Small Is Beautiful—Or At Least, Weird

… among small groups there will be greater variation …
—Howard Wainer and Harris Zwerling.

The central concept of allopatric speciation is that new species can arise only when a small local population becomes isolated at the margin of the geographic range of its parent species.
—Stephen Jay Gould and Niles Eldredge.

If you flipped a coin a thousand times, you were more likely to end up with heads or tails roughly half the time than if you flipped it ten times.
—Michael Lewis. 

No humanist intellectual today is a “reductionist.” To Penn State English professor Michael Bérubé for example, when the great biologist E.O. Wilson speculated—in 1998’s Consilience: The Unity of Knowledge—that “someday … even the disciplines of literary criticism and art history will find their true foundation in physics and chemistry,” Wilson’s claim was (Bérubé wrote) “almost self-parodic.” Nevertheless, despite the withering disdain of English professors and such, examples of reductionism abound: in 2002, journalist Malcolm Gladwell noticed that a then-recent book—Randall Collins’ The Sociology of Philosophies—argued that French Impressionism, German Idealism, and Chinese neo-Confucianism, among other artistic and philosophic movements, could all be understood by the psychological principle that “clusters of people will come to decisions that are far more extreme than any individual member would have come to on his own.” Collins’ claim, of course, is sure to call down the scorn of professors of the humanities like Bérubé for ignoring what literary critic Victor Shklovsky might have called the “stoniness of the stone”; i.e., the specificity of each movement’s work in its context, and so on. Yet from a political point of view (and despite both the bombastic claims of certain “leftist” professors of the humanities and their supposed political opponents) the real issue with Collins’ (and Gladwell’s) “reductionism” is not that they attempt to reduce complex artistic and philosophic movements to psychology—nor even, as I will show, to biology. Instead, the difficulty is that Collins (and Gladwell) do not reduce them to mathematics.  

Yet, to say that neo-Confucianism (or, to cite one of Gladwell’s examples, Saturday Night Live) can be reduced to mathematics first begs the question of what it means to “reduce” one sort of discourse to another—a question still largely governed, Kenneth Schaffner wrote in 2012, by Ernest Nagel’s “largely unchanging and immensely influential analysis of reduction.” According to Nagel’s 1961 The Structure of Science: Problems in the Logic of Scientific Explanation, a “reduction is effected when the experimental laws of the secondary science … are shown to be the logical consequences of the theoretical assumptions … of the primary science.” Gladwell for example, discussing “the Lunar Society”—which included Erasmus Darwin (grandfather to Charles), James Watt (inventor of the steam engine), Josiah Wedgwood (the pottery maker), and Joseph Priestly (who isolated oxygen)—says that this group’s activities bears all “the hallmarks of group distortion”: someone proposes “an ambitious plan for canals, and someone else tries to top that [with] a really big soap factory, and in that feverish atmosphere someone else decides to top them all with the idea that what they should really be doing is fighting slavery.” In other words, to Gladwell the group’s activities can be explained not by reference to the intricacies of thermodynamics or chemistry, nor even the political difficulties of the British abolitionist movement—or even the process of heating clay. Instead, the actions of the Lunar Society can be understood in somewhat the same fashion that, in bicycle racing, the peloton (which is not as limited by wind resistance) can reach speeds no single rider could by himself. 

Yet, if it is so that the principle of group psychology explains, for instance, the rise of chemistry as a discipline, it‘s hard to see why Gladwell should stop there. Where Gladwell uses a psychological law to explain the “Blues Brothers” or “Coneheads,” in other words, the late Harvard professor of paleontology Stephen Jay Gould might have cited a law of biology: specifically, the theory of “punctuated equilibrium”—a theory that Gould, along with his colleague Niles Eldredge, first advanced in 1972. The theory that the two proposed in “Punctuated Equilibria: an Alternative to Phyletic Gradualism” could, thereby, be used to explain the rise of the Not Ready For Prime Time Players as equally well as the psychological theory Gladwell advances.    

In that early 1970s paper, the two biologists attacked the reigning idea of how new species begin: what they called the “picture of phyletic gradualism.” In the view of that theory, Eldredge and Gould  wrote, new “species arise by the transformation of an ancestral population into its modified descendants.” Phyletic gradualism thusly answers the question of why dinosaurs went extinct by replying that they didn’t: dinosaurs are just birds now. More technically, under this theory the change from one species to another is a transformation that “is even and slow”; engages “usually the entire ancestral population”; and “occurs over all or a large part of the ancestral species’ geographic range.” For nearly a century after the publication of Darwin’s Origin of Species, this was how biologists understood the creation of new species. To Gould and Eldredge however that view simply was not in accordance with how speciation usually occurs. 

Instead of ancestor species gradually becoming descendant species, they argued that new species are created by a process they called “the allopatric theory of speciation”—a theory that might explain how Hegel’s The Philosophy of Right and Chevy Chase’s imitation of Gerald Ford could be produced by the same phenomena. Like Gladwell’s use of group psychology (which depends on the competition within a set of people who all know each other), where “phyletic gradualism” thinks that speciation occurs over a wide area to a large population, the allopatric theory thinks that speciation occurs in a narrow range to a small population: “The central concept of allopatric speciation,” Gould and Eldredge wrote, “is that new species can arise only when a small local population becomes isolated at the margin of the geographic range of its parent species.” Gould described this process for a non-professional audience in his essay, “The Golden Rule: A Proper Scale for Our Environmental Crisis,” from his 1982 book, Eight Little Piggies: Reflections on Natural History—a book that perhaps demonstrates just how considerations of biological laws might show why John Belushi’s “Samurai Chef,” or Gilda Radner’s “Roseanne Rosannadanna” succeeded. 

The Pinaleno Mountains, in New Mexico, house a population of squirrel called the Mount Graham Red Squirrel, which “is isolated from all other populations and forms the southernmost extreme of the species’s range.” The Mount Graham subspecies can survive in those mountains despite being so far south of the rest of its species because the Pinalenos are “‘sky islands,’” as Gould calls them: “patches of more northern microclimate surrounded by southern desert.” It’s in such isolated places, the theory of allopatric speciation holds, that new species develop: because the Pinalenos are “a junction of two biogeographic provinces” (the Nearctic “by way of the Colorado Plateau“ and the Neotropical “via the Mexican Plateau”), they are a space where new kinds of selection pressures can work upon a subpopulation than are available on the home range, and therefore places where subspecies can make the kinds of evolutionary “leaps” that can allow such new populations, after success in such “nurseries,” to return to the original species’ home range and replace the ancestral species. Such a replacement, of course, does not involve the entire previous population, nor does it occur over the entire ancestral range, nor is it even and slow, as the phyletic gradualist theory would suggest.

The application to the phenomena considered by Gladwell then is fairly simple. What was happening at 30 Rockefeller Center in New York City in the autumn of 1975 might not have been an example of “group psychology” at work, but instead an instance where a small population worked at the margins of two older comedic provinces: the new improvisational space created by such troupes as Chicago’s Second City, and the older tradition of live television created by such shows as I Love Lucy and Your Show of Shows. The features of the new form thereby forged under the influence of these pressures led, ultimately, to the extinction of older forms of television comedy like the standard three-camera situation comedy, and the eventual rise of single-camera shows like Seinfeld and The Office. Or so, at least, it can be imagined that the story might be told, rather than in the form of Gladwell’s idea of group psychology. 

Yet, it isn’t simply possible to explain a comedic phenomenon or a painting movement in terms of group psychology, instead of the terms familiar to scholars of the humanities—or even, one step downwards in the explanatory hierarchy, in terms of biology instead of psychology. That’s because, as the work of Israeli psychologists Daniel Kahneman and Amos Tversky suggests, there is something odd, mathematically, about small groups like subspecies—or comedy troupes. That “something odd” is this: they’re small. Being small has (the two pointed out in their 1971 paper, “Belief in the Law of Large Numbers”) certain mathematical consequences—and, perhaps oddly, those consequences may help to explain something about the success of Saturday Night Live. 

That’s anyway the point the two psychologists explored in their 1971 paper, “Belief in the Law of Large Numbers”—a paper whose message would, perhaps oddly, later be usefully summarized by Gould in a 1983 essay, “Glow, Big Glowworm”: “Random arrays always include some clumping … just as we will flip several heads in a row quite often so long as we can make enough tosses.” Or—as James Forbes of Edinburgh University noted in 1850—it would be absurd to expect to find “on 1000 throws [of a fair coin] there should be exactly 500 heads and 500 tails.” (In fact, as Forbes went on to remark, there’s less than a 3 percent chance of getting such a result.) But human beings do not usually realize that reality: in “Belief,” Kahneman and Tversky reported G.S. Tune’s 1964 study that found that when people “are instructed to generate a random sequence of hypothetical tosses of a fair coin … they produce sequences where the proportion of heads in any short segment stays far closer to .50 than the laws of chance would predict.” “We assume”—as Atul Gawande summarized the point of “Belief” for the New Yorker in 1998—“that a sequence of R-R-R-R-R-R is somehow less random than, say, R-R-B-R-B-B,” while in reality “the two sequences are equally likely.” Human beings find it difficult to understand true randomness—which may be why it may be so difficult to see how this law of probability might apply to, say, the Blues Brothers.

Yet, what the two psychologists were addressing in “Belief” was the idea expressed by statisticians Howard Wainer and Harris Zwerling in a 2006 article later cited by Kahneman in his recent bestseller, Thinking: Fast and Slow: the statistical law that “among small groups there will be greater variation.” In their 2006 piece, Wainer and Zwerling illustrated the point by observing that, for example, the lowest-population counties in the United States tend to have the highest kidney cancer rates per capita, or the smallest schools disproportionately appear on lists of the best-performing schools. What they mean is that a “county with, say, 100 inhabitants that has no cancer deaths would be in the lowest category” of kidney cancer rates—but “if it has one cancer death it would be among the highest”—while similarly, examining the Pennsylvania System of School Assessment for 2001-02 found “that, of the 50 top-scoring schools (the top 3%), six of them were among the 50 smallest schools (the smallest 3%),” which is “an overrepresentation by a factor of four.” “When the population is small,” they concluded, “there is wide variation”—but when “populations are large … there is very little variation.” Or, it may not be that small groups push each member to achieve more, it’s that small groups of people tend to have high amounts of variation, and (every so often) one of those groups varies so much that somebody invents the discipline of chemistry—or invent the Festrunk Brothers.

The $64,000 question, from this point of view, isn’t the groups that created a new way of painting—but instead all of the groups that nobody has ever heard of that tried, but failed, to invent something new. Yet as a humanist intellectual like Bérubé would surely point out, to investigate this question in this way is to miss nearly everything about Impressionism (or the Land Shark) that makes it interesting. Which, perhaps, is so—but then again, isn’t the fact that such widely scattered actions and organisms can be united under one theoretical lens interesting? Taken far enough, what matters to Bérubé is the individual peculiarities of everything in existence—an idea that recalls what Jorge Luis Borges once described as John Locke’s notion of “an impossible idiom in which each individual object, each stone, each bird and branch had an individual name.” To think of Bill Murray in the same frame as a New Mexican squirrel is, admittedly, to miss the smell of New York City at dawn on a Sunday morning after a show the night before—but it also involves a gain, and one that is applicable to many other situations besides the appreciation of the hard work of comedic actors. Although many in the humanities then like to attack what they call reductionism for its “anti-intellectual” tendencies, it’s well-known that a large enough group of trees constitutes more than a collection of individual plants. There is, I seem to recall, some kind of saying about it.  

Ribbit

 “‘The frog is almost five hundred million years old. Could you really say with much certainty that America, with all its strength and prosperity, with its fighting man that is second to none, and with its standard of living that is the highest in the world, will last as long as … the frog?’”
—Joseph Heller. Catch-22. (1961).

 … the fall of empires which aspired to universal dominion could be predicted with very high probability by one versed in the calculus of chance.
—Laplace. Theórie Analytique des Probabilities. (1814).

 

If sexism exists, how could it be proved? A recent lawsuit—Chen-Oster v. Goldman Sachs, Inc., filed in New York City on 19 May, 2014—aims to do just that. The suit makes four claims: that Goldman’s women employees make less than men at the same positions; that a “disproportionate” number of men have been promoted “over equally or more qualified women”; that women employees’ performance was “systematic[ally] underval[ued]”; and that “managers most often assign the most lucrative and promising opportunities to male employees.” The suit, then, echoes many of the themes developed by feminists over the past two generations, and in a general sense may perhaps be accepted, or even cheered, by those Americans sensitive to feminism. But those Americans may not be aware of the potential dangers of the second claim: dangers that threaten not merely the economic well-being of the majority of Americans, including women, but also America’s global leadership. Despite its seeming innocuousness, the second claim is potentially an existential threat to the future of the United States.

That, to be sure, is a broad assertion, and seems disproportionate, you might say, to the magnitude of the lawsuit: it hardly seems likely that a lawsuit over employment law, even one involving a firm so important to the global financial machinery as Goldman Sachs, could be so important as to threaten the future of the United States. Yet few today would deny the importance of nuclear weapons—nor that they pose an existential threat to humanity itself. And if nuclear weapons are such a threat, then the reasoning that led to those weapons must be at least as, if not more so, as important than the weapons themselves. As I will show, the second claim poses a threat to exactly that chain of reasoning.

That, again, may appear a preposterous assertion: how can a seemingly-minor allegation in a lawsuit about sexism have anything to do with nuclear weapons, much less the chain of logic that led to them? One means of understanding how requires a visit to what the late Harvard biologist Stephen Jay Gould called “the second best site on the standard tourist itinerary of [New Zealand’s] North Island—the glowworm grotto of Waitomo Cave.” Upon the ceiling of this cave, it seems, live fly larvas whose “illuminated rear end[s],” Gould tells us, turn the cave into “a spectacular underground amphitheater”—an effect that, it appears, mirrors the night sky. But what’s interesting about the Waitomo Cave is that it does this mirroring with a difference: upon observing the cave, Gould “found it … unlike the heavens” because whereas stars “are arrayed in the sky at random,” the glowworms “are spaced more evenly.” The reason why is that the “larvae compete with, and even eat, each other—and each constructs an exclusive territory”: since each larva has more or less the same power as every other larva, each territory is more or less the same size. Hence, as Gould says, the heaven of the glowworms is an “ordered heaven,” as opposed to the disorderly one visible on clear nights around the the world—a difference that not only illuminates just what’s wrong with the plaintiff’s second claim in Chen-Oster v. Goldman Sachs, Inc, but also how that claim concerns nuclear weapons.

Again, that might appear absurd: how can understanding a Southern Hemispheric cavern help illuminate—as it were—a lawsuit against the biggest of Wall Street players? To understand how requires another journey—though this one is in time, not space.

In 1767, an English clergyman named John Michell published a paper with the unwieldy title of “An Inquiry into the Probable Parallax, and Magnitude of the Fixed Stars, from the Quantity of Light Which They Afford us, and the Particular Circumstances of Their Situation.” Michell’s purpose in the paper, he wrote, was to inquire whether the stars “had been scattered by mere chance”—or, instead, by “their mutual gravitation, or to some other law or appointment of the Creator.” Since (according to Michell’s biographer, Russell McCommach), Michell assumed “that a random distribution of stars is a uniform distribution,” he concluded that—since the night sky does not resemble the roof of the Waitomo Cave—the distribution of stars must be the result of some natural law. Or even, he hinted, the will of the Creator himself.

So things might have stayed had Michell’s argument “‘remained buried in the heavy quartos of the Philosophical Transactions”—as James Forbes, the Professor of Natural Philosophy at Edinburgh University, would write nearly a century later. But Michell’s argument hadn’t; several writers, it seems, took his argument as evidence for the existence of the supernatural. Hence, Forbes felt obliged to refute an argument that, he thought, is “‘too absurd to require refutation.’” To think—as Michell did—that “a perfectly uniform and symmetrical disposition of the stars over the sky,” as Forbes wrote, “could alone afford no evidence of causation” would be “palpably absurd.” The reason Forbes thought that way, in turn, is the connection both to the Goldman lawsuit—and nuclear weapons.

Forbes made his point by an analogy to flipping a coin: to think that the stars had been distributed randomly because they were evenly spaced across the sky, he wrote, would be as ridiculous as the chances that “on 1000 throws [of a fair coin] there should be exactly 500 heads and 500 tails.” In fact, the Scotsman pointed out, mathematics demonstrates that in such a case of 1000 throws “there are almost forty chances to one [i.e., nearly 98%], that some one of the other possible events shall happen instead of the required one.” In 1000 throws of a fair coin, there’s less than a three percent chance that the flipper will get exactly 500 heads: it’s simply a lot more likely that there will be some other number of heads. In Gould’s essay about the Waitomo Cave, he put the same point like this: “Random arrays always include some clumping … just as we will flip several heads in a row quite often so long as we can make enough tosses.” Because the stars clump together, Forbes argued, that is evidence that they are randomly distributed—not of a benevolent Creator, like Michell thought. Forbes’ insight, in turn, about how to detect randomness, or chance, in astronomical data had implications far beyond the stars: in a story that would take much more space than this essay to tell, it eventually led a certain Swiss patent clerk to take up the phenomena called “Brownian motion.”

The clerk, of course, was Albert Einstein; the subject of his 1905 paper, “On the Movement of Small Particles Suspended In a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat,” was the tendency—“easily observed in a microscope,” Einstein remarks—for tiny particles to move in an apparently-spontaneous manner. What Einstein realized (as physicist Leonard Mlodinow put it in his 2008 book, The Drunkard’s Walk: How Randomness Rules Our Lives) was that the “jiggly” motion of dust particles and so on results from collisions between them and even smaller particles, and so “there was a predictable relationship between factors such as the size, number, and speed of the molecules and the observable frequency and magnitude of the jiggling.” In other words, “though the collisions [between the molecules and the larger particles] occur very frequently, because the molecules are so light, those frequent isolated collisions have no visible effects” for the most part—but once in a while, “when pure luck occasionally leads to a lopsided preponderance of hits from some particular direction,” there are enough hits to send the particle moving. Or, to put it another way, when the flip of a 1000 coins all come up heads, the particle will move. Put in that fashion, to be sure, Einstein’s point might appear obscure at best—but as Mlodinow goes on to say, it is no accident that this seemingly-minor paper became the great physicist’s “most cited work.” That’s because the ultimate import of the paper was to demonstrate the existence … of the atom. Which is somewhat of a necessity for building an atom bomb.

The existence of the atomic bomb, then, can be said to depend on the insight developed by Forbes: just how significant the impact of chance can be in the formation of both the very large (the universe itself, according to Forbes), and the very small (the atom, according to Einstein). The point both men attempted to make, in turn, is that the existence of order is something very rare in this universe, at any rate (whatever may be the case in others). Far more common, then, is the existence of disorder—which brings us back to Goldman Sachs and the existence of sexism.

It is the contention of the second point in the plaintiffs’ brief in Chen-Oster v. Goldman Sachs, Inc., remember, that there exists (as University of Illinois English professor Walter Benn Michaels has noted) a “‘“stark” underrepresentation’ [of women] in management” because “‘just 29 percent of vice presidents, 17 percent of managing directors, and 14 percent of partners’” are women. Goldman Sachs, as it happens, has roughly 35,000 employees—which, it turns out, is about 0.001% of the total population of the United States, which is 323 million. Of those 323 million, as of the 2010 Census women number about 157 million, compared to around 151 million men. Hence, the question to be asked about the Goldman Sachs lawsuit (and I write this as someone with little sympathy for Goldman Sachs) is—if the reasoning Einstein followed to demonstrate the existence of the atom is correct—then if the chances of landing exactly 500 heads, when tossing a coin 1000 times, is less than three percent, how much less likely is it that a sample of 35,000 people will exactly mirror the proportions of 323 million? The answer, it would seem, is rather low: it’s simply a lot more likely that Goldman Sachs would have something other than a proportionate ratio of men to women than the reverse, just as it it’s a lot more likely that stars should clump together than be equally spaced like the worms in the New Zealand cave. And that is to say that the disproportionate number of men in leadership in positions at Goldman Sachs is merely evidence of the absence of a pro-woman bias at Goldman Sachs, not evidence of the existence of a bias against women.

To which it might be replied, of course, that the point isn’t the exact ratio, but rather that it is so skewed toward one sex: what are the odds, it might be said, that all three categories of employee should all be similarly bent in one direction? Admittedly, that is an excellent point. But it’s also a point that’s missing from the plaintiffs’ brief: there is no mention of a calculation respecting the particular odds in the case, despite the fact that the mathematical techniques necessary to do those calculations have been known since long before the atomic bomb, or even Einstein’s paper on the existence of the atom. And it’s that point, in turn, that concerns not merely the place of women in society—but ultimately the survival of the United States.

After all, the reason that the plaintiffs in the Goldman Sachs suit do not feel the need to include calculations of the probability of the disproportion they mention—despite the fact that it is the basis of their second claim—is that the American legal system is precisely structured to keep such arguments at bay. As Oliver Roeder observed in FiveThirtyEight last year, for example, the justices of the U.S. Supreme Court “seem to have a reluctance—even an allergy—to taking math and statistics seriously.” And that reluctance is not limited to the justices alone: according to Sanford Levinson, a University of Texas professor of law and government interviewed by Roeder in the course of reporting his story, “top-level law schools like Harvard … emphasize … traditional, classical legal skills” at the expense of what Levinson called “‘genuine familiarity with the empirical world’”—i.e., the world revealed by techniques pioneered by investigators like James Forbes. Since, as Roeder observes, all nine current Supreme Court justices attended either Harvard or Yale, that suggests that the curriculum followed at those schools has a connection to the decisions reached by their judicial graduates.

Still, that exclusion might not be so troublesome were it limited merely to the legal machinery. But as Nick Robinson reported last year in the Buffalo Law Review, attorneys have “dominated the political leadership of the United States” throughout its history: “Since independence,” Robinson pointed out there, “more than half of all presidents, vice presidents, and members of Congress have come from a law background.” That then implies that if the leadership class of the United States is derived from American law schools, and American law schools train students to disdain mathematics and the empirical world, then it seems plausible to conclude that much of the American leadership class is specifically trained to ignore both the techniques revealed by Forbes and the underlying reality they reveal: the role played by chance. Hence, while such a divergence may allow plaintiffs like those in the Goldman case to make allegations of sexism without performing the hard work of actually demonstrating how it might be possible mathematically, it might also have consequences for actual women who are living, say, in a nation increasingly characterized by a vast difference between the quantifiable wealth of those at the top (like people who work for Goldman Sachs) and those who aren’t.

And not merely that. For decades if not centuries, Americans have bemoaned the woeful lack of performance of American students in mathematics: “Even in Massachusetts, one of the country’s highest-performing states,” Elizabeth Green observed in the latest of one of these reports in the New York Times in 2014, “math students are more than two years behind their counterparts in Shanghai.” And results like that, as the journalist Michael Lewis put the point several years ago in Vanity Fair, risk “ceding … technical and scientific leadership to China”—and since, as demonstrated, it’s knowledge of mathematics (and specifically knowledge of the mathematics of probability) that made the atomic bomb possible, that implies conversely that ignorance of the subject is a serious threat to national existence. Yet, few Americans have, it seems, considered whether the fact that students do not take mathematics (and specifically probability) seriously may have anything to do with the fact that the American leadership class explicitly rules such topics, quite literally, out of court.

Of course, as Lewis also pointed out in his recent book, The Undoing Project: A Friendship that Changed Our Minds, American leaders may not be particularly alone in ignoring the impact of probabilistic reasoning: when, after the Yom Kippur War—which had caught Israel’s leaders wholly by surprise—future Nobel Prize winner Daniel Kahneman and intelligence officer Zvi Lanir attempted to “introduce a new rigor in dealing with questions of national security” by replacing intelligence reports written “‘in the form of essays’” with “probabilities, in numerical form,” they found that “the Israeli Foreign Ministry was ‘indifferent to the specific probabilities.’” Kahneman suspected that the ministry’s indifference, Lewis reports, was due to the fact that Israel’s leaders’ “‘understanding of numbers [was] so weak that [the probabilities did not] communicate’”—but betting that the leadership of other countries continues to match the ignorance of our own does not particularly appear wise. Still, as Oliver Roeder noted for FiveThirtyEight, not every American is willing to continue to roll those dice: University of Texas law professor Sanford Levinson, Roeder reported, thinks that the “lack of rigorous empirical training at most elite law schools” requires the “long-term solution” of “a change in curriculum.” And that, in turn, suggests that Chen-Oster v. Goldman Sachs, Inc. might be more than a flip of a coin over the existence of sexism on Wall Street.

Stayin’ Alive

And the sun stood still, and the moon stayed,
until the people had avenged themselves upon their enemies.
—Joshua 10:13.

 

“A Sinatra with a cold,” wrote Gay Talese for Esquire in 1966, “can, in a small way, send vibrations through the entertainment industry and beyond as surely as a President of the United States, suddenly sick, can shake the national economy”; in 1994, Nobel laureate economist Paul Krugman mused that a “commitment to a particular … doctrine” can eventually set “the tone for policy-making on all issues, even those which may seem to have nothing to do with that doctrine.” Like a world leader—or a celebrity—the health of an idea can have unforeseen consequences; for example, it is entirely possible that the legal profession’s intellectual bias against mathematics has determined the nation’s racial policy. These days after all, as literary scholar Walter Benn Michaels observed recently, racial justice in the United States is held to what Michaels calls “the ideal of proportional inequality”—an ideal whose nobility, it so happens that Nobel Prize-winner Daniel Kahneman and his colleague Amos Tversky have demonstrated, is matched only by its mathematical futility. The law, in short, has what Oliver Roeder of FiveThirtyEight recently called an “allergy” to mathematics; what I will argue is that, as a consequence, minority policy in the United States has a cold.

“The concept that mathematics can be relevant to the study of law,” law professor Michael I. Meyerson observed in 2002’s Political Numeracy: Mathematical Perspectives on Our Chaotic Constitution, “seems foreign to many modern legal minds.” In fact, he continued, to many lawyers “the absence of mathematics is one of law’s greatest appeals.” The strength of that appeal was on display recently in the 2011 Wisconsin case discussed by Oliver Roeder, Gill v. Whitford—a case that, as Roeder says, “hinges on math” because it involves the invention of a mathematical standard to measure “when a gerrymandered [legislative] map infringes on voters’ rights.” In oral arguments in Gill, Roeder observed, Chief Justice John Roberts said, about the mathematical techniques that are the heart of the case, that it “may be simply my educational background, but I can only describe [them] as sociological gobbledygook”—a derisory slight that recalls 19th-century Supreme Court Justice Joseph Story’s sneer concerning what he called “men of speculative ingenuity, and recluse habits.” Such statements are hardly foreign in the annals of the Supreme Court: “Personal liberties,” Justice Potter Stewart wrote in a 1975 opinion, “are not rooted in the law of averages.” (Stewart’s sentence, perhaps incidentally, uses a phrase—“law of averages”—found nowhere in the actual study of mathematics). Throughout the history of American law, in short, there is strong evidence of bias against the study and application of mathematics to jurisprudence.

Yet without the ability to impose that bias on others, even conclusive demonstrations of the law’s skew would not matter—but of course lawyers, as Nick Robinson remarked just this past summer in the Buffalo Law Review, have “dominated the political leadership of the United States.” As Robinson went on to note, “more than half of all presidents, vice presidents, and members of Congress have come from a law background.” This lawyer-heavy structure has had an effect, Robinson says: for instance, he claims “that lawyer-members of Congress have helped foster the centrality of lawyers and courts in the United States.” Robinson’s research then, which aggregates many studies on the subject, demonstrates that the legal profession is in a position to have effects on the future of the country—and if lawyers can affect the future of the country in one fashion, it stands to reason that they may have affected it in others. Not only then may the law have an anti-mathematical bias, but it is clearly positioned to impose that bias on others.

That bias in turn is what I suspect has led the Americans to what Michaels calls the theory of “proportional representation” when it comes to justice for minority populations. This theory holds, according to Michaels, that a truly just society would be a “society in which white people were proportionately represented in the bottom quintile [of income] (and black people proportionately represented in the top quintile)”—or, as one commenter on Michaels’ work has put it, it’s the idea that “social justice is … served if the top classes at Ivy League colleges contain a percentage of women, black people, and Latinos proportionate to the population.” Within the legal profession, the theory appears to be growing: as Michaels has also observed, the theory of the plaintiffs in the “the recent suit alleging discrimination against women at Goldman Sachs” complained of the ‘“stark” underrepresentation’ [of women] in management” because women represented “‘just 29 percent of vice presidents, 17 percent of managing directors, and 14 percent of partners’”—percentages that, of course, vary greatly from the roughly 50% of the American population who are women. But while the idea of a world in which the population of every institution mirrors the population as a whole may appear plausible to lawyers, it’s absurd to any mathematician.

People without mathematical training, that is, have wildly inaccurate ideas about probability—precisely the point of the work of social scientists Daniel Kahneman and Amos Tversky. “When subjects are instructed to generate a random sequence of hypothetical tosses of a fair coin,” wrote the two psychologists in 1971 (citing an earlier study), “they produce sequences where the proportion of heads in any short segment stays far closer to .50 than the laws of chance would predict.” In other words, when people are asked to write down the possible results of tossing a coin many times, they invariably give answers that are (nearly) half heads and half tails despite the fact that—as Brian Everitt observed in his 1999 book Chance Rules: An Informal Guide to Probability, Risk, and Statistics—in reality “in, say, 20 tosses of a fair coin, the number of heads is unlikely to be exactly 10.” (Everitt goes on to note that “an exact fifty-fifty split of heads and tails has a probability of a little less than 1 in 5.”) Hence, a small sample of 20 tosses has less than a twenty percent chance of being ten heads and ten tails—a fact that may appear yet more significant when it is noted that the chance of getting exactly 500 heads when flipping a coin 1000 times is less than 3%. Approximating the ideal of proportionality, then, is something that mathematics tells us is not simple or easy to do even once, and yet, in the case of college admissions, advocates of proportional representation suggest that colleges, and other American institutions, ought to be required to do something like what baseball player Joe DiMaggio did in the summer of 1941.

In that year in which “the Blitzkrieg raged” (as the Rolling Stones would write later), the baseball player Joe DiMaggio achieved what Gould says is “the greatest and most unattainable dream of all humanity, the hope and chimera of all sages and shaman”: the New York Yankee outfielder hit safely in 56 games. Gould doesn’t mean, of course, that all human history has been devoted to hitting a fist-sized sphere, but rather that while many baseball fans are aware of DiMaggio’s feat, what few are aware of is that the mathematics of DiMaggio’s streak shows that it was “so many standard deviations above the expected distribution that it should not have occurred at all.” In other words, Gould cites Nobel laureate Ed Purcell’s research on the matter.

What that research shows is that, to make it a better-than-even money proposition “that a run of even fifty games will occur once in the history of baseball,” then “baseball’s rosters would have to include either four lifetime .400 batters or fifty-two lifetime .350 batters over careers of one thousand games.” There are, of course, only three men who ever hit more than .350 lifetime (Cobb, Hornsby, and, tragically, Joe Jackson), which is to say that DiMaggio’s streak is, Gould wrote, “the most extraordinary thing that ever happened in American sports.” That in turn is why Gould can say that Joe DiMaggio, even as the Panzers drove a thousand miles of Russian wheatfields, actually attained a state chased by saints for millennia: by holding back, from 15 May to 17 July, 1941, the inevitable march of time like some contemporary Joshua, DiMaggio “cheated death, at least for a while.” To paraphrase Paul Simon, Joe DiMaggio fought a duel that, in every way that can be looked at, he was bound to lose—which is to say, as Gould correctly does, that his victory was in postponing that loss all of us are bound to one day suffer.

Woo woo woo.

What appears to be a simple baseball story, then, actually has a lesson for us here today: it tells us that advocates of proportional representation are thereby suggesting that colleges ought to be more or less required not merely to reproduce Joe DiMaggio’s hitting streak from the summer of 1941, but to do it every single season—a quest that in a practical sense is impossible. The question then must be how such an idea could ever have taken root in the first place—a question that Paul Krugman’s earlier comment about how a commitment to bad thinking about one issue can lead to bad thinking about others may help to answer. Krugman suggested in that essay that one reason why people who ought to know better might tolerate “a largely meaningless concept” was “precisely because they believe[d] they [could] harness it in the service of good policies”—and quite clearly, proponents of the proportional ideal have good intentions, which may be just why it has held on so long despite its manifest absurdity. But good intentions are not enough to ensure the staying power of a bad idea.

“Long streaks always are, and must be,” Gould wrote about DiMaggio’s feat of survival, “a matter of extraordinary luck imposed upon great skill”—which perhaps could be translated, in this instance, by saying that if an idea survives for some considerable length of time it must be because it serves some interest or another. In this case, it seems entirely plausible to think that the notion of “proportional representation” in relation to minority populations survives not because it is just, but instead because it allows the law, in the words of literary scholar Stanley Fish, “to have a formal existence”—that is, “to be distinct, not something else.” Without such a distinction, as Fish notes, the law would be in danger of being “declared subordinate to some other—non-legal—structure of concern,” and if so then “that discourse would be in the business of specifying what the law is.” But the legal desire Fish dresses up in a dinner jacket, attorney David Post of The Volokh Conspiracy website suggests, may merely be the quest to continue to wear a backwards baseball cap.

Apropos of Oliver Roeder’s article about the Supreme Court’s allergy to mathematics, Post says in other words, not only is there “a rather substantial library of academic commentary on ‘innumeracy’ at the court,” but “it is unfortunately well within the norms of our legal culture … to treat mathematics and related disciplines as kinds of communicable diseases with which we want no part.” What’s driving the theory of proportional representation, then, may not be the quest for racial justice, or even the wish to maintain the law’s autonomy, but instead the desire of would-be lawyers to avoid mathematics classes. But if so, then by seeking social justice through the prism of the law—which rules out of court at the outset any consideration of mathematics as a possible tool for thinking about human problems, and hence forbids (or at least, as in Gill v. Whitford, obstructs) certain possible courses of action to remedy social issues—advocates for African-Americans and others may be unnecessarily limiting their available options, which may be far wider, and wilder, than anyone viewing the problems of race through the law’s current framework can now see.

Yet—as any consideration of streaks and runs must, eventually, conclude—just because that is how things are at the moment is no reason to suspect that things will remain that way forever: as Gould says, the “gambler must go bust” when playing an opponent, like history itself, with near-infinite resources. Hence, Paul Simon to the contrary, the impressive thing about the Yankee Clipper’s feat in that last summer before the United States plunged into global war is not that after “Ken Keltner made two great plays at third base and lost DiMaggio the prospect of a lifetime advertising contract with the Heinz ketchup company” Joe DiMaggio left and went away. Instead, it is that the great outfielder lasted as long as he did; just so, in Oliver Roeder’s article he mentions that Sanford Levinson, a professor of law at the University of Texas at Austin and one of the best-known American legal scholars, has diagnosed “the problem [as] a lack of rigorous empirical training at most elite law schools”—which is to say that “the long-term solution would be a change in curriculum.” The law’s streak of avoiding mathematics, in other words, may be like all streaks. In the words of the poet of the subway walls,

Koo-koo …

Ka-choo.

No Justice, No Peace

 

‘She’s never found peace since she left his arms, and never will again till she’s as he is now!’
—Thomas Hardy. Jude the Obscure. (1895).

“Done because we are too menny,” writes little “Father Time,” in Thomas Hardy’s Jude the Obscure—a suicide note that is meant to explain why the little boy has killed his siblings, and then hanged himself. The boy’s family, in other words, is poor, which is why Father Time’s father Jude (the titular obscurity) is never able, as he wished, to become the scholar he once dreamed of becoming. Yet, although Jude is a great tragedy, it is also something of a mathematical textbook: the principle taught by little Jude instructs not merely about why his father does not get into university, but perhaps also about just why, as Natasha Warikoo remarked in last week’s London Review of Books blog, “[o]ne third of Oxford colleges admitted no black British students in 2015.” Unfortunately, Warikoo never considers that possibility suggested by Jude: although Warikoo considers a number of reasons why black British students do not go to Oxford, she does not consider what we might call, in honor of Jude, the “Judean Principle”: that minorities simply cannot be proportionately represented everywhere always. Why? Well, because of the field goal percentages of the 1980-81 Philadelphia 76ers—and math.

“The Labour MP David Lammy,” wrote Warikoo, “believes that Oxford and Cambridge are engaging in social apartheid,” while “others have blamed the admissions system.” These explanations, Warikoo suggests, are incorrect: due to interviews with “15 undergraduates at Oxford who were born in the UK to immigrant parents, and 52 of their white peers born to British parents,” she believes that the reason for the “massive underrepresentation” of black British students is “related to a university culture that does not welcome them.” Or in other words, the problem is racism. But while it’s undoubtedly the case that many people, even today, are prejudiced, is prejudice really adequate to explain the case here?

Consider, after all, what it is that Warikoo is claiming—beginning with the idea of “massive underrepresentation.” As Walter Benn Michaels of the University of Illinois at Chicago has pointed out, the goal of many on the political “left” these days appears to be a “society in which white people were proportionately represented in the bottom quintile (and black people proportionately represented in the top quintile)”—in other words, a society in which every social strata contained precisely the same proportion of minority groups. In line with that notion, Warikoo assumes that, because Oxford and Cambridge do not contain the same proportion of black British people as the larger society does, that necessarily implies the racism of the system. But such an argument betrays an ignorance of how mathematics works—or more specifically, as MacArthur grant-winning psychologist Amos Tversky and his co-authors explained more than three decades ago, how basketball works.

In “The Hot Hand in Basketball: On the Misperception of Random Sequences,” Tversky and company investigated an entire season’s worth of shooting data from the NBA’s Philadelphia 76ers in order to discover whether there was evidence “that the performance of a player during a particular period is significantly better than expected on the basis of the player’s own record”—that is, whether players sometimes shot better (or “got hot”) than their overall shot record would predict. Prior to the research, it seems, everyone involved in basketball—fans, players, and coaches—appeared to believe that sometimes players did “get hot”—a belief that seems to predict that, sometimes, players have a better chance of making the second basket of a series than they did the first one:

Consider a professional basketball player who makes 50% of his shots. This player will occasionally hit four or more shots in a row. Such runs can properly be called streak shooting, however, only if their length or frequency exceeds what is expected on the basis of chance alone.

In other words, if a player really did get “hot,” or was “clutch,” then that fact would be reflected in the statistical record by a showing that sometimes players made second and third (and so on) baskets at a rate higher than that player’s chance of making a first basket: “the probability of a hit should be greater following a hit than following a miss.” If the “hot hand” existed, in other words, there should be evidence for it.

Unfortunately—or not—there was no such evidence, the investigators found: after analyzing the data for the nine players who took the vast majority of the 76ers shots for the 1980-81 season, Tversky and company found that “for eight of the nine players the probability of a hit is actually lower following a hit … than following a miss,” which is clearly “contrary to the hot-hand hypothesis.” (The exception is Daryl Dawkins, who played center—and was best known, as older fans may recall, for his backboard shattering dunks; i.e., a high-percentage shot.) There was no such thing as the “hot hand,” in short. (To use an odd turn of phrase with regards to the NBA.)

Yet, what has that to do with the fact that there were no black British students at one third of Oxford’s colleges in 2015? After all, not many British people play basketball, black or not. But as Tversky and his co-authors argue in “The Hot Hand,” the existence of the belief in a “hot hand” intimates that people’s “intuitive conception of randomness depart systematically from the laws of chance.” That is, when faced with a coin flip for example “people expect even short sequences of heads and tails to reflect the fairness of a coin and contain roughly 50% heads and 50% tails.” Yet, in reality, “the occurrence of, say, four heads in a row … is quite likely in a sequence of 20 tosses.” In just the same way, in other words, professional basketball players (who are obviously quite skilled at shooting baskets) are likely to make several baskets in a row—not because of any special quality of “heat” they possess, but instead simply because they are good shooters. It’s this inability to perceive randomness, in other words, that may help explain the absence of black British students at many Oxford colleges.

As we saw above, when Warikoo asserts that black students are “massively underrepresented” at Oxford colleges, what she means is that the proportion of black students at Oxford is not the same as the percentage of black people in the United Kingdom as a whole. But as “The Hot Hand” shows, to “expect [that] the essential characteristics of a chance process to be represented not only globally in the entire sequence, but also locally, in each of its parts” is irrational: in reality, a “locally representative sequence … deviates systematically from chance expectation.” Since Oxford colleges, after all, are much smaller population samples than the United Kingdom as a whole is, it would be absurd to believe that their populations could somehow exactly replicate precisely the same proportions as the larger population.

Maybe though you still don’t see why, which is why I’ll now call on some backup: professors of statistics Howard Wainer and Harris Zwerling. In 2006, the two observed that, during the 1990s, many became convinced that smaller schools were the solution to America’s “education crisis”—the Bill and Melinda Gates Foundation, they note, became so convinced of the fact that they spent $1.7 billion on it. That’s because “when one looks at high-performing schools, one is apt to see an unrepresentatively large proportion of smaller schools.” But while that may be so, the two say, in reality “seeing a greater than anticipated number of small schools” in the list of better schools “does not imply that being small means having a greater likelihood of being high performing.” The reason, they say, is precisely the same reason that you don’t have a higher risk of kidney cancer by living in the American South.

Why might you think that? Turns out, Wainer and Zwerling say, that U.S. counties with the highest apparent risk of kidney cancer are all “rural and located in the Midwest, the South, and the West.” So, should you avoid those parts of the country if you are afraid of kidney cancer? Not at all—because the U.S. counties with the lowest apparent risk of kidney cancer are all “rural and located in the Midwest, the South, and the West.” The county characteristics that tend to have both the highest and lowest rates of cancer are precisely the same.

What Wainer and Zwerling’s example shows is precisely the same as that shown by Tversky and company’s work on the field goal rates of the Philadelphia 76ers. It’s a “same” that can be expressed with the words of journalist Michael Lewis, who recently authored a book about Amos Tversky and his long-time research partner (and Nobel Prize-winner) Daniel Kahneman called The Undoing Project: A Friendship That Changed Our Minds: “the smaller the sample, the lower the likelihood that it would mirror the broader population.” As Brian S. Everitt notes in 1999’s Chance Rules: An Informal Guide to Probability, Risk, and Statistics, “in, say, 20 tosses of a fair coin, the number of heads is unlikely to be exactly 10”—the probability, in fact, is “a little less than 1 in 5.” In other words, a sample of 20 tosses is much more likely to come up biased towards either heads or tails—and much, much more likely to be heavily biased towards one or the other than a larger population of coin flips is. Getting extreme results is much more likely in smaller populations.

Oxford colleges are, of course, very small samples of the population of the United Kingdom, which is about 66 million people. Oxford University as a whole, on the other hand, contains about 23,000 students. There are 38 colleges (as well as some other institutions), and some of these—like All Souls, for example—do not even admit undergraduate students; those that that do consist largely of a few hundred students each. The question then that Natasha Warikoo ought to ask first about the admission of black British students to Oxford colleges is, “how likely is it that a sample of 300 would mirror a population of 66 million?” The answer, as the work of Tversky et al. demonstrates,  is “not very”—it’s even less likely, in other words, than the likelihood of throwing exactly 2 heads and 2 tails when throwing a coin four times.

Does that mean that racism does not exist? No, certainly not. But Warikoo says that “[o]nly when Oxford and Cambridge succeed in including young Britons from all walks of life will they be what they say they are: world-class universities.” In fact, however, the idea that institutional populations ought to mirror the broader population is not only not easy to obtain—but flatly absurd. It isn’t that that a racially proportionate society is a difficult goal, in other words—it is that it is an impossible one. To get 300 people, or even 23,000, to reflect the broader population would require, essentially, rewiring the system to such an extent that it’s possible that no other goals—like, say, educating qualified students—could also be achieved; it would require so much effort fighting the entropy of chance that the cause would, eventually, absorb all possible resources. In other words, Oxford can either include “young Britons from all walks of life”—or it can be a world-class university.  It can’t, however, be both; which is to say that Natasha Warikoo—like one character says about little “Father Time’s” stepmother, Sue, at the end of Jude the Obscure—will never find peace.

Forked

He had already heard that the Roman armies were hemmed in between the two passes at the Caudine Forks, and when his son’s courier asked for his advice he gave it as his opinion that the whole force ought to be at once allowed to depart uninjured. This advice was rejected and the courier was sent back to consult him again. He now advised that they should every one be put to death. On receiving these replies … his son’s first impression was that his father’s mental powers had become impaired through his physical weakness. … [But] he believed that by taking the course he first proposed, which he considered the best, he was establishing a durable peace and friendship with a most powerful people in treating them with such exceptional kindness; by adopting the second he was postponing war for many generations, for it would take that time for Rome to recover her strength painfully and slowly after the loss of two armies.
There was no third course.
—Titus Livius. Ab Urbe Condita. Book IX 

 

“Of course, we want both,” wrote Lee C. Bollinger, the president of Columbia University, in 2012, about whether “diversity in post-secondary schools should be focused on family income rather than racial diversity.” But while many might wish to do both, is that possible? Can the American higher educational system serve two masters? According to Walter Benn Michaels of the University of Illinois at Chicago, Bollinger’s thought that American universities can serve both economic goals and racial justice has been the thought of “every academic” with whom he’s ever discussed the subject—but Michaels, for his part, wonders just how sincere that wish really is. American academia, he says, has spent “twenty years of fighting like a cornered raccoon on behalf of the one and completely ignoring the other”; how much longer, he wonders, before “‘we want both’ sounds hollow not only to the people who hear it but to the people who say it?” Yet what Michaels doesn’t say is just why, as pious as that wish is, it’s a wish that is necessarily doomed to go unfulfilled—something that is possible to see after meeting a fictional bank teller named Linda.

“Linda”—the late 1970s creation of two Israeli psychologists, Amos Tversky and Daniel Kahneman—may be the most famous fictional woman in the history of the social sciences, but she began life as a single humble paragraph:

Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Following that paragraph, there were a series of eight statements describing Linda—but as the biologist Stephen Jay Gould would point out later, “five are a blind, and only three make up the true experiment.” The “true experiment” wouldn’t reveal anything about Linda—but it would reveal a lot about those who met her. “Linda,” in other words, is like Nietzsche’s abyss: she stares back into you.

The three pointed statements of Kahneman and Tversky’s experiment are these: “Linda is active in the feminist movement; Linda is a bank teller; Linda is a bank teller and is active in the feminist movement.” The two psychologists would then ask their test subjects to guess which of the three statements was more likely. Initially, these test subjects were lowly undergraduates, but as Kahneman and Tversky performed and then re-performed the experiment, they gradually upgraded: using graduate students with a strong background in statistics next—and then eventually faculty. Yet, no matter how sophisticated the audience to which they showed this description, what Kahneman and Tversky found was that virtually everyone always thought that the statement “Linda is a bank teller and active in the feminist movement” was more likely than the statement “Linda is a bank teller.” But as only a little thought requires, that is impossible.

I’ll let the journalist Michael Lewis, who recently published a book about the work of the pair of psychologists entitled The Undoing Project: A Friendship That Changed Our Minds, explain the impossibility:

“Linda is a bank teller and is active in the feminist movement” could never be more probable than “Linda is a bank teller.” “Linda is a bank teller and is active in the feminist movement” was just a special case of “Linda is a bank teller.” “Linda is a bank teller” included “Linda is a bank teller and is active in the feminist movement” along with “Linda is a bank teller and likes to walk naked through Serbian forests” and all other bank-telling Lindas. One description was entirely contained by the other.

“Linda is a bank teller and is active in the feminist movement” simply cannot be more likely than “Linda is a bank teller.” As Louis Menand of Harvard observed about the “Linda problem” in The New Yorker in 2005, thinking that “bank teller and feminist” is more likely than the “bank teller” description “requires two things to be true … rather than one.” If the one is true so is the other; that’s why, as Lewis observed in an earlier article on the subject, it’s “logically impossible” to think otherwise. Kahneman and Tversky’s finding is curious enough on its own terms for what it tells us about human cognition, of course, because it exposes a reaction that virtually every human being ever encountering it has made. But what makes it significant in the present context is that it is also the cognitive error Lee C. Bollinger makes in his opinion piece.

“The Linda problem,” as Michael Lewis observed in The Undoing Project, “resembled a Venn diagram of two circles, but with one of the circles wholly contained by the other.” One way to see the point, perhaps, is in relation to prison incarceration. As political scientist Marie Gottschalk of the University of Pennsylvania has observed, although the

African-American incarceration rate of about 2,300 per 100,000 people is clearly off the charts and a shocking figure … [f]ocusing so intently on these racial disparities often obscures the fact that the incarceration rates for other groups in the United States, including whites and Latinos, is also comparatively very high.

While the African-American rate of imprisonment is absurdly high, in other words, the “white incarceration rate in the United States is about 400 per 100,000,” which is at least twice the rate of “the most punitive countries in Western Europe.” What that means is that, while it is possible to do something regarding, say, African-American incarceration rates by lowering the overall incarceration rates, it can’t be done the other way.“Even,” as Gottschalk says, “if you released every African American from US prisons and jails today, we’d still have a mass incarceration crisis in this country.” Releasing more prisoners means fewer minority prisoners, but releasing minority prisoners still means a lot of prisoners.

Which, after all, is precisely the point of the “Linda problem”: just as “bank teller” contains both “bank teller” and any other set of descriptors that could be added to “bank teller,” so too does “prisoner” include any other set of descriptors that could be added to it. Hence, reducing the prison population will necessarily reduce the numbers of minorities in prison—but reducing the numbers of minority prisoners will not do (much) to reduce the number of prisoners. “Minority prisoners” is a circle contained within the circle of “prisoners”—saying you’d like to reduce the numbers of minority prisoners is essentially to say that you don’t want to do anything about prisons.

Hence, when Hillary Clinton asked her audience during the recent presidential campaign “If we broke up the big banks tomorrow … would that end racism?” and “Would that end sexism?”—and then answered her own question by saying, “No,” what she was effectively saying was that she would do nothing about any of those things, racism and sexism included. (Which, given that this was the candidate who asserted that politicians ought to have “both a public and a private position,” is not out of the question.) Wanting “both,” or an alleviation of economic inequality and discrimination—as Lee Bollinger and “every academic” Walter Benn Michaels has ever talked to say they want—is simply the most efficient way of not getting either. As Michaels says, “diversity and antidiscrimination have done and can do [emp. added] nothing whatsoever to mitigate economic inequality.” The sooner that Americans realize that Michaels isn’t kidding—that anti-discrimination, identity politics is not an alternative solution, but in fact no solution—and why he’s right, the sooner that something could be done about America’s actual problems.

Assuming, of course, that’s something anyone really wants.

Size Matters

That men would die was a matter of necessity; which men would die, though, was a matter of circumstance, and Yossarian was willing to be the victim of anything but circumstance.
—Catch-22.

I do not pretend to understand the moral universe; the arc is a long one, my eye reaches but little ways; I cannot calculate the curve and complete the figure by the experience of sight; I can divine it by conscience. And from what I see I am sure it bends towards justice.
Things refuse to be mismanaged long.
—“Of Justice and the Conscience.”

 

The Casino at Monte Carlo

 

 

Once, wrote the baseball statistician Bill James, there was “a time when Americans” were such “an honest, trusting people” that they actually had “an unhealthy faith in the validity of statistical evidence”–but by the time James wrote in 1985, things had gone so far the other way that “the intellectually lazy [had] adopted the position that so long as something was stated as a statistic it was probably false.” Today, in no small part because of James’ work, that is likely no longer as true as it once was, but nevertheless the news has not spread to many portions of academia: as University of Virginia historian Sophia Rosenfeld remarked in 2012, in many departments it’s still fairly common to hear it asserted—for example—that all “universal notions are actually forms of ideology,” and that “there is no such thing as universal common sense.” Usually such assertions are followed by a claim for their political utility—but in reality widespread ignorance of statistical effects is what allowed Donald Trump to be elected, because although the media spent much of the presidential campaign focused on questions like the size of Donald Trump’s … hands, the size that actually mattered in determining the election was a statistical concept called sample size.

First mentioned by the mathematician Jacob Bernoulli made in his 1713 book, Ars Conjectandi, sample size is the idea that “it is not enough to take one or another observation for such a reasoning about an event, but that a large number of them are needed.” Admittedly, it might not appear like much of an observation: as Bernoulli himself acknowledged, even “the most stupid person, all by himself and without any preliminary instruction,” knows that “the more such observations are taken into account, the less is the danger of straying from the goal.” But Bernoulli’s remark is the very basis of science: as an article in the journal Nature put the point in 2013, “a study with low statistical power”—that is, few observations—“has a reduced chance of detecting a true effect.” Sample sizes need to be large enough to be able to eliminate chance as a possible factor.

If that isn’t known it’s possible to go seriously astray: consider an example drawn from the work of Israeli psychologists Amos Tversky (MacArthur “genius” grant winner) and (Nobel Prize-winning) Daniel Kahneman—a study “of two toys infants will prefer.” Let’s say that in the course of research our investigator finds that, of “the first five infants studied, four have shown a preference for the same toy.” To most psychologists, the two say, this would be enough for the researcher to conclude that she’s on to something—but in fact, the two write, a “quick computation” shows that “the probability of a result as extreme as the one obtained” being due simply to chance “is as high as 3/8.” The scientist might be inclined to think, in other words, that she has learned something—but in fact her result has a 37.5 percent chance of being due to nothing at all.

Yet when we turn from science to politics, what we find is that an American presidential election is like a study that draws grand conclusions from five babies. Instead of being one big sample—as a direct popular national election would be—presidential elections are broken up into fifty state-level elections: the Electoral College system. What that means is that American presidential elections maximize the role of chance, not minimize it.

The laws of statistics, in other words, predict that chance will play a large role in presidential elections—and as it happens, Tim Meko, Denise Lu and Lazaro Gamio reported for The Washington Post three days after the election that “Trump won the presidency with razor-thin margins in swing states.” “This election was effectively decided,” the trio went on to say, “by 107,000 people”—in an election in which more than 120 million votes were cast, that means that election was decided by less than a tenth of one percent of the total votes. Trump won Pennsylvania by less than 70,000 votes of nearly 6 million, Wisconsin by less than 30,000 of just less than three million, and finally Michigan by less than 11,000 out of 4.5 million: the first two by just more than one percent of the total vote each—and Michigan by a whopping .2 percent! Just to give you an idea of how insignificant these numbers are by comparison with the total vote cast, according to the Michigan Department of Transportation it’s possible that a thousand people in the five largest counties were involved in car crashes—which isn’t even to mention people who just decided to stay home because they couldn’t find a babysitter.

Trump owes his election, in short, to a system that is vulnerable to chance because it is constructed to turn a large sample (the total number of American voters) into small samples (the fifty states). Science tells us that small sample sizes increase the risk of random chance playing a role, American presidential elections use a smaller sample size than they could, and like several other presidential elections, the 2016 election did not go as predicted. Donald Trump could, in other words, be called “His Accidency” with even greater justice than John Tyler—the first vice-president to be promoted due to the death of his boss in office—was. Yet, why isn’t that point being made more publicly?

According to John Cassidy of The New Yorker, it’s because Americans haven’t “been schooled in how to think in probabilistic terms.” But just why that’s true—and he’s essentially making the same point Bill James did in 1985, though more delicately—is, I think, highly damaging to many of Clinton’s biggest fans: the answer is, because they’ve made it that way. It’s the disciplines where many of Clinton’s most vocal supporters make their home, in other words, that are most directly opposed to the type of probabilistic thinking that’s required to see the flaws in the Electoral College system.

As Stanford literary scholar Franco Moretti once observed, the “United States is the country of close reading”: the disciplines dealing with matters of politics, history, and the law within the American system have, in fact, more or less been explicitly constructed to prevent importing knowledge of the laws of chance into them. Law schools, for example, use what’s called the “case method,” in which a single case is used to stand in for an entire body of law: a point indicated by the first textbook to use this method, Christopher Langdell’s A Selection of Cases on the Law of Contracts. Other disciplines, such as history, are similar: as Emory University’s Mark Bauerlein has written, many such disciplines depend for their very livelihood upon “affirming that an incisive reading of a single text or event is sufficient to illustrate a theoretical or historical generality.” In other words, it’s the very basis of the humanities to reject the concept of sample size.

What’s particularly disturbing about this point is that, as Joe Pinsker documented in The Atlantic last year, the humanities attract a wealthier student pool than other disciplines—which is to say that the humanities tend to be populated by students and faculty with a direct interest in maintaining obscurity around the interaction between the laws of chance and the Electoral College. That doesn’t mean that there’s a connection between the architecture of presidential elections and the fact that—as Geoffrey Harpham, former president and director of the National Humanities Center, has observed—“the modern concept of the humanities” (that is, as a set of disciplines distinct from the sciences) “is truly native only to the United States, where the term acquired a meaning and a peculiar cultural force that it does not have elsewhere.” But it does perhaps explain just why many in the national media have been silent regarding that design in the month after the election.

Still, as many in the humanities like to say, it is possible to think that the current American university and political structure is “socially constructed,” or in other words could be constructed differently. The American division between the sciences and the humanities is not the only way to organize knowledge: as the editors of the massive volumes of The Literary and Cultural Reception of Darwin in Europe pointed out in 2014, “one has to bear in mind that the opposition of natural sciences … and humanities … does not apply to the nineteenth century.” If that opposition that we today find so omnipresent wasn’t then, it might not be necessary now. Hence, if the choice of the American people is between whether they ought to get a real say in the affairs of government (and there’s very good reason to think they don’t), or whether a bunch of rich yahoos spend time in their early twenties getting drunk, reading The Great Gatsby, and talking about their terrible childhoods …well, I know which side I’m on. But perhaps more significantly, although I would not expect that it happens tomorrow, still, given the laws of sample size and the prospect of eternity, I know how I’d bet.

Or, as another sharp operator who’d read his Bernoulli once put the point:

“The arc of the moral universe is long, but it bends towards justice.”

 

Lex Majoris

The first principle of republicanism is that the lex majoris partis is the fundamental law of every society of individuals of equal rights; to consider the will of the society enounced by the majority of a single vote, as sacred as if unanimous, is the first of all lessons in importance, yet the last which is thoroughly learnt. This law once disregarded, there is no other but that of force, which ends necessarily in military despotism.
—Thomas Jefferson. Letter to Baron von Humboldt. 13 June 1817.

Since Hillary Clinton lost the 2016 American presidential election, many of her supporters have been quick to cry “racism” on the part of voters for her opponent, Donald Trump. According to Vox’s Jenée Desmond-Harris, for instance, Trump won the election “not despite but because he expressed unfiltered disdain toward racial and religious minorities in the country.” Aside from being the easier interpretation, because it allows Clinton voters to ignore the role their own economic choices may have played in the broad support Trump received throughout the country, such accusations are counterproductive even on their own terms because—only seemingly paradoxically—they reinforce many of the supports racism still receives in the United States: above all, because they weaken the intellectual argument for a national direct election for the presidency. By shouting “racism,” in other words, Hillary Clinton’s supporters may end up helping to continue racism’s institutional support.

That institutional support begins with the method by which Americans elect their president: the Electoral College—a method that, as many have noted, is not used in any other industrialized democracy. Although many scholars and others have advanced arguments for the existence of the college through the centuries, most of these “explanations” are, in fact, intellectually incoherent: while the most common of the traditional “explanations” concerns the differences between the “large states” and the “small,” for instance, in the actual United States—as James Madison, known as the “Father of the Constitution,” noted at the time—there had not then, and has not ever been since, a situation in American history that involved a conflict between larger-population and smaller-population states. Meanwhile, the other “explanations” for the Electoral College do not even rise to this level of incoherence.

In reality there is only one explanation for the existence of the college, and that explanation has been most forcefully and clearly made by law professor Paul Finkelman, now serving as a Senior Fellow at the University of Pennsylvania after spending much of his career at obscure law schools like the University of Tulsa College of Law, the Cleveland-Marshall College of Law, and the Albany Law School. As Finkelman has been arguing for decades (his first papers on the subject were written in the 1980s), the Electoral College was originally invented by the delegates to the Constitutional Convention of 1787 in order to protect slavery. That such was the purpose of the College can be known, most obviously, because the delegates to the convention said so.

When the means of electing a president were first debated, it’s important to remember that the convention had already decided, for the purposes of representation in the newly-created House of Representatives, to count black slaves by the means of the infamous three-fifths ratio. That ratio, in turn, had its effect when discussing the means of electing a president: delegates like James Madison argued, as Finkelman notes, that the existence of such a college—whose composition would be based on each state’s representation in the House of Representatives—would “guarantee that the nonvoting slaves could nevertheless influence the presidential election.” Or as Hugh Williamson, a delegate from North Carolina, observed during the convention, if American presidents were elected by direct national vote the South would be shut out of electing a national executive because “her slaves will have no suffrage”—that is, because in a direct vote all that would matter is the number of voters, the Southern states would lose the advantage the three-fifths ratio gave them in the House. Hence, the existence of the Electoral College is directly tied to the prior decision to grant Southern slave states an advantage in Congress, and so the Electoral College is another in a string of institutional decisions made by convention delegates to protect domestic slavery.

Yet, assuming that Finkelman’s case for the racism of the Electoral College is true, how can decrying the racism of the American voter somehow inflict harm on the case for abolishing the Electoral College? The answer goes back to the very justifications of, not only presidential elections, but elections in general—the gradual discovery, during the eighteenth century Enlightenment, of what is today known as the Law of Large Numbers.

Putting the law in capital letters, I admit, tends to mystify it, but anyone who buys insurance already understands the substance of the concept. As New Yorker writer Malcolm Gladwell once explained insurance, “the safest and most efficient way to provide insurance” is “to spread the costs and risks of benefits over the biggest and most diverse group possible.” In other words, the more people participating in an insurance plan, the greater the possibility that the plan’s members will be protected. The Law of Large Numbers explains why that is.

That reason is the same as the reason that, as Peter Bernstein remarks in Against the Gods: The Remarkable Story of Risk, if we toss a coin enough times that “will correspondingly increase the probability that the ratio of heads thrown to total throws” will decrease. Or, the reason that—as physicist Leonard Mlodinow has pointed out—in order really to tell which baseball team is better than another a World Series would have to be at least 23 games long (if one team were much better than the other), and possibly as long as 269 games (between two closely-matched opponents). Only by playing so many games can random chance be confidently excluded: as Carl Bialik of FiveThirtyEight once pointed out, usually “in sports, the longer the contest, the greater the chance that the favorite prevails.” Or, as Israeli psychologists Daniel Kahneman and Amos Tversky put the point in 1971, “the law of large numbers guarantees that very large samples will indeed be representative”: it’s what scientists rely upon to know that, if they have performed enough experiments or poured over enough data, they know enough to exclude idiosyncratic results. The Law of Large Numbers asserts, in short, that the more times we repeat something, the closer we will approach its true value.

It’s for just that reason that many have noted the connection between science and democratic government: “Science and democracy are powerful partners,” as the website for the Union of Concerned Scientists has put it. What makes these two objects such “powerful” partners is that the Law of Large Numbers is what underlies the act of holding elections: as James Surowiecki put the point in his book, The Wisdom of Crowds, the theory of democracy is that “the larger the group, the more reliable its judgment will be.” Just as scientists think that, by replicating an experiment, they can more readily trust in its results, so too does a democratic government implicitly think that, by including more people in the decision-making process, the government can the more readily arrive at the “correct” solution: as James Madison put it in The Federalist No. 10, if you “take in a greater variety of parties and interests,” then “you make it less probable that a majority of the whole will have a common motive for invading the rights of other citizens.” Without such a belief, after all, there would be no reason not to trust, say, a ruling caste to make decisions for society—or even a single, perhaps orange-toned, individual. Without some concept of the Law of Large Numbers—some belief that increasing the numbers of trials, or increasing the number of inputs, will make for better results—there is no reason for democratic government at all.

That’s why, when people criticize the Electoral College, they are implicitly invoking the Law of Large Numbers. The Electoral College divides the pool of American voters into fifty smaller pools, but a national popular vote would collect all Americans into a single lump—a point that some defenders of the College sometimes seek to make into a virtue, instead of the vice it is. In the wake of the 2000 election, for example, Senator Mitch McConnell wrote that the “Electoral College served to center the post-election battles in Florida,” preventing the “vote recounts and court battles in nearly every state of the Union” that, McConnell assures us, would have occurred in the college’s absence. But as Timothy Noah pointed out in The New Republic in 2012, what McConnell’s argument “fails to realize is that when you’re assembling one big count rather than a lot of little ones it’s a lot less clear what’s to be gained from rigging any of the little ones.” If what matters is the popular vote, what happens in any one location doesn’t matter so much; hence, stealing votes in downstate Illinois won’t allow you to steal the entire state—just as, with enough samples or experiments run, the fact that the lab assistant was drowsy at the time she recorded one set of results won’t matter so much. Or why deliberately losing a single game in July hardly matters so much as tanking a game of the World Series.

Put in such a way, it’s hard to see how anyone without a vested stake in the construction of the present system could defend the Electoral College—yet, as I suspect we are about to see, the very people now ascribing Donald Trump’s victory to the racism of the American voter will soon be doing just that. The reason will be precisely the same reason that such advocates want to blame racism, rather than the ongoing thievery of economic elites, for the rejection of Clinton: because racism is a “cultural” phenomenon, and most left-wing critics of the United States now obtain credentials in “cultural,” rather than scientific, disciplines.

If, in other words, Donald Trump’s victory was due to a complex series of renegotiations of the global contract between capital and labor, then that would require experts in economic and other, similar, disciplines to explain it; if his victory was due to racism, however—racism being considered a cultural phenomenon—then that will call forth experts in “cultural” fields. Because those with “liberal” or “leftist” political leanings now tend to gather in “cultural” fields, those with those political leanings will (indeed, must) now attempt to shift the battleground towards their areas of expertise. That shift, I would wager, will in turn lead those who argue for “cultural” explanations for the rise of Trump against arguments for the elimination of the Electoral College.

The reason is not difficult to understand: it isn’t too much to say, in fact, that one way to define the study of the humanities is to say it comprises the disciplines that largely ignore, or even oppose, the Law of Large Numbers both as a practical matter and as a philosophic one. As literary scholar Franco Moretti, now of Stanford, observed in his Atlas of the European Novel, 1800-1900, just as “silver fork novels”—a genre published in England between the 1820s and the 1840s—do not “show ‘London,’ but only a small, monochrome portion of it,” so too does the average student of literature not really study her ostensible subject matter. “I work on west European narrative between 1790 and 1930, and already feel like a charlatan outside of Britain and France,” Moretti confesses in an essay entitled “Distant Reading”—and even then, he only works “on its canonical fraction, which is not even 1 percent of published literature.” As Joshua Rothman put the point in a New Yorker profile of Moretti a few years ago, Moretti instead insists that “if you really want to understand literature, you can’t just read a few books or poems over and over,” but instead “you have to work with hundreds or even thousands of texts at a time”—that is, he insists on the significance of the Law of Large Numbers in his field, an insistence whose very novelty demonstrates how literary study is a field that has historically resisted precisely that recognition.

In order to proceed, in other words, disciplines like literary study or art history—or even history itself—must argue for the representativeness of a given body of work: usually termed, at least in literary study, “the Canon.” Such disciplines are already, simply by their very nature, committed to the idea that it is not necessary to read all of what Moretti says is the “thirty thousand nineteenth-century British novels out there” in order to arrive at conclusions about the nineteenth-century British novel: in the first place, “no one really knows” how many there really are (there could easily be twice as many), and in the second “no one has read them [all], [and] no one ever will.” In order to get off the ground, such disciplines must necessarily deny the Law of Large Numbers: as Moretti says, “you invest so much in individual texts only if you think that very few of them really matter”—a belief with an obvious political corollary. Rejection of the Law of Large Numbers is thusly, as Moretti also observes, “an unconscious and invisible premiss” for most who study such fields—which is to say that although students of the humanities often make claims for the political utility of their work, they sometimes forget that the enabling presuppositions of their fields are inherently those of the pre-Enlightenment ancien régime.

Perhaps that’s why—as Joe Pinsker observed in a fascinating, but short, article for The Atlantic several years ago—studies of college students find that those “from lower-income families tend toward ‘useful’ majors, such as computer science, math, and physics,” while students “whose parents make more money flock to history, English, and the performing arts”: the baseline assumptions of those disciplines are, no matter the particular predilections of a given instructor, essentially aristocratic, not democratic. To put it most baldly, the disciplines of the humanities must reject the premise of the Law of Large Numbers, which says that as more examples are added, the closer we approach to the truth—a point that can be directly witnessed when, for instance, English professor Michael Bérubé of Pennsylvania State University observes that the “humanists at [his] end of the [academic] hallway roundly dismissed” Harvard biologist E.O. Wilson’s book, Consilience: The Unity of Knowledge for arguing that “all human knowledge can and eventually will be unified under the rubric of the natural sciences.” Rejecting the Law of Large Numbers is foundational to the very operation of the humanities: without making that rejection, they cannot exist.

In recent decades, of course, presumably Franco Moretti has not been the only professor of the humanities to realize that their disciplines stood on a collision course with the Law of Large Numbers—it may perhaps explain why disciplines like literature and others have, for years, been actively recruiting among members of minority groups. The institutional motivations of such hiring, in other words, ought to be readily apparent: by making such hires, departments of the humanities could insulate themselves from charges from the political left—while at the same time continuing the practices that, without such cover, might have appeared increasingly anachronistic in a democratic age. Minority hiring, that is, may not be so politically “progressive” as its defenders sometimes argue: it may, in fact, have prevented the intellectual reforms within the humanities urged by people like Franco Moretti for a generation or more. Of course, by joining such departments, members of minority groups also may have, consciously or not, tied their own fortunes to a philosophic rejection of concepts like the Law of Large Numbers—as African-American sportswriter Michael Wilbon, of ESPN fame, wrote this past May, black people supposedly have some kind of allergy to statistical analysis: “in ‘BlackWorld,’” Wilbon solemnly intoned, “never is heard an advanced analytical word.” I suspect then that many who claim to be on the political left will soon come out to defend the Electoral College. If that happens, then in one last cruel historical irony the final defenders of American slavery may end up being precisely those slavery meant to oppress.

Noble Lie

With a crew and good captain well seasoned,
They left fully loaded for Cleveland.
—“The Wreck of the Edmund Fitzgerald.” 1976.

The comedian Bill Maher began the “panel” part of his show Real Time the other day—the last episode before the election—by noting that virtually every political expert had dismissed Donald Trump’s candidacy at every stage of the past year’s campaign. When Trump announced he was running, Maher observed, the pundits said “oh, he’s just saying that … because he just wants to promote his brand.” They said Trump wouldn’t win any voters, Maher noted—“then he won votes.” And then, Maher went on, they said he wouldn’t win any primaries—“then he won primaries.” And so on, until Trump became the Republican nominee. So much we know, but what was of interest about the show was the response one of Maher’s guests: David Frum, a Canadian who despite his immigrant origins became a speechwriter for George W. Bush, invented the phrase “axis of evil,” and has since joined the staff of the supposedly liberal magazine, The Atlantic. The interest of Frum’s response was not only how marvelously inane it was—but also how it had already been decisively refuted only hours earlier, by men playing a boy’s game on the Lake Erie shore.

Maybe I’m being cruel however: like most television shows, Real Time with Bill Maher is shot before it is aired, and this episode was released last Friday. Frum then may not have been aware, when he said what he said, that the Chicago Cubs won the World Series on Wednesday—and if he is like most people, Frum is furthermore unaware of the significance of that event, which goes (as I will demonstrate) far beyond matters baseball. Still, surely Frum must have been aware of how ridiculous what he said was, given that the conversation began with Maher reciting the failures of the pundit class—and Frum admitted to belonging to that class. “I was one of those pundits that you made fun of,” Frum confessed to Maher—yet despite that admission, Frum went on to make a breathtakingly pro-pundit argument.

Trump’s candidacy, Frum said, demonstrated the importance of the gatekeepers of the public interest—the editors of the national newspapers, for instance, or the anchors of the network news shows, or the mandarins of the political parties. Retailing a similar  argument to one made by, among others, Salon’s Bob Cesca—who contended in early October that “social media is the trough from which Trump feeds”—Frum proceeded to make the case that the Trump phenomena was only possible once apps like Facebook and Twitter enabled presidential candidates to bypass the traditional centers of power. To Frum, in other words, the proper response to the complete failure of the establishment (to defeat Trump) was to prop up the establishment (so as to defeat future Trumps). To protect against the failure of experts Frum earnestly argued—with no apparent sense of irony—that we ought to give more power to experts.

There is, I admit, a certain schadenfreude in witnessing a veteran of the Bush Administration tout the importance of experts, given that George W.’s regime was notable for, among other things, “systematically chang[ing] and supress[ing] … scientific reports about global warming” (according to the British Broadcasting Corporation)—and not even to discuss how Bush cadres torpedoed the advice of the professionals of the CIA vis á vis the weapons-buying habits of a certain Middle Eastern tyrant. But the larger issue, however, is that the very importance of “expert” knowledge has been undergoing a deep interrogation for decades now—and that the victory of the Chicago Cubs in this year’s World Series has brought much of that critique to the mainstream.

What I mean can be demonstrated by a story told by the physicist Freeman Dyson—a man who never won a Nobel Prize, nor even received a doctorate, but nevertheless was awarded a place at Princeton’s Institute of Advanced Study at the ripe age of thirty by none other than Robert Oppenheimer (the man in charge of the Manhattan Project) himself. Although Dyson has had a lot to say during his long life—and a lot worth listening to—on a wide range of subjects, from interstellar travel to Chinese domestic politics, of interest to me in connection to Frum’s remarks on Donald Trump is an article Dyson published in The New York Review of Books in 2011, about a man who did win the Nobel Prize: the Israeli psychologist Daniel Kahneman, who won the prize for economics in 2002. In that article, Dyson told a story about himself: specifically, what he did during World War II—an experience, it turns out, that leads by a circuitous path over the course of seven decades to the epic clash resolved by the shores of Lake Erie in the wee hours of 3 November.

Entitled “How to Dispel Your Illusions,” Dyson there tells the story of being a young statistician with the Royal Air Force’s Bomber Command in the spring of 1944—a force that suffered, according to the United Kingdom’s Bomber Command Museum, “a loss rate comparable only to the worst slaughter of the First World War trenches.” To combat this horror, Dyson was charged with discovering the common denominator between the bomber crews that survived until the end of their thirty-mission tour of duty (about 25% of all air crews). Since they were succeeding when three out of four of their comrades were failing, Dyson’s superiors assumed that those successful crews were doing something that their less-successful colleagues (who were mostly so much less successful that they were no longer among the living) were not.

Bomber Command, that is, had a theory about why some survived and some died: “As [an air crew] became more skillful and more closely bonded,” Dyson writes that everyone at Bomber Command thought, “their chances of survival would improve.” So Dyson, in order to discover what that something was, plunged in among the data of all the bombing missions the United Kingdom had run over Germany since the beginning of the war. If he could find it, maybe it could be taught to the others—and the war brought that much closer to an end. But despite all his searching, Dyson never found that magic ingredient.

It wasn’t that Dyson didn’t look hard enough for it: according to Dyson, he “did a careful analysis of the correlation between the experience of the crews and their loss rates, subdividing the data into many small packages so as to eliminate effects of weather and geography.” Yet, no matter how many different ways he looked at the data, he could not find evidence that the air crews that survived were any different than the ones shot down over Berlin or lost in the North Sea: “There was no effect of experience,” Dyson’s work found, “on loss rate.” Who lived and who died while attempting to burn Dresden or blow up Hamburg was not a matter of experience: “whether a crew lived or died,” Dyson writes, “was purely a matter of chance.” The surviving crews possessed no magical ingredient. They couldn’t—perhaps because there wasn’t one.

Still, despite the conclusiveness of Dyson’s results his studies had no effect on the operations of Bomber Command: “The crews continued to die, experienced and inexperienced alike, until Germany was overrun and the war finally ended.” While Dyson’s research suggested that dying in the stratosphere over Lübeck had no relation to skill, no one at the highest levels wanted to admit that the survivors weren’t experts—that they were instead just lucky. Perhaps, had the war continued, Dyson’s argument might eventually have won out—but the war ended, fortunately (or not) for the air crews of the Royal Air Force, before Bomber Command had to admit he was right.

All of that, of course, might appear to have little to do with the Chicago Cubs—until it’s recognized that the end of their century-long championship drought had everything to do with the eventual success of Dyson’s argument. Unlike Bomber Command, the Cubs have been at the forefront of what The Ringer’s Rany Jazayerli calls baseball’s “Great Analytics War”—and unlike the contest between Dyson and his superiors, that war has had a definite conclusion. The battle between what Jazayerli calls an “objective, data-driven view” and an older vision of baseball “ended at 48 minutes after midnight on November 3”—when the Cubs (led by a general manager who, like Dyson, trusted to statistical analysis) recorded the final out of the 2016 season.

That general manager is Theo Epstein—a man who was converted to Dyson’s “faith” at an early age. According to ESPN, Epstein, “when he was 12 … got his first Bill James historical abstract”—and as many now recognize, James pioneered applying the same basic approach Dyson used to think about how to bomb Frankfurt to winning baseball games. An obscure graduate of the University of Kansas, after graduation James took a job as a night security guard at the Stokely-Van Camp pork and beans cannery in Kansas City—and while isolated in what one imagines were the sultry (or wintry) Kansas City evenings of the 1970s, James had plenty of time to think about what interested him. That turned out to be somewhat like the problem Dyson had faced a generation earlier: where Dyson was concerned with how to win World War II, James was interested in what appeared to be the much-less portentous question of how to win the American League. James thereby invented an entire field—what’s now known as sabermetrics, or the statistical study of baseball—and in so doing, the tools James invented have become the keys to baseball’s kingdom. After all, Epstein—employed by a team owner who hired James as a consultant in 2003—not only used James’ work to end the Cubs’ errand in baseball’s wilderness but also, as all the world knows, constructed the Boston Red Sox championship teams of 2004 and 2007.

What James had done, of course, is shown how the supposed baseball “experts”—the ex-players and cronies that dominated front offices at the time—in fact knew very little about the game: they did not know, for example, that the most valuable single thing a batter can do is to get on base, or that stolen bases are, for the most part, a waste of time. (The risk of making an out, as per for example David Smith’s “Maury Wills and the Value of a Stolen Base,” is more significant than the benefit of gaining a base.) James’ insights had not merely furnished the weaponry used by Epstein; during the early 2000s another baseball team, the Oakland A’s, and their manager Billy Beane, had used James-inspired work to get to the playoffs four consecutive years (from 2000 to 2003), and won twenty consecutive games in 2002—a run famously chronicled by journalist Michael Lewis’ book, Moneyball: The Art of Winning an Unfair Game, which later became a Hollywood movie starring Brad Pitt. What isn’t much known, however, is that Lewis has noticed the intellectual connection between this work in the sport of baseball—and the work Dyson thought of as similar to his own work as a statistician for Bomber Command: the work of psychologist Kahneman and his now-deceased colleague, Amos Tversky.

The connection between James, Kahneman, and Tversky—an excellent name for a law firm—was first noticed, Lewis says, in a review of his Moneyball book by University of Chicago professors Cass Sunstein, of the law school, and Richard Thaler, an economist. When Lewis described the failures of the “old baseball men,” and conversely Beane’s success, the two professors observed that “Lewis is actually speaking here of a central finding in cognitive psychology”: the finding upon which Kahneman and Tversky based their careers. Whereas Billy Beane’s enemies on other baseball teams tended “to rely on simple rules of thumb, on traditions, on habits, on what other experts seem to believe,” Sunstein and Thaler pointed out that Beane relied on the same principle that Dyson found when examining the relative success of bomber pilots: “Statistics and simple arithmetic tell us more about ourselves than expert intuition.” While Bomber Command in other words relied on the word of their “expert” pilots, who perhaps might have said they survived a run over a ball-bearing plant because of some maneuver or other, baseball front offices relied for decades on ex-players who thought they had won some long-ago game on the basis of some clever piece of baserunning. Tversky and Kahneman’s work, however—like that of Beane and Dyson—suggested that much of what passes as “expert” judgment can be, for decades if not centuries, an edifice erected on sand.

That work has, as Lewis found after investigating the point when his attention was drawn to it by Sunstein and Thaler’s article, been replicated in several fields: in the work of the physician Atul Gawande, for instance, who, Lewis says, “has shown the dangers of doctors who place too much faith in their intuition.” The University of California, Berkeley finance professor Terry Odean “examined 10,000 individual brokerage accounts to see if stocks the brokers bought outperformed stocks they sold and found that the reverse was true.” And another doctor, Toronto’s Donald Redelmeier—who studied under Tversky—found “that an applicant was less likely to be admitted to medical school if he was interviewed on a rainy day.” In all of these cases (and this is not even to bring up the subject of, say, the financial crisis of 2007-08, a crisis arguably brought on precisely by the advice of “experts”), investigation has shown that “expert” opinion may not be what it is cracked up to be. It may in fact actually be worse than the judgment of laypeople.

If so, might I suggest, then David Frum’s “expert” suggestion about what to do to avoid a replay of the Trump candidacy—reinforce the rule of experts, a proposition that itself makes several questionable assumptions about the nature of the events of the past two years, if not decades—stops appearing to be a reasonable proposition. It begins, in fact, to appear rather more sinister: an attempt by those in Frum’s position in life—what we might call Eastern, Ivy League-types—to will themselves into believing that Trump’s candidacy is fueled by a redneck resistance to “reason,” along with good old-fashioned American racism and sexism. But what the Cubs’ victory might suggest is that what could actually be powering Trump is the recognition by the American people that many of the “cures” dispensed by the American political class are nothing more than snake oil proffered by cynical tools like David Frum. That snake oil doubles down on exactly the same “expert” policies (like freeing capital to wander the world, while increasingly shackling labor) that, debatably, is what led to the rise of Trump in the first place—a message that, presumably, must be welcome to Frum’s superiors at whatever the contemporary equivalent of Bomber Command is.

Still, despite the fact that the David Frums of the world continue to peddle their nonsense in polite society, even this descendant of South Side White Sox fans must allow that Theo Epstein’s victory has given cause for hope down here at the street-level of a Midwestern city that for has, for more years than the Cubs have been in existence, been the plaything of Eastern-elite labor and trade policies. It’s a hope that, it seems, now has a Ground Zero.

You can see it at the intersection of Clark and Addison.

Luck of the Irish

 … I hear him mock
The luck of Caesar, which the gods give men
To excuse their after wrath.
—Antony and Cleopatra V, ii

Stephanie Wei, the ex-Yalie golf blogger, recently got her press credentials revoked for the crime of filming tour players during a non-televised Monday practice round at the WGC-Match Play using a live-stream video app. According to her own account, the tour said that her “live-streaming of behind-the-scenes content had violated the Tour’s media regulations.” Wei has admitted that the tour did have a right to take away her credentials (it’s in her contract), but she argued in response that her work produced “fresh, interesting and different content,” and thus enhanced the value of the tour’s product. Wei’s argument however, as seductive as it might be, is a great example of someone manipulating what Thomas Frank has called “the titanic symbolic clash of hip and square” for their own ends: Wei wants to be “hip”—but her actual work is not only just as “square” as any old-school sportswriter who didn’t see fit to mention that Ty Cobb was one of the meanest and most racist men in America, or that Mickey Mantle was a nihilistic drunk, but in fact might be even more harmful.

As Thomas Frank was writing so long ago as the 1990s, the new digital economy has been sold as an “economic revolution,” celebrating “artists rather than commanders, wearers of ponytails and dreamers of cowboy fantasies who proudly proclaim their ignorance of ‘rep ties.’” In contrast to the old world of “conformity, oppression, bureaucracy, meaninglessness, and the disappearance of individualism”—in a word, golf—the new would value “creativity” and “flexibility.” It’s the bright new world we live in today.

So inevitable does that narrative appear that of course Deadspin, the hipsters’ ESPN, jumped on it. “It’s not surprising,” proclaimed Samer Kalaf, “that the PGA Tour, a stuffy organization for a stuffy sport, is being truculent over something as inconsequential as this, but that doesn’t make it any less ridiculous.” The part of Judge Smails (Caddyshack’s prototypical stuffed shirt) is played in this drama by the PGA Tour’s Ty Votaw, who told Golf.com that in the eyes of the tour, what Wei did was “stealing.” On the theory of the tour, what Wei did extracted value from the tour’s product.

Wei herself, to be sure, had a different theory about her actions. Wei wrote that her purpose in transmitting the “raw, alternative footage”—excellent use of buzzwords!—was to “spread fanfare.” In other words, Wei was actually doing the PGA Tour a favor because of her hip, new kind of journalism. It’s an argument you are probably familiar with, because it is the same one the venues that don’t pay bands, or the companies that tell you to take an internship, or people who tell you to “get on YouTube” make: think of the exposure, man!

Yet while Wei pleads her case on the basis of her hepcat, app-using new jive journo-ing, in fact her stuff isn’t much, if any, different from the bad old days of sports reporting, when writers like Grantland Rice were more interested in palling around with the athletes (and, more worryingly, the owners) than with the audience. The telling detail can be found in her coverage of Rory McIlroy’s win at the very same tournament she got busted at: the Match Play.

The Match Play, obviously, is conducted under match play rules and not stroke play, which meant that, to win, Rory McIlroy had to win seven consecutive matches. In several of those matches, McIlroy came from behind to win, which prompted the following from Wei: “What I found the most interesting [what? Wei is missing a noun here] about McIlroy’s victory,” Wei wrote, “and his route to the winner’s circle was the way he found another gear when he was losing late in the match.” This McIlroy is not the same McIlroy as the one “we knew two years ago”—he is “a more mature one that knows how to dig deep.” Wei thusly repeats one of the most standard sorts of sportswriting cliche.

What of it? Well, the difficulty with this particular cliche, the reason why it is not “on a par” with those jolly old-school fellows who didn’t mention that a lot of ball players took speed, or cheated on their wives, or beat them, or that the owners were chiseling everyone for pennies on the dollar while looking the other way as men’s brains were slowly battered into jello—oh wait, that still happens—is that it justifies a species of rhetoric that gets repeated in many other arenas of life. (The most important of them being, of course, the economic.) That is the rhetoric of “toughness,” the “intangibles,” and so on—you know, the ghosts that don’t exist but are awfully handy when justifying why nobody’s getting a raise.

The belief in a player’s “toughness” or whatever words a given sportswriter can invent—the invention of such terms being largely what sportswriting is about—has been at best questionable, and at worst a knowing cynicism, ever since Gilovich’s, Tversky’s, and Vallone’s landmark 1985 paper, “The Hot Hand in Basketball: On the Misperception of Random Sequences.” The “hot hand,” the three proved, is merely a product of cognitive bias: when people are asked, for instance, to predict sequences of coin tosses, they inevitably expect the tosses to be half heads and half tails—even though such an even breakdown, no matter how many tosses are made, is nearly impossible.

So too in sports: writers continually ask their audience to believe that an athlete has “matured,” or “dug deep,” or what have you, when the more likely explanation is just that the athlete’s inherent talent level eventually expressed itself—or, in the case of a losing effort, the other side “got lucky.” Outcomes in sports are determined by skill (and the lack of it), not by “grit” or “will.” Rory won because he is a better golfer than nearly anyone on the planet, and while that skill can be masked by chance, over time it is more likely to expose the other player’s relative lack of skill.

Rory McIlroy won his tournament because he is a good golfer, not because he has some kind of psychological strength the rest of us lack. The fact that Stephanie Wei participates in this age-old sporting charade demonstrates that, for all her pretensions to the contrary, there isn’t a great deal different between her “new school” approach and that of her “stuffy” opponents. There is, perhaps, even reason to cheer for the PGA Tour in this dispute: at least they, unlike many in the age of the New Economy, believe people ought to get paid.

Windy Orders

Time flies like an arrow; fruit flies like a banana.
–Modern Saying

There’s a story told at Royal Troon, site of the “Postage Stamp” par-three hole, about the lady golfer, playing into an extreme wind, who was handed her driver by her caddie. After she hit the shot, as the ball fell helplessly short against the gale, she shouted reproachfully, “You underclubbed me!” It’s a story that has a certain resonance for me—perhaps obviously—but also, more immediately, due to my present work at a golf course in South Carolina, where I have repaired following the arrival of snow in Chicago. It’s easy enough to imagine something similar occurring at Chechessee Creek’s 16th hole—which, if it did, might not furnish the material for a modest laugh so much as, in concurrence with the golf course’s next hole, demonstrate something rather more profound. 

     Chechessee Creek, the golf course where I am spending this late fall, is a design of the Coore/Crenshaw operation, and it’s very well known that Ben Crenshaw, one of the principals of the firm, considers Chicago Golf Club to be the epitome of good course design. It’s reflected in a number of features of the course: the elevated greens, the various “dunes” strewn about for no apparent reason. But it’s also true that Chicago Golf is, despite its much greater age, by far the more daring of the two courses: it has blind shots and incredibly risky greens where putts can not only fall off the green, but go bounding down the fairway twenty yards or more. There are places where at times it is better to hit a putt off the green deliberately—because that is the only way to get the ball to stop near the hole. Chechessee Creek, for good or ill, has none of these features.

     What it does have, however, is a sense of what David Mihm, writer of the EpicGolf website, calls “pacing.” “Golf is a game,” he points out, “that is experienced chronologically”—that is, it isn’t just the quality of the holes that is important, but also their situation within the golf course as a whole. “By definition,” he says, “part of a hole’s greatness must depend on where it falls in the round.” 

     Chicago Golf Club has that quality of pacing in abundance, starting with the very first hole, Valley. By means of a trompe l’oeil the hole, in reality a 450 yard monster of a par four, appears to be a quite sedate, much-shorter hole. It’s only upon seeing his drive “disappear” (into the concealed vale that gives the hole its name) that the golfer realizes that his eye has misled him. It’s a trick, sure, that would be fantastic on any hole—but is particularly appropriate on the first, since it signals to the golfer immediately—on the first shot of the day—that this is a different kind of golf course, and that he cannot trust what he sees. 

     I would not say that Chechessee Creek exemplifies that notion to the same degree; it may not be too much to wonder whether South Carolina, or at least the Lowcountry, Tidewater parts of it, might not be too level of a countryside really to lend itself to golf. (“All over the world,” says Anita Harris, the geologist turned tour guide in John McPhee’s monumental Annals of the Former World, “when people make golf courses they are copying glacial landscapes.” South Carolina, needless to say, did not experience the devastations of an ice sheet during the last Ice Age, or any other time.) Still, there is one set of holes that does exhibit what Mihm is talking about—and perhaps something more besides. 

     The sixteenth hole at Chechessee is, as perhaps might be put together, a long par three hole; so long, in fact, that it isn’t unlikely that a short hitter might use a driver there. But, of course, there is the small matter of pride to contend with—few (male) golfers ever want to concede that they needed a driver on a “short” hole. It’s something I saw often working at Medinah, when coming to the thirteenth hole—almost inevitably, someone would not hit the correct club because he took as it an affront to suggest hitting a driver or even a three wood. Fair enough, one supposes; these days, the long par three might be close to becoming a design cliche (and in any case, all iconic courses I have seen have one: Olympia Fields, Chicago Golf, and Butler do, as does Riviera). 

     Just having a long par three isn’t enough, obviously, to satisfy Mihm’s criteria, and it isn’t that alone that makes Chechessee unique or even interesting. What makes the course go is the hole that follows the sixteenth, the seventeenth (duh). It’s an intriguing design in its own right, because it is an example of a “Leven” hole. According to A Disorderly Compendium of Golf (and what better source?), Leven holes are modeled on the 7th at the Leven Links, a hole that no longer exists. The idea of it is simple: it is a short hole with an enormous hazard on one side of the fairway; at Chechessee, the hazard is a long-grassed and swampy depression. Thus, the question posed is, how much of the hazard will you dare? Bailing out to the side leaves the player with a poor, often obstructed view of the green; at Chechessee, that function is furnished by an enormous pine tree.

     Yet that dilemma alone isn’t the real crux of the matter—what matters is that the seventeenth follows the sixteenth. After all, at the sixteenth the golfer is tempted, by his own ego, not to hit enough club. Conversely, at the seventeenth, the golfer is tempted to hit too much club. The quandary posed at each tee, in short, is precisely the mirror of the other: failing to reach for a driver on the sixteenth can cause the player to demand it on the seventeenth—with disastrous consequences in each case. And that is interesting enough merely in terms of golf, to be sure. But what is likely far more intriguing about it is that the placing of these holes could not be better situated to illustrate—nay, perform—what two psychologists said about how the human mind actually works.  

      The psychologists were Daniel Kahneman and Amos Tversky—Kahneman recently received the Nobel Prize for his work with Tversky, who couldn’t receive the award because he died in 1996. What their work did was to uncover, by means of various experiments, some of the hidden pathways of the human mind: the “cognitive shortcuts” taken by the brain. One of these discoveries was the fact that human beings are “loss averse”—or, as Jonah Lehrer put it not long ago in the New Yorker, that for human beings “losses hurt more than gains feel good.” Kahneman and Tversky called this idea “prospect theory.” 

     The effect has been measured in golf. In a paper entitled “Is Tiger Woods Loss Averse? Persistent Bias In the Face of Experience, Competition, and High Stakes” two Wharton professors found that, for PGA Tour golfers, “the agony of a bogey seems to outweigh the thrill of a birdie.” What their data (from the PGA Tour’s ShotLink system, which measures the distance of every shot hit on tour) demonstrated was that tour players “make their birdie putts approximately two percentage points less often than they make comparable par putts.” Somehow, when pros are faced with a par putt instead of a birdie putt—even though they might be identical putts—they make the former slightly more than the latter. What that translates into is one stroke left on the table per tournament—and that leaves $1.2 million per year in prize money being given away by the top twenty players.

     It’s a phenomenon that’s been found again and again in many disparate fields: investors hold on to too many low-risk bonds, for instance, while condos stay on the market far too long (because their owners won’t reduce their price even during economic downturns), and NFL coaches will take the “sure thing” of a field goal even when it might actually hurt their chances of winning the game. This last, while being about sports, has also another dimension of application to golf: the way in which what can be called “social expectations” guides human decision-making. That is, how our ideas about how others judge us plays a role in our decisions.

     In the case of the NFL, studies have shown that coaches far more likely to make the decision to kick the ball—to punt or attempt a field goal—than they are to attempt a first down or a touchdown. This is so even in situations (such as on the opponent’s 2 yard line) where, say, scoring a field goal actually leaves the opponent in a better position: if the team doesn’t get the touchdown or first down, the opponent is pinned against his own goal line, whereas a field goal means a kickoff that will likely result in the opponent starting at the twenty yard line at least. NFL coaches, in other words, aren’t making these decisions entirely rationally. To some, it suggests that they are attempting to act conventionally: that is, by doing what everyone else does, each coach can “hide” better.

     What that suggests is just why golfers, faced with the sixteenth hole, are averse to select what’s actually the right club. Each golfer is, in a sense, engaged in an arms race with every other golfer: by taking more club than another, that implicitly cedes something to the player taking less. This, despite the fact that rationally speaking selecting a different club than another golfer does nothing towards the final score of each. Taking less club becomes a kind of auction—or as we might term it, a bidding war—but one where the risk of “losing face” is seen as more significant than the final score. 

     The same process is, if it exists at all, also at work on the seventeenth hole. But this time there’s an additional piece of information playing out in the golfer’s mind: whatever happened on the last hole. One plausible scenario—I’ve seen it happen—is that the player doesn’t take enough club on the sixteenth, and comes up short of the hole. Having made that decision, and been wrong, the golfer determines on the next hole to make the “sensible” choice, and lays up away from the hazard—leaving a difficult second shot to a small green. But here’s the thing: the “carry” on the tee shot on seventeen, which I’ve withheld until now, is only about 210 yards—which is about the same as that of the sixteenth hole. In other words, the reality is that—evaluated dispassionately—golfers should probably hit about the same club on each hole. If they don’t, it’s probably due to a collision between “prospect theory” and “pacing”—which is to say that the Coore and Crenshaw design of Chechessee Creek is, all things considered, clubbed about right.