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.

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