How not to be Wrong

Highlights

Page 68

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.

Page 156

One thing the American defense establishment has traditionally understood very well is that countries don’t win wars just by being braver than the other side, or freer, or slightly preferred by God. The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost. That’s not the stuff war movies are made of, but it’s the stuff wars are made of. And there’s math every step of the way.

Page 186

the dead funds together with the surviving ones, the rate of return dropped down to 134.5%, a much more ordinary 8.9% per year.

Page 230

The specialized language in which mathematicians converse with each other is a magnificent tool for conveying complex ideas precisely and swiftly. But its foreignness can create among outsiders the impression of a sphere of thought totally alien to ordinary thinking. That’s exactly wrong.

Page 409

Laffer’s conjecture that lower tax rates would raise tax revenue. When Reagan cut taxes after he was elected, the result was less tax revenue, not more. Revenue from personal income taxes (per person, adjusted for inflation) fell by 9 percent from 1980 to 1984, even though average income (per person, adjusted for inflation) grew by 4 percent over this period. Yet once the policy was in place, it was hard to reverse.

Page 415

Friedman’s famous slogan on taxation is “I am in favor of cutting taxes under any circumstances and for any excuse, for any reason, whenever it’s possible.”

Page 422

Mankiw also points out that the very richest people—the ones who’d been paying 70% on the top tranche of their income— did contribute more tax revenue after Reagan’s tax cuts.* That leads to the somewhat vexing possibility that the way to maximize government revenue is to jack up taxes on the middle class, who have no choice but to keep on working, while slashing rates on the rich; those guys have enough stockpiled wealth to make credible threats to withhold or offshore their economic activity, should their government charge them a rate they deem too high. If that story’s right, a lot of liberals will uncomfortably climb in the boat with Milton Friedman: maybe maximizing tax revenue isn’t so great after all.

Page 444

It could be the case that lowering taxes will increase government revenue;        I want it to be the case that lowering taxes will increase government revenue;        Therefore, it is the case that lowering taxes will increase government revenue.

Page 547

Now here’s the conceptual leap. Newton said, look, let’s go all the way. Reduce your field of view until it’s infinitesimal—so small that it’s smaller than any size you can name, but not zero. You’re studying the missile’s arc, not over a very short time interval, but at a single moment. What was almost a line becomes exactly a line. And the slope of this line is what Newton called the fluxion, and what we’d now call the derivative. That’s a kind of jump Archimedes wasn’t willing to make. He understood that polygons with shorter sides got closer and closer to the circle— but he would never have said that the circle actually was a polygon with infinitely many infinitely short sides.

Page 560

so naturally to us: our intuition about time and motion is formed by the phenomena we observe in the world. Even before Newton codified his laws, something in us knew that things like to move in straight lines, unless given a reason to do otherwise.

Page 633

some people said yes, including the Italian mathematician/priest Guido Grandi, after whom the series 1 − 1 + 1 − 1 + 1 − 1 + … is usually named; in a 1703 paper, he argued that the sum of the series is 1/2, and moreover that this miraculous conclusion represented the creation of the universe from nothing. (Don’t worry, I don’t follow that last step either.)

Page 770

There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

Page 780

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. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel. And let’s be frank: that really is what many of our math courses are doing.

Page 873

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.

Page 008

If you want to make the error bar half as big, you need to survey four times as many people.

Note: Bochra agrees

Page 040

That’s how the Law of Large Numbers works: not by balancing out what’s already happened, but by diluting what’s already happened with new data, until the past is so proportionally negligible that it can safely be forgotten.

Page 081

The slogan to live by here is: Don’t talk about percentages of numbers when the numbers might be negative.

Page 603

It’s not enough that the data be consistent with your theory; they have to be inconsistent with the negation of your theory, the dreaded null hypothesis. I may assert that I possess telekinetic abilities so powerful that I can drag the sun out from beneath the horizon—if you want proof, just go outside at about five in the morning and see the results of my work! But this kind of evidence is no evidence at all, because, under the null hypothesis that I lack psychic gifts, the sun would come up just the same.

Page 742

If only we could go back in time to the dawn of statistical nomenclature and declare that a result passing Fisher’s test with a p-value of less than 0.05 was “statistically noticeable” or “statistically detectable” instead of “statistically significant”! That would be truer to the meaning of the method, which merely counsels us about the existence of an effect but is silent about its size or importance. But it’s too late for that. We have the language we have.*

Page 811

A statistical study that’s not refined enough to detect a phenomenon of the expected size is called underpowered—the equivalent of looking at the planets with binoculars.

Page 865

Computer scientists Kevin Korb and Michael Stillwell worked out exactly that in a 2003 paper. They generated simulations with a hot hand built in: the simulated player’s shooting percentage leaped up all the way to 90% for two ten-shot “hot” intervals over the course of the trial. In more than three-quarters of those simulations, the significance test used by GVT reported that there was no reason to reject the null hypothesis—even though the null hypothesis was completely false. The GVT design was underpowered, destined to report the nonexistence of the hot hand even if the hot hand was real.

Page 309

For Neyman and Pearson, the purpose of statistics isn’t to tell us what to believe, but to tell us what to do. Statistics is about making decisions, not answering questions. A significance test is no more or less than a rule, which tells the people in charge whether to approve a drug, undertake a proposed economic reform, or tart up a website. It sounds crazy at first to deny that the goal of science is to find out what’s true, but the Neyman-Pearson philosophy is not so far from reasoning we use in other spheres. What’s the purpose of a criminal trial? We might naively say it’s to find out whether the defendant actually committed the crime they’re on trial for. But that’s obviously wrong. There are rules of evidence, which forbid the jury from hearing testimony obtained improperly, even if it might help them accurately determine the defendant’s innocence or guilt. The purpose of a court is not truth, but justice. We have rules, the rules must be obeyed, and when we say that a defendant is “guilty” we mean, if we are careful about our words, not that he committed the crime he’s accused of, but that he was convicted fair and square according to those rules. Whatever rules we choose, we’re going to let some criminals go free and imprison some of the blameless. The less you do of the first, the more you’re likely to do of the second. So we try to design the rules in whatever way society thinks we best handle that fundamental trade-off.

Page 335

But Fisher certainly understood that clearing the significance bar wasn’t the same thing as finding the truth. He envisions a richer, more iterated approach, writing in 1926: “A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance.” Not “succeeds once in giving,” but “rarely fails to give.” A statistically significant finding gives you a clue, suggesting a promising place to focus your research energy. The significance test is the detective, not the judge.

Page 494

You might well think that Facebook would never cook up a list of potential terrorists (or tax cheats, or pedophiles) or make the list public if they did. Why would they? Where’s the money in it?

Page 647

First of all: when we call Bayes’s Theorem a theorem it suggests we are discussing incontrovertible truths, certified by mathematical proof. That’s both true and not. It comes down to the difficult question of what we mean when we say “probability.” When we say that there’s a 5% chance that RED is true, we might mean that there actually is some vast global population of roulette wheels, of which exactly one in twenty is biased to fall red 3/5 of the time, and that any given roulette wheel we encounter is randomly picked from the roulette wheel multitude. If that’s what we mean, then Bayes’s Theorem is a plain fact, akin to the Law of Large Numbers we saw in the last chapter; it says that, in the long run, under the conditions we set up in the example, 12% of the roulette wheels that come up RRRRR are going to be of the red-favoring kind. But this isn’t actually what we’re talking about. When we say that 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.

Page 767

“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.

Page 848

Defining religion as ‘belief in gods’ is also problematic. We tend to say that a devout Christian is religious because she believes in God, whereas a fervent communist isn’t religious because communism has no gods. However, religion is created by humans rather than by gods, and it is defined by its social function rather than by the existence of deities. Religion is any all-encompassing story that confers superhuman legitimacy on human laws, norms and values. It legitimises social structures by arguing that they reflect superhuman laws.

Page 916

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. 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.

Page 353

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.

Page 376

It might look like the state’s losing money that day, but that’s taking a limited view. Those millions never belonged to Massachusetts; they were earmarked as prize money from the beginning. The state takes its 80 cents out of each ticket and gives back the rest. The more tickets sold, the more revenue comes in. The state doesn’t care who wins. The state just cares how many people play. So when the betting cartels cashed in the fat profits on their roll-down bets, they weren’t taking money from the state. They were taking it from the other players, especially the ones who made the bad decision to play the lottery on days without a roll-down. The cartels weren’t beating the house. They were the house.

Note: Feel very happy that I figured this out

Page 387

for Harvey and the other high-volume bettors, excitement wasn’t the point. Their approach was governed by a simple maxim: if gambling is exciting, you’re doing it wrong.

Page 552

Voltaire dubbed Pascal “the sublime misanthrope” and devoted a long essay to knocking down the gloomy Pensées piece by piece. His attitude toward Pascal is that of the popular smart kid toward the bitter and nonconforming nerd.

Note: Haha

Page 706

In the decision-theory literature, the former kind of unknown is called risk, the latter uncertainty.

Note: I figured this out and used the same word - uncertainty

Page 754

A risky investment can make sense even if you don’t have the money to cover your losses—as long as you have a backup plan. A certain market move might come with a 99% chance of making a million dollars and a 1% chance of losing $50 million. Should you make that move? It has a positive expected value, so it seems like a good strategy. But you might also balk at the risk of absorbing such a big loss—especially because small probabilities are notoriously hard to be certain about.* The pros call moves like this “picking up pennies in front of a steamroller”—most of the time you make a little money, but one small slip and you’re squashed.

Page 768

Financial firms are not human, and most humans, even rich humans, don’t like uncertainty. The rich investor might happily take the 50-50 bet with an expected value of $50,000, but would probably prefer to take the $50,000 outright. The relevant term of art is variance, a measure of how widely spread out the possible outcomes of a decision are, and how likely one is to encounter the extremes on either end. Among bets with the same expected dollar value, most people, especially people without limitless liquid assets, prefer the one with lower variance. That’s why some people invest in municipal bonds, even though stocks offer higher rates of return in the long run. With bonds, you’re sure you’re going to get your money. Invest in stocks, with their greater variance, and you’re likely to do better—but you might end up much worse.

Page 028

Shannon, in the paper that launched the theory of information, identified the basic tradeoff that engineers still grapple with today: the more resistant to noise you want your signal to be, the slower your bits are transmitted. The presence of noise places a cap on the length of a message your channel can reliably convey in a given amount of time; this limit was what Shannon called the capacity of the channel. Just as a pipe can only handle so much water, a channel can only handle so much information.

Page 103

What makes the Hamming code work? To understand this, you have to come at it from the other direction, asking: What would make it fail? Remember, the bête noire of an error-correcting code is a block of digits that’s simultaneously close to two different code words.

Page 208

And if machine intelligences of the future can take over from us much of the work we know as research now? We’ll reclassify that research as “computation.” And whatever we quantitatively minded humans are doing with our newly freed-up time, that’s what we’ll call “mathematics.”

Page 217

you’re naturally inclined to think an error-correcting code is a very special thing, designed and engineered and tweaked and retweaked until every pair of code words has been gingerly nudged apart without any other pair being forced together. Shannon’s genius was to see that this vision was totally wrong. Error-correcting codes are the opposite of special. What Shannon proved—and once he understood what to prove, it was really not so hard—was that 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.

Page 327

Pascal sees the pleasures of gambling as contemptible. And enjoyed to excess, they can of course be harmful. The reasoning that endorses lotteries also suggests that methamphetamine dealers and their clients enjoy a similar win-win relationship. Say what you want about meth, you can’t deny it is broadly and sincerely enjoyed.*

Note: Can’t deny

Page 334

That’s the nature of entrepreneurship: you balance a very, very small probability of making a fortune against a modest probability of eking out a living against a substantially larger probability of losing your pile, and for a large proportion of potential entrepreneurs, when you crunch the numbers, the expected financial value, like that of a lottery ticket, is less than zero. Typical entrepreneurs (like typical lottery customers) overrate their chance of success. Even businesses that survive typically make their proprietors less money than they’d have drawn in salary from an existing company. And yet society benefits from a world in which people, against their wiser judgment, launch businesses. We want restaurants, we want barbers, we want smartphone games. Is entrepreneurship “a tax on the stupid”? You’d be called crazy if you said so. Part of that is because we esteem a business owner more highly than we do a gambler; it’s hard to separate our moral feelings about an activity from the judgments we make about its rationality. But part of it—the biggest part—is that the utility of running a business, like the utility of buying a lottery ticket, is not measured only in expected dollars. The very act of realizing a dream, or even trying to realize it, is part of its own reward.

Page 421

Galton made his case by means of a detailed study of British men of achievement, from clerics to wrestlers, arguing that notable Englishmen* tend to have disproportionately notable relatives.

Note: What a muppet

Page 572

“The thesis of the book,” he writes in response, “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.”

Note: Bah gawd, that man had a family

Page 587

Biologists are eager to think regression stems from biology, management theorists like Secrist want it to come from competition, literary critics ascribe it to creative exhaustion—but it is none of these. It is mathematics.

Page 594

“All those with rapid times slowed down towards 48 hours … those with medium length transits showed no change … and those with slow transit times tended to speed up towards 48 hours. Thus bran tended to modify both slow and fast initial transit times towards a 48-hour mean.” This, of course, is precisely what you’d expect if bran had no effect at all. To put it delicately, we all have our fast days and our slow days, whatever our underlying level of intestinal health. And an unusually quick transit on Monday is likely to be followed by a more average transit time on Tuesday, bran or no bran.*

Note: Where was the control group ffs

Page 680

When the son’s height is completely unrelated to those of the parents, as in the second scatterplot above, Galton’s ellipses are all circles, and the scatterplot looks roughly round. When the son’s height is completely determined by heredity, with no chance element involved, as in the first scatterplot, the data lies along a straight line, which one might think of as an ellipse that has gotten as elliptical as it possibly can. In between, we have ellipses of various levels of skinniness. That skinniness, which the classical geometers called the eccentricity of the ellipse, is a measure of the extent to which the height of the father determines that of the son. High eccentricity means that heredity is powerful and regression to the mean is weak; low eccentricity means the opposite, that regression to the mean holds sway. Galton called his measure correlation, the term we still use today. If Galton’s ellipse is almost round, the correlation is near 0; when the ellipse is skinny, lined up along the northeast-southwest axis, the correlation comes close to 1.

Note: Bhaijaan, this is so elegant

Page 708

Kepler showed (although it took the astronomical community some decades to catch on) that the planets traveled in elliptical orbits, not circular ones as had been previously thought. Now, the very same curve arises as the natural shape enclosing heights of parents and children. Why? It’s not because there’s some hidden cone governing heredity which, when lopped off at just the right angle, gives Galton’s ellipses. Nor is it that some form of genetic gravity enforces the elliptical form of Galton’s charts via Newtonian laws of mechanics. The answer lies in a fundamental property of mathematics—in a sense, the very property that has made mathematics so magnificently useful to scientists. In math there are many, many complicated objects, but only a few simple ones. So if you have a problem whose solution admits a simple mathematical description, there are only a few possibilities for the solution. The simplest mathematical entities are thus ubiquitous, forced into multiple duty as solutions to all kinds of scientific problems. The simplest curves are lines. And it’s clear that lines are everywhere in nature, from the edges of crystals to the paths of moving bodies in the absence of force. The next simplest curves are those cut out by quadratic equations,* in which no more than two variables are ever multiplied together. So squaring a variable, or multiplying two different variables, is allowed, but cubing a variable, or multiplying one variable by the square of another, is strictly forbidden. Curves in this class, including ellipses, are still called conic sections out of deference to history; but more forward-looking algebraic geometers call them quadrics.† Now there are lots of quadratic equations: any such is of the form A x2 + B xy + C y2 + D x + E y + F = 0 for some values of the six constants A, B, C, D, E, and F. (The reader who feels so inclined can check that no other type of algebraic expression is allowed, subject to our requirement that we are only allowed to multiply two variables together, never three.) That seems like a lot of choices—infinitely many, in fact! But these quadrics turn out to fall into three main classes: ellipses, parabolas, and hyperbolas.*

Page 907

What can I say? Mathematics is a way not to be wrong, but it isn’t a way not to be wrong about everything. (Sorry, no refunds!) Wrongness is like original sin; we are born to it and it remains always with us, and constant vigilance is necessary if we mean to restrict its sphere of influence over our actions. There is real danger that, by strengthening our abilities to analyze some questions mathematically, we acquire a general confidence in our beliefs, which extends unjustifiably to those things we’re still wrong about. We become like those pious people who, over time, accumulate a sense of their own virtuousness so powerful as to make them believe the bad things they do are virtuous too.

Page 925

Were I to write down Pearson’s formula right now, or were you to go look it up, you would see a mess of square roots and ratios, which, unless you have Cartesian geometry at your fingertips, would not be very illuminating. But in fact, Pearson’s formula has a very simple geometric description. Mathematicians ever since Descartes have enjoyed the wonderful freedom to flip back and forth between algebraic and geometric descriptions of the world. The advantage of algebra is that it’s easier to formalize and to type into a computer. The advantage of geometry is that it allows us to bring our physical intuition to bear on the situation, particularly when you can draw a picture.

Page 958

In the same way, a point in three-dimensional space is described by a list of three coordinates (x,y,z). And nothing except habit and craven fear keeps us from pushing this further. A list of four numbers can be thought of as a point in four-dimensional space, and a list of ten numbers, like the California temperatures in our table, is a point in ten-dimensional space. Better yet, think of it as a ten-dimensional vector.

Page 963

That’s the dirty little secret of advanced geometry. It may sound impressive that we can do geometry in ten dimensions (or a hundred, or a million …), but the mental pictures we keep in our mind are two- or at most three-dimensional.

Page 977

And this is Pearson’s formula, in geometric language. The correlation between the two variables is determined by the angle between the two vectors. If you want to get all trigonometric about it, the correlation is the cosine of the angle.

Page 990

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.)

Page 018

But statistician Andrew Gelman found that the story is more complicated than the Brooksian portrait of a new breed of latte-sipping, Prius-driving liberals with big tasteful houses and NPR tote bags full of cash.

Note: Poetry

Page 022

In some states, like Texas and Wisconsin, richer counties tend to vote more Republican. In others, like Maryland, California, and New York, the richer counties are more Democratic. Those last states happen to be the ones where many political pundits live. In their limited worlds, the rich neighborhoods are loaded with rich liberals, and it’s natural for them to generalize this experience to the rest of the country.

Page 086

As the voters get more informed, they don’t get more Democratic or more Republican, but they do get more polarized: lefties go farther left, right-wingers get farther right, and the sparsely populated space in the middle gets even sparser. In the lower half of the graph, the less-informed voters tend to adopt a more centrist stance. The graph reflects a sobering social fact, which is by now commonplace in the political science literature. Undecided voters, by and large, aren’t undecided because they’re carefully weighing the merits of each candidate, unprejudiced by political dogma. They’re undecided because they’re barely paying attention.

Page 092

Keep this in mind when you’re told that two phenomena in nature or society were found to be uncor-related. It doesn’t mean there’s no relationship, only that there’s no relationship of the sort that correlation is designed to detect.

Page 340

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. The negative correlation between looks and personality in your dating pool is absolutely real. But if you try to improve your boyfriend’s complexion by training him to act mean, you’ve fallen victim to Berkson’s fallacy. Literary snobbery works the same way. You know how popular novels are terrible? It’s not because the masses don’t appreciate quality. It’s because there’s a Great Square of Novels, and the only novels you ever hear about are the ones in the Acceptable Triangle, which are either popular or good. If you force yourself to read unpopular novels chosen essentially at random—I’ve been on a literary prize jury, so I’ve actually done this— you find that most of them, just like the popular ones, are pretty bad.

Page 370

It turns out the things the U.S. government spends money on are things people kind of like. A Pew Research poll from February 2011 asked Americans about thirteen categories of government spending: in eleven of those categories, deficit or no deficit, more people wanted to increase spending than dial it down. Only foreign aid and unemployment insurance—which, combined, accounted for under 5% of 2010 spending—got the ax. That, too, agrees with years of data; the average American is always eager to slash foreign aid, occasionally tolerant of cuts to welfare or defense, and pretty gung ho for increased spending on every single other program our taxes fund. Oh, yeah, and we want small government.

Page 390

Two out of three people want to cut spending; so in a poll that asks “Should we cut spending or raise taxes?” the cutters are going to win by a massive 67–33 margin. So what to cut? If you ask, “Should we cut the defense budget?” you’ll get a resounding no: two-thirds of voters—the tax raisers joined by the Medicare cutters—want defense to keep its budget. And “Should we cut Medicare?” loses by the same amount. 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.

Note: Nicely stated but obvious to me at least. Maybe because I’ve heard people complain about typical Redditors for years

Page 399

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 ones. In large part, that’s because most Americans think the programs that benefit them personally are the ones that must, at all costs, be preserved. (I didn’t say we weren’t selfish, I just said we weren’t stupid!)

Page 405

President Obama’s signature domestic policy accomplishment, the Affordable Care Act. In an October 2010 poll of likely voters, 52% of respondents said they opposed the law, while only 41% supported it. Bad news for Obama? Not once you break down the numbers. Outright repeal of health care reform was favored by 37%, with another 10% saying the law should be weakened; but 15% preferred to leave it as is, and 36% said the ACA should be expanded to change the current health care system more than it currently does. That suggests that many of the law’s opponents are to Obama’s left, not his right. 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.* The incoherence of the majority creates plentiful opportunities to mislead. 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!

Page 430

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. Safe to say it’s the public’s opinion that terrorism is bad and The Big Bang Theory is a great show. But cutting the deficit is a different story. The majority preferences don’t meld into a definitive stance. If there’s no such thing as the public opinion, what’s an elected official to do? The simplest answer: when there’s no coherent message from the people, do whatever you want. As we’ve seen, simple logic demands that you’ll sometimes be acting contrary to the will of the majority. If you’re a mediocre politician, this is where you point out that the polling data contradicts itself. If you’re a good politician, this is where you say, “I was elected to lead—not to watch the polls.”

Page 473

Livermore’s nightmare came true; we do not now cut people’s ears off, even if they were totally asking for it, and what’s more, we hold that the Constitution forbids us from doing so. Eighth Amendment jurisprudence is now governed by the principle of “evolving standards of decency,” first articulated by the Court in Trop v. Dulles (1958), which holds that contemporary American norms, not the prevailing standards of August 1789, provide the standard of what is cruel and what unusual.

Page 493

The majority’s ruling does the math differently. By their reckoning, there are thirty states that prohibit execution of the mentally retarded: the eighteen mentioned by Scalia and the twelve that prohibit capital punishment entirely. That makes thirty out of fifty, a substantial majority.

Note: I wish I had found this flaw in Scalia’s opinion

Page 533

Scalia is right to be troubled by a system in which the whims of one generation of Americans end up constitutionally binding our descendants. But it’s clear his objection is more than legal; his concern is an America that loses the habit of punishment through enforced disuse, an America that is not only legally barred from killing mentally retarded murderers but that, by virtue of the court’s lenient ratchet, has forgotten that it wants to. Scalia—much like Samuel Livermore two hundred years earlier—foresees and deplores a world in which the populace loses by inches its ability to impose effective punishments on wrongdoers. I can’t manage to share their worry. The immense ingenuity of the human species in devising ways to punish people rivals our abilities in art, philosophy, and science. Punishment is a renewable resource; there is no danger we’ll run out.

Note: Poetry

Page 636

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,” and slime molds are not the only creatures subject to it. Biologists have found jays, honeybees, and hummingbirds acting in the same seemingly irrational way.

Page 706

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.

Page 781

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?

Page 918

In the purest version of this view, mathematics becomes a kind of game played with symbols and words. A statement is a theorem precisely if it follows by logical steps from the axioms. But what the axioms and theorems refer to, what they mean, is up for grabs. What is a Point, or a Line, or a frog, or a kumquat? It can be anything that behaves the way the axioms demand, and the meaning we should choose is whichever one suits our present needs. A purely formal geometry is a geometry you can in principle do without ever having seen or imagined a point or a line; it is a geometry in which it’s irrelevant what points and lines, understood in the usual way, are actually like.

Page 045

But it was no less obvious to the ancient Greeks that a geometric magnitude must be a ratio of two whole numbers; that’s how their notion of measurement worked, until the whole framework got mugged by the Pythagorean Theorem and the stubbornly irrational square root of 2.

Note: I remember being unimpressed with this result. So what, I thought. But no doubt it was earth shattering to the Greeks

Page 132

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 ones whose contributions matter. We don’t treat any other subject that way! I’ve never heard a student say, “I like Hamlet, but I don’t really belong in AP English—that kid who sits in the front row knows all the plays, and he started reading Shakespeare when he was nine!” Athletes don’t quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was “ahead” of them. We lose a lot of math majors this way. Thus, we lose a lot of future mathematicians; but that’s not the whole of the problem. I think we need more math majors who don’t become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators. But we won’t get there until we dump the stereotype that math is only worthwhile for kid geniuses. 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. But the ability to work hard—to keep one’s whole attention and energy focused on a problem, systematically turning it over and over and pushing at everything that looks like a crack, despite the lack of outward signs of progress—is not a skill everybody has. Psychologists nowadays call it “grit,” and it’s impossible to do math without it.

Page 191

It’s not wrong to say Hilbert was a genius. But it’s more right to say that what Hilbert accomplished was genius. Genius is a thing that happens, not a kind of person.

Page 222

“What subjects are you lecturing on this semester?” Hilbert asked.        “I do not lecture anymore,” Blumenthal gently reminded him.        “What do you mean, you do not lecture?”        “I am not allowed to lecture anymore.”        “But that is completely impossible! This cannot be done. Nobody has the right to dismiss a professor unless he has committed a crime. Why do you not apply for justice?”

Note: :(

Page 306

And yet—when Roosevelt says, “The closet philosopher, the refined and cultured individual who from his library tells how men ought to be governed under ideal conditions, is of no use in actual governmental work,” I think of Condorcet, who spent his time in the library doing just that, and who contributed more to the French state than most of his time’s more practical men. And when Roosevelt sneers at the cold and timid souls who sit on the sidelines and second-guess the warriors, I come back to Abraham Wald, who as far as I know went his whole life without lifting a weapon in anger, but who nonetheless played a serious part in the American war effort, precisely by counseling the doers of deeds how to do them better. He was unsweaty, undusty, and unbloody, but he was right. He was a critic who counted.

Page 342

The paladin of principled uncertainty in our time is Nate Silver, the online-poker-player-turned-baseball-statistics-maven-turned-political-analyst whose New York Times columns about the 2012 presidential election drew more public attention to the methods of probability theory than they have ever before enjoyed. I think of Silver as a kind of Kurt Cobain of probability. Both were devoted to cultural practices that had previously been confined to a small, inward-looking cadre of true believers (for Silver, quantitative forecasting of sports and politics, for Cobain, punk rock). And both proved that if you carried their practice out in public, with an approachable style but without compromising the source material, you could make it massively popular.

Note: Haha, best use of Paladin I’ve ever seen

Page 429

The fetish of perfect precision affects elections, not just in the fevered poll-watching period but after the election takes place. The Florida 2000 election, remember, rode on a difference of a few hundred votes between George W. Bush and Al Gore, a hundredth of a percent of the total votes cast. It was of critical importance, by our law and custom, to determine which candidate it was who could claim a few hundred more ballots than the other. But as a way of thinking about who Floridians wanted to be president, this is absurd; the imprecision caused by ballots spoiled, ballots lost, ballots miscounted, is much greater than the tiny difference in the final count. We don’t know who got more votes in Florida. The difference between judges and mathematicians is that judges have to find a way to pretend we know, while mathematicians are free to tell the truth.

Page 473

As F. Scott Fitzgerald said, “The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.”

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