Additional thoughts on the Ted Hill paper

First, I’d like to thank the large number of commenters on my previous post for keeping the discussion surprisingly calm and respectful given the topic discussed. In that spirit, and to try to practise the scientific integrity that I claimed to care about, I want to acknowledge that my views about the paper have changed somewhat as a result of the discussion. My understanding of the story of what happened to the paper has changed even more now that some of those attacked in Ted Hill’s Quillette article have responded, but about that I only want to repeat what I said in one or two comments on the previous post: that my personal view is that one should not “unaccept” or “unpublish” a paper unless something was improper about the way it was accepted or published, and that that is also the view of the people who were alleged to have tried to suppress Ted Hill’s paper on political grounds. I would also remark that whatever happened at NYJM would not have happened if all decisions had to be taken collectively by the whole editorial board, which is the policy on several journals I have been on the board of. According to Igor Rivin, the policy at NYJM is very different: “No approval for the full board is required, or ever obtained. The approval of the Editor in Chief is not required.” I find this quite extraordinary: it would seem to be a basic safeguard that decisions should be taken by more than one person — ideally many more.

To return to the paper, I now see that the selectivity hypothesis, which I said I found implausible, was actually quite reasonable. If you look carefully at my previous post, you will see that I actually started to realize that even when writing it, and it would have been more sensible to omit that criticism entirely, but by the time it occurred to me that ancient human females could well have been selective in a way that could (in a toy model) be reasonably approximated by Hill’s hypothesis, I had become too wedded to what I had already written — a basic writer’s mistake, made in this case partly because I had only a short window of time in which to write the post. I’m actually quite glad I left the criticism in, since I learnt quite a lot from the numerous comments that defended the hypothesis.

I had a similar experience with a second criticism: the idea of dividing the population up into two subpopulations. That still bothers me somewhat, since in reality we all have large numbers of genes that interact in complicated ways and it is not clear that a one-dimensional model will be appropriate for a high-dimensional feature space. But perhaps for a toy model intended to start a discussion that is all right.

While I’m at it, some commenters on the previous post came away with the impression that I was against toy models. I agree with the following words, which appeared in a book that was published in 2002.

There are many ways of modelling a given physical situation and we must use a mixture of experience and further theoretical considerations to decide what a given model is likely to teach us about the world itself. When choosing a model, one priority is to make its behaviour correspond closely to the actual, observed behaviour of the world. However, other factors, such as simplicity and mathematical elegance, can often be more important. Indeed, there are very useful models with almost no resemblance to the world at all …

But that’s not surprising, since I was the author of the book.

But there is a third feature of Hill’s model that I still find puzzling. Some people have tried to justify it to me, but I found that either I understood the justifications and found them unconvincing or I didn’t understand them. I don’t rule out the possibility that some of the ones I didn’t understand were reasonable defences of this aspect of the model, but let me lay out once again the difficulty I have.

To do this I’ll briefly recall Hill’s model. You have two subpopulations P and Q of, let us say, the males of a species. (It is not important for the model that they are male, but that is how Hill hopes the model will be applied.) The distribution of desirability of subpopulation P is more spread out than that of subpopulation Q, so if the females of the species choose to reproduce only with males above a rather high percentile of desirability, they will pick a greater proportion of subpopulation P than of subpopulation Q.

A quick aside is that what I have just written is more or less the entire actual content (as opposed to surrounding discussion) of Hill’s paper. Of course, he has to give a precise definition of “more spread out”, but it is very easy to come up with a definition that will give the desired conclusion after a one-line argument, and that is what he does. He also gives a continuous-time version of the process. But I’m not sure what adding a bit of mathematical window dressing really adds, since the argument in the previous paragraph is easy to understand and obviously correct. But of course without that window dressing the essay couldn’t hope to sell itself as a mathematics paper.

The curious feature of the model, and the one that I still find hard to accept, is that Hill assumes, and absolutely needs to assume, that the only thing that can change is the sizes of the subpopulations and not the distributions of desirability within those populations. So if, for example, what makes a male desirable is height, and if the average heights in the two populations are the same, then even though females refuse to reproduce with anybody who isn’t unusually tall, the average height of males remains the same.

The only way this strange consequence can work, as far as I can see, is if instead of there being a gene (or combination of genes) that makes men tall, there is a gene that has some complicated effect of which a side-effect is that the height of men is more variable, and moreover there aren’t other genes that simply cause tallness.

It is hard to imagine what the complicated effect might be in the case of height, but it is not impossible to come up with speculations about mathematical ability. For example, maybe men have, as has been suggested, a tendency to be a bit further along the autism spectrum than women, which causes some of them to become very good at mathematics and others to lack the social skills to attract a mate. But even by the standards of evolutionary just-so stories, that is not a very good one. Our prehistoric ancestors were not doing higher mathematics, so we would need to think of some way that being on the spectrum could have caused a man at that time to become highly attractive to women. One has to go through such contortions to make the story work, when all along there is the much more straightforward possibility that there is some complex mix of genes that go towards making somebody intelligent, and that if prehistoric women went for intelligent men, then those genes would be selected for. But if that is what happened, then the proportion of less intelligent men would go down, and therefore the variability would go down.

While writing this, I have realized that there is a crucial assumption of Hill’s, the importance of which I had not appreciated. It’s that the medians of his two subpopulations are the same. Suppose instead that the individuals in male population P are on average more desirable than the individuals in male population Q. Then even if population P is less variable than population Q, if females are selective, it may very well be that a far higher proportion of population P is chosen than of population Q, and therefore a tendency for the variability of the combined population to decrease. In fact, we don’t even need to assume that P is less variable than Q: if the population as a whole becomes dominated by P, it may well be less variable than the original combination of populations P and Q.

So for Hill’s model to work, it needs a fairly strange and unintuitive combination of hypotheses. Therefore, if he proposes it as a potential explanation for greater variability amongst males, he needs to argue that this combination of hypotheses might actually have occurred for many important features. For example, if it is to explain greater variability for males in mathematics test scores, then he appears to need to argue (i) that there was a gene that made our prehistoric male ancestors more variable with respect to some property that at one end of the scale made them more desirable to females, (ii) that this gene had no effect on average levels of desirability, (iii) that today this curious property has as a side-effect greater variability in mathematics test scores, and (iv) this tendency to increase variability is not outweighed by reduction of variability due to selection of other genes that do affect average levels. (Although he explicitly says that he is not trying to explain any particular instance of greater variability amongst males, most of the references he gives concerning such variability are to do with intellectual ability, and if he can’t give a convincing story about that, then why have all those references?)

Thus, what I object to is not the very idea of a toy model, but more that with this particular toy model I have to make a number of what seem to me to be highly implausible assumptions to get it to work. And I don’t mean the usual kind of entirely legitimate simplifying assumptions. Rather, I’m talking about artificial assumptions that seem to be there only to get the model to do what Hill wants it to do. If some of the hypotheses above that seem implausible to me have in fact been observed by biologists, it seems to me that Hill should have included references to the relevant literature in his copious bibliography.

As with my previous post, I am not assuming that everything I’ve just written is right, and will be happy to be challenged on the points above.


150 Responses to “Additional thoughts on the Ted Hill paper”

  1. Alex Jones Says:

    Great points! Quick comment and question:

    What does “ust” mean? Also, why do you think the gender imbalance in math needs to be “corrected” under the assumption that males have more raw mathematical brainpower, which certainly has a positive correlation with what it takes to be a mathematician, even if it isn’t sufficient.

    • David Roberts Says:

      My guess is that ‘ust’ is a typo for ‘must’.

    • Anonymous Says:

      He probably realized it should have been “must” half way through but then became too wedded to using “ust”.

    • gowers Says:

      Thank you for pointing out that typo (now corrected). As for your question, it’s a bit strange. I described two theories. The first is that the gender imbalance in mathematics is mainly due to biological factors. The second is that it is mainly due to societal factors. I said that the difference between the two theories is that if the second theory is correct, then it makes sense to try to change society (as has happened, for example, in the field of law). You are, bizarrely, assuming that I think the same thing under the first hypothesis.

      Actually, even if males have more raw brainpower than females, it is hardly likely that societal factors have no role to play, so it still makes sense to correct for whatever such factors there might be.

    • Alex Jones Says:

      Thank you for your response. I was assuming that you think the same thing under the first hypothesis because in your recent blog post, you said:

      So even if it turned out that the genetic variability hypothesis was correct and could be applied to something called raw mathematical brainpower, a conclusion that would be very hard to establish convincingly (it’s certainly not enough to point out that males find it easier to visualize rotating 3D objects in their heads), that still wouldn’t imply that it is pointless to try to correct the underrepresentation of women amongst the higher ranks of mathematicians.

      You said this after arguing that raw mathematical brainpower is not sufficient or necessary to becoming a mathematician, suggesting that you think that even if we accept this discrepancy of raw mathematical brainpower in the genders, there still shouldn’t be a gender difference in math. I don’t agree with this suggestion, for I think raw mathematical brainpower has a positive correlation with ability to become a successful mathematician.

      I agree that we should correct societal factors at play, but I just wanted to say that I think a gender imbalance would naturally result from a difference in raw mathematical brainpower.

    • gowers Says:

      I’m about to put forward an argument that may be nonsense, since I’ve only just thought of it. But rather than searching for flaws, I’d rather let the blogosphere do its usual quick and efficient scrutiny.

      1. It is, as you say, highly likely that raw brainpower is helpful for mathematicians and will be positively correlated with mathematical success.

      2. To simplify, let’s suppose that the population is divided into two subpopulations, which I will label as “extraordinary” (people who have a very unusual level of raw brainpower) and “ordinary” (people who may be intelligent, but are not freakishly so).

      3. Also to simplify, let’s also assume (i) that mathematical ability is normally distributed, (ii) that the mean ability for extraordinary people is higher than it is for ordinary people and (iii) that the variances are the same. (Actually, what I’m going to say works even better if the variance for extraordinary people is higher.)

      4. Then if the mean is only a little bit higher — something like half a standard deviation, say — we find that in the top centile, the kind of level where one would expect professional mathematicians to be, a huge majority of people should be extraordinary. And yet this seems not to be the case: a significant proportion of very successful mathematicians appear to be more like ordinarily intelligent people who love mathematics, have worked hard, have a certain kind of maturity, and so on.

      5. To be clear, I think that extraordinary people are certainly overrepresented amongst successful mathematicians, but not to the extent that one might expect from the naive model above. That suggests that maybe the advantage conferred by raw brainpower is not as great as one might think.

    • Billemy Says:

      Tim, one reason is that the proportion of extraordinary people working in math is not enough to cover all directions of research. If there was a Terry Tao working in my field, I am sure much of it would have been developed by him before others had a chance. Another reason is that, in my observations, people who have extraordinary raw power use it in a way that hurts them in a long run. I see people who are so good that they are tempted to work on many different problems that are hard but that are within reach. As a result, they produce many good results but don’t spend enough time thinking about truly difficult problems over long periods of time, and someone less extraordinary but more patient will get lucky.

    • Anonymous Says:

      Let me offer two points of scrutiny. They are rather weak in pertaining only to the kind of notions involved.

      First I’d ask about the assertion of over-representation. We might for the sake of argument just presume extraordinary people are easy to identify, or generally adhere to some pattern, involving (for example) a track record at the IMO or exceptionally young age of entry into profession. Even if so, their representation among top mathematicians, it seems to me, is not all that great, them being rather in a minority no matter how to choose the pattern (other than exceptional work everyone at this level does). The important counterpoint is it’s not an insignificant minority, but it is harder to compare this significance with other fields because of not that clear-cut numbers involved.

      Second, the difference in effects between brain power or intelligence, and willpower or the ability to work hard in a driven or focused way is hard to discern to the point of the distinction being artificial. We may as well name the property of innate ability either way and use willpower as definition of intelligence or other way around as long as the basic notion is posited or hypothesized to be innate. This makes discussing such model as the above very hard.

    • Alex Jones Says:

      I understand your point now, Tim. I just didn’t know the fact you said about mathematicians not being as extraordinary as would be expected. I will definitely trust you, as you have more experience with mathematicians than I do (after all, I’m just a radio show host). Thanks a lot!

    • aiaas Says:

      Look like you cannot understand (apparently naturally, due to your own raw brainpower) what is a motivation’s part in the intelligence of a living systems and why AI can solve a task, but cannot invent it, i.e. give birth to it.

  2. Olav Says:

    I agree that the assumption that the distributions remain unchanged from generation to generation seems unrealistic (although I’m not an expert on this issue). However, in my view what Hill has done is show how sexual selection could plausibly lead to higher variability being favored over lower variability, all other factors being held fixed. It’s a highly idealized model that singles out a specific factor. Of course, in the real world, all other factors aren’t fixed, and natural selection would presumably try to optimize any relevant feature of the distribution, including the variance and mean, (and maybe skew and who knows what else), within the limits of what’s biologically possible. These various selection pressures might interact with each other in unpredictable ways, but it’s still interesting to see what their individual effects plausibly would be when isolated.

  3. Noah Snyder Says:

    I really wish you wouldn’t do this. A bunch of mathematicians speculating about stuff they know nothing about is not a good way to get to the truth. If you really want to do some modeling of evolutionary biology, then find some experts to collaborate or at least spend a year learning some background. Idle uninformed speculation of non-experts is how we got into this mess in the first place.

    • erasmuse Says:

      Please, Mr. Snyder let’s be scholars. It’s fun to figure things out. We may be reinventing the wheel here, but that’s OK. Moreover, a lot of the time a better way to understand an idea is to work it out yourself rather than read how someone else has done it. The main downside is looking ignorant and making silly mistakes, but that’s just a hit to our pride, so we should try not to worry about that.
      Of course, if we were making national policy, we’d want to use a different approach, but this is a blog.

    • Noah Snyder Says:

      In a situation that has literally lead to a hoard of terrorist internet trolls making death threats, I think it’s pretty naive to think that the main potential downside is a hit to our pride for saying something silly. But at any rate, my point is exactly that we should be scholars, and part of that is understanding that half-baked speculations not based in serious engagement with science is not scholarly. If we want to try to learn some basic evolutionary modeling by working it out ourselves, why don’t we go with something basic, interesting, and established like trying to understand Fisher’s principle explaining why sex ratios in populations are typically close to equal. Then we’d at least be learning something.

    • Gladly anonymous Says:

      This discussion as of now succeeded in attracting a troll and fraudster banned even from Twitter while commenters incorrecting eachother encited in Sir Gowers a state he mistakes with understanding ina evening the relation of selectivity to genetics studies.

    • dilettante289 Says:

      Exploring ideas is a perfect use for a blog. Of course, serious studying and/or consulting relevant experts would seem to be necessary before publishing in a research journal. (And if you say that just these particular ideas should not be naively discussed, then you are playing into the hands of the right-wing trolls.)

    • P Says:

      > This discussion as of now succeeded in attracting a troll and fraudster banned even from Twitter

      That is definitely not the real Alex Jones.

    • Javier Says:

      I second this comment of Noah Snyder’s. Toy models make sense when they are caricatures of a well grounded model that capture some essential point (Ising model, phase transition). They can be worse than useless when they are based on thought experiments.

    • Anonymous Says:

      Noah: Even if one leaves aside the fact that the paper was considered by a serious mathematical journal, which should make it fair game for mathematicians to comment on, I simply cannot agree with your point of view.

      Your response is essentially “go away and study for a year and then we’ll talk”, which is patronising and unhelpful. It contributes to the idea that any open discussion about science can only be had by experts in the field, and that any other viewpoints are not just pointless but in fact harmful. This is a damaging and insular point of view.

      Gowers has used his blog, on which he is free to express opinions which people are in turn (more than) welcome to disagree with, to openly discuss what he finds problematic with the paper. Rather than simply trash the quality of maths in the paper, he has attempted to engage with the area and the model. He has done so in a polite way that shows his willingness to be corrected.

      If an eminent biologist were to try and engage in the same way with one of Gowers’ papers, even if they came to a conclusion that the work was mediocre, then I am certain that Gowers’ response would be to try and clarify things and correct misunderstandings in a constructive way. His response would not simply be “if you’re not an expert then I don’t care what you think”.

      Finally, I should add that I agree completely that trolling and death threats are a digraceful blight on rational discussion, but their existence should not be used to silence genuine debate, and moreover it does not seem that Gowers’ posts have attracted either.

    • aiaas Says:

      You sound like a monk being unhappy that the flock may start questioning the priest and canon of the Scripture.

      «We have little solid science on the origin of life, in large part because there is virtually no physical record, but also because we still have gaps in our understanding of what constitute the fundamental principles of life.

      All we know is that natural genetic engineering is ubiquitous in living cells today. Like cells themselves, when and how NGE originated is unknown.» James Shapiro.

  4. Ian Walker Says:

    This is fantastic, this is where we get to see the meta process of publication, and true groundbreaking research. It was what I was trying to get at all those years ago when trying to incept Ted Nelson’s DynaBook concept for markup of what I called ‘the pinky-green cloud of unwritten lore’, the context of discovery and paradigm shift, the Bruno to the Galileo. A subset of SGML was suggested and called HTML, the browser was posited, and javascript prototyped. I got to put my initial in http://WWW... TBL and others got the cudos. And the moolah… The irony of all of it does not escape me. I did get to say “this is my gift to you all” amongst a lot of other things at a rather large event… So certain folks do know, and it is possible to elicit a certain grudging acknowledgement from them from time to time, ‘eh? All of which is rather resonant in the context of this post, no? Apollo v Dionysus, Cathedral v Bazaar, canon v fodder, bokeh v zit, Yin v Yang. What was the cause of me incepting so much? I kicked off the social web, and twitter especially, as a means (hah!) of getting at the truth. Jury (of one) is still out on that. Beamforming via twitter will have to wait… Others have paid attention – HeroX and IARPA’s Global Forecasting Challenge are but one example, Augur and etherium are another. Meaningless rambles I the context of this post? Think again…

  5. Anonymous Says:

    >According to Igor Rivin, the policy at NYJM is very different: “No >approval for the full board is required, or ever obtained. The >approval of the Editor in Chief is not required.” I find this quite >extraordinary: it would seem to be a basic safeguard that decisions >should be taken by more than one person — ideally many more.

    While the above quote is not factually incorrect, I think it would be more fair to include the context in which it was used:

    “The paper was properly refereed at NYJM, and went through MORE than the usual vetting (NYJM requires one report, we got two; no approval for the full board is required, or ever obtained. The approval of the Editor in Chief is not required (but was obtained in this instance).”

    • Anonymous Says:

      Scrutinizing this quote gets to one aspect of the affair that I find poignant. Rivin is to be pitied here. He believes that not scraping the bottom of what the journal allowed means that he conducted a rigorous review. It was captious to appoint him to the editorial board of a serious math journal that allows unilateral action, and the result is unfortunate.

    • Anonymous Says:

      Well, it’s hard for me to judge how genuine was Rivin’s belief (if any) on whether the review process was objective and sufficient.

      What seems important to me is that, even according Sir Timothy’s opinion, the removal (not retraction!!) of a published paper can only happen if “something was improper about the way it was accepted or published”. Because there were essentially no precedents like this so far, I assume that “improper” means improper to an extremely high level, never seen before in academic publishing.

      Regarding the paper, it would be hard to make a case of “extremely improper”. Rivin went at least two steps above the bare minimum sufficient for publication. Three weeks for a review is not a short time. In fact, most of us do not spend more than a few hours on refereeing a short paper, and it becomes more of a question of being able to find time to conduct the review.

      If one pretends to remove all politics from the discussion of this paper, it’s pretty clear that it’s much less severe case than Sergeev’s one, and is not very far away from the level of depth of other papers on the same subject (and I don’t think that it can be unanimously called pseudoscientific).

      No one has ever seriously considered “unpublishing” Sergeev’s paper, even though it was a textbook case of academic dishonesty. It wasn’t even officially retracted.

      In general, the line of argument “this paper was really bad and trivial anyway, so nothing wrong with removing it” is toxic, and has already been used many times in the last 100 years to justify all kinds of censorship.

    • Igor Rivin (@igriv) Says:

      While I do not appreciate your pity (every step of the process was conducted under the direction of the Editor in Chief, and both I and he believed (and still do) that the paper would be very beneficial to the journal), the last coment in the series is quite a propos, and I agree completely.

    • Anonymous Says:

      That being said, it is still not clear why the referee reports were not provided for the editorial board for almost three months, despite repeated requests. This question was asked twice in the previous thread, with no answer.

      I do not see a reason not to share reports with the editorial board. It is a common standard, based on the fact that the editorial board is ultimately responsible for the quality of peer review.

      It’s possible that Farb did not share all the details, but, if no further details are provided by either party, the three months delay does look like an act of bad faith.

    • Anonymous Says:

      > Re: “this paper was really bad and trivial anyway, so nothing wrong with removing it” is toxic, and has already been used many times in the last 100 years to justify all kinds of censorship.

      I am unaware of the many historical examples of this kind of censorship. I expect you to provide them so we can compare the structure of the arguments with the present.

      I am however aware of your objection always coming up from the accused in cases of misconduct I personally pursued. It usually also comes together with argument Igor used of somehow exclusive right of the referees he choose to judge the paper’s merit (and berating Tim Gowers for using his right to do so too).

    • Anonymous Says:

      If you read Solzhenitsyn, you will also find examples of the government sending “corrected” versions of encyclopedia articles to the subscribers, with the instructions to cut the old version of the article, throw it away, and replace with the other one. Or sometimes literally replace it by a different article of the same length. Of course, the usual reason was that the public person, once admired by the government, is now in prison or executed.

      In case of demand, I can find exact references to the articles that were replaced in this way by the above reason.

      I will let the readers decide whether this example is good.

    • Anonymous Says:

      Sorry, but these are not cases in science.

      I’d be glad to discuss examples matching the current situation, not some vague associations to journalism in Soviet Union (comparing us to weakens your argument).

    • Anonymous Says:

      By “toxic”, I mean justifying and illegal and academically unprecedented act of quietly replacing one paper with another, by appealing to the low quality of the paper.

      There is an established academic procedure of dealing with published papers of bad or questionable quality, including situations where publication of the paper was the result of misconduct of the editors. Sergeev’s paper is an illustrative example.

    • Anonymous Says:

      Please stay on topic, and provide analogues you talked about as going back a hundred years.

    • Anonymous Says:

      >I’d be glad to discuss examples matching the current situation, not some vague associations to journalism in Soviet Union (comparing us to weakens your argument).

      Publishing of an encyclopedias is closer to academic publishing than to journalism. The articles are expository, but are usually written by experts in the field, mostly academic.

      This is the closest matching example I could find. I’m not familiar with situations of quietly replacing one article by another in modern academic publishing.

    • Anonymous Says:

      I observe you evading the question thrice already replacing it with one that suits you better.

      I was not asking about delisting but your assertion that argument of the kind “this paper was really bad and trivial anyway, so nothing wrong with removing it” is “toxic” and has been used many times over long time spans. If so, producing an example should be easy.

    • Anonymous Says:

      >I was not asking about delisting but your assertion that argument of the kind “this paper was really bad and trivial anyway, so nothing wrong with removing it” is “toxic” and has been used many times over long time spans. If so, producing an example should be easy.

      Most times an encyclopedia article was delisted in USSR, the government and a portion of general public were justifying this by referring to the fact that the person described in the article was (or turned out to be), not a good person, and does not deserve an encyclopedia article.

      If you don’t like this comparison of think it is not accurate enough (no comparison is perfect), I am happy to leave it at that. You can consider it as a personal opinion, as everything else I said.

    • Willard Says:

      > both I and he believed (and still do) that the paper would be very beneficial to the journal

      Evidence for that claim is sorely lacking, Igor, at least insofar as Mark’s mind states are concerned. It’s far from clear that the responsibility for this episode doesn’t fall squarely on your own shoulders, and that you did not simply exploited a very loose editorial process:

      On October 13, a lifeline appeared. Igor Rivin […] got in touch with me. He had learned about the article from my erstwhile co-author, read the archived version, and asked me if I’d like to submit a newly revised draft for publication. Rivin said that Mark Steinberger, the NYJM’s editor-in-chief, was also very positive and that they were confident the paper could be refereed fairly quickly.

      Let’s hope the reviewer wasn’t the “erstwhile co-author.”


      Further, in a written statement, we learn that:

      At the request of several editors, the editor-in-chief pulled the paper temporarily on 11/9/17 so that the entire editorial board could discuss these concerns. A crucial component of such a discussion are the reports by experts judging the novelty and quality of the mathematics in Hill’s paper. The editor who handled the paper was asked to share these reports with the entire board. My doubts about the paper – and the process – grew when repeated requests for the reports went unanswered. Nearly 3 months passed until the two reports were finally shared with the entire board on 2/7/18. The reports themselves were not from experts on the topic of the paper. They did not address our concerns about the substantive merit of the paper.

      I assume you were the “editor who handled the paper.” If that’s correct, and assuming the statement as being truthful, then the following questions beg to be asked:

      (1) why did you pick non experts on the topic?

      (2) why did it take three months?

      Thank you for your consideration,


    • swanson Says:

      Why don’t we all accept the evidence of our own eyes: NYJM is some kind of freewheeling bumper car in the guise of a peer reviewed math journal. A single editor can solicit a paper and quickly guide it to publication without telling anyone on the “distinguished” editorial board under whose “direction” refereeing supposedly takes place. And then the editorial board can unpublish the paper. These gyrations don’t mean the world is coming to an end, just that we shouldn’t assume fancy names mean a trustworthy journal.

  6. N Says:

    “The only way this strange consequence can work, as far as I can see, is if instead of there being a gene (or combination of genes) that makes men tall, there is a gene that has some complicated effect of which a side-effect is that the height of men is more variable, and moreover there aren’t other genes that simply cause tallness.”

    Alleles that simply cause tallness are just sexually selected for and become more common in the population over time. But these are not the alleles Hill’s paper is about.

    I think it’s important to distinguish between two types of variability: genetic and phenotipic. The paper should have made this distinction clear, but probably failed to do.

    Genetic variability in a population means having a more variable distribution of alleles at a given locus.
    Phenotipic variability means having a more variable phenotype for a given trait.
    In general, neither necessarily implies the other, but for the purpose of our analysis we can ignore the case where multiple alleles cause the same (or similar) phenotypes because these alleles don’t cause observable effects and are not selectable.

    By assumption subpopulation P has more phenotipic variability than subpopulation Q. If this is due to genetic variability then, as you figured out, Hill’s argument doesn’t work.

    E.g. 50% of men in P have allele A_p1 which causes height 1.60 m and the other 50% have allele A_p2 which causes height 1.90 m, while 100% of men in Q have allele A_q which causes height 1.75 m. Initially, P and Q are the same size, resulting in an overall mean height of 1.75 m and std 0.075 m (if I did my math right).
    The women indeed prefer the men in P, but not all of them, only the ones with allele A_p2, thus over time both A_p1 and A_q will die out, causing the average height of all men to approach 1.90 m with vanishing std, hence reducing the phenotipic variability.

    But if instead genetic variability is low, then Hill’s argument works:
    E.g. 100% of men in P have allele A_p which causes height 1.60 m with 50% probability and height 1.90 m otherwise, while 100% of men in Q have allele A_q as above. The women prefer the tall men in P, but these have the same A_p allele as the short men in P, therefore over time only A_q dies out, thus mean height remains 1.75 m but the std increases to ~0.106 m.

    You seem to think that an allele that works like A_p is biologically implausible. I don’t think it is. Lots of things in biology are random or chaotic, from embryonic development, to wear, to the external environment. In general there are costs incurred in reducing variability even before taking into account sexual selection: metabolic costs of the homeostatic mechanisms that counter low-level randomness, and missed opportunity costs of forfeiting high-risk high-reward opportunities in the environment, e.g. picking up berries (low-risk, low-reward) vs chasing buffaloes (high-risk, high-reward).
    Therefore we should expect in a population a mix of low-variance and high-variance alleles for any given trait. Sexual selection can act to make the high-variance alleles more common in men, up to other tradeoffs.

    Now you may object: since women also inherit alleles from their fathers, if the variability of a trait increses in men, it should also increase in women. This can be true for some traits, but not all of them, depending on which chromosome the locus is on, and other considerations:

    – Loci on the Y chromosomes only affect men. The Y chromosome is small, though, so only a few loci are there.

    – Loci on the X chromosome typically affect men more extremely than women: say that there is only one height locus on the X chromosome, a man has only one of A_p, A_p1, A_p2 or A_q defined as above, a woman has two of them, if their phenotypic expression averages out (in practice it’s more complicated, but this holds to a first approximation), then women will have less phenotipic variance.

    – The expression of the loci located on any chromosome may be mediated by sex hormones. If there is a general tendency to favor higher variance in men but not in women (caused by many instances of Hill’s mechanism, for instance), then a general mechanism to cause this for many traits may evolve.

    In conclusion, if an allele can cause higher variance in males of a sexually desirable trait then it can be sexually selected by females and become more common.

    “Our prehistoric ancestors were not doing higher mathematics, so we would need to think of some way that being on the spectrum could have caused a man at that time to become highly attractive to women.”

    Math is arguably the most g-loaded of all human activities. Clearly there was a selective pressure to increase the general intelligence factor in the history of our species, given that we are much more intelligent than the other animals. Even today men with higher g-factor tend to have more children. A female preference for smarter men over time increases the average intelligence of men, but can also increase the variance of it if alleles like A_p exist for intelligence.

    • araybold Says:

      “Therefore we should expect in a population a mix of low-variance and high-variance alleles for any given trait.”

      Excellent! Instead of interminable speculative arguments from both sides, we can just point to the experimental evidence that has accumulated for this proposition.

    • Eli Rabett Says:

      You appear to believe that courtship marriage is the rule for humans. It is not, even today, most marriages in the world are arranged by parents and this true even into prehistoric times. Disappointing as it may be to you there is no point in trying to use Hill’s model to talk about humans.

    • Igor Rivin (@igriv) Says:

      Eli Rabett: Firstly, this “courtship marriage” thing you are bringing up (on, I believe, at least three blogs) is a canard. Do peacocks practice courtship marriage? Do cats? Which species does (if any)? and yet, I will call your attention to this post: (it has been mentioned elsewhere, but you have obviously missed the mention).

  7. Anonymous Says:

    “general intelligence factor”

    This is a rare invocation in this discussion of something of actual mathematical content. By Frobenius–Perron one is always guaranteed a dominating factor in positively-correlated studies, thus g-factor has zero scientific content being a mathematical artifact of unknown relation to reality.

    • N Says:

      There is no a priori mathematical necessity for all cognitive abilities to be positively correlated with each other.
      It could have been possible for some of them to be uncorrelated (the multiple intelligences hypothesis) or even negatively correlated (as a result of some tradeoffs).

      Instead we empirically observe that the correlation is always positive. This is an important scientific discovery. The g factor accounts for this shared variability.

      It’s not entirely known what physical differences in brain anatomy and physiology contribute to the g factor, in principle it could be many of them with an additive effect. But even if the g factor is an average of many things it doesn’t make it any less real, just like the fact that temperature is an average of kinetic energies of particles doesn’t make it any less real.

    • Anonymous Says:

      Sorry, but this is muddying the issue on your part. These studies are just set up the way you can have only positive correlation, and if you have a matrix with nonnegative entries, you get a dominant factor by theorem of Perron and Frobenius regardless of any justification you want to invent for it. I am not aware of any study with negative correlations that gives rise to anything alike g-factor.

    • Anonymous Says:

      g does not “account for” the correlations between cognitive abilities. It summarizes those correlations.

      There are an indefinite number of causal structures besides a single common cause that can result in positively correlated factors. The fact that we can calculate a dominant component follows from the mere existence of positive correlations; it does not discriminate between the different possibilities for the latent causal structure that produces those correlations.

    • N Says:

      “These studies are just set up the way you can have only positive correlation, ”

      This is obviously false. There is nothing in the design of, say, verbal ability tests that forces them to positively correlate with mathematical ability tests or spatial reasoning tests, or anything else.

      “There are an indefinite number of causal structures besides a single common cause that can result in positively correlated factors.”

      Yes, a large number that common causes, which would imply that g factor is an emergent attribute, just like temperature is an emergent attribute of the kinetic energy of many molecules. Is temperature not real?

    • P Says:

      “These studies are just set up the way you can have only positive correlation”

      That is not true at all. As N already pointed out, there’s no a priori reason to expect (for example) working memory and reaction speed to correlate positively or negatively.

    • Anonymous Says:

      “… would imply that g factor is an emergent attribute, just like temperature is an emergent attribute of the kinetic energy of many molecules. Is temperature not real?”

      No one would claim that the kinetic energy of an individual particular is caused by the temperature of the collection of particles to which it belongs, but you are claiming that g is an attribute that explains individual cognitive achievement.

    • Anonymous Says:

      “There is nothing in the design of, say, verbal ability tests that forces them to positively correlate with mathematical ability tests or spatial reasoning tests”

      But of course yest there is if you only score on positive axes. Please read up on design of experiments, you have to null-hypothesize a special way to to research negative correlations.

    • N Says:

      The first step of factor analysis (PCA) consists in subtracting from each observable variable its population mean. This is done before the eigendecomposition, therefore Frobenius-Perron does not apply.

      The matrix that is eigendecomposed is in fact the covariance matrix, which can have both positive and negative entries.

      Seriously, you might want to read up on factor analysis if you want to productively contribute to this discussion.

    • Anonymous Says:

      For what it’s worth, I’d like to point out that there are two people commenting in this subthread as Anonymous. I wrote [1] and [2]. I don’t know what the other Anonymous is going on about.


  8. michaelgreinecker Says:

    If one sees this as a piece of disinterested academic work in mathematical biology, using the model to explain variability in math ability in humans is probably the last application one should come up with. I’m sure there are phenomena where this model applies (I don’t think exact equality of medians is that important), but I’d rather look for examples in less complex species. Even in humans, there are cases where the model fits better. For example, testosterone helps with a number of generally desirable qualities but seems to have a negative effect on the immune system and can, therefore, increase the risk of sharp falls in quality. The application to math ability looks more like a political stunt than serious scientific inquiry.

    • Mark C. Wilson Says:

      Might it not also have some kind of application to evolution of artificial life? I don’t see why the editors of NYJM did not insist on removal of some of the speculative parts and focus on the mathematics and possible applications.

  9. erasmuse Says:

    “thus g-factor has zero scientific content being a mathematical artifact of unknown relation to reality.”
    I generally don’t like factor analysis either, but g turns out to have an amazing number of correlations with other variables such as who marries whom, success in school, and job performance. So it really does seem to capture what we mean by “intelligence”.

    • Anonymous Says:

      This strikes me as circular argument if then such circumstantial notion of intelligence is applied back to explain the circumstances.

    • Anonymous Says:

      “g turns out to have an amazing number of correlations with other variables”

      It’s not amazing at all that the principal component of the covariance matrix is correlated with the things the factors are correlated with; it’s guaranteed by construction. Dimension reduction is not a tool of causal discovery.

      You might find this discussion by Cosma Shalizi useful:

    • Olav Says:

      Anonymous: it’s not amazing that g is correlated with the factors out of which it is constructed. But the very fact that there is such a large manifold of distinct but positively correlated factors is a genuine — and important — discovery, which the g factor summarizes.

    • Anonymous Says:

      What “summary” this is if it’s a tautological restatement?

    • N Says:

      It is not tautological that the results of many different cognitive tests are all positively correlated.

      And the g factor has been found to be positively correlated even with things other than those used to define it. Essentially all the measures of personal achievement (academic, professional, not going to jail, etc.) are positively correlated with the g factor.

      This is strong evidence that the g factor is not a mere statistical artifact but instead reflects an underlying “brainpower” which is used in all cognitive tasks.

    • Anonymous Says:

      “Even with things other than used to define it” is rather silly kind of achievement, more of a reason for scrutiny about abundance of all the other cases.

    • Anonymous Says:

      “Essentially all the measures of personal achievement (academic, professional, not going to jail, etc.) are positively correlated with the g factor. This is strong evidence that the g factor is not a mere statistical artifact”

      It’s not strong evidence. These measures you allude to are correlated with IQ, and IQ tests are correlated with each other. From the fact that IQ tests are correlated with each other it follows that we can calculate a factor like g, because g just is a way of summarizing that correlation. g, thus calculated, is also going to correlate with the behavioral outcomes in question, but this is because the correlation of g with those outcomes is a way of summarizing the correlation of IQ tests with those outcomes. It’s not a discovery of the underlying cause of cognitive performance.

    • N Says:

      “These measures you allude to are correlated with IQ, and IQ tests are correlated with each other. ”

      Which means that these IQ tests measure something real, which does not depend on the specifics of the test and has predictive power w.r.t. life outcomes.

      Sure, there is no known mechanistic explanation for intelligence, if there was then artificial intelligence would be a solved problem, but we don’t need a mechanistic explanation to predict that if somebody has a low IQ they are more likely to go to jail instead of becoming a math professor.

      Or that if a group of people (men) has a higher IQ variance than another group of people (women), the former will probably have both more inmates and more math professors than the latter (which is what we observe).

  10. Anonymous Says:

    That’s exactly my point; g is descriptive, not explanatory. The correlations between g and other phenomena are not evidence of the reality of g as attribute that causes the correlations enjoyed by the factors out of which it’s constructed.

    Height is positively correlated with income and reproductive success. IQ is also positively correlated with income. Therefore the sum of height and IQ–call it h–is positively correlated with those things. It might be useful for me to have a single number summarizing my data instead of two, but that’s no reason at all to think that this h is an attribute of a person that explains income or reproductive outcomes.

    g is real in the way that h is. That one person consistently outperforms another on different IQ tests is a fact that can be summarized as the former possessing more g, but it’s a fallacy to turn that around and take possession of greater g to be what explains the differential performance.

  11. Winston Says:

    As a mathematician with a strong interest in evolutionary biology, two things struck out in particular from the comments of the previous post.

    The first is, how ignorant most mathematicians are of research both in evolutionary psychology as well as social psychology.

    For example, it is now a very mainstream idea in evolutionary psychology, with tremendous experimental evidence (particularly from separated identical twin studies) that virtually anything that can be measured about the mind, is strongly influenced by genetics. (No I am not saying that environment does not make a big difference, merely that genetics influences who we are to an astoundingly large degree, much larger than most people here realize). This includes not just brainpower/IQ but virtually every aspect of personality one can think of. Yet, most mathematicians seem to largely subscribe to a primarily environmental hypothesis, with genetics only very little contribution. This goes completely against what we know.

    Regarding social psychology, let me give one example. Most of you have heard of the study where student evaluations depend on the gender of the name of the instructor (this was done in an online course, with male and female names switched). The study found that students gave better feedback to male-named instructors, regardless of whether males or females were teaching the course. I think this is a very nice study, and demonstrates likely bias. You have probably heard of similar studies in hiring applications, and in many related areas. You have heard of studies that say that female candidates face a higher barrier when they apply for STEM positions. Each time you see such an article, it confirms your view. Yet I bet that most of you haven’t heard of the study that showed that female candidates are 2 times likelier to be hired in top STEM positions than men ) If you have, you have probably immediately tried to find faults in the research, and have succeeded.
    Of course, most social science research in these areas have some flaws. For example, the online evaluation study which is so famous is not even double-blind. That does not mean it is not valuable. If a similar bar is applied to all studies of this sort, over the long run they will allow us to perform meta-analyses, and ultimately bring us closer to the truth, regardless of minor flaws in each individual paper. However, right now, the bar is much higher for studies that oppose the equality/diversity/feminist ideology, which makes it harder to reach the truth and easier for mathematicians to live in a bubble.

    The second thing that struck me is, how uncommitted most mathematicians are to values of academic freedom and free speech and how ready they are to embrace censorship (even if in a mild form).

    If the day comes (as it may in the next 20 to 30 years) when our understanding of neuroscience and evolutionary biology has advanced to the point where it becomes undeniable that biological factors play a primary role (at least 50%, probably more) in the gender gap that we see today in achievements at the highest levels of fields such as mathematics and theoretical physics, how would you react? Would you give up your current support for positive action that relies implicitly on the environmental assumption? Or would you maintain your support but find other grounds for it, which do not rely on such assumptions? If so, would it not be better to find those grounds today? If not, why not?

    • hailongdao Says:

      Thank you for the interesting post. You mentioned that “it is now a very mainstream idea in evolutionary psychology, with tremendous experimental evidence (particularly from separated identical twin studies) that virtually anything that can be measured about the mind, is strongly influenced by genetics.” Can you give me some references to that?

    • Anonymous Says:

      First, there is no serious field of evolutionary psychology, what can charitably be supplied for the GP’s argument instead is the field of behavioural genetics. It is very far from having a single mainstream consensus on, it seems, anything. Two of main diverging factions revolve around the highly publicized work of Plomin and negating it but less known work of Turkheimer. Recently Plomin fending off another bath of critique, argued in Nature that his work is respectable because it has no worse levels of signal than social sciences. I think this settles where the field is situated despite “genetics” part of its name suggesting something air-tight about it.

    • Winston Says:

      To hailongdao: Yes there are plenty. For starters, you can look at this paper and if interested, at the references therein. There’s a gazillion other papers on this topic, and the consensus is especially strong among those who describe themselves as being in the cross-disciplinary field of evolutionary psychology, though the consensus is not as overwhelming, as say, the consensus among number theorists that RH is likely true (I deliberately chose an example where no proof is available).

      However my strong recommendation, if you wish to get a survey level view of the subject from a more popular viewpoint, is to read the book “The blank Slate” by Pinker.

      To anonymous: My use of the term “evolutionary psychology” rather than “behavioural genetics” was deliberate. Obviously the two fields are closely related, but the former can be more properly thought of as a cross-disciplinary field emerging from psychology, rather than belonging to genetics. I find your claim that there is no serious field with this name mystifying. If this is a factual claim, then it is patently false, as the sheer number of papers, the number of researchers who use this term, and a Wikipedia entry, can testify. Perhaps you instead mean that you have no respect for the field, and likely have criticisms similar to what are summarised in the Wikipedia page. (This would also explain your misrepresentation of twin studies) Those criticisms have been rebutted in countless places and it does not make sense for me to engage with you on that front to rehash that debate here.

    • Javier Says:

      It is a mistake to assume that those posting here are mathematicians, or that they are representative of mathematicians.

    • Anonymous Says:

      Sorry but it doesn’t make sense to rely on Wikipedia on highly controversial topics.

    • Anonymous Says:

      Also, I said nothing of twin studies. I may however say you misrepresent work of Pinker as science he’s certainly not an expert on and has no standing in either evolutionary biology nor even behavioural genetics communities.

    • M Says:

      It is very far from having a single mainstream consensus on, it seems, anything. Two of main diverging factions revolve around the highly publicized work of Plomin and negating it but less known work of Turkheimer.

      On the contrary, there’s widespread consensus on many foundational issues in behavior genetics. Much of the consensus is summarized in the three “laws” of behavior genetics:

      First Law: All human behavioural traits are heritable.

      Second Law: The effect of being raised in the same family is smaller than the effect of the genes.

      Third Law: A substantial portion of the variation in complex human behavioural traits is not accounted for by the effects of genes or families.

      You might think that a contrarian like Turkheimer would disagree with such sweeping claims about human behavior, but in fact the three laws were promulgated by Turkheimer himself; they are Turkheimer’s three laws of behavior genetics which he stated a couple of decades ago but which continue to be a valid way to characterize the origins of most human differences.

    • Anonymous Says:

      I think you are mistaken in you reading of the three laws. They in sum say that the necessary distinction between environment and genetics in the notion of heritability istelf is undiscernable with methods so far. I can thusly see the lack of consensus on what any of this research really implies. Plomin says it’s nature no doubt, Turkheimer says it’s environment as well.

      Especially in context of this discussion the recently added fourth law is illuminating
      “A typical human behavioral trait is associated with very many genetic variants, each of which accounts for a very small percentage of the behavioral variability.”
      So you don’t have a this-or-that gene, but an extremely complex parallel self-interaction in which environment also fully takes part.

    • Anonymous Says:

      Also “all” in “all is heritable” is really meant as everything.

    • M Says:

      None of that is incompatible with what someone like Plomin says. Plomin has written about gene-environment interplay since the 1970s and polygenicity has always been the working assumption in behavior genetic (even though the minuteness of individual effect sizes certainly took just about everyone by surprise).

    • Anonymous Says:

      Incompatibility is a major one in that for one the interplay is discernible and for other it is not. As evidenced in the first law. The whole definition of heritability hinges on that distinction.

      Turkheimer expands on this:

      “The prescribed cure for the vanishingly small effect sizes typical of genomics has always been more statistical power, in the form of ever-larger sample sizes. But at some point, the field is going to have to grapple with the possibility
      that the difficulty is not statistical power at all, and therefore cannot be remedied by enormous sample sizes and stringent p levels. […] social science has learned a bitter lesson: the explanation of behavior is difficult not because the relevant causes, though countable and essentially additive, are small and difficult to detect; rather, social science is difficult because causes are innumerable and essentially non-additive.”

      “The endless repetitions of genome scans that identify a few weak-to moderate signals which then don’t replicate very well in the next study is simply a rediscovery on the molecular level of what I (Turkheimer, 2000) have called the First Law of Behavior Genetics: everything is heritable. Everything is heritable because of weak biologism, GWAS is always bound to produce a few “results” because everything is heritable, and heritability is instantiated in the genome, in the same not very useful sense that cognition is instantiated in the brain.”

    • Anonymous Says:

      When social constructs start to turn out heritable it loses explanatory meaning. They are both quite careful with their language, with Plomin just alluding to explanations that aren’t there and Turkheimer calling him out in most charitable terms imaginable.

      A typical exchange:

      “if the authors think their data support the hypothesis that socioeconomic educational differences are simply the result of pre-existing genetic differences among the students assigned to different schools, that is their right. But I would prefer that they explicitly defended the idea, instead of letting it float implicitly in the background. The data they report here do nothing to actually make the case in one direction or the other.”

    • M Says:

      That 2012 paper by Turkheimer is already badly dated. Here’s the latest preprint from Plomin’s lab. Large samples and stringent p levels continue to bear fruit.

      “Polygenic scores for educational attainment and intelligence are the most powerful predictors in the behavioural sciences and exceed predictions that can be made from parental phenotypes such as educational attainment and occupational status.”

    • Anonymous Says:

      That paper’s from 2008 actually, and he’s been repeating these thing since mid-90’s after riding on his youthful enthusiasm in the 70’s and 80’s just like Plomin. So that’s a quarter century waiting for these fruits to appear in ever-newer-wider studies waiting around the corner.

      Then there are people waking up to the realities of use(lessness) of this all for anything (especially medical journals no longer accept “heritability” at face value).

    • M Says:

      Eppur si muove. The GWAS revolution keeps rolling on, confirming the views that Plomin et al. have held for decades, while polygenic scores are getting better all the time and will be a standard research tool in the near future (and may have clinical utility as well). His detractors have been forced to update their views. (I don’t think Turkheimer is a great offender here as his views have always been more subtle than those of genetic denialists.)

  12. Bill Says:

    Hi Tim, you should read the latest post of Lubos Motl to see if he answers some of your questions. His explanations seem very clear to me. I hope you can point out at which step you disagree with him:

    • Pyrrho Says:

      I stopped reading this when he completely misread the first passage he quotes from Tim. Tim’s description of which narrative he’s referring to comes in the ellipsis, but Motl goes on for several paragraphs as if the narrative was described in the sentence he quotes. Does it get any better after that?

    • Bill Says:

      Of course, you have to ignore his insults and misrepresentations and focus only on his explanations of the issues discussed. I think his explanations are nice and clear.

  13. Centrocercus urophasianus Says:

    Regarding the plausibility of the idea that an allele could make some males more desirable to females, but make males more variable with respect to some property – can I suggest that mathematicians unfamiliar with evolutionary biology take a look at some of the large body of work in that field on the issue of “honest handicaps” that attempts to resolve what has been termed the “paradox of the lek”.

    There are lots of good – plausible – theoretical models that explain the maintenance of male variability in the face of consistent female preferences (e.g. the Hamilton-Zuk model or Zahavi’s).

    This discussion is not helped by people making pronouncements about what is and isn’t common in the animal kingdom, and what is or isn’t plausible, based on their intuitions.

    There are many fascinating outstanding problems in evolutionary biology – it would be great if a by-product of this furore was that some really good mathematicians got interested in these problems and contributed to solving them.

  14. Centrocercus urophasianus Says:

    P.S. being on the evolutionary biology side of things – I would have been astonished to see Hill’s paper turn up with no mention of leks, Hamilton-Zuk or Zahavi.

  15. M Says:

    I would think it likely that many gene or genes that confer greater trait variability on males would have across-the-board effects, influencing many traits rather than just one, such as math ability. For example, this Norwegian study found that males are generally more variable in birth weight as well as in adult weight, height, BMI and blood parameters, running performance and aptitude test scores. It seems unlikely that all of these variance differences are socially caused. It also seems unlikely that the ones that have genetic causes all have different causes. More likely, sex differentiation early in the development causes males to have more noisy or variable gene expression across many traits.

    The thesis is then that high-variance-inducing sex-limited genetic variants have become common or reached fixation because a high-risk, high-reward strategy has been successful for males in a Darwinian sense; in the long evolutionary timescale, women have disproportionately had children with men who have extreme (positive) values for any of the traits whose variance is affected by the genetic variants in question.

    • M Says:

      Another theory is that the heterogametic sex (the one with two different sex chromosomes), which is the male in mammals, will be the more variable one:

      The Y‐chromosome in mammals is almost devoid of genes besides the male sex‐determining factors, whereas the X‐chromosome has about its fair share of genes. Binomial sampling of the large X chromosomes leads to the intuitive prediction that males should show larger variation. In females, the traits that are influenced by X‐chromosomal genes will be under the average influence of the two parental copies, whereas in males, the effect of the single X‐chromosome will not be averaged. As a result, male mammals are expected to show larger variability than females in all traits that are, at least to some extent, influenced by X‐chromosomal alleles.

  16. Polymath Says:

    You’re getting closer to seeing your error.

    Medians are assumed the same because it is much easier to change the variance than the mean, it’s easier in investment to go for beta rather than alpha. Evolutionarily, it can be quite expensive to increase general fitness-enhancing features (like intelligence, to take an example of a trait which is never bad in and of itself although achieving more of it may require tradeoffs). It can be much cheaper and easier to roll the dice, for example by an adaptation which made the organism’s fitness more sensitive to environmental factors (like abundance of a particular prey or absence of freezing temperatures).

  17. Matematica pretestuosa - Ocasapiens - Blog - Says:

    […] Gowers si rimbocca le maniche e fa la peer-review anche se "virtualmente non c'è nulla di matematico" (lo dice lui), elenca assunti rozzi e […]

  18. Скандал вокруг статьи Хиля – Alexa Project Says:

    […] математика Тимоти Гауэрса с продолжением. Замечу, что Гауэрс, пожалуй, наиболее авторитетный […]

  19. [Thing] Says:

    A minor complaint: Throughout this discussion I keep noticing people attributing the fact that a smaller percentage of males than females successfully reproduce to greater female selectivity in mate choice, and not mentioning any other factors. While sexual selection undoubtedly plays a role, we should acknowledge that individual males and females do not always enjoy full agency in choosing their mates. For example, is it more plausible that thegenetic bottleneck in male lineages following the invention of agriculture happened because women suddenly became much pickier, or because a highly successful minority of men suddenly became much more effective at imposing their preferences on women and less successful men?

    I don’t think this affects the analysis in the Hill paper; what matters is that there is greater variability in male reproductive success, not whether this is attributable to female sexual selection or direct competition between males. But sexual agency is a touchy subject politically, so let’s try not to misrepresent it.

  20. Mathematics matters | Bits of DNA Says:

    […] removed paper from the New York Journal of Mathematics, Has an uncomfortable truth been suppressed, Additional thoughts on the Ted Hill paper) because some mathematician wrote a paper titled “An Evolutionary Theory for the Variability […]

  21. Anima Says:

    I am hugely disappointed at the amount of time and energy that so-called brilliant minds are spending debating something that should be a non-starter. We look back at the racist and sexist theories of previous century measuring the brain size to intelligence and think we are more enlightened but the comments here and in your previous thread prove otherwise.
    Dr. Gowers, you are enabling all the sexist comments by providing a platform and not vehemently denying the hypothesis that under-representation of women in math is due to innate biological differences. By calling it a valid hypothesis you are lending credence to it. Why not debate another hypothesis: that men are brutes who have subjugated women for millennia and robbed humanity of brilliant discoveries that could have come about. Why is no one writing papers on this hypothesis? This is obviously a flawed hypothesis because not all men are brutes.
    I know the hell that women have to go through to be a researcher in STEM and it has nothing to do with biology or genetics. If you really want to be an ally for women in STEM, please read Lior Pachter’s blog on this

    • Darij Grinberg Says:

      > Dr. Gowers, you are enabling all the sexist comments by providing a platform and not vehemently denying the hypothesis that under-representation of women in math is due to innate biological differences.

      Sorry but this is exactly the kind of backwards reasoning that this controversy is highlighting (apart, of course, from the lots of dirty laundry and academic infighting that comes along for the right). Science is about truth, and truth is not downstream from ethics. (Nor the other way round.) “For, he reasons pointedly, that which must not, that cannot be”. (Morgenstern, in a surprisingly good translation: )

      > that men are brutes who have subjugated women for millennia and robbed humanity of brilliant discoveries that could have come about.

      That is a mainstream view you can find in most history books (which is likely where you got it from). And if you don’t interpret “men” to be “all men”, it has good credence too, but its descriptiveness has its limits, particularly as far as the 21st century is concerned.

      Greater male variability is actually *not* the alt-right’s theory of choice; they prefer to go with cruder stuff.

    • Anima Says:

      Darij: I am not going to respond anymore than this one. I agree science is about truth and not about bias. So that means the dominant group should not try to use pseudo-science to further bully the under-represented classes. I never said that male variability is alt-right. Sexism doesn’t just exist in alt-right but it is pervasive. I encourage you to talk to women about their terrible experiences in trying to make a career in STEM rather than lecturing me about “truthiness”. It is shameful that you try to defend sexism

    • Winston Says:

      Anima, things don’t become pseudo-science just because the conclusions offend you or oppose your deeply held values.

      Apart from obviously environmental factors (bias/discrimination, lack of role models, structural inequalities, etc) our current state of knowledge suggests that there are three key other reasons for the gender gap we see in mathematics.

      One is greater male variability. What I want to emphasize again here is how mainstream and uncontroversial this hypothesis is among people who actually study it. For example, this remarkable recent paper on sex-differences in brain structure: contains the sentence “the best-studied human phenotype in this context has been cognitive ability: almost universally, studies have found that males show greater variance in this trait”. Note the word “universally”. (By the way, I encourage everyone to read this paper, which focusses on actual difference in brain structure).

      The second is the “interest in persons vs interest in things” difference. This is a well known difference between the sexes, and if anything, has even more scientific backing than variability. It is worth noting that this difference occurs throughout the spectrum of cognitive abilities rather than onky at the ends.

      There is a tremendous amount of indirect evidence, and some direct evidence, that the factors mentioned above are of biological origin. However, the science is not settled.

      The third factor is less known, but possibly equally important. This is the notion of tilts, as explained in various papers of Wai. See for example the summary paper “Sex differences in ability tilt in the right tail of cognitive abilities: A 35-year examination Jonathan Wai, Jaret Hodges, Matthew C. Makel.” The rough idea is that females with high math ability also tend to have high verbal ability, which leads them often to choose non-math fields. The same is not true for men.

      I think (and so do a lot of people who study these) that the three factors mentioned above are biological, rooted in evolution. Direct and indirect evidence for this comes from a wide range of studies, which I am happy to discuss further. Based on this, it is a reasonable position to take that a large part (probably higher than 50%) of the maths gender gap is of biological origin. I subscribe to this proposition. However, I will be first to agree that the science is *not currently fully settled* in this regard, and I can respect those who look at the evidence and say “Hey, we simply do not have enough evidence for the biological hypothesis; furthermore for XYZ reasons, I think environment plays a primary role”. I can respect this, and this is how knowledge progresses. Most currently accepted scientific theories went through phases where it was somewhat reasonable to be on either side of them.

      What I *cannot* respect is someone who thinks the science is *settled* in favour of the environmental hypothesis (i.e., believes it is an established scientific fact that differences are overwhelmingly or totally attributable to environmental factors).

      Perhaps Gowers realizes that despite his own predisposition towards the environmental hypothesis, the science is most certainly not settled in that direction, and perhaps this is why he will not “vehemently deny” biological differences or declare them invalid?

    • Darij Grinberg Says:

      @Anima: Let me leave it at this: I suspect most women I know in academia think that denouncing their colleagues in front of funding bodies or employers is worse than a paper whose main result is considered harmful. That said, I am rather unpleasantly surprised that Ted Hill, too, has indulged in this behavior (and more directly than his adversaries).

    • Yemon Choi Says:

      Darij, without wishing to pile on: the fact that you say you are “unpleasantly surprised” may indicate that academia, submitting and refereeing maths articles, and “truth-telling” do not actually work in practice the way that you wish/hope they do. (I know very little about biology, mathematical or otherwise, but I have some familiarity with the shape and intent of things that Quillette encourage, and I increasingly set my priors not to assume good faith.)

      Outside of mathematics, most things are not “well-defined”, and hence syllogistic reasoning only gets us so far. I don’t in any way class myself as pomo, but the topics that Hill’s article is clearly motivated by are extremely loaded ones, and I think that it is actually appropriate in principle to interrogate the context and assumptions. (Note that once again I am not defending the actions at the Intelligencer, and I think I’ve already made known my position on the NYJM events in earlier comments.)

      FWIW, I hope that you and Anima will in due course find that you actually have more common ground than the medium of comments-in-blogs tends to display.

    • Anonymous Says:

      You are doing politics and you don’t even know it. Here is a question: can a conservative learn mathematics, talk about mathematics, and publish papers in mathematics without being sexist? The answer would be something like “conservative=sexist” therefore NO.

      T. Hill paper shows what happens when you take left arguments at face value. It is time we realise that radical leftist have taken control of academic institutions, they don’t know they are doing politics. T. Hill was addressing a theoretical issue in a theoretical manner, he was attacked politically for political reasons.

      By the way, one of the activist censoring the paper was the Chair of Diversity and Climate. This suggest “Diversity and Climate” have ideological components…or maybe it’s just T. Hill being a mean old white man

  22. Darij Grinberg Says:

    Some subjective notes from skimming :

    – The controversy started out as a (superficially, at least) friendly discussion. Some colleague (probably from WIM) invited Sergei Tabachnikov to a WIM lunch to discuss it, but warning him that there will be “strong disagreements” (pp. 7–8). However, the conversation turned sour; this colleague later mentioned “a group of outraged faculty”, whom he/she agreed with; Tabachnikov responded that the lunch was looking more and more like a tribunal and apparently declined the invitation (pp. 8–10). There seem to be emails missing from between these stages.

    – Tabachnikov reported significant backlash at PSU (pp. 10). At a faculty meeting, he was accused of bias by colleagues and the article was claimed to be harmful.

    – Diane Henderson and Nate Brown asked the NSF about “a paper that concerns [them] greatly and acknowledges NSF support”, which they claim “appears to promote pseudoscientific ideas that are detrimental to the advancement of women in science, and are at odds with the values of the NSF”. They ask whether the NSF actually “supports the work in this paper”. They stop short of making any specific requests or concrete accusations. (This is on p. 13; the email was FOIAed, which seems to be a really good approach to peeking behind the NSF curtain.)

    – Marjorie Senechal rescinded the acceptance of the paper at the Intelligencer because she “received concerned messages from several colleagues, warning of extremely strong reactions to
    the accepted version of your paper” (p. 14). This is a fairly textbook instance of the heckler’s veto. She proposed a debate, and publishing the results of the debate “as a special issue of The Mathematical Intelligencer”.

    – Leland Wilkinson’s criticism of the paper (p. 17) is apolitical and looks constructive.

    – Cornered on the exact motives for the decision to rescind the acceptance, Marjorie Senechal claimed that it were private messages that “concerned not the substance of your article but the very real possibility of extremely damaging fallout” (p. 19). It is never really clear who these messages come from.

    – Igor Rivin heard about the controversy from Tabachnikov, and contacted Ted Hill about the possibility of publishing the paper in the NYJM instead (pp. 23–24).

    – Only one referee report (pp. 24) was obtained at NYJM. The report is probably best summarized by the sentence “the math is straightforward and correct; the issue is whether or not the model proposed is a good model for the real world”; the reviewer does recuse him/herself from the latter issue and suggests contacting a mathematical biologist for their opinion. He says that the paper “should be published if correct”. I’m afraid my brief experience with publishing doesn’t give me a real idea of whether the referee is trying to say “accept” but hedging against embarrassment, or conversely, trying to say “decline” but offering cop-outs if the editor thinks otherwise. It is not exactly a typical report, but then again this is not a typical paper.

    – Benson Farb wrote a caustic mail (p. 26) to the NYJM editor-in-chief, in which he claimed to be “sickened”, accused Rivin of being “well-known as a person with extremist views who likes to pick fights with people via inflammatory statements”, and the paper of being “politically charged”, “filled with pseudo-science” and a “non-math piece of crap”, and clearly demands: “Whatever your policy at NYJM, this is a case where you need to pull this paper ASAP”. This does not technically contradict Farb’s statement, but utterly belies its tone and implicature.

    – Asked about why the article was removed from NYJM, Mark Steinberger wrote to Ted Hill (p. 27): “Ted, my board made it clear that half of them would walk if it stayed up and that, moreover they would harass the journal until it died.”. The “half of them” here sounds like a typical exaggeration, but the statement is fairly clear.

    – Amie Wilkinson indeed started attacking the Intelligencer on Facebook (p. 29) for accepting the paper, even after the acceptance was pulled. She furthermore accused Senechal of revealing her name to the authors, from which she claimed to have received “an obnoxious email”. I have not found any obnoxiousness in my cursory read of the emails so far, but of course it might have been a different email. Further Facebook and email messages from third parties (p. 30) suggest that Amie Wilkinson implied that “she was going to unfriend anyone who didn’t unfriend” Igor(?), but then had taken down the post where she said that. Once again, this does not formally contradict Amie Wilkinson’s public statement, but suggest that at least for the timespan of a day or more, she exhibited rather allergic and panic behavior that was nowhere near warranted by the situation. Nevertheless, there are no signs of the behavior persisting beyond that short timeframe.

    – Ted Hill mailed a formal complaint to President Zimmer of the University of Chicago (p. 32) about Amie Wilkinson and Benson Farb for “unprofessional, uncollegial, and unethical conduct damaging to my professional reputation and to the reputation of the University of Chicago”. Laying out more or less the same claims as in the Quillette article, Ted Hill requested “that you reprimand both Dr. Amie Wilkinson and Dr. Benson Farb, and have them make public apologies for conduct unbecoming University of Chicago professors”. This is probably the highest level of escalation that this conflict has reached, at least to the extent I can see it: It is not exactly a symmetric response to the Henderson-and-Brown letter to the NSF (as the latter made no specific requests); also, unlike Amie Wilkinson’s “unfriending campaign”, this was done through official channels. Furthermore, apparently due to lack of a response, Hill followed up with another mail to Zimmer, where he cited the University of Chicago’s official Statement on Principles of Free Expression. This technically doesn’t contradict Hill’s Quillette article, but heavily damages his implied self-presentation as a radically honest scientist uninterested in political games; at the very least, he could have mentioned that he had stooped down to the level of his accusers if not further below!

    – The communication between Hill and Jason Merchant (vice provost at UChicago) goes on for several pages; Merchant considered Farb’s and Wilkinson’s actions to be legitimate advocacy and laid the blame on the “unpublishing” of the Tabachnikov-Hill paper at the feet of the editors (p. 39). The conversation led nowhere.

    • longtail Says:

      I think you’re misreading the Facebook post. The post appeared on the same day Hill sent his email to her. It seems likely that the fact that her identity was exposed to Hill by the editor-in-chief was what got her upset. The correspondence with the editor about the paper had occurred more than two weeks before that.

      The unfriending incident took place almost two months after the final disposition of Hill’s paper by NYJM in February. Hill’s documents include this: “Amie sent me (and a number of others) a message that she and Benson felt deeply hurt by something you [i.e. Igor Rivin] did.” You’re assuming you know what the situation was when you say that “she exhibited rather allergic and panic behavior that was nowhere near warranted by the situation.” It’s clear there’s a lot of history between the players here, and it’s not all about the paper.

    • longtail Says:

      It’s also worth noting that Hill describes Farb’s email to Steinberger as “forwarded to IR.” The email itself says “Please note that Igor is not on the list of people to whom I sent the email. It was only a subset of the board.” There were at least two breaches of confidence required for this email to reach Hill, one of which was presumably due to IR.

    • Pyrrho Says:

      Farb says the board was eventually given two reports. Is it possible the second report was obtained after the paper was rescinded, and that explains the three month delay before reports were given to members of the board who had asked for them?

    • Darij Grinberg Says:

      @longtail: I’m not assuming I know the full history (as I said, I am not sure whether the PDF contains all the relevant emails, even those authored by Ted Hill), but I’ve seen enough of what Ted Hill has written. Occam’s razor suggests to me that Amie Wilkinson’s lashing out at Hill and Senechal was unwarranted. Whatever legitimate reasons she had to be afraid of or just disgusted by Hill should have appeared in Wilkinson’s or Farb’s statements; after all, neither Farb nor Wilkinson have shown much restraint in lobbing accusations at third parties.

      @Pyrrho: Oh, that would explain something that seemed off to me when I read the PDF.

    • M Says:

      Leland Wilkinson’s criticism of the paper (p. 17) is apolitical and looks constructive

      I don’t think it’s either apolitical or constructive. From the get-go, he claims that the topic is disreputable and that Hill’s article “oversimplifies the issues to the point of embarrassment.”

      Then he goes on to dismissively discuss Galton’s and Pearson’s early work on the topic, nonsequitously referring to them as eugenicists. He suggests that non-experimental research is unscientific and should be rejected, demonstrating an alarming level of ignorance about not only evolutionary biology but of science in general. And so on.

    • Anonymous Says:

      Embarrassment it is for statistician. Here is another take, from an actual geneticist “It is a pile of hot garbage. Creationist trade magazines publish articles with more evolutionary content”

    • M Says:

      I don’t think Hill’s paper is good, but Leland Wilkinson’s comments are silly.

    • Igor Rivin (@igriv) Says:

      A slight correction: two referees reports were obtained, one a little (a couple of weeks) after the other, since the Editor in Chief wanted to cover all bases. The paper was published only after the receipt of the second report. I do not know (and don’t especially care, to be honest) why Professor Hill did not include the second report in his document dump.

      As for the “hot garbage” remark, please find my last post with a link to a blog post that sets things straight.

    • David Roberts Says:

      @Professor Rivin

      So why not release that second referee report yourself? People are laying their cards on the table after all. I think everyone would agree the usual rules are out the window at this point.

    • Igor Rivin (@igriv) Says:

      Dear Professor Roberts,

      The only person who can ethically release the report is the author (I am not 100% sure than even the author can, but I am about 90% sure). I am also not entirely sure that I understand your ‘people are laying their cards on the table” remark. Which people? And are we playing Texas Hold-em? What is your interest in this, other than idle curiosity? (this is not a rhetorical question, nor sarcasm, but an expression of genuine interest).

      Professor Hill’s actions are understandable: he alleged gross malfeasance by certain parties, and then he backed up his allegations with documentation.

      As a side effect, Professor Hill’s documentation indicates that A referee’s report was obtained, and that the handling editor (me) and the editor in chief (the late Mark Steinberger) were satisfied, so the journal’s policies were followed. The second report was obtained, but this exceeded the journal’s requirements, so in any formal sense, the existence of the report is superfluous to demonstrate that the procedures were followed.

      I look forward to your explanation.

    • David Roberts Says:

      Thank you for your reply and questions.

      a) I’m not a Professor, but thank you for the promotion 🙂

      b) “The only person who can ethically release the report is the author” well, I’ve read that at least in some cases the journal holds copyright to referee’s reports, but this is an underexplored area of law. As the handling editor, you presumably have some authority in this case. I don’t imagine the other editors would object, but of course they could be consulted if you were worried, likewise so could Hill, in case you were worried he might not want you to release it (but, I don’t see why, since it apparently supported the publication of the paper).

      c) “people” = Hill, Wilkinson, Farb, yourself

      d) my interest in this is that releasing the second referee report would at least vindicate you as having done your duty as an editor (and, as you’ve said elsewhere, gone above the minimum requirements by asking for *two* reports) by getting solid support from experts for the paper. If I were in a similar position I would like to put these concerns to rest.

      e) “backed up his allegations with documentation” – but, curiously, with one omission: the second report. I don’t presume to guess how or why this happened. I agree that to document his interactions with the other people in this case he has indeed presented pieces of relevance, but the first referee report is not evidence of his interactions with Senechal, Wilkinson, Wilkinson or Farb. So why include it and not the other one?

      Perhaps the second report was by a mathematical biologist with experience in this area. Perhaps it was by an actual biologist. Perhaps it was by a mathematician that was damning the paper with faint praise. Perhaps this person inadvertently identified themselves in the report and they need their identity kept secret. Who knows? My assumption is that you have nothing to hide, since that is what you have been saying. I’m also forming an opinion here based on what I see as expert opinion where I am not capable of judging, and otherwise the evidence of the case, and I think it would help others do the same if we had more expert opinion/documentation, of which the second referee report is a key piece.

      Sunlight is the best disinfectant, no?

      Thanks again for being willing to discuss this.

  23. james c (@james1071) Says:

    Tim, two points. Again, I make the disclosure that I am very much the general reader and could be wrong.

    Point 1: you asked why there was an equilibrium between the two sub populations. That doesn’t seem, in principle to be a stumbling block – biology is all about different equilibria. We of course, only notice the sub-populations that exist side by side and ignore those that have been wiped out.

    Point 2:you asked about the link between genes, their selection (in prehistoric times) and mathematical ability. Yes, that is a tricky one!
    We might eventually be able to untangle genetic aptitude for maths (say) from various environmental factors, but that must be years away.

    Finally, what do actual biologists (such as Richard Dawkins) have to say on the matter?

    Thanks for an excellent discussion.

    James C

  24. Anonymous Says:

    Pure mathematics consists entirely of our hallucinations, and is “precise” in their own terms. In science and applied math, we try to adapt our hallucinations to what we observe, and any “imprecision” is the discrepancy between what really happens, and what we are capable of hallucinating about it. Given this, I am surprised to see people criticize an attempt to match the hallucinations with reality in the context of “precision” of pure hallucinations. It merely indicates to me that they are unable to comprehend what the work is about.

  25. Anonymous Says:

    The real issue is that if you disagree with SJWs your career is in jeopardy and appearing conservative is unofficially illegal. The second author was told to remove the reference to the grant that funds his usual research! This is a canary in the cold mine. And is not the first one. Stop ignoring the real issue!

    Also, If we are gonna judge papers of applied mathematics by the same standards we do in abstract mathematics very little would get publish. I’d like to reference S. Lovejoy’s paper Scaling fluctuation analysis and statistical hypothesis testing of anthropogenic warming This paper has the same level of mathematics behind it. It could also be described as “assuming what you want to prove” in the sense that the model is picked to lead the conclusion.

    • GM Says:

      if you disagree with SJWs your career is in jeopardy and appearing conservative is unofficially illegal

      It should perhaps be pointed out that those are also not necessarily the same thing — one needs not be a conservative in order to disagree with SJWs.

      What few people seem to realize is that there is really nothing “left” about the SJWs, the fact that they have been labeled so is merely a testament to how dysfunctional the US political system is, where all you have is a right wing and an ultra right wing of the same corporatist party, and no actual left whatsoever.

      The “left” is supposed to be about defending the interests of the poor and working class people. Judith Butler is neither poor nor working class, and the verbal garbage that has come out from under her fingers not only has very little relevance to their concerns and issues, but is totally repulsive and disgusting to many of them (which is one reason why they voted for Trump in the absence of a better alternative).

  26. PC Says:

    It seems to me that there is a simpler possible explanation for greater male variability than the one discussed in the blogpost. One can think that, historically, dominant males had more access to women than non dominant ones. This implies that their genes had a greater chance to be transmitted which induced a bias towards aggressivity in male genes. The effect of this is that males spend all their time competing (mainly against each other) and this, alone, would suffice to explain a greater variability: the winners get a boost and the loosers can end up giving up the competition altogether and fall very low.

    A rather recent phenomena is that men also have to compete with women in many aspects of modern life. If the above is correct it could imply that the distribution of men will become more and more skewed towards the bottom, the top being relatively unchanged. I don’t know if there is data supporting this conclusion (else than recent political events…).

    • GM Says:

      There is an even simpler one — males are the heterogametic sex in mammals and many other animals, which means that there is no buffering of the effects on alleles on the sex chromosomes by the presence of another allele.

    • Shoudda been referee #3 Says:

      Yup. Ginger tabbies.

  27. namae nanka Says:

    “I would guess that one of the most important is that the message is sent in various ways to young girls that mathematics is more of a boy’s pursuit: there seems to be plenty of evidence that from a certain age (about 6 if I remember correctly)”

    The data shows that girls get better grades than boys even in maths, do more homework and have better language skils which no doubt also plays into their favor.

    Despite these shortcomings boys do better on maths tests.

    Girls might come to think that way when they see boys doing better in maths despite their working harder at it.

    “It seems, from the link Krzysztof provides, that in order to get to the point where women outnumbered men in law schools, there first had to be a campaign of positive discrimination”

    The amusing thing about all this is that boys might be interested in other fields if they had better language skills. The college going population in developed countries seem to be easily majority female and yet, it’s not enough if they aren’t at least equal in every field.

    • Ian Walker Says:

      I had a majority of female teachers until university, never saw any form of discrimination against females, at all, ever. In fact, it was the other way! Females were never discouraged from doing anything! They always did what they wanted to do! So talk of positive discrimination must be designed as an overt subversion of ability in western society… I repeat, I have never seen females discriminated against, overtly or covertly, in any educational institution or place of employment that I have experience of. Talk of deliberately crippling anyones chances in the name of ‘positive discrimination’ is IMHO gaming the system to gain advantage for one’s self. For purposes yet to be revealed. But easy to understand. I have seen what happens when an incompetent person is put into a position they are not suited for by forces of ‘positive discrimination’ It is ugly and damaging, and that is the desired effect. And it is not possible to protest without being branded a bigot…

  28. Igor Rivin (@igriv) Says:

    To the merit of the paper, this blog post seems to put it all in place:

    • gowers Says:

      It’s not clear to me that that blog post is to the merit of the paper. It seems to suggest that Hill’s idea was already present in the biology literature. And the “model” isn’t adding anything nontrivial to the basic idea — it’s just giving some sufficient conditions for it to work that any competent mathematician could work out in five minutes.

    • Igor Rivin (@igriv) Says:

      Gowers: I am a bit mystified by where your thinking is on this, perhaps you can clarify: you spend two blog posts accusing the paper of fostering pseudo-science, making no sense, no, wait, making some sense, etc. Then, when the evidence is presented that the model actually makes a lot of sense, you dismiss it as trivial.

      Notice that the paper is clearly independent, so Hill (who is not a biologist but a mathematician) discovered the idea independently. What is more, the result is not (or was not) known to even most evolutionary biologists (certainly not those who have been accusing the paper of being “hot garbage”). So, at worst, Hill rediscovered something. Are you suggesting unpublishing every mathematics paper which rediscovered something? That will reduce mathematics literature by a factor of around 3 (conservatively).

      So, I repeat, where are you on this?

    • gowers Says:

      First of all, I didn’t use phrases like “fostering pseudo-science” — you’re probably confusing me with someone else. I found the model highly implausible in various ways, and later realized that some, but not all, of those ways were more plausible than I had realized, so I wrote a second post to make that clear, and also to make clear what my remaining doubts were, which still persist. In my recent comment, I was arguing simply that that blog post surely doesn’t strengthen the case for the paper as you claimed, since the main contribution of the paper — suspending judgment for the moment about its plausibility — is the idea of what might cause greater male variability and not in the mathematical model that supposedly backs it up but in fact is just a different language for saying the same thing. And when the main contribution of a paper is an idea that is present in the literature already, that is not usually considered a plus point — that’s all I’m saying above. I’ve always thought that the mathematical part of Hill’s paper was trivial, and have made that clear in my posts, so that’s not some new criticism I’m suddenly bringing up. I have also carefully not said that I thought the paper should have been unpublished. But I did say that I thought it shouldn’t have been published in NYJM in the first place, and I stand by that.

    • Igor Rivin (@igriv) Says:

      OK, thanks for the elucidation!

    • gowers Says:

      Just to add one further point of clarification, my carefully not saying that the paper should have been unpublished by NYJM doesn’t mean that I’m arguing that it shouldn’t have been unpublished. It just means that I don’t have enough information to feel comfortable making a public judgment on that issue.

    • Igor Rivin (@igriv) Says:

      I think that given your prominence in both mathematics and mathematics publishing, it would be of great interest to all of us to know what would render a paper a good candidate for unpublishing.

  29. Hot garbage – posttenuretourettes Says:

    […] be worthless from the point of view of biology. Brilliant or not, the paper has prompted a vigorous discussion among leading mathematicians — which, ipso facto, would seem to indicate its mathematical […]

  30. xerxes Says:

    (This is intended as a reply to gowers September 26, 2018 at 3:41 am but the wordpress threading doesn’t permit that.)

    Let’s be clear about the meaning of “unpublish”: it is to delete a paper and replace it with another of the same length, without any editorial statement. To kill and bury in an unmarked grave, in other words. We only know about this case because the author noticed it.
    If it had happened to a paper from the past then maybe nobody would ever have noticed, despite the problems caused for subsequent papers that cite it and thereby for science in general.

    Gowers, can you foresee circumstances in which you would tolerate (that is, not argue against) unpublication in your new journal of combinatorics (sorry, I don’t recall the exact title)?

    • gowers Says:

      I agree that the manner of “unpublication” in this case was bizarre. (One thing I don’t know is who decided to do things in that way — it may not have been quite what those on the board who voted for unpublication were advocating.) If I were unlucky enough to find myself in a situation where Advances in Combinatorics published a paper and then came to think that the decision was a disaster (it’s hard to imagine such circumstances — but perhaps if the paper was plagiarized and nobody had noticed, or something ilke that) then I think what we would probably do is a bit different. Instead of taking the paper down, we would replace the editorial introduction (see the Discrete Analysis to find out what those are) by a statement giving our reasons for not wanting to give our stamp of approval to the paper. Or if the paper had been there for only a very short time, then perhaps we would remove it, but I agree that in such a case it would still be appropriate to make a statement giving the reasons. A situation like the one at NYJM is very unlikely to occur, because it is the policy at Discrete Analysis and Advances in Combinatorics for decisions to be justified to the entire board by the handling editor and made only once everyone has had a chance to comment.

  31. David Roberts Says:

    People may enjoy reading this study

    just published In Nature Comms. The abstract:

    “Fewer women than men pursue careers in science, technology, engineering and mathematics (STEM), despite girls outperforming boys at school in the relevant subjects. According to the ‘variability hypothesis’, this over-representation of males is driven by gender differences in variance; greater male variability leads to greater numbers of men who exceed the performance threshold. Here, we use recent meta-analytic advances to compare gender differences in academic grades from over 1.6 million students. In line with previous studies we find strong evidence for lower variation among girls than boys, and of higher average grades for girls. However, the gender differences in both mean and variance of grades are smaller in STEM than non-STEM subjects, suggesting that greater variability is insufficient to explain male over-representation in STEM. Simulations of these differences suggest the top 10% of a class contains equal numbers of girls and boys in STEM, but more girls in non-STEM subjects.”

    Note the acknowledgement of the GMVH, but the data, according to this study, doesn’t support it explaining the distributions of girls vs boys at the top end. If Hill wishes to update his paper, this is worth a read…

    • Anonymous Says:

      This is a vivid example of an SJW propaganda. Fig. 1 unambiguously shows far greater boys variability in STEM, with the right tail much thicker than that of the girls. It is sad that Nature has fallen to this level.

    • Igor Rivin (@igriv) Says:

      A couple of comments: First, Anonymous is not correct – the first graph is a hypothetical “this is what we might expect”. The rest of the paper sets out to show that this is not the reality.

      On the other hand, the propaganda aspect of the paper is more subtle. They speak in the abstract of the top 10%, but this is obviously irrelevant for representation in academia – the cutoff is too low. In figure 3, they give the “true” distribution, and you see that for the top 1%, males outnumber females 11 to 9. Even that, however, corresponds to IQ level (I know the paper is not about IQ, but this makes it a little simpler to talk about) of 127, which is not very high. As an example, Hunter High School claims to take the top 0.1% of applicants. At that level we should (given the results given in the paper) expect overwhelming male dominance in STEM fields.

    • Anonymous Says:

      Could it be that the standardized tests do not accurately measure the actual brain functions needed for STEM activities, and are instead devised specifically to equalize girls with boys? For example, modern IQ tests are specifically formulated so that males and females score similarly on average. Given the current PC climate, I would not be surprised if it was also the case with various school tests.

    • Igor Rivin (@igriv) Says:

      @Anonymous: The tests are so designed, and this is not a secret, neither is the male dominance in some cognitive areas and female dominance in others:
      Any attempt to claim that men and women (as groups) are the same is absurd.

      This, of course, has nothing to do with the abilities (or proclivities) of any specific person.

    • Anonymous Says:

      Then this work goes roughly as follows. “The Gravity Hypothesis predicts that dense celestial bodies tend to have round shape. We studied the data from some measurements, designed to ignore the curvature as much as possible, and what we found suggests that the Earth is flat.” I do not demand that this paper be silently deleted, but wouldn’t it be prudent of the editors to include a note with apologies at least?

    • Igor Rivin (@igriv) Says:

      The funny thing about the data (and I suppose a hard look at the raw data is needed) is this: hey claim, in STEM fields, that women’s mean is 0.25 standard deviation (or thereabouts) higher than men, the women’s standard deviation is 3% lower than men, so that means that women should dominate up to some eight standard deviations out (corresponding to the unusual IQ of 220). But in fact, men reverse the numbers at about 1.8 SD (if I am reading correctly), which would indicate that the distribution of the scores has very subgaussian tails, which I find very hard to swallow. This may suggest that, as @Anonymous suggests, the fudging of the mean was not done very adroitly, or maybe something more profound.

  32. xerxes Says:

    @gowers September 26, 2018 at 9:47 am:
    So you tolerate unpublication in NYJM but not in Advances in Combinatorics. Why the difference?

    • gowers Says:

      That is a misreading of what I wrote. As I have said, first, I do not want to judge whether withdrawal of the paper was appropriate, since that depends on information I do not have, and secondly, I think the manner in which the withdrawal took place — the strange “disappearance” and replacement by another paper — was unsatisfactory. And finally, there are two important differences between NYJM and Advances in Combinatorics, as I also pointed out. First, the editorial procedures are different, so any accepted paper in AiC will have been approved by everyone on the editorial board, and secondly the Advances in Combinatorics website is (or rather will be when papers start to appear) designed in a way that would make it easy to make an editorial statement about a paper that turned out after publication to be highly problematic.

  33. Igor Rivin (@igriv) Says:

  34. True Colors Says:

    Prof. Gowers, Could you please bring this level of critical thinking to gender studies or to climate research?

    It will be very interesting to hear your opinion on circular arguments such as the work of Prof. Lovejoy at McGill

  35. erasmuse Says:

    I’ve written a long essay on the Hill affair, with lots of links and footnotes. Corrections welcomed. See

    –Eric Rasmusen, economist, Indiana University

  36. A follow-up to Hill’s paper or – is free speech dead in the West? – Deep thoughts on all subjects Says:

    […] Ted Hill’s article, and much discussion in the various blogs (Tao’s, Gowers’ and others) an even bigger fight has broken out.  Alessandro Strumia, a theoretical physicist at […]

  37. George Says:

    I think a lot of the arguments trying to explain the sex ratios in STEM fields focus too much on ability. That is, whether there is more variation in males vs females in math ability for example.

    Seems to me that the answer is a little more simple. Academia is rough, and not in a nice way. Before a tenure-track position, the pay is low and there is great job uncertainty. It doesn’t seem very welcoming to either males or females. For many the stress is so great that it’s actually physically strenuous. I think women are just less likely to become profs simply because they are unwilling to go through the torturous path to becoming one.

  38. Anonymous Says:

    Although this is at a slight tangent to the main discussion, I found it interesting that that you mentioned spatial reasoning in your original post. The apparent gender disparity in this ability is often linked to mathematical ability. However, there is some evidence that this ability is strongly linked to self perception, to the point that in one particular study a simple visualisation exercise (imagine yourself as a stereotypical man) removed the disparity all together.

  39. gowers Says:

    Sabine Hossenfelder has written an interesting post about the Strumia affair, and about how statistics that seem to indicate that men outperform women in high-energy physics are in fact extremely misleading. I won’t summarize her arguments here, because that won’t be an adequate substitute for reading her post.

    • Igor Rivin (@igriv) Says:

      Hers is an interesting post, but your characterization of it seems, in turn, rather misleading. She does not refute Strumia’s analysis, but rather suggests a mechanism for why it might be true (that being, that women are more likely to leave the academia). I can think of many mechanisms, in turn, for why that might be true, and it all comes down to the banal observation that men are different from women, and it is counterproductive to force women to be men (and vice versa).

  40. Ian Walker Says:

    Mathematicians:What is ‘true’,
    Physicists:What is ‘real’,
    Engineers:What can be ‘done’…

    Mathematics – is not a monolith, there are different elements within, and corresponding sex biases, ditto the sciences; science too has different elements – all? profess to cleave to the realm of the ‘testable by experiment’. Engineering *used* to mean using mathematics and physics to describe and realise means of ameliorating the human condition.

    The human condition: the detritus of eons of hard struggle. In stark relief, you live and you reproduce – and this was the test for any female who birthed. A very stringent test indeed. Not really matched for males until the era of universal industrialised warfare.

    Hunter gatherers: males hunt in silence, mostly, better in a small group, strength, courage, agility, endurance, knowledge of terrain for both tactics and strategy, knowledge of prey/opponent/self, imagination used to develop all as needed for unforeseen/insurmountable foes (David v Goliath). Use and development of tools and weapons. Systematisation of such as a better means of countering same in opponents.

    Hunter gatherers: females gather in groups, chattering (in the main, communicating nuance of interaction), Use baskets/bags for carrying, use senses differently to males – sight (tetrachromaticity), taste (is this berry/[meat/fish] good), smell (is fruit here, is it good, does he/she smell sick/like me), touch (ripe/not ripe…). Need some way of being productive in teeth of reproductive biology. Have huge selection pressures on ability to attract mate, birth child. Why do female mammals want to reproduce in the teeth of dangerous viviparous reproductive biology? Technology has mostly ameliorated the danger of viviparous birth. But through the course of human evolution this was not the case. A very stringent test indeed.

    These are generalisations, of course. Easy to find exceptions to the norm, especially after agriculture/rise of cities/politics and rise of ‘elites’. Why was Archimedes subordinate to Heron? It is these exceptions that point to variability, circumstances that rub out the norm will amplify the variant… Get stuck in a rut (pun intended) and others will rise to out-compete due to changes in circumstances.

    Out compete means live to reproduce, btw. And the big mistake is to think that evolution is some kind of global optimisation problem… Advantage in one scenario can be a disadvantage in another. Unless one has the ‘gift’ of prescience, and the agency to act, any individual can only ever be locally optimised. And how often do we see that a disadvantaged individual has just the right secret sauce to overcome a vastly (on the face of it) superior opponent?

    And here lies the rub, how to quantify ability? Is it the number of papers produced, the number of offspring sprang, or the ‘quality’ of the product? Defined as the simple ability of that offspring to survive and reproduce, doesn’t really seem to cut it as a measure. Nobel prize winners are typically heretical (adherence to canon does not begat new knowledge) and typically perform at the credit level when tested, which can be attributed to “WTF, I don’t care who you are and what you say, this just does not seem right” and it shows when quantified by the elemental canonical swots that run the tribe… And in the past the elucidation of new truths generally was met by tying to a stick and roasting by the normative types.

    So what if a paper is really terrible on a number of counts (spelling, grammar, scholarship, reasoning, adherence to canon) that are easy to quantify because the metrics are extant, yet exposes an unknown aspect of the natural world, or realm of mathematics, what then? What if the referee does not have the intellectual capacity to see through the ineptidude and thus rejects a paper that contains discoveries of such import that a new era of enlightenment can be ushered in. What then? Tangentially, why didn’t the ancient Egyptians have an industrial revolution? The Chinese, the Romans, the Greeks, the Mayans…

    IMHO: quantifying worth is a dangerous game in the face of a capricious and dangerous universe. Often done for political reasons, to garner ‘unfair’ advantage, it ultimately serves no-one very well. Because the only proof of that pudding was not being eaten! The arguments about who deserves more of less based on sex or social status or some other metric miss the point completely on how progress is made, and often the people actually making the discoveries are cheated of recognition by the more fancied players. So IMHO, if the ‘mediocre’ (as gauged by some ultimately wrong FOM/metric) want into the game, then find a space for them… You just do not know what will turn up. This applies to any gender, race, caste, age… If you normalise based on ‘positive discrimination’ then you lose the variability that you profess to seek.

    Perhaps instead of denying opportunity to some now vilified section of the community in the name of ‘positive discrimination’, we afforded better access to all who want to play… And let the real measure of worth be in real terms of better outcomes for all… As in better quality of life.

    I think Einstein would agree.

    As it is, all the current foment seems shaped at getting more chomps into the funding pie, with a covert agenda for divide and conquer. And all shaped at creating more wealth for a very successful few. Why wasn’t Archimedes king? How did Boudicca fail? Did Hypatia overzealously fine the late returns, or mis-manage her sources of funding?

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