorem. He wants to create a method of coding, but he doesn’t know what to do so he makes a random code. Then he is stuck. And then he asks the impossible question, “What would the average random code do?” He then proves that the average code is arbitrarily good, and that therefore there must be at least one good code. Who but a man of infinite courage could have dared to think those thoughts? That is the characteristic of great scientists; they have courage. They will go forward under incredible circumstances; they think and continue to think.

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Codes that have a lot of structure, like the Hamming codes, tend to be easy to decode. But these very special codes, it turns out, are usually not as efficient as the completely random ones that Shannon studied!

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In order to play the lottery without risk, it’s not enough to play hundreds of thousands of tickets; you have to play the right hundreds of thousands of tickets.

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Their “prospect theory,” for which Kahneman later won the Nobel Prize, is now seen as the founding document of behavioral economics, which aims to model with the greatest possible fidelity the way people do act, not the way that, according to an abstract notion of rationality, they should. In the Kahneman-Tversky theory, people tend to place more weight on low-probability events than a person obedient to the von Neumann-Morgenstern axioms would; so the allure of the jackpot exceeds what a strict expected utility calculation would license.

### PART IV: Regression

#### Fourteen. THE TRIUMPH OF MEDIOCRITY

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Charles Darwin wrote Galton in a kind of intellectual frenzy, not even waiting until he’d finished the book: DOWN, BECKENHAM, KENT, S.E. December 23rd MY DEAR GALTON, — I have only read about 50 pages of your book (to Judges), but I must exhale myself, else something will go wrong in my inside. I do not think I ever in all my life read anything more interesting and original — and how Well and clearly you put every point! George, who has finished the book, and who expressed himself in just the same terms, tells me that the earlier chapters are nothing in interest to the later ones!

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To be fair, Darwin might have been biased, being Galton’s first cousin.

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When a six-foot-two man and a five-foot-ten woman get married, their sons and daughters are likely to be taller than average. But now here is Galton’s remarkable discovery: those children are not likely to be as tall as their parents. The same goes for short parents, in the opposite direction; their kids will tend to be short, but not as short as they themselves are. Galton had discovered the phenomenon now called regression to the mean. His data left no doubt that it was real.

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“it is theoretically a necessary fact, * and one that is clearly confirmed by observation, that the Stature of the adult offspring must on the whole, be more mediocre than the stature of their Parents.”

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Galton was observing the same phenomenon that Secrist would uncover in the operations of business. Excellence doesn’t persist; time passes, and mediocrity asserts itself.

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But there’s one big difference between Galton and Secrist. Galton was, in his heart, a mathematician, and Secrist was not. And so Galton understood why regression was taking place, while Secrist remained in the dark.

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That’s what causes regression to the mean: not a mysterious mediocrity-loving force, but the simple workings of heredity intermingled with chance. That’s why Galton writes that regression to the mean is “theoretically a necessary fact.”

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Why is the second novel by a breakout debut writer, or the second album by an explosively popular band, so seldom as good as the first? It’s not, or not entirely, because most artists only have one thing to say. It’s because artistic success is an amalgam of talent and fortune, like everything else in life, and thus subject to regression to the mean.

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Ballplayers get this. Derek Jeter, when bugged about being on pace to break Pete Rose’s career hit record, told the New York Times, “One of the worst phrases in sports is ‘on pace for.’ ” Wise words!

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“when correctly interpreted, is essentially trivial. .. . To ‘prove’ such a mathematical result by a costly and prolonged numerical study of many kinds of business profit and expense ratios is analogous to proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals. The performance, though perhaps entertaining, and having a certain pedagogical value, is not an important contribution either to zoölogy or mathematics.”

#### Fifteen. GALTON’S ELLIPSE

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Galton had invented the type of graph we now call a scatterplot.

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scatterplot along which the density of points was roughly constant. Curves of this kind are called isopleths, and they’re very familiar to you, if not under that tongue-twisting name.

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The isopleth wasn’t Galton’s invention; the first published isoplethic map was produced in 1701 by Edmond Halley,

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The curves on Halley’s map were isogons, showing sailors where the discrepancy between magnetic north and true north was constant.

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The data was based on measurements Halley made aboard the Paramore, which crossed the Atlantic several times with Halley himself at the helm. (This guy really knew how to keep busy between comets.)

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Galton called his measure correlation, the term we still use today.

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Bertillon was appalled by the unsystematic and haphazard way in which French police identified criminal suspects. How much better and more modern it would be, Bertillon reasoned, to attach to each miscreant Frenchman a series of numerical measurements: the length and breadth of the head, the length of fingers and feet, and so on. In Bertillon’s system, each arrested suspect was measured and his data filed on cards and stored away for future use. Now, if the same man were nabbed again, identifying him was a simple matter

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of getting out the calipers, taking his numbers, and comparing them with the cards on file. “Aha, Mr. 15-6-56-42, thought you’d get away, didn’t you?” You can replace your name by an alias, but there’s no alias for the shape of your head.

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To study this phenomenon, Galton made another scatterplot, this one of height versus “cubit,” the distance from the elbow to the tip of the middle finger.

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As it happens, Galton’s invention of correlation didn’t lead to the institution of a vastly improved Bertillon system. That was largely thanks to Galton himself, who championed a competing system, dactyloscopy — what we now call fingerprinting.

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It’s a neat trick, keeping track of something complicated like the shape of a human being with a short string of symbols. And the trick isn’t limited to human physiognomy.

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A similar system, called the Parsons code, * is used to classify musical melodies. Here’s how it goes. Take a melody — one we all know, like Beethoven’s “Ode to Joy,” the glorious finale of the Ninth Symphony. We mark the first note with a *. And for each note thereafter, you mark down one of three symbols: u if the note at hand goes up from the previous note, d if it goes down, or r if it repeats the note that came before. The first two notes of Ode to Joy are the same, so you start out with * r. Then a higher note followed by a still higher one: * ruu. Next you repeat the top note, and then follow with a string of four descents: so the code for the whole opening segment is * ruurdddd.

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Galton’s great insight was that the same thing applies even if finger length and cubit length aren’t identical, but only correlated.

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Darwin showed that one could meaningfully talk about progress without any need to invoke purpose. Galton showed that one could meaningfully talk about association without any need to invoke underlying cause.

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And this is Pearson’s formula, in geometric language. The correlation between the two variables is determined by the angle between the two vectors.

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Slowly this usage is creeping out of the geekolect into the wider language. You can just about see it happening in a recent Supreme Court oral argument: MR. FRIEDMAN: I think that issue is entirely orthogonal to the issue here because the Commonwealth is acknowledging — CHIEF JUSTICE ROBERTS: I’m sorry. Entirely what? MR. FRIEDMAN: Orthogonal. Right angle. Unrelated. Irrelevant. CHIEF JUSTICE ROBERTS: Oh. JUSTICE SCALIA: What was that adjective? I like that. MR. FRIEDMAN: Orthogonal. JUSTICE SCALIA: Orthogonal? MR. FRIEDMAN: Right, right. JUSTICE SCALIA: Ooh. (Laughter.) I’m rooting for orthogonal to catch on. It’s been a while since a mathy word really broke out into demotic English. Lowest common denominator has by now lost its mathematical flavor almost entirely, and exponentially — just don’t get me started on exponentially.

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But there seems to be a paradox here. Being rich is positively correlated with being from a rich state, more or less by definition. And being from a rich state is positively correlated with voting for Democrats. Doesn’t that mean being rich has to be correlated with voting Democratic? Geometrically: if vector 1 is at an acute angle to vector 2, and vector 2 is at an acute angle to vector 3, does vector 1 have to be at an acute angle to vector 3? No! Proof by picture:

#### Sixteen. DOES LUNG CANCER MAKE YOU SMOKE CIGARETTES?

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A typical way to express that fact is to say “If you’re a smoker, you’re less likely to be married.” But one small change makes the meaning very different: “If you were a smoker, you’d be less likely to be married.” It seems strange that changing the sentence from the indicative to the subjunctive mood can change what it says so drastically. But the first sentence is merely a statement about what is the case. The second concerns a much more delicate question: What would be the case if we changed something about the world?

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Their financial success is correlated, but it’s not because Tim’s fund is causing Sara’s to take off, or the reverse. It’s because there’s a mystery factor, the Honda stock, that affects both Tim and Sara. Clinical researchers call this the surrogate endpoint problem. It’s time consuming and expensive to check whether a drug improves average life span, because in order to record someone’s life span you have to wait for them to die.

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Our old friend R. A. Fisher, the founding hero of modern statistics, was a vigorous skeptic of the tobacco-cancer link on exactly those grounds.

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Fisher was the natural intellectual heir to Galton and Pearson; in fact, he succeeded Pearson in 1933 as the Galton Chair of Eugenics at University College, London. (In deference to modern sensibilities, the position is now called the Galton Chair of Genetics.)

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One sees here both a brilliant and rigorous statistician’s demand that all possibilities receive fair consideration, and a lifelong smoker’s affection for his habit.

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The epidemiologist Jan Vandenbroucke wrote of Fisher’s articles on tobacco, “To my surprise, I found extremely well-written and cogent papers that might have become textbook classics for their impeccable logic and clear exposition of data and argument if only the authors had been on the right side.”

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Just how fierce the dispute remained, even within the scientific establishment, is made clear by the remarkable work of historian of medicine Jon Harkness.

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Makers of public policy don’t have the luxury of uncertainty that scientists do. They have to form their best guesses and make decisions on the basis thereof. When the system works — as it unquestionably did in the case of tobacco — the

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And if we held ourselves to a stricter evidentiary standard, declining to issue any of these recommendations because we weren’t sure we were right? Then the lives we would have saved would be lost instead.

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It’s a lot like George Stigler’s advice about missing planes. If you never give advice until you’re sure it’s right, you’re not giving enough advice.

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Correlations can also come from common effects. This phenomenon is known as Berkson’s fallacy, after the medical statistician Joseph Berkson, who back in chapter 8 explained how blind reliance on p-values could lead you to conclude that a small group of people including an albino consisted of nonhumans.

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The two conditions are negatively correlated, but that’s not because one causes the absence of the other. It’s also not because there’s a hidden factor that both raises your blood pressure and helps regulate your insulin. It’s because the two conditions have a common effect — namely, they put you in the hospital.

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To put it in words: if you’re in the hospital, you’re there for a reason. If you’re not diabetic, that makes it more likely the reason is high blood pressure. So what looks at first like a causal relationship between high blood pressure and diabetes is really just a statistical phantom.

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But only 18 of the 28 nondiabetics, or 64 %, have high blood pressure. That makes it seem that high blood pressure makes you more likely to have diabetes. But again, it’s an illusion; all we’re measuring is the fact that the set of people who end up in the hospital is anything but a random sample of the population.

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You may have noticed that, among the men * in your dating pool, the handsome ones tend not to be nice, and the nice ones tend not to be handsome. Is that because having a symmetrical face makes you cruel? Or because being nice to people makes you ugly? Well, it could be. But it doesn’t have to be. I present below the Great Square of Men:

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So inside the Great Square is a Smaller Triangle of Acceptable Men: And now the source of the phenomenon is clear. The handsomest men in your triangle run the gamut of personalities, from kindest to cruelest. On average, they’re about as nice as the average person in the whole population, which, let’s face it, is not that nice. And by the same token, the nicest men are only averagely handsome. The ugly guys you like, though — they make up a tiny corner of the triangle, and they are pretty darn nice — they have to be, or they wouldn’t be visible to you at all.

### PART V: Existence

#### Seventeen. THERE IS NO SUCH THING AS PUBLIC OPINION

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“The most plausible reading of this data is that the public wants a free lunch,” economist Bryan Caplan wrote. “They hope to spend less on government without touching any of its main functions.”

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That’s the familiar self-contradicting position we see in polls: We want to cut! But we also want each program to keep all its funding! How did we get to this impasse? Not because the voters are stupid or delusional. Each voter has a perfectly rational, coherent political stance. But in the aggregate, their position is nonsensical.

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the infantile “average American,” who wants to cut spending but demands to keep every single program, doesn’t exist. The average American thinks there are plenty of non-worthwhile federal programs that are wasting our money and is ready and willing to put them on the chopping block to make ends meet. The problem is, there’s no consensus on which programs are the worthless

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The “majority rules” system is simple and elegant and feels fair, but it’s at its best when deciding between just two options. Any more than two, and contradictions start to seep into the majority’s preferences.

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There are (at least) three choices here: leave the health care law alone, kill it, or make it stronger. And each of the three choices is opposed by most Americans.

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Here’s how Fox News might report the poll results above: Majority of Americans oppose Obamacare! And this is how it might look on MSNBC: Majority of Americans want to preserve or strengthen Obamacare!

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I think the right answer is that there are no answers. Public opinion doesn’t exist. More precisely, it exists sometimes, concerning matters about which there’s a clear majority view.

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The mathematical buzzword in play here is “independence of irrelevant alternatives.” That’s a rule that says, whether you’re a slime mold, a human being, or a democratic nation, if you have a choice between two options, A and B, the presence of a third option, C, shouldn’t affect which of A and B you like better. If you’re deciding whether you’d rather have a Prius or a Hummer, it doesn’t matter whether you also have the option of a Ford Pinto. You know you’re not going to choose the Pinto. So what relevance could it have? Or, to keep it closer to politics: in place of an auto dealership, put the state of Florida. In place of the Prius, put Al Gore. In place of the Hummer, put George W. Bush. And in place of the Ford Pinto, put Ralph Nader. In the 2000 presidential election, George Bush got 48.85 % of Florida’s votes and Al Gore got 48.84 %. The Pinto got 1.6 %.

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In other words: the slime mold likes the small, unlit pile of oats about as much as it likes the big, brightly lit one. But if you introduce a really small unlit pile of oats, the small dark pile looks better by comparison; so much so that the slime mold decides to choose it over the big bright pile almost all the time. This phenomenon is called the “asymmetric domination effect,”

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The appeal of instant-runoff voting (or “preferential voting,” as they call it in Australia) is obvious. People who like Ralph Nader can vote for him without worrying that they’re throwing the race to the person they like least. For that matter, Ralph Nader can run without worrying about throwing the race to the person he likes least. * Instant-runoff voting (IRV) has been around for more than 150 years. They use it not only in Australia but in Ireland and Papua New Guinea. When John Stuart Mill, who always had a soft spot for math, heard about the idea, he said it was “among the very greatest improvements yet made in the theory and practice of government.”

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In other words, a majority of voters liked the centrist candidate Montroll better than Kiss, and a majority of voters liked Montroll better than Wright. That’s a pretty solid case for Montroll as the rightful winner — and yet Montroll was tossed in the first round. Here you see one of IRV’s weaknesses. A centrist candidate who’s liked pretty well by everyone, but is nobody’s first choice, has a very hard time winning. To sum up: Traditional American voting method: Wright wins Instant-runoff method: Kiss wins Head-to-head matchups: Montroll wins

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Vexing circles like this are called Condorcet paradoxes, after the French Enlightenment philosopher who first discovered them in the late eighteenth century.

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Marie-Jean-Antoine-Nicolas de Caritat, Marquis de Condorcet, was a leading liberal thinker in the run-up to the French Revolution, eventually becoming president of the Legislative Assembly. He was an unlikely politician — shy and prone to exhaustion, with a speaking style so quiet and hurried that his proposals often went unheard in the raucous revolutionary chamber. On the other hand, he became quickly exasperated with people whose intellectual standards didn’t match his own. This combination of timidity and temper led his mentor Jacques Turgot to nickname him “le mouton enragé,” or “the rabid sheep.”

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It’s a little like the World Series. If the Phillies and the Tigers are facing off, and we agree that the Phillies are a bit better than the Tigers — say, they have a 51 % chance of winning each game — then the Phillies are more likely to win the Series 4 − 3 than to lose by the same margin. If the World Series were best of fifteen instead of best of seven, Philadelphia’s advantage would be even greater. Condorcet’s so-called “jury theorem” shows that a sufficiently large jury is very likely to arrive at the right outcome, as long as the jurors have some individual bias toward correctness, no matter how small.

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When it came to voting, Condorcet was every inch the mathematician. A typical person might look at the results of Florida 2000 and say, “Huh, weird: a more left-wing candidate ended up swinging the election to the Republican.” Or they might look at Burlington 2009 and say, “Huh, weird: the centrist guy who most people basically liked got thrown out in the first round.” For a mathematician, that “Huh, weird” feeling comes as an intellectual challenge. Can you say in some precise way what makes it weird? Can you formalize what it would mean for a voting system not to be weird?

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Condorcet thought he could. He wrote down an axiom — that

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If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people’s choice.

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In the pie chart above, Condorcet’s axiom says Montroll cannot be elected, because he loses the head-to-head matchup to Wright. The same goes for Wright, who loses to Kiss, and for Kiss, who loses to Montroll. There is no such thing as the people’s choice. It just doesn’t exist.

#### Eighteen. “OUT OF NOTHING I HAVE CREATED A STRANGE NEW UNIVERSE”

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The proper business of the state is not to count the votes as accurately as possible — to know what actually happened — but to obey the formal protocol that tells us, in Hardy’s terms, who the winner should be defined to be.

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Hilbert is so revered that any work that touches even tangentially on one of his problems takes on a little extra shine, even a hundred years later. I once met a historian of German culture in Columbus, Ohio, who told me that Hilbert’s predilection for wearing sandals with socks is the reason that fashion choice is still noticeably popular among mathematicians today. I could find no evidence this was actually true, but it suits me to believe it, and it gives a correct impression of the length of Hilbert’s shadow.

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Hilbert saw formalism as a way of starting over clean, building on a foundation so basic as to be completely incontrovertible.

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Russell closes the letter by expressing regret that Frege had not yet published the second volume of his Grundgesetze (“Foundations”). In fact, the book was finished and already in press when Frege received Russell’s letter. Despite the respectful tone (“I have encountered a difficulty,” not “Hi, I’ve just borked your life’s work”), Frege understood at once what Russell’s paradox meant for his version of set theory. It was too late to change the book, but he hurriedly appended a postscript recording Russell’s devastating insight. Frege’s explanation is perhaps the saddest sentence ever written in a technical work of mathematics: “Einem wissenschaftlichen Schriftsteller kann kaum etwas Unerwünschteres begegnen, als dass ihm nach Vollendung einer Arbeit eine der Grundlagen seines Baues erschüttert wird.” Or: “A scientist can hardly encounter anything more undesirable than, just as a work is completed, to have its foundation give way.”

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One of the most painful parts of teaching mathematics is seeing students damaged by the cult of the genius. The genius cult tells students it’s not worth doing mathematics unless you’re the best at mathematics, because those special few are the only

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The cult of the genius also tends to undervalue hard work. When I was starting out, I thought “hardworking” was a kind of veiled insult — something to say about a student when you can’t honestly say they’re smart.

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What you learn after a long time in math — and I think the lesson applies much more broadly — is that there’s always somebody ahead of you, whether they’re right there in class with you or not.

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Nobody ever looks in the mirror and says, “Let’s face it, I’m smarter than Gauss.” And yet, in the last hundred years, the joined effort of all these dummies-compared-to-Gauss has produced the greatest flowering of mathematical knowledge the world has ever seen.

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But the problem wasn’t the voters; it was the math. Condorcet, we now understand, was doomed to failure from the start. Kenneth Arrow, in his 1951 PhD thesis, proved that even a much weaker set of axioms than Condorcet’s, a set of requirements that seem as hard to doubt as Peano’s rules of arithmetic, leads to paradoxes.

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but it surely would have disappointed Condorcet, just as Gödel’s Theorem had disappointed Hilbert.

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Just as Hilbert’s style of mathematics persisted despite the destruction of his formal program by Gödel, Condorcet’s approach to politics survived his demise.

### HOW TO BE RIGHT

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This sounds weird, but as a logical deduction it’s irrefutable; drop one tiny contradiction anywhere into a formal system and the whole thing goes to hell. Philosophers of a mathematical bent call this brittleness in formal logic ex falso quodlibet, or, among friends, “the principle of explosion.”

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In fact, it’s a common piece of folk advice — I know I heard it from my PhD adviser, and presumably he from his, etc. — that when you’re working hard on a theorem you should try to prove it by day and disprove it by night. (The precise frequency of the toggle isn’t critical; it’s said of the topologist R. H. Bing that his habit was to split each month between two weeks trying to prove the Poincaré Conjecture and two weeks trying to find a counterexample. *)

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For if you have really understood what’s keeping you from disproving the theorem, you very likely understand, in a way inaccessible to you before, why the theorem is true.

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Scott Fitzgerald was talking about, by the way. His endorsement of holding contradictory beliefs comes from “The Crack-Up,” his 1936 essay about his own irreparable brokenness. The opposing ideas he has in mind there are “the sense of futility of effort and the sense of the necessity to struggle.” Samuel Beckett later put it more succinctly: “I can’t go on, I’ll go on.”

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It’s not clear how much higher math Beckett knew, but in his late prose piece Worstward Ho, he sums up the value of failure in mathematical creation more succinctly than any professor ever has: Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.