And the sun stood still, and the moon stayed,
until the people had avenged themselves upon their enemies.
“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,