Hypothesis as thought-crime

Are women worse at maths than men? An American professor gives his view on the dispute engulfing Harvard.

Norman Levitt

Topics Politics

Harvard University, the oldest in the USA and the wealthiest in the world, thinks very well of itself. Within its precincts, to speak of ‘the President’ is to speak of the potentate who rules from Massachusetts Hall, not the upstart jackass, whoever it may be, who lives in exile at the White House.

In recent decades, however, Harvard has bestowed the office on a succession of rather colourless fellows; earnest, conscientious, and ostentatiously fair-minded, to be sure, but deeply averse to controversy and far too pallid to set the world of intellectual dispute afire when they spoke out.

That all changed when the current President, Lawrence Summers, was installed by Harvard’s mysterious archons a few years ago. Summers, the first non-alumnus since the seventeenth century to be given the post, is an MIT-trained economist, the scion of a family of first-rate economists, and famous as a whiz kid in that field. Venturing beyond the academic Eden, he was, with notable success, Secretary of the Treasury during the last years of the Clinton Administration.

It is an open secret that he was handed the helm at Harvard out of a growing sense that the place had grown stale, complacent, and narcissistic. Too many Harvard professors had settled into the habit of assuming that any old doctrine, opinion, or casual observation they chanced to utter was, ipso facto, profound and epochal merely because it issued from the great faux-Georgian citadel on the Charles. In truth, the place had grown somewhat dowdy, intellectually speaking, and, even worse, had proved itself susceptible to the vagaries of academic fashion, including that most risible of fashions, ‘political correctness’.

In some areas, Harvard had not only tolerated trendy mediocrity, but actively embraced it. Summers’ task, then, was to shake things up and to restore a relentlessly meritocratic ethic to the process of hiring and rewarding faculty where mere piety and sentimentality had previously been permitted to call the shots.

Summers lost no time in taking up the challenge. Early in his regime, he notoriously confronted Black Studies eminence Cornel West, essentially accusing him of goofing off with flashy and trivial projects (like voice-overs on hip-hop CDs) rather than turning out scholarly work of real substance. The touchy West promptly picked up his marbles and headed for Princeton, where a certain soft-heartedness still reigns.

Many Harvard students, bred on the platitudes of ‘diversity’ and greatly susceptible to West’s showmanship, were outraged. So, too, were a number of faculty, if only because they were terrified by the notion that a President might have the right or even the duty to challenge the scholarly worth of an exalted professor. But though some still blame Summers for ‘losing’ West, the prevailing opinion – most often stated anonymously, of course – is that Summers did the university a favour by cleverly easing out a dubious academic ‘superstar.’

An equally noisy flap arose a couple of years ago during the height of the bloodshed instigated by the ‘Second Intifada’ in Israel. Summers, speaking out forcefully on a political issue where his predecessors would probably have been more equivocal, bluntly suggested that a good part of the vociferous criticism of Israel then inundating Harvard and other campuses was tainted with outright anti-Semitism. Again, outrage erupted from a number of students and faculty, particularly those who had passionately adopted the Palestinian cause. But when the dust settled, it was pretty clear that most observers felt that Summers had made a valid point precisely when it most needed to be made.

Now, however, a new brouhaha has erupted and it seems impossible that Summers will emerge from this one without serious erosion of his moral authority. The trigger was a statement he made at a conference, suggesting that the reason there are more men than women in the mathematical sciences at top-flight institutions has to do with a small statistical difference in inate ability, which becomes a pretty large disparity when one looks at the ‘high end’ of the respective distribution curves.

The fatal words did not set forth his main theme, but merely constituted a brief aside, thoroughly hedged and qualified. Nonetheless, they touched off a firestorm of indignation, the most striking aspect of which was the intemperate response of a number of feminist scientists, who offered no counter-arguments, but simply declared the whole idea misogynistic and therefore forbidden intellectual territory.

In order to get across what is at issue here, it is necessary to delve into the grim world of statistical theory to explain the so-called ‘tail end effect’. Suppose, then, that we have two large populations, A and B, of roughly the same size, and a certain quantitatively measurable property X that has a specific value for any member of the combined population. Think, for instance, of A and B as Czechs and Hungarians respectively, and think of X as height. X, of course, will vary from individual to individual within each population, so that A and B will exhibit (quite possibly different) statistical distributions for X. I want, now, to assume that both these distributions are ‘normal’, that is to say, graphed by the famous bell-shaped curve. We shall then have two similar-looking but typically unidentical distributions for A and B.

Now for the key hypothesis; to begin with, assume that both the mean and variance of X are at least as great for B as for A. The mean is simply the value for which the height of the bell curve is maximal; the variance is a measure of how flat, as opposed to ‘spiky’, the bell curve appears.

Assume, further, that at least one of the following statements holds:

1. The mean of X is actually slightly greater for B than for A.

2. The ‘variance’ of X is actually slightly greater for B than for A.

Under this hypothesis, we now examine the right-hand ‘tail ends’ of both distributions, that is to say, the sub-populations of A and B that score extremely high for the parameter X. The fact is that if we place the ‘cut-off point’ sufficiently far to the right, the high-scoring subpopulation of B will be considerably larger than the high-scoring subpopulation of A.

This is merely a mathematical fact of life, independent of what the populations actually are, and independent of what is being measured. It should go without saying that it doesn’t say very much about the value of X for a randomly chosen member of either population. Quite obviously, it comes nowhere near saying that a given member of B will invariably have a higher X than a given member of A. Nor does it automatically preclude a member of A from having a very high X indeed.

Still, it does have significance. If, for instance, A, B, and X are Czechs, Hungarians, and (normally distributed) height respectively, and if the mean height for Hungarians is, say, a centimetre greater than that for Czechs while the variance is the same, one may confidently predict that the ratio of Hungarians taller than two metres to equally tall Czechs will be pretty large.

With this under our belt, we turn to the issue that got President Summers into trouble. He had been discussing the undeniable fact that at first-rate universities, male professors greatly outnumber females in the highly mathematical sciences – pure and applied mathematics, theoretical physics, and so forth. This holds despite the fact that for several decades there have been serious efforts to recruit and retain women in these challenging areas of science. The percentage of women mathematicians and physicists at elite departments has crept up only fitfully and slowly, in contrast to sciences like biology, where parity between the sexes is much closer to realisation.

Feminists in and out of the sciences are understandably eager to remedy what appears to them the defective performance of the mathematically intense disciplines, and to devise strategies for swiftly increasing women’s representation therein. The most militant demand a strict quota system based on a 50-50 split (at least!) of available appointments even at the most prestigious schools. Summers, however, had the temerity to point out that the feminist assumption that the current skewed distribution results purely from misogynist prejudice and the play of inimical social forces still lacks conclusive proof. There are competing hypotheses, and Summers was reckless enough to mention the leading one.

In a nutshell, this holds that the relative paucity of women at the top levels of mathematical achievement is, in fact, a tail-end effect. This is quite plausible. There is widespread and persistent evidence that, insofar as we can measure mathematical talent quantitatively, men have a slightly higher mean as well as a greater variance. Moreover, it seems reasonable to assume that genetic differences, acting via neurophysiology, are chiefly responsible for the observed differences in performance. (In other words, to use the fatal word that tripped up Summers, these may be ‘innate’ differences.)

Note, however, that these conditions entail a large ratio of men to women at the very highest level of ability. Since elite institutions, by definition, recruit their maths and physics faculty from this level – a very small pool indeed – it follows that, even without misogyny of any sort playing a role, even with every sort of encouragement given to young women of talent, many more men than women will continue to occupy those desirable posts. (It goes without saying that this does not preclude the ascent of truly superb women mathematicians, who certainly exist in significant numbers.)

Ironically, the very phenomenon that feminists decry – that is, the numerical disparity between the sexes that characterises maths and physics faculties at Harvard, Princeton, Cambridge, Oxford and the like – is, in itself, pretty good evidence for the validity of the tail-end explanation offered for it. Clearly, to assume that misogyny is behind the imbalance is to assume that mathematicians and physicists are simply more misogynistic than, say, biologists and psychologists. But this is plainly not the case.

Broadly speaking, scientists (as opposed to engineers), independent of particular field, tend to have liberal-to-left views on social and political issues, and to be generally strongly supportive of women’s rights. I am a mathematician at a pretty good research-oriented university. Over 30 years, I have sat in on many hiring and promotion decisions. My own experience convinces me that women in my field are simply not the victims of unfair neglect or disparagement. That being so, it is natural to look to other explanations for the skewed demographics of the field.

The tail-end explanation is definitely a leading candidate. That is not to say that this explanation is iron-clad. The question of what leads a person into the strange byways of higher mathematics has no simple answer at this point. Genetics is almost certainly involved, but so are child-rearing practices, cultural assumptions, social expectations, political attitudes, and economic facts of life, and these all intertwine in complex ways. Perhaps further evidence and a sharper analysis will expose the tail-end hypothesis as entirely illusory. I’m personally inclined to think that it’s right, but it would not surprise me to be proved wrong.

In any case, Summers did not speak as a committed proponent of this particular theory; indeed, he said he hoped it was wrong. He merely mentioned it en passant as one possibility, among a list of alternative hypotheses, that ought to be looked at in the process of formulating a fair and sensible policy for recruiting more women into elite academic science. Nonetheless, a torrent of rage and bile immediately struck him in the face.

At the conference itself, one prominent woman academic walked out in a huff as soon as the mere possibility of cognitive differences stemming from genetics was alluded to. Subsequently, a horde of academic sexual egalitarians, men as well as women, scientists as well as non-scientists, rose to denounce Summers as a monstrous combination of misogynist and genetic determinist, bent on driving woman scientists into the wilderness. Within a few days, the poor fellow was forced to grovel shamelessly and to tender, again and again, apologies of the most cringing sort.

This is a horror story primarily because it discloses a pathological cultural and political mood – at Harvard, throughout the American academic establishment, and doubtless in Europe as well. We see it in action here as it reaches forth its talons to eviscerate the reputation a sensible and well-meaning man – all for the unspeakable crime of making a fair, objective, and level-headed survey of a difficult issue. This episode reveals, in particular, that many scholars and students are committed to an egalitarian doctrine that is, in its way, as dogmatic and immune to contradictory evidence as the biblical literalism of fundamentalists.

The real disgrace is that Summers’ assailants, particularly the scientists among them, feel no obligation to come up with a reasoned refutation of the tail-end hypothesis or to produce any substantive evidence relevant to the issue. They accept it as a Revealed Truth that innate statistical differences between the sexes simply cannot extend to the cognitive realm, no matter what the evidence seems to show. In their view, the contrary idea is not debatable; it must be ruthlessly squelched outright!

They rigidly reject the possibility that nature may have neglected to insure a strict 50-50 allocation of rare intellectual talents to the respective sexes. They thirst for impossibly perfect justice. That thirst cannot be slaked by the obvious and unexceptionable principle that each individual ought to be judged in accordance with that person’s specific combination of talents, achievements, and potential.

One can sympathise with the critics to a limited extent. It would be silly to smugly assume that misogyny and sex prejudice could never take a hand in the processes that make or break scientific careers. In mathematics, we are still haunted by the spectres of geniuses like Emmy Noether and Julia Robinson, women shamefully cheated out of deserved jobs and adequate salaries merely because of their sex. Nonetheless, one cannot condone the barbarity of refusing to examine a theory on its merits simply because it goes against the grain of one’s favourite pieties. If the guilty party is a scientist, then the offence is especially deplorable.

Summers has found a few supporters. Some, like Harvard cognitive theorist Steven Pinker, are quite forthright, defending the plausibility of the tail-end hypothesis with solid arguments and even more vigorously denouncing the crime of suppressing ideas without even giving them a hearing. More than a few are feminist scientists who have the good grace to be embarrassed by doctrinaire sisters who insist that belief in social and professional equality for women entails a priori acceptance of the postulate that the sexes are absolutely indistinguishable on the neurophysiological level.

Others, alas, reluctantly concede that Summers was substantially correct, but do so in hedged, apologetic language that reveals how frightened they are of getting caught on the ‘wrong’ side of the ideological divide. But it hardly matters now; Summers has committed the grave sin of proposing that mere common sense might have something to say in the face of relentless feminist dogma. He has deeply offended the sanctimonious. Their thought-police are on the march, rendering mere logic irrelevant. The President’s painful atonement has only just begun. Veritas, indeed!

Norman Levitt is professor of mathematics at Rutgers University and author of Prometheus Bedeviled: Science and the Contradictions of Contemporary Culture.

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Topics Politics


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