Some commenters to my earlier post reason:
It [presumably discussion of possible innate gender differences in cognition] shouldn't be suppressed for political reasons. I think that a scientific issue, though, shouldn't be taught if there's not good scientific evidence for it. And I haven't seen good scientific evidence for this theory
I think the words "without proof" are implied from the second paragraph.
I.E. "when faculty tell their students that they are innately inferior, without proof, based on race, religion, gender or sexual orientation, they are crossing a line that should not be crossed"
I can see that as being fair.
Yet not every step in the scientific process can (or should) contain "proof" in any strong sense of the word. Scientific debates often present conjectures based on ambiguous evidence. The conjectures lead to responses, often equally conjectural. Evidence on one side or the other grows or shrinks. We often get something approaching "proof" only after decades of unproven conjectures have led to more evidence-gathering and more discussion.
This is especially so in areas (including biology and social science) where the evidence tends to be suggestive, not dispositive, and confounding factors are always potentially present. Even in math, though, where "proof" is indeed the touchstone, unproven conjectures are often made. If you couldn't say anything without "proof," whether logical proof, proof beyond a reasonable doubt, or even proof by a preponderance of the evidence as determined by some campus committee, scientific debate would be much curtailed.
I agree with Frank that professors who are teaching classes shouldn't teach as true statements that haven't been adequately proven. But Summers wasn't teaching a class, nor did he assert his claims as having been proven true. If professors can't express such conjectures — yet presumably their rivals would be quite free to present contrary conjectures, unless we're to completely eliminate public conjecture in science — then what sort of debate would we have?
Now perhaps conjectures that are entirely unbacked by any evidence, or backed by asserted evidence or reasoning that has been conclusively disproven through decades of open debate, are sufficiently implausible that we'd fault people who make them. But there is indeed a hot debate on the subject, involving some pretty serious people on both sides. There is evidence that some say points towards biological differences, and that others think is not probative enough (since it's so hard to isolate biological causes from social causes). There is the unquestioned reality that men and women are materially different chromosomally, hormonally, and physically, and that male and female animals of other species (where the explanation must presumably be biology and not "culture" divorced from biology) are often different in temperament and behavior. This at least suggests that looking for possible biological cognitive and temperamental differences and differences in distribution of various traits, alongside the chromosomal, hormonal, and physical differences, is not obviously a fool's errand — and thus is plausible fodder for conjecture and discussion of suggestive evidence, even in the absence of conclusive proof.
We have, as best I can tell from my layman's perspective, a serious and potentially intellectually exciting debate here. Shutting off one side, imposing on one side requirements of "proof" that bar the proposal of scientific conjectures, or for that matter imposing on both sides such requirements, strikes me as bad both for our knowledge of this subject and for scientific debate more generally.
UPDATE: In the post in which I first mentioned Prof. Barres' article, I linked to the article and also to an Edge.org debate between Steven Pinker and Elizabeth Spelke.
I probably should have also, for balance, linked to an article that takes the view contrary to Prof. Barres'; since I suspect that the attention on this thread has shifted from that post to this more recent one, I decided to add that link here -- it's to Kingsley Browne, Women in Science: Biological Factors Should Not Be Ignored, 11 Cardozo Women's Law Journal 509 (2005). Browne's piece is just a short version of some much longer work he's done, but I figure that shorter is often better for at least a first look at the matter. I stress again that I'm not claiming I know what the right answer is; I'm just trying to pass along pointers to both sides of the argument.
Related Posts (on one page):
- Prof. Ben Barres' Response:
- Credentials and Interdisciplinary Work:
- Scientific Debate, Proof, and Conjecture:
- Should Speech About Gender Cognitive Differences "Not Be Tolerated" on Campus, and Instead Treated as "Verbal Violence" Rather Than "Free Speech"?
- Be Careful Trusting Data, Even in Nature:
- Gender and Science:
- More on Sex and Supreme Court Clerks:
- Data on Women in Legal Academia:
- Why So Few Women Supreme Court Clerks?
Is that even an accurate statement of what the gender-cognition argument is? I don't think so. Gender differences in cognition do not, of themselves imply superiority or inferiority, and the Summers suggestion wasn't even about cognitive differences per se, as far as I know, but about differing distributions of cognitive abilities in male and female populations. To jump from that to a statement about the innate superiority or inferiority of any individual student (much less a pool of students pre-selected for cognitive ability, as one would imagine the student population of Harvard to be) is a leap too far.
To clarify, I first think that professors (or presidents) should be free to conjecture anything they want.
I second think that it is irresponsible to conjecture without at least a minimal amount of supporting evidence. There are a million conjectures that can be made and one can distinguish the good from the bad.
There are biological differences between men and women. There are nonbiological differences between men and women. The latter are far more profound, as is evident from historical change.
A professor should not tell his student that the student is innately inferior, even with proof! But telling a student that, say, some scholars believe that men have an innately wider bell curve than women in math ability, resulting in more male geniuses in math than women says nothing about the innate (or otherwise) inferiority of any given student, male or female. For that matter, it doesn't say anything about the innate inferiority of women or men as a group. That's true even if the issue is whether men are slightly better than women in math. How does one group being better at math make the other group "innately inferior?" Is there some group of math chauvinists going around claiming that math ability is the one mark of superiority? If it's suggested that women are (much!) less "innately" prone to aggressive violence than are men (which seems to be true), does that make men "inferior" to women? Differences of these are not a question of inferiority or superiority, they just "are". It's a salient and unfortunate feature of political correctness that this is beyond the comprehension of too many people these days, and reminds me in it's no-knowthingism of "don't teach me evolution--your mother may have descended from an ape, but mine didn't."
I didn't want to press the topic further, but that is how the other side has framed the debate. What does one do after such an encounter?
This is very difficult to measure scientifically, but I was struck by one of the Pinker slides. It reported the meta-analysis of 254 studies that found certain genetic differences in adolescence but not in childhood. That doesn't sound genetic.
Do most people agree that the hypothesis "black people are intrinsically more lazy than white people" should be out of bounds?
Let's say I try to frame it academically: "Perhaps the underperformance of black students relative to whites is attributable in part to a degree of intrinsic laziness, and I think we ought to conduct some rigorous scientific studies to confirm or disprove this hypothesis."
"Laziness" may be an amorphous term but I'm sure we can test for levels of motivation much like we test for any number of other loosely-defined qualities.
If people think my hypothesis should be out of bounds, then it's worthwhile to discuss where the distinction lies.
If people think my hypothesis is fair game as well, I would ask whether there is any point on this slippery slope at which they would draw the line.
If you're going to, as an influential scholar, publicly make a claim that taints an entire race/gender, is it really too much to ask that you have some evidence backing you up?
What ever happened to actually designing an experiment and testing hypotheses before announcing them to the whole world? In fact, I will go so far as saying that making random public conjectures goes against the whole grain of science.
(Also, could you break up the two quotes? The current formatting makes them look like one quote.)
Similarly, public statements in formal contexts, like Summers' statements, are statements which might reasonably be constrained. When a high official of a university speaks in public . . . well, he may not be taken particularly seriously the way his predecessors might have been fifty or an hundred years ago. But he is taken somewhat seriously, as expressing something like an expert consensus. With highly controversial issues, one real danger is that the outside public will then come to rely on that false consensus, exploiting it for rhetorical and political advantage, before it's really been tested. So it might be reasonable to restrain members of the academic class from these kinds of public pronouncements. Again, this doesn't actually shut down the debate -- it just leaves the debate less accessible to the laiety, while the debate continues.
Of course, these mild constraints wouldn't work against cases in which laymen trawl the academic literature cherry picking expert sources to support their preferred interpretation on a given controversy. And in some cases (e.g. questions regarding differences in IQ distributions between genders, races, etc.), the possibility itself is hurtful (although I think a large part of that comes from statistical innumeracy among the public). But at least, in that case, there's not the sense that the academy is putting its imprimatur on the pronouncement. There is no expectation that an individual scholar is going to reflect the weight of the consensus.
So essentially, I think it's not outrageous to impose informal constraints on what's acceptable for a professor to say in the classroom, or for what a high official of the university is permitted to say in general speeches, or on press releases issued in the name of a university or other academic institution. Restraining individual scholars from proposing certain possibilities in the academic literature would be bad. But most of the concerns people have with the expression of these opinions could be addressed by something far short of a general restraint on academic expression.
Summer's seemed to accept the idea that the average cognitive ability of men and women was the same. His hypothesis was (more or less):
(i) The populations of men and women have different standard deviations in their cognitive ability.
(ii) The standard deviation of men being higher.
This is an easily testable hypothesis. Data on cognitive abilities and a rudimentary understanding of statistics would suffice.
MnZ, your test would identify a difference but tell us nothing about whether it was genetic.
i see no such extraordinary evidence here. nonetheless, verbal violence this is not.
Taeyoung: Even with undergraduates, it's important to inculcate critical thinking as well as impart information. Bringing up controversial ideas can be a good way to do that -- though it's important to do so in context and to make clear that an idea is not an assertion.
Steve: No, that topic should not be out of bounds, in or out of a classroom. However, if you're going to criticize as aspect of people over which they have some control (motivation, as opposed to genetics), you can expect an even more hostile reaction. My view would be, if you advance serious arguments in a serious way, go ahead. If not, you'd just be being an asshole.
Well, for what? For scientific inquiry? Probably not.
I think to the extent we can formulate a reliable and testable measure, then it's a meaningful thing to study. It's hard to think of a hypothesis that, if formulated in testable form, wouldn't be meaningful to pursue. I mean, if it's testable, why not? Go ahead and test it.
The problem in many cases is isolating the relevant variable. In many cases, I don't think we have a good enough understanding to do it.
In certain cases (like research on cognitive differences) we're certainly at the point where we can run ANOVAs and MANOVAs across populations. But I don't think we're yet at the point where we can trace the outcomes of those ANOVAs and MANOVAs to individual causes reliably. We may be in the future, though, and when we are, we may want to have this kind of data handy. Figuring out ways to discern the causes underlying statistical differences is also something that would advance science, and a goal which does not seem wholly out of reach. So I think that would be worthwhile research to pursue.
One of the benefits we might expect from such research is also that as our knowledge and understanding of these particular between-gender-population and between-race-population differences deepens, we might acquire information leading to new categories that account more directly for the distribution of particular higher order attributes across the population as a whole. We might not, of course. But we also might -- the point is to find out (and to find out what they are). So why not?
The same is true of laziness.
(1) No legitimate scientist (or social scientist) is going to present findings from this research that yield conclusions about the "inferority" of women. Even if there is a signficiant mean difference in IQ between men and women, this says nothing about superiority or inferiority.
(2) The debate centers mostly around differential variance in IQ between genders - not a differential mean. In other words, theory is that males are overrepresented in the highest strata of the IQ distribution because IQ is more variable among males - not because males, on average, have a higher IQ than females.
(3) I strongly agree with Professor Volokh that "proof" is an unreasonable standard to present scientific evidence to lay audiences. The most appropriate way to present research findings is to detail the most important findings to date and let the audience weigh the evidence for themselves.
That is quite true. However, it also means that you cannot "refute" Summers on the basis of average test scores. Bizarrely, I have seen some people try to do this. Moreover, policy perscriptions to fix a gap in the standard deviations are likely to be different from the policy perscriptions to fix a gap in the means.
But there is evidence. A lot. Maybe not enough to settle the issue decisivly, but certainly enough that it behooves anyone commenting on the issue to have at least a passing familiarty with it.
I understand that Summers' point was complex and subtle. He was not saying that women should not practice science. But, nonetheless, his argument amounts to an excuse for the institutions of science for what could be discrimination. Further, his argument could easily be part of cultural norms that push women away from science, or that discourage scientific institutions from taking female scientists as seriously as male scientists. I agree that none of this should short-circuit scientific inquiry on this issue, and liberals are wrong if they try to crush all examination of this issue. But we need to be very carefull how we treat this kind of inquiry, and in that context, Larry Summers' comments were reckless and irresponsible.
But . . . academics do make that conjecture, don't they? All the time in fact. And activists (e.g. the people at National Review, on their education blog) will inevitably point to it gleefully, as evidence in their quest to undermine the authority of the universities. But I think the reaction of most people (most conservatives, at least) is to roll their eyes.
I think so long as the context is provided -- so long as the professor is not presenting the theory magisterially in his capacity as lecturer -- that shouldn't raise concerns.
nonetheless, i think claims that there aren't many conservatives in academia because conservatives are just dumber than lefties are wholly unsupported by evidence and are (almost?) exclusively the result of an irrational hostility to conservatives.
I don't think it's a very plausible hypothesis but as with all hypotheses, it should be studied and discussed by open-minded individuals without the interference of a political or social agenda.
And to echo my previous point, "inferiority" is a subjective term you have used to refer to "lower mean IQ." Concluding that individuals with lower IQ are inferior to those with higher IQs is a value judgement that you are making.
And again .... the theory has to do with variance not means.
I expect no one would be truly wounded by such a statement, because conservatives don't spend their lives being told they're intellectually inferior by birth. (Also, it seems extremely silly, whereas the alleged intellectual inferiority of non-Asian minorities and mathematical incompetence of women is a familiar stereotype. So the "conjectures" in line with stereotypes seem a lot less silly, even if they don't have a lot more hard evidence.)
Just like those who conjecture genetic gender differences without good evidence are wrong.
Maybe the "cognitive ability" reference is too broad. However, one could certainly look at standard deviations in test scores.
You're talking now about the weight we should accord "people spouting off half-cocked conjecture on topics they aren't familiar with". But what you initially said was that "we wouldn't countenance" it. I countenance all kinds of things that I think are worthless; I don't shout them down or try to get the speakers fired.
Especially when there's a good deal of evidence favoring their suggestion.
Since everyone studying science at MIT, Stanford etc. is far out in the tail of the distribution, I doubt our potential scientists care where the distribution is centered. The question you ask (e.g. when applying to a top grad school) is whether you could be among the best in their field, which is a source of anxiety unless you're extremely self-confident (which is, not coincidentally, more likely if you're male).
Personally, I don't think that an extra ounce of raw intelligence beyond a certain point much matters in terms of achievement except in pure math, theoretical physics and the like, but most seem to disagree.
But as best I can tell from my layman's view, the suggestive evidence regarding cognitive differences between men and women -- who, I stress again, different in 1/46th of their chromosomes, potentially a very large difference, even taking into account the relatively vestigial nature of the Y chromosome -- is strong enough to make for an eminently legitimate debate. Steven Pinker is not a dope or a loon; neither, in our field, is Kingsley Browne, who has read a lot of the scientific literature and written about it. There's ample evidence to base plausible conjecture here, even though (as I keep stressing) it may well be that after enough debate we conclude that the evidence on balance weighs heavily against the conjectures. The conjectures are thus hardly ridiculous or lacking at least a minimum of supporting evidence. (Naturally they lack dispositive evidence, but that's common when we're speculating about nature vs. nurture -- the contrary views lack dispositive evidence, too -- and is just another way of saying that there's no proof.)
We are talking about people who are able to reach the pinnacle of their field - tenure at a leading research university - in fields like math, physics, chemistry etc. I think it's pretty plausible that these individuals are largely drawn from the top 1-2% of the population.
The Left could used (and has used) bad science to rationalize bad, wasteful, and even unjust social policy as well.
How about my hypothetical from the other thread. Suppose a person conjectured that the disparity in liberals in academia is due to the genetic inferiority of conservatives? Would you react the same way to that conjecture?
I would react by pointing out that you have begun with a faulty premise that being a relatively low paid acedemic is superior to being, say, an overpaid corporate manager. Some would say that wasting one's abilities on a lower paying pursuit is a genetic inferiority. Others would argue that an enjoyable pursuit is more important. So there is not really inferiority or superiority, just DIFFERENCES.
He was speaking in his position as president of harvard. Now, I don't think people should get fired for that, but they do deserved to be pilloried, or at the very least, ridiculed for it.
Also, all this talk of "as with all hypotheses, it should be studied and discussed by open-minded individuals without the interference of a political or social agenda" is incredibly disingenuous. Science isn't, and never has been an apolitical activity. Why do you think this became a political issue? Of course the debate over gendered cognitive differences is political. Pretending that all you want is a fair and open debate is a time honored political tradition. If you can stomouch all the stupid Marxism, the first part of Max Horkheimers essay "Traditional and critical theory" gives a pretty good account of how it happens.
Oh, those individuals are entirely drawn from the top few percent of the population. When I say that "I don't think that an extra ounce of raw intelligence beyond a certain point much matters", I'd place that threshold somewhere in the 99th percentile. Note that's still a heck of a lot of people, and most will accomplish nothing of note--for reasons that often have little to do with their level of pure brilliance.
Naturally science has never been an entirely apolitical activity. But it should be. We shouldn't seek to strive for a research environment that is less than ideal simply because science has been clouded by politics in the past.
Partisan individuals will inevitably use research into gender differences to score political points for their pet policies. But that is neither here nor there. As a social scientist, I consider myself duty-bound to examine the facts as objectively as I know how. And while I do not expect the same from politicians or the media, university faculty should be open-minded enough to seek the truth regardles sof the political fallout.
There is a point that has been made repeatedly, if intermittently, that no one seems to answer: Why does being better than me at math make someone superior to me in any respect? Perhaps I'm able to maintain my equanimity because I'm not even at the narrow portion of the distribution where you're going to see more men than women. (although Feynman's IQ was only measured at 125, and mine's certainly higher than that, so who knows?)
But I'd just like to point out that while I was in college I got laid a lot. A lot.
So? What's wrong with being wrong? We'll never get any progress if people all shy away from saying things that turn out, ultimately, to be wrong. While I think most of what has been put out on the subject so far is a kind of pseudoscience arrived at by manipulating definitions and test population demographics, that doesn't mean that in the future, there might not be some surprising and useful result about the intersection of cognitive abilities and political opinions that comes out in the future.
Again, "superiority" is a value judgement. There is no reason why a lower IQ would necessarily make an individual "inferior" to another individual. I certainly think a 125 IQ and getting laid a lot in college is superior to a 140 IQ and getting laid less often.
It certainly suggests that women who join professional leagues are different from their male counterparts, not necessarily inferior. But it could also suggest that the audience who enjoy sports prefer watching other men play, which would say nothing about being better or worse.
On another topic, I think dearieme's point about genetic differences in average height is important. If cognitive ability and whatever else is useful for an academic career were perfectly measurable, then we wouldn't be having this conversation - individuals should be evaluated on their own terms, and not on what the statistical properties of each subpopulation they belong to are. If we want to stop undue discrimination, then we need more research into cognitive differences, so that they can be more accurately assessed.
There's nothing innately superior about high intelligence. Indeed, from a purely Darwinian standpoint, his anecdotal evidence suggests, if you read between the lines, that it is a nontrivial evolutionary disadvantage.
There is a great deal of evidence suggesting that this conjecture is inconsistent with the evidence. There is a wealth of data showing that differences in gender outcomes is attributable to culture. As I have suggested a couple of Pinker's slides actually seem to support this hypothesis. Given this data, why would one make a conjecture that is unsupported? Is there any evidence of the genetic explanation? Maybe I'm just unaware of the evidence. So I'll ask again what it is.
Then what, should we be suspicious of frankcross's suggestion that these differences might be attributable to culture, because of our long history of demeaning other cultures as inferior and using it to justify forcing native peoples to adopt the majority culture at gunpoint (or just wiping them out)? I don't think a history of bigotry really inclines in any particular way. The history of using biology to justify notions of superiority and inferiority is comparatively recent, I think, since modern biology (esp. genetics) is comparatively recent. On the other hand, people have been setting other people on fire based on notions of superiority and inferiority underpinned by cultural differences for thousands of years. That dates back to the Romans.
Sorry, that shouldn't be "majority" culture -- that should be "our" culture, whoever "we" happen to be at the moment: Imperial China, Imperial Rome, or France, or the British Empire or whoever. Not always the majority.
By this reasoning, Huntington's disease, inherited forms of breast and ovarian cancer, and certain forms of muscular dystrophy which are first manifes in adolescence are not genetic either.
Provided you could define some criteria for laziness that would (1) be widely acceptable, and (2) be susceptible to reasonably sbjective measurement
A nice goal. But, though application of scientific methods need not be political, the choice of subjects to study or fund inevitably are. The choice to study climatology or agronomy or psychometrics-- or a given problem within those fields-- involves a decision by the investigator regarding what is important. Example: research in gynecologic surgery during the 1950's led to bigger and more expensive operations, such as radical cancer operations and infertility procedures. In the 1990's, investigation led to diminution of the scope and physiological impact of gynecologic procedures-- laparoscopy and medical treatments for diseases treated surgically. In both eras, the efficacy of the new procedures was indubitable, but the trend of research was shaped by the zeitgeist. Other surgical fields experienced the same trend-- mega-operations after WWII, and minimally invasive surgery in recent times.
Your position is similar to that of a letter to Science magazine signed by 79 (mostly) scientists, who asserted that “there is little evidence that those scoring at the very top of the range in standardized tests are likely to have more successful careers in the sciences.” Carol B. Muller et al., Gender Differences and Performance in Science, Science 307, 1043 (2005, February 18).
However, that turns out not to be a correct conclusion, with respect to science or professional accomplishments generally. Wai, Lubinski and Benbow did a follow-up study at age 33 of people who had been identified as being intellectually precocious (top 1%) at age 13. They compared those who were in the top quarter of the top 1% at age 13 with those in the bottom quarter of the top 1% and found substantial differences in accomplishments, not just in science but in other professions as well. They concluded that “the data reported here on secured doctorates, math–science PhDs, income, patents, and tenure track positions at top U.S. universities collectively falsify the idea that after a certain point more ability does not matter.”
Wai, Lubinski &Benbow (2005). Creativity and Occupational Accomplishments Among Intellectually Precocious Youth: An Age 13 to Age 33 Longitudinal Study, Journal of Educational Psychology, 97:484-492, p
Curiosity then demands its own proof for the case at hand. That's what it's curious about.
Modern organizational science if anything suppresses curiosity, going for consensus and funding, so probably ought not to be the model.
It is an observed fact that men are more likely to be of extremely high, or extremely low, intelligence, than women are. This is true across cultures. Even when all variables are controlled for, the gender differences are still detectable. The statement "men are inherently more likely to be geniuses than women are" is well-supported by the available scientific evidence, as are the statements "men are inherently more likely to be of extremely low intelligence than women are", "men are more likely to be aggressive", "men are more likely to be promiscuous", etc.
So, no, it is definitely not the case that all gender-based mental differences are attributable to culture. Indeed, it is an observed fact that NOT all gender-based mental differences are due to culture.
It's worth noting that, as to questions having to do with at least some gender or racial disparities, our society has tried to fashion appropriate remedial measures, measures which implicate some pretty significant issues regarding the rights of all individuals in general. What if it turns out that after a fuller scientific investigation we conclude that these remedial measures are misguided (or that others might be better, cheaper or more efficient)? Isn't that a good thing?
Sherlock Holmes used to tell Watson to eliminate the impossible and whatever is left, however improbable, must be the truth. To the improbable I would add the unpopular.
Perhaps women just value "success" in a different fashion than men do. Maybe they've taken more to heart the adage about "No one ever dies wishing she spent more time at the office", while men may be more focussed on the "He who dies with the most toys wins."
Why is it male behavior that is normed? Why should a successful woman have to emulate male behavior?
Frankly, I'm less amazed that women don't do well (or elect to not participate) in cut-throat corporate or research academic cultures that demand 60+ hour work weeks than I am by the fact that many men accept that as the going rate, without protest and, apparently, without second thought. Why?
I recall seeing studies that say that most highly successful female executives (at the upper reaches) are either unmarried or childless, while most successful male executives are both married and have children. Why should women have to give up having families to achieve success equal to what men have (assuming they want to stay within that culture)?
Also, while it may be rational on an individual couple basis for a woman to sacrifice her career for her spouse's if he's making more money or is already several steps ahead of her, perhaps what is necessary is for women to only date and marry men who are their age or younger, so the woman will be several steps ahead of the man, and thus more likely to have the favored career within the couple.
I can think of many reasons why women are underrepresented without having to jump to biological inferiority as my first conjecture.
I took the liberty of running some of the numbers (relax, this won't hurt.. much). First of all, how many standard deviations of the whole population above the whole population's mean should we count in "the elite". Five standard deviations is above 99.99994% of the population, leaving 180 in the elite -- a bit too small. Four, however, is above 99.99366%, leaving about 20,000 in the elite. So let's say we're talking about the top 20,000 mathematicians in the country.
Let's also assume that there are as many men as women in America (roughly true) and that each is normally distributed around the same mean with two different standard deviations. Call the standard deviation for women s, and that for men ks, for some multiplier k.
We can calculate the standard deviation for the population at large as
sqrt((1+k^2)/2)*s
So we want to consider the probability that a member of each population is above four times this level. Amazingly, this doesn't depend on s at all, but only on k! For the women:
1/2 * erfc(2*sqrt(1+k^2))
For the men:
1/2 * erfc(2*sqrt(1+k^2)/k)
(erfc is the complementary error function)
So now we get the probability that a random member of the top 20,000 is a woman by calculating
erfc(2*sqrt(1+k^2)) / (erfc(2*sqrt(1+k^2)) + erfc(2*sqrt(1+k^2)/k))
which is a function of the ratio of the standard deviations. Obviously if k is 1 (the two deviations are the same) this probability is 1/2 (a random member of the top 20,000 is a woman half the time).
So, for what value of k do we get 1/5? That is, what ratio of standard deviations? 1.085. That is, the standard deviation for males need only be 9% higher than that for females to explain an elite female population under 20% on the basis of this effect alone. With the help of sociological factors is may be far less. The supposed biological difference we're talking about here is much more slight than I think many of those opposed to posing the question really appreciate.
But why assume that men's drive for status is "nonbiological" when in fact there is much evidence that it is an evolutionarily derived feature of males, and not just human ones? Over history, status has been widely associated with reproductive success in males, so that "he who dies with the most toys" was also likely to be the one who died with the most offspring.
There is certainly the tendency that you describe, but one shouldn't assume that this is "nonbiological" either, as dominant, assertive women tend to be high in testosterone (and exposed to high levels of testosterone in utero). It is also a biological fact that high testosterone in females is associated with fertility problems (although also a high libido).
They don't if they are willing to accept "house husbands," but most women -- especially "high status" women -- want high-status men. Status for men is largely based upon their position in male status hierarchies and is not achieved by staying home with the children. Certainly there are men who would be more than happy to be supported by a female executive, there just don't seem to be many female executives who want to support them.
Women don't have to "give up families" any more than men do, if they are willing, as men are, to have someone else raise their children.
I think it is significant that the only people who suggest that this is about "inferiority" are the people who claim that biology plays no role. Staying home with your children is only "inferior" to being a corporate executive if you value it less. If women, on average, have different priorities (whether or not the priorities are affected by biology), then I don't know why one would want to say that women who maximize their utility are somehow inferior to men who maximize theirs.
Nearly all academic mathematicians do their significant work early in their careers, and then do little or no importnat work for the rest of their lives. Nearly all academic mathematicians are service teachers, with a handful at most of research publications, none major. All but a handful in each subfield of mathmatics will make no significant contribution to mathematics research in their lives.
SO why all these mathematicians? It's not a scam: they are teaching mathematics to those who will use it in applied fields. The recently maligned NSA employs directlyb or indirectly a significant portion of all the Math PhD in the country, but the NSA is not interested (primarily) in contriubitions to scientific mathematics (Thank God!)
That said, I think I'm in a position to say with confidence that the number of top-flight mathematicians in the United States numbers in the thousands rather than the hundreds (check the email).