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.  

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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.

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 LiviusAb 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.

Don Thumb

Then there was the educated Texan from Texas who looked like someone in Technicolor and felt, patriotically, that people of means—decent folk—should be given more votes than drifters, whores, criminals, degenerates, atheists, and indecent folk—people without means.
Joseph Heller. Catch-22. (1961).

 

“Odd arrangements and funny solutions,” the famed biologist Stephen Jay Gould once wrote about the panda’s thumb, “are the proof of evolution—paths that a sensible God would never tread but that a natural process, constrained by history, follows perforce.” The panda’s thumb, that is, is not really a thumb: it is an adaptation of another bone (the radial sesamoid) in the animal’s paw; Gould’s point is that the bamboo-eater’s thumb is not “a beautiful machine,” i.e. not the work of “an ideal engineer.” Hence, it must be the product of an historical process—a thought that occurred to me once again when I was asked recently by one of my readers (I have some!) whether it’s really true, as law professor Paul Finkelman has suggested for decades in law review articles like “The Proslavery Origins of the Electoral College,” that the “connection between slavery and the [electoral] college was deliberate.” One way to answer the question, of course, is to pour through (as Finkelman has very admirably done) the records of the Constitutional Convention of 1787: the notes of James Madison, for example, or the very complete documents collected by Yale historian Max Farrand at the beginning of the twentieth century. Another way, however, is to do as Gould suggests, and think about the “fit” between the design of an instrument and the purpose it is meant to achieve. Or in other words, to ask why the Law of Large Numbers suggests Donald Trump is like the 1984 Kansas City Royals.

The 1984 Kansas City Royals, for those who aren’t aware, are well-known in baseball nerd circles for having won the American League West division despite being—as famous sabermetrician Bill James, founder of the application of statistical methods to baseball, once wrote—“the first team in baseball history to win a championship of any stripe while allowing more runs (684) than they scored (673).” “From the beginnings of major league baseball just after the civil war through 1958,” James observes, no team ever managed such a thing. Why? Well, it does seem readily apparent that scoring more runs than one’s opponent is a key component to winning baseball games, and winning baseball games is a key component to winning championships, so in that sense it ought to be obvious that there shouldn’t be many winning teams that failed to score more runs than their opponents. Yet on the other hand, it also seems possible to imagine a particular sort of baseball team winning a lot of one-run games, but occasionally giving up blow-out losses—and yet as James points out, no such team succeeded before 1959.

Even the “Hitless Wonders,” the 1906 Chicago White Sox, scored more runs than their opponents  despite hitting (according to This Great Game: The Online Book of Baseball) “a grand total of seven home runs on the entire season” while simultaneously putting up the American League’s “worst batting average (.230).” The low-offense South Side team is seemingly made to order for the purposes of this discussion because they won the World Series that year (over the formidable Chicago Cubs)—yet even this seemingly-hapless team scored 570 runs to their opponents’ 460, according to Baseball Reference. (A phenomenon most attribute to the South Siders’ pitching and fielding: that is, although they didn’t score a lot of runs, they were really good at preventing their opponents’ from scoring a lot of runs.) Hence, even in the pre-Babe Ruth “dead ball” era, when baseball teams routinely employed “small ball” strategies designed to produce one-run wins as opposed to Ruth’s “big ball” attack, there weren’t any teams that won despite scoring fewer runs than their opponents’.

After 1958, however, there were a few teams that approached that margin: the 1959 Dodgers, freshly moved to Los Angeles, scored only 705 runs to their opponents’ 670, while the 1961 Cincinnati Reds scored 710 to their opponents 653, and the 1964 St. Louis Cardinals scored 715 runs to their opponents’ 652. Each of these teams were different than most other major league teams: the ’59 Dodgers played in the Los Angeles Coliseum, a venue built for the 1932 Olympics, not baseball; its cavernous power alleys were where home runs went to die, while its enormous foul ball areas ended many at-bats that would have continued in other stadiums. (The Coliseum, that is, was a time machine to the “deadball” era.) The 1961 Reds had Frank Robinson and virtually no other offense until the Queen City’s nine was marginally upgraded through a midseason trade. Finally, the 1964 Cardinals team had Bob Gibson (please direct yourself to the history of Bob Gibson’s career immediately if you are unfamiliar with him), and second they played in the first year after major league baseball’s Rules Committee redefined the strike zone to be just slightly larger—a change that had the effect of dropping home run totals by ten percent and both batting average and runs scored by twelve percent. In The New Historical Baseball Abstract, Bill James calls the 1960s the “second deadball era”; the 1964 Cardinals did not score a lot of runs, but then neither did anyone else.

Each of these teams was composed of unlikely sets of pieces: the Coliseum was a weird place to play baseball, the Rule Committee was a small number of men who probably did not understand the effects of their decision, and Bob Gibson was Bob Gibson. And even then, these teams all managed to score more runs than their opponents, even if the margin was small. (By comparison, the all-time run differential record is held by Joe DiMaggio’s 1939 New York Yankees, who outscored their opponents by 411 runs: 967 to 556, a ratio may stand until the end of time.) Furthermore, the 1960 Dodgers finished in fourth place, the 1962 Reds finished in third, and the 1965 Cards finished seventh: these were teams, in short, that had success for a single season, but didn’t follow up. Without going very deeply into the details then, suffice it to say that run differential is—as Sean Forman noted in the The New York Times in 2011—“a better predictor of future win-loss percentage than a team’s actual win-loss percentage.” Run differential is a way to “smooth out” the effects of chance in a fashion that the “lumpiness” of win-loss percentage doesn’t.

That’s also, as it happens, just what the Law of Large Numbers does: first noted by mathematician Jacob Bernoulli in his Ars Conjectandi of 1713, that law holds that “the more … observations are taken into account, the less is the danger of straying from the goal.” It’s the principle that is the basis of the insurance industry: according to Caltech physicist Leonard Mlodinow, it’s the notion that while “[i]ndividual life spans—and lives—are unpredictable, when data are collected from groups and analyzed en masse, regular patterns emerge.” Or for that matter, the law is also why it’s very hard to go bankrupt—which Donald Trump, as it so happens, has—when running a casino: as Nicholas Taleb commented in The Black Swan: The Impact of the Highly Improbable, all it takes to run a successful casino is to refuse to allow “one gambler to make a massive bet,” and instead “have plenty of gamblers make series of bets of limited size.” More bets equals more “observations,” and the more observations the more likely it is that all those bets will converge toward the expected result. In other words, one coin toss might be heads or might be tails—but the more times the coin is thrown, the more likely it is that there will be an equal number of both heads and tails.

How this concerns Donald Trump is that, as has been noted, although the president-elect did win the election, he did not win more votes than the Democratic candidate, Hillary Clinton. (As of this writing, those totals now stand at 62,391,335 votes for Clinton to Trump’s 61,125,956.) The reason that Clinton did not win the election is because American presidential elections are not won by collecting more votes in the wider electorate, but rather through winning in that peculiarly American institution, the Electoral College: an institution in which, as Will Hively remarked remarkably presciently in a Discover article in 1996, a “popular-vote loser in the big national contest can still win by scoring more points in the smaller electoral college.” Despite how weird that bizarre sort of result actually is, however, according to some that’s just what makes the Electoral College worth keeping.

Hively was covering that story in 1996: his Discovery story was about how, in the pages of the journal Public Choice that year, mathematician Alan Natapoff tried to argue that the “same logic that governs our electoral system … also applies to many sports”—for example, baseball’s World Series. In order “to become [World Series] champion,” Natapoff noticed, a “team must win the most games”—not score the most runs. In the 1960 World Series, the mathematician wrote, the New York Yankees “scored more than twice as many total runs as the Pittsburgh Pirates, 55 to 27”—but the Yankees lost game 7, and thus the series. “Runs must be grouped in a way that wins games,” Natapoff thought, “just as popular votes must be grouped in a way that wins states.” That is, the Electoral College forces candidates to “have broad appeal across the whole nation,” instead of playing “strongly on a single issue to isolated blocs of voters.” It’s a theory that might seem, on its face, to have a certain plausibility: by constructing the Electoral College, the delegates to the constitutional convention of 1787 prevented future candidates from winning by appealing to a single, but large, constituency.

Yet, recall Stephen Jay Gould’s remark about the panda’s thumb, which suggests that we can examine just how well a given object fulfills its purpose: in this case, Natapoff is arguing that, because the design of the World Series “fits” the purpose of identifying the best team in baseball, so too does the Electoral College “fit” the purpose of identifying the best presidential candidate. Natapoff’s argument concerning the Electoral College presumes, in other words, that the task of baseball’s playoff system is to identify the best team in baseball, and hence it ought to work for identifying the best president. But the Law of Large Numbers suggests that the first task of any process that purports to identify value is that it should eliminate, or at least significantly reduce, the effects of chance: whatever one thinks about the World Series, presumably presidents shouldn’t be the result of accident. And the World Series simply does not do that.

“That there is”—as Nate Silver and Dayn Perry wrote in their ESPN.com piece, “Why Don’t the A’s Win In October?” (collected in Jonah Keri and James Click’s Baseball Between the Numbers: Why Everything You Know About the Game Is Wrong)—“a great deal of luck involved in the playoffs is an incontrovertible mathematical fact.” It’s a point that was


argued so early in baseball’s history as 1904, when the New York Giants refused to split the gate receipts evenly with what they considered to be an upstart American League team (Cf. “Striking Out” https://djlane.wordpress.com/2016/07/31/striking-out/.). As Caltech physicist Leonard Mlodinow has observed, if the World Series were designed—by an “ideal engineer,” say—to make sure that one team was the better team, it would have to be 23 games long if one team were significantly better than the other, and 269 games long if the two teams were evenly matched—that is, nearly as long as two full seasons. In fact, since it may even be argued that baseball, by increasingly relying on a playoff system instead of the regular season standings, is increasing, not decreasing, the role of chance in the outcome of its championship process: whereas prior to 1969, the two teams meeting in the World Series were the victors of a paradigmatic Law of Large Numbers system—the regular season—now many more teams enter the playoffs, and do so by multiple routes. Chance is playing an increasing role in determining baseball’s champions: in James’ list of sixteen championship-winning teams that had a run differential of less than 1.100: 1, all of the teams, except the ones I have already mentioned, are from 1969 or after. Hence, from a mathematical perspective the World Series cannot be seriously argued to eliminate, or even effectively reduce, the element of chance—from which it can be reasoned, as Gould says about the panda’s thumb, that the purpose of the World Series is not to identify the best baseball team.

Natapoff’s argument, in other words, has things exactly backwards: rather than showing just how rational the Electoral College is, the comparison to baseball demonstrates just how irrational it is—how vulnerable it is to chance. In the light of Gould’s argument about the panda’s thumb, which suggests that a lack of “fit” between the optimal solution (the human thumb) to a problem and the actual solution (the panda’s thumb) implies the presence of “history,” that would then intimate that the Electoral College is either the result of a lack of understanding of the mathematics of chance with regards to elections—or that the American system for electing presidents was not designed for the purpose that it purports to serve. As I will demonstrate, despite the rudimentary development of the mathematics of probability at the time at least a few—and these, some of the most important—of the delegates to the Philadelphia convention in 1787 were aware of those mathematical realities. That fact suggests, I would say, that Paul Finkelman’s arguments concerning the purpose of the Electoral College are worth much more attention than they have heretofore received: Finkelman may or may not be correct that the purpose of the Electoral College was to support slavery—but what is indisputable is that it was not designed for the purpose of eliminating chance in the election of American presidents.

Consider, for example, that although he was not present at the meeting in Philadelphia, Thomas Jefferson possessed not only a number of works on the then-nascent study of probability, but particularly a copy of the very first textbook to expound on Bernoulli’s notion of the Law of Large Numbers: 1718’s The Doctrine of Chances, or, A Method of Calculating the Probability of Events in Play, by Abraham de Moivre. Jefferson also had social and intellectual connections to the noted French mathematician, the Marquis de Condorcet—a man who, according to Iain McLean of the University of Warwick and Arnold Urken of the Stevens Institute of Technology, applied “techniques found in Jacob Bernoulli’s Ars Conjectandi” to “the logical relationship between voting procedures and collective outcomes.” Jefferson in turn (McLean and Urken inform us) “sent [James] Madison some of Condorcet’s political pamphlets in 1788-9”—a connection that would only have reaffirmed a connection already established by the Italian Philip Mazzei, who sent a Madison a copy of some of Condorcet’s work in 1786: “so that it was, or may have been, on Madison’s desk while he was writing the Federalist Papers.” And while none of that implies that Madison knew of the marquis prior to coming to Philadelphia in 1787, before even meeting Jefferson when the Virginian came to France to be the American minister, the marquis had already become a close friend, for years, to another man who would become a delegate to the Philadelphia meeting: Benjamin Franklin. Although not all of the convention attendees, in short, may have been aware of the relationship between probability and elections, at least some were—and arguably, they were the most intellectually formidable ones, the men most likely to notice that the design of the Electoral College is in direct conflict with the Law of Large Numbers.

In particular, they would have been aware of the marquis’ most famous contribution to social thought: Condorcet’s “Jury Theorem,” in which—as Norman Schofield once observed in the pages of Social Choice Welfare—the Frenchman proved that, assuming “that the ‘typical’ voter has a better than even chance of choosing the ‘correct’ outcome … the electorate would, using the majority rule, do better than an average voter.” In fact, Condorcet demonstrated mathematically—using Bernoulli’s methods in a book entitled Essay on the Application of Analysis to the Probability of Majority Decisions (significantly published in 1785, two years before the Philadelphia meeting)—that adding more voters made a correct choice more likely, just as (according to the Law of Large Numbers) adding more games makes it more likely that the eventual World Series winner is the better team. Franklin at the least then, and perhaps Madison next most-likely, could not but have been aware of the possible mathematical dangers an Electoral College could create: they must have known that the least-chancy way of selecting a leader—that is, the product of the design of an infallible engineer—would be a direct popular vote. And while it cannot be conclusively demonstrated that these men were thinking specifically of Condorcet’s theories at Philadelphia, it is certainly more than suggestive that both Franklin and Madison thought that a direct popular vote was the best way to elect a president.

When James Madison came to the floor of Independence Hall to speak to the convention about the election of presidents for instance, he insisted that “popular election was better” than an Electoral College, as David O. Stewart writes in his The Summer of 1787: The Men Who Invented the Constitution. Meanwhile, it was James Wilson of Philadelphia—so close to Franklin, historian Lawrence Goldstone reports, that the infirm Franklin chose Wilson to read his addresses to the convention—who originally proposed direct popular election of the president: “Experience,” the Scottish-born Philadelphian said, “shewed [sic] that an election of the first magistrate by the people at large, was both a convenient & successful mode.” In fact, as William Ewald of the University of Pennsylvania has pointed out, “Wilson almost alone among the delegates advocated not only the popular election of the President, but the direct popular election of the Senate, and indeed a consistent application of the principle of ‘one man, one vote.’” (Wilson’s positions were far ahead of their time: in the case of the Senate, Wilson’s proposal would not be realized until the passage of the Seventeenth Amendment in 1913, and his stance in favor of the principle of “one man, one vote” would not be enunciated as part of American law until the Reynolds v. Sims line of cases decided by the Earl Warren-led U.S. Supreme Court in the early 1960s.) To Wilson, the “majority of people wherever found” should govern “in all questions”—a statement that is virtually identical to Condorcet’s mathematically-influenced argument.

What these men thought, in other words, was that an electoral system that was designed to choose the best leader of a nation would proceed on the basis of a direct national popular vote: some of them, particularly Madison, may even have been aware of the mathematical reasons for supposing that a direct national popular vote was how an American presidential election would be designed if it were the product of what Stephen Jay Gould calls an “ideal engineer.” Just as an ideal (but nonexistent) World Series would be at least 23, and possibly so long as 269 games—in order to rule out chance—the ideal election to the presidency would include as many eligible voters as possible: the more voters, Condorcet would say, the more likely those voters would be to get it right. Yet just as with the actual, as opposed to ideal, World Series, there is a mismatch between the Electoral College’s proclaimed purpose and its actual purpose: a mismatch that suggests researchers ought to look for the traces of history within it.

Hence, although it’s possible to investigate Paul Finkelman’s claims regarding the origins of the Electoral College by, say, trawling through the volumes of the notes taken at the Constitutional Convention, it’s also possible simply to think through the structure of the Constitution itself in the same fashion that Stephen Jay Gould thinks about, say, the structure of frog skeletons: in terms of their relation to the purpose they serve. In this case, there is a kind of mathematical standard to which the Electoral College can be compared: a comparison that doesn’t necessarily imply that the Constitution was created simply and only to protect slavery, as Finkelman says—but does suggest that Finkelman is right to think that there is something in need of explanation. Contra Natapoff, the similarity between the Electoral College and the World Series does not suggest that the American way of electing a head of state is designed to produce the best possible leader, but instead that—like the World Series—it was designed with some other goal in mind. The Electoral College may or may not be the creation of an ideal craftsman, but it certainly isn’t a “beautiful machine”; after electing the political version of the 1984 Kansas City Royals—who, by the way, were swept by Detroit in the first round—to the highest office in the land, maybe the American people should stop treating it that way.

Beams of Enlightenment

And why thou beholdest thou the mote that is in thy brother’s eye, but considerest not the beam that is in thine own eye?
Matthew 7:3

 

“Do you know what Pied Piper’s product is?” the CEO of the company, Jack Barker, asks his CTO, Richard, during a scene in HBO’s series Silicon Valley—while two horses do, in the background, what Jack is (metaphorically) doing to Richard in the foreground. Jack is the experienced hand brought in to run the company Richard founded as a young programmer; on the other hand, Richard is so ingenuous that Jack has to explain to him the real point of everything they are doing: “The product isn’t the platform, and the product isn’t your algorithm either, and it’s not even the software. … Pied Piper’s product is its stock. Whatever makes the value of that stock go up, that is what we’re going to make.” With that, the television show effectively dramatizes the case many on the liberal left have been trying to make for decades: that the United States is in trouble because of something called “financialization”—or what Kevin Phillips (author of 1969’s The Emerging Republican Majority) has called, in one of the first uses of the term, “a prolonged split between the divergent real and financial economies.” Yet few on that side of the political aisle have considered how their own arguments about an entirely different subject are, more or less, the same as those powering “financialization”—how, in other words, the argument that has enhanced Wall Street at the expense of Main Street—Eugene Fama’s “efficient market hypothesis”—is precisely the same as the liberal left’s argument against the SAT.

That the United States has turned from an economy largely centered around manufacturing to one that centers on services, especially financial ones, can be measured by such data as the fact that the total fraction of America’s Gross Domestic Product consumed by the financial industry is now, according to economist Thomas Philippon of New York University, “around 9%,” while just more than a century ago it was under two. Most appear to agree that this is a bad thing: “Our economic illness has a name: financialization,” Time magazine columnist Rana Foroohar argues in her Makers and Takers: The Rise of Finance and the Fall of American Business, while Bruce Bartlett, who worked in both the Reagan and George H.W. Bush Administrations (which is to say that he is not exactly the stereotypical lefty), claimed in the New York Times in 2013 that “[f]inancialization is also an important factor in the growth of income inequality.” In a 2007 Bloomberg News article, Lawrence E. Mitchell—a professor of law at George Washington Law School—denounced how “stock market considerations” have come “to trump those that improve the actual workings of a business.” The consensus view appears to be that it is bad for a business to be, as Jack is on Silicon Valley, more concerned with its stock price than on what it actually does.

Still, if it is such a bad idea, why do companies do it? One possible answer might be found in the timing, which seems to have happened some time after the 1960s: as John Bellamy Foster put it in a 2007 Monthly Review article entitled “The Financialization of Capitalism,” the “fundamental issue of a gravitational shift toward finance in capitalism as a whole … has been around since the late 1960s.” Undoubtedly, that turn was conditioned by numerous historical forces, but it’s also true that it was during the 1960s that the “efficient market hypothesis,” pioneered above all by the research of Eugene Fama of the University of Chicago, became the dominant intellectual force in the study of economics and in business schools—the incubators of the corporate leaders of today. And Fama’s argument was—and is—an intellectual cruise missile aimed at the very idea that the value of a company might be separate from its stock price.

As I have discussed previously (“Lions For Lambs”), Eugene Fama’s 1965 paper “The Behavior of Stock Market Prices” demonstrated that “the future path of the price level of a security is no more predictable than the path of a series of cumulated random numbers”—or in other words, that there was no rational way to beat the stock market. Also known as the “efficient market hypothesis,” the idea is largely that—as Fama’s intellectual comrade Burton Malkiel observed in his book, A Random Walk Down Wall Street (which has gone through more than five editions since its first publication in 1973),“the evidence points mainly toward the efficiency of the market in adjusting so rapidly to new information that it is impossible to devise successful trading strategies on the basis of such news announcements.” Translated, that means that it’s essentially impossible to do better than the market by paying close attention to what investors call a company’s “fundamental value.”

Yet, if there’s never a divergence between a company’s real worth and the price of its stock, that means that there’s no other means to measuring a company’s real worth than by its stock. From Fama’s or Malkiel’s perspective, “stock market considerations” simply are “the actual workings of a business.” They argued against the very idea that there even could be such a distinction: that there could be something about a company that is not already reflected in its price.

To a lot of educated people on the liberal-left, of course, such an argument will affirm many of their prejudices: against the evils of usury, and the like. At the same time, however, many of them might be taken aback if it’s pointed out that Eugene Fama’s case against fundamental economic analysis is the same as the case many educators make, when it comes to college admissions, against the SAT. Take, for example, a 1993 argument made in The Atlantic by Stanley Fish, former chairman of the English Department at Duke University and dean of the humanities at the University of Illinois at Chicago.

In “Reverse Racism, or, How the Pot Got to Call the Kettle Black,” the Miltonist argued against noted conservative Dinesh D’Souza’s contention, in 1991’s Illiberal Education, that affirmative-action in college admissions tends “‘to depreciate the importance of merit criteria.’” The evidence that D’Souza used to advance that thesis is, Fish tells us, the “many examples of white or Asian students denied admission to colleges and universities even though their SAT scores were higher than the scores of some others—often African-Americans—who were admitted to the same institution.” But, Fish says, the SAT has been attacked as a means of college admissions for decades.

Fish cites David Owen’s None of the Above: Behind the Myth of Scholastic Aptitude as an example. There, Owen says that the

correlation between SAT scores and college grades … is lower than the correlation between height and weight; in other words, you would have a better chance of predicting a person’s height by looking at his weight than you would of predicting his freshman grades by looking only at his SAT scores.”

As Fish intimates, most educational professionals these days would agree that the only way to judge a student these days is not by SAT, but by GPA—grade point average.

To judge students by grade point average, however, is just what the SAT was designed to avoid: as Nicholas Lemann describes in copious detail in The Big Test: The Secret History of the American Meritocracy, the whole purpose of the SAT was to discover students whose talents couldn’t be discerned by any other method. The premise of the test’s designers, in short, was that students possessed, as Lemann says, “innate abilities”—and that the SAT could suss those abilities out. What the SAT was designed to do, then, was to find those students stuck in, say, some lethargic, claustrophobic small town whose public schools could not, perhaps, do enough for them intellectually and who stagnated as a result—and put those previously-unknown abilities to work in the service of the nation.

Now, as Lemann remarked in an interview with PBS’ Frontline,  James Conant (president of Harvard and chief proponent of the SAT at the time it became prominent in American life, in the early 1950s) “believed that you would look out across America and you would find just out in the middle of nowhere, springing up from the good American soil, these very intelligent, talented people”—if, that is, America adopted the SAT to do the “looking out.” The SAT would enable American universities to find students that grade point averages did not—a premise that, necessarily, entails believing that a student’s worth could be more than (and thus distinguishable from) her GPA. That’s what, after all, “aptitude” means: “potential ability,” not “proven ability.” That’s why Conant sometimes asked those constructing the test, “Are you sure this is a pure aptitude test, pure intelligence? That’s what I want to measure, because that is the way I think we can give poor boys the best chance and take away the advantage of rich boys.” The Educational Testing Service (the company that administered the SAT), in sum, believed that there could be something about a student that was not reflected in her grades.

To use an intellectual’s term, that means that the argument against the SAT is isomorphic with the “efficient market hypothesis.” In biology, two structures are isomorphic with each other if they share a form or structure: a human eye is isomorphic with an insect’s eye because they both take in visual information and transmit it to the brain, even though they have different origins. Hence, as biologist Stephen Jay Gould once remarked, two arguments are isomorphic if they are “structurally similar point for point, even though the subject matter differs.” Just as Eugene Fama argued that a company could not be valued other than by its stock price—which has had the effective consequence of meaning that a company’s product is now not whatever superficial business it is supposedly in, but its stock price—educational professionals have argued that the only way to measure a student’s value is to look at her grades.

Now, does that mean that the “financialization” of the United States’ economy is the fault of the liberal left, instead of the usual conservative suspects? Or, to put it more provocatively, is the rise of the 1% at the expense of the 99% the fault of affirmative action? The short answer, obviously, is that I don’t have the slightest idea. (But then, neither do you.) What it does mean, I think, is that at least some of what’s happened to the United States in the past several decades is due to patterns of thought common to both sides of the American political congregation: most perniciously, in the related notions that all value is always and everywhere visible, or that it takes no time and patience for value to manifest itself—and that at least some of the erosion of those ideas is due to the efforts of those who meant well. Granted, it’s always hardest to admit wrongdoing when not only were your intentions pure, but even the immediate effects were also—but it’s also very much more powerful. The point, anyway, is that if you are trying to persuade, it’s probably best to avoid that other four-lettered word associated with horses.

 

 

Double Vision

Ill deeds are doubled with an evil word.
The Comedy of Errors. III, ii

The century just past had been both one of the most violent ever recorded—and also perhaps the highest flowering of civilized achievement since Roman times. A great war had just ended, and the danger of starvation and death had receded for millions; new discoveries in agriculture meant that many more people were surviving into adulthood. Trade was becoming more than a local matter; a pioneering Westerner had just re-established a direct connection with China. As well, although most recent contact with Europe’s Islamic neighbors had been violent, there were also signs that new intellectual contacts were being made; new ideas were circulating from foreign sources, putting in question truths that had been long established. Under these circumstances a scholar from one of the world’s most respected universities made—or said something that allowed his enemies to make it appear he had made—a seemingly-astonishing claim: that philosophy, reason, and science taught one kind of truth, and religion another, and that there was no need to reconcile the two. A real intellect, he implied, had no obligation to be correct: he or she had only to be interesting. To many among his audience that appeared to be the height of both sheer brainpower and politically-efficacious intellectual work—but then, none of them were familiar with either the history of German auto-making, or the practical difficulties of the office of the United States Attorney for the Southern District of New York.

Some literary scholars of a previous generation, of course, will get the joke: it’s a reference to then-Johns Hopkins University Miltonist Stanley Fish’s assertion, in his 1976 essay “Interpreting ‘Interpreting the Variorum,’” that, as an interpreter, he has no “obligation to be right,” but “only that [he] be interesting.” At the time, the profession of literary study was undergoing a profound struggle to “open the canon” to a wide range of previously-neglected writers, especially members of minority groups like African-Americans, women, and homosexuals. Fish’s remark, then, was meant to allow literary scholars to study those writers—many of whom would have been judged “wrong” according to previous notions of literary correctness. By suggesting that the proper frame of reference was not “correct/incorrect,” or “right/wrong,” Fish implied that the proper standard was instead something less rigid: a criteria that thusly allowed for the importation of new pieces of writing and new ideas to flourish. Fish’s method, in other words, might appear to be an elegant strategy that allowed for, and resulted in, an intellectual flowering in recent decades: the canon of approved books has been revamped, and a lot of people who probably would not have been studied—along with a lot of people who might not have done the studying—entered the curriculum who might not have had the change of mind Fish’s remark signified not have become standard in American classrooms.

I put things in the somewhat cumbersome way I do in the last sentence because of course Fish’s line did not arrive in a vacuum: the way had been prepared in American thought long before 1976. Forty years prior, for example, F. Scott Fitzgerald had claimed, in his essay “The Crack-Up” for Esquire, that “the test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.” In 1949 Fitzgerald’s fellow novelist, James Baldwin, similarly asserted that “literature and sociology are not the same.” And thirty years after Fish’s essay, the notion had become so accepted that American philosopher Richard Rorty could casually say that the “difference between intellectuals and the masses is the difference between those who can remember and use different vocabularies at the same time, and those who can remember only one.” So when Fish wrote what he wrote, he was merely putting down something that a number of American intellectuals had been privately thinking for some time—a notion that has, sometime between now and then, become American conventional wisdom.

Even some scientists have come to accept some version of the idea: before his death, the biologist Stephen Jay Gould promulgated the notion of what he called “non-overlapping magisteria”: the idea that while science might hold to one version of truth, religion might hold another. “The net of science,” Gould wrote in 1997, “covers the empirical universe,” while the “net of religion extends over questions of moral meaning and value.” Or, as Gould put it more flippantly, “we [i.e., scientists] study how the heavens go, and they [i.e., theologians] determine how to go to heaven.” “Science,” as medical doctor (and book reviewer) John Carmody put the point in The Australian earlier this year, “is our attempt to understand the physical and biological worlds of which we are a part by careful observation and measurement, followed by rigorous analysis of our findings,” while religion “and, indeed, the arts are, by contrast, our attempts to find fulfilling and congenial ways of living in our world.” The notion then that there are two distinct “realms” of truth is a well-accepted one: nearly every thinking, educated person alive today subscribes to some version of it. Indeed, it’s a belief that appears necessary to the pluralistic, tolerant society that many envision the United States is—or should be.

Yet, the description with which I began this essay, although it does in some sense apply to Stanley Fish’s United States of the 1970s, also applies—as the learned knew, but did not say, at the time of Fish’s 1976 remark—to another historical era: Europe’s thirteenth century. At that time, just as during Fish’s, the learned of the world were engaged in trying to expand the curriculum: in this case, they were attempting to recoup the work of Aristotle, largely lost to the West since the fall of Rome. But the Arabs had preserved Aristotle’s work: “In 832,” as Arthur Little, of the Jesuits, wrote in 1947, “the Abbaside Caliph, Almamun,” had the Greek’s work translated “into Arabic, roughly but not inaccurately,” in which language Aristotle’s works “spread through the whole Moslem world, first to Persia in the hand of Avicenna, then to Spain where its greatest exponent was Averroes, the Cordovan Moor.” In order to read and teach Aristotle without interference from the authorities, Little tells us, Averroes (Ibn Rushd) decided that “Aristotle’s doctrine was the esoteric doctrine of the Koran in opposition to the vulgar doctrine of the Koran defended by the orthodox Moslem priests”—that is, the Arabic scholar decided that there was one “truth” for the masses and another, far more subtle, for the learned. Averroes’ conception was, in turn, imported to the West along with the works of Aristotle: if the ancient Greek was at times referred to as the Master, his Arabic disciple was referred to as the Commentator.

Eventually, Aristotle’s works reached Paris, and the university there, sometime towards the end of the twelfth century. Gerard of Cremona, for example, had translated the Physics into Latin from the Arabic of the Spanish Moors sometime before he died in 1187; others had translated various parts of Aristotle’s Greek corpus either just before or just afterwards. For some time, it seems, they circulated in samizdat fashion among the young students of Paris: not part of the regular curriculum, but read and argued over by the brightest, or at least most well-read. At some point, they encountered a young man who would become known to history as Siger of Brabant—or perhaps rather, he encountered them. And like many other young, studious people, Siger fell in love with these books.

It’s a love story, in other words—and one that, like a lot of other love stories, has a sad, if not tragic, ending. For what Siger was learning by reading Aristotle—and Averroes’ commentary on Aristotle—was nearly wholly incompatible with what he was learning in his other studies through the rest of the curriculum—an experience that he was not, as the experience of Averroes before him had demonstrated, alone in having. The difference, however, is that whereas most other readers and teachers of the learned Greek sought to reconcile him to Christian beliefs (despite the fact that Aristotle long predated Christianity), Siger—as Richard E. Rubenstein puts it in his Aristotle’s Children—presented “Aristotle’s ideas about nature and human nature without attempting to reconcile them with traditional Christian beliefs.” And even more: as Rubenstein remarks, “Siger seemed to relish the discontinuities between Aristotelian scientia and Christian faith.” At the same time, however, Siger also held—as he wrote—that people ought not “try to investigate by reason those things which are above reason or to refute arguments for the contrary position.” But assertions like this also left Siger vulnerable.

Vulnerable, that is, to the charge that what he and his friends were teaching was what Rubenstein calls “the scandalous doctrine of Double Truth.” Or, in other words, the belief that “a proposition [that] could be true scientifically but false theologically, or the other way round.” Whether Siger and his colleagues did, or did not, hold to such a doctrine—there have been arguments about the point for centuries now— isn’t really material, however: as one commenter, Vincent P. Benitez, has put it, either way Siger’s work highlighted just how the “partitioning of Christian intellectual life in the thirteenth century … had become rather pronounced.” So pronounced, in fact, that it suggested that many supposed “intellectuals” of the day “accepted contradictories as simultaneously true.” And that—as it would not to F. Scott Fitzgerald later—posed a problem to the medievals, because it ran up against a rule of logic.

And not just any rule of logic: it’s one that Aristotle himself said was the most essential to any rational thought whatever. That rule of logic is usually known by the name the Law of Non-Contradiction, usually placed as the second of the three classical rules of logic in the ancient world. (The others being the Law of Identity—A is A—and the Law of the Excluded Middle—either A is A or it is not-A.) As Aristotle himself put it, the “most certain of all basic principles is that contradictory propositions are not true simultaneously.” Or—as another of Aristotle’s Arabic commenters, Avicenna (Ibn-Sina) put it in one of its most famous formulations—that rule goes like this: “Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.” In short, a thing cannot be both true and not true at the same time.

Put in Avicenna’s way, of course, the Law of Non-Contradiction will sound distinctly horrible to most American undergraduates, perhaps particularly those who attend the most exclusive colleges: it sounds like—and, like a lot of things, has been—a justification for the worst kind of authoritarian, even totalitarian, rule, and even torture. In that sense, it might appear that attacking the law of non-contradiction could be the height of oppositional intellectual work: the kind of thing that nearly every American undergraduate attracted to the humanities aspires to do. Who is not, aside from members of the Bush Administration legal team (for that matter, nearly every regime known to history) and viewers of the television show 24, against torture? Who does not know that black-and-white morality is foolish, that the world is composed of various “shades of gray,” that “binary oppositions” can always be dismantled, and that it is the duty of the properly educated to instruct the lower orders in the world’s real complexity? Such views might appear obvious—especially if one is unfamiliar with the recent history of Volkswagen.

In mid-September of 2015, the Environmental Protection Agency of the United States issued a violation notice to the German automaker Volkswagen. The EPA had learned that, although the diesel engines Volkswagen built were passing U.S. emissions tests, they were doing it on the sly: each car’s software could detect when the car’s engine was being tested by government monitors, and if so could reduce the pollutants that engine was emitting. Just more than six months later, Volkswagen agreed to pay a settlement of 15.3 billion dollars in the largest auto-related class-action lawsuit in the history of the United States. That much, at least, is news; what interests me, however,  about this story in relation to this talk about academics and monks was a curious article put out by The New Yorker in October of 2015. Entitled “An Engineering Theory of the Volkswagen Scandal,” Paul Kedrosky—perhaps significantly—“a venture investor and a former equity analyst,” explains these events as perhaps not the result of “engineers … under orders from management to beat the tests by any means necessary.” Instead, the whole thing may simply have been the result of an “evolution” of technology that “subtly and stealthily, even organically, subverted the rules.” In other words, Kedrosky wishes us to entertain the possibility that the scandal ought to be understood in terms of the undergraduate’s idea of shades of gray.

Kedrosky takes his theory from a book by sociologist Diane Vaughn, about the Challenger space shuttle disaster of 1986. In her book, Vaughn describes how, over nine launches from 1983 onwards, the space shuttle organization had launched Challenger under colder and colder temperatures, until NASA’s engineers had “effectively declared the mildly abnormal normal,” Kedrosky says—and until, one very frigid January morning in Florida, the shuttle blew into thousands of pieces moments after liftoff. Kedrosky’s attempt at an analogy is that maybe the Volkswagen scandal developed similarly: “Perhaps it started with tweaks that optimized some aspect of diesel performance and then evolved over time.” If so, then “at no one step would it necessarily have felt like a vast, emissions-fixing conspiracy by Volkswagen engineers.” Instead—as this story goes—it would have felt like Tuesday.

The rest of Kedrosky’s thrust is relatively easy to play out, of course—because we have heard a similar story before. Take, for instance, another New Yorker story; this one, a profile of the United States Attorney for the Southern District of New York, Preet Bharara. Mr. Bharara, as the representative of the U.S. Justice Department in New York City, is in charge of prosecuting Wall Street types; because he took office in 2009, at the crest of the financial crisis that began in 2007, many thought he would end up arresting and charging a number of executives as a result of the widely-acknowledged chicaneries involved in creating the mess. But as Jeffrey Toobin laconically observes in his piece, “No leading executive was prosecuted.” Even more notable, however, is the reasoning Bharara gives for his inaction.

“Without going into specifics,” Toobin reports, Bharara told him “that his team had looked at Wall Street executives and found no evidence of criminal behavior.” Sometimes, Bharara went on to explain, “‘when you see a bad thing happen, like you see a building go up in flames, you have to wonder if there’s arson’”—but “‘sometimes it’s not arson, it’s an accident.’” In other words, to Bharara, it’s entirely plausible to think of the entire financial meltdown of 2007-8, which ended three giant Wall Street firms (Bear Stearns, Merrill Lynch, and Lehman Brothers) and two arms of the United States government (Fannie Mae and Freddie Mac), and is usually thought to have been caused by predatory lending practices driven by Wall Street’s appetite for complex financial instruments, as essentially analogous to Diane Vaughn’s view of the Challenger disaster—or Kedrosky’s view of Volkswagen’s cavalier thoughts about environmental regulation. To put it in another way, both Kedrosky and Bharara must possess, in Fitzgerald’s terms, “first-rate intelligences”: in Kedrosky’s version of Volkswagen’s actions or Bharara’s view of Wall Street, crimes were committed, but nobody committed them. They were both crimes and not-crimes at the same time.

These men can, in other words, hold opposed ideas in their head simultaneously. To many, that makes these men modern—or even, to some minds, “post-modern.” Contemporary intellectuals like to cite examples—like the “rabbit-duck” illusion referred to by Wittgenstein, which can be seen as either a rabbit or a duck, or the “Schroedinger’s Cat” thought experiment, whereby the cat is neither dead nor alive until the box is opened, or the fact that light is both a wave and a particle—designed to show how out-of-date the Law of Noncontradiction is. In that sense, we might as easily blame contemporary physics as contemporary work in the humanities for Kedrosky or Bharara’s difficulties in saying whether an act was a crime or not—and for that matter, maybe the similarity between Stanley Fish and Siger of Brabant is merely a coincidence. Still, in the course of reading for this piece I did discover another apparent coincidence in Arthur Little’s same article I previously cited. “Unlike Thomas Aquinas,” the Jesuit wrote 1947, “whose sole aim was truth, Siger desired most of all to find the world interesting.” The similarity to Stanley Fish’s 1976 remarks about himself—that he has no obligation to be right, only to be interesting—are, I think, striking. Like Bharara, I cannot demonstrate whether Fish knew of this article of Little’s, written thirty years before his own.

But then again, if I have no obligation to be right, what does it matter?