]]>

According to a new psychology paper, our political passions can even undermine our very basic reasoning skills. More specifically, the study finds that people who are otherwise very good at math may totally flunk a problem that they would otherwise probably be able to solve, simply because giving the right answer goes against their political beliefs.

I was reminded of it while reading the article “Does Diversity Trump Ability? An Example of the Misuse of Mathematics in the Social Sciences” in the Notices of the AMS. The author, Abigail Thompson, takes on a well known and widely cited paper:

“Diversity” has become an important concept in the modern university, affecting admissions, faculty hiring, and administrative appointments. In the paper “Groups of diverse problem solvers can outperform groups of high-ability problem solvers” [1], L. Hong and S. Page claim to prove that “To put it succinctly, diversity trumps ability.” We show that their arguments are fundamentally flawed.

Why should mathematicians care? Mathematicians have a responsibility to ensure that mathematics is not misused. The highly specialized language of mathematics can be used to obscure rather than reveal truth. It is easy to cross the line between analysis and advocacy when strongly held beliefs are in play. Attempts to find a mathematical justification for “diversity” as practiced in universities provide an instructive example of where that line has been crossed.

Thompson proceeds to shred both the “mathematical theorem” and the numerical examples from the Hong-Page paper. The actual paper is available here, and I have satisfied myself that Thompson is not unfair in her mathematical analysis. Her article, however, does not exist in a vacuum. It will be read in mathematics departments, organizations and committees where “diversity” is viewed as a bureaucratic imposition made on them by distant administrators who don’t understand research, even as their few women faculty often find themselves alienated and sidelined. That’s why I would like to add a few things.

First, there are many sound reasons for diversity that have nothing to do the article in question. (I will restrict this post to the benefits of diversity *per se*, independently of how that diversity was achieved. Affirmative action has its own reasons and will get its own post soon.) It should be common sense, not a mathematical theorem, that there are advantages in having a wider perspective and more than one problem-solving approach. In business, the lack of diversity among designers and decision-makers courts a mismatch between the product and its clientele, from the non-functionality of women’s clothing to drugs tested primarily on white men and “health apps” oblivious to the basics of women’s health. Women, too, have been guilty of not looking past their own noses. The cosmetics industry has long had women leaders, going back to Elizabeth Arden who died in 1966, yet it had little or nothing to offer to women of colour before Iman created her own product line in 1989. It should not be rocket science that catering to the preferences of your actual clientele, and not just your own, makes good business sense.

You’ll say that diversity does not matter in mathematics because arithmetics and algebra do not depend on gender or skin colour. I will reply that solving mathematical problems is only one part of what we do. We also teach undergraduates, supervise graduate students and postdocs, perform administrative functions and participate in committees. In all of these, diversity does matter. Women and minority students benefit from having women and minority professors. An administrator who is a woman or a person of colour is more likely to be aware of the sexism and racism in the faculty ranks. But even if you don’t believe that sexism and racism exist, and if solving mathematical problems is all you care about, you might still want to read this:

Katherine W. Phillips, senior vice dean at Columbia Business School, and her colleagues gave three-person groups complex murder mysteries, and then asked them to work together on solutions. Each member received clues that her or his companions didn’t possess, giving an edge to groups with the ability to share information. Racially diverse teams significantly outperformed those with similar members. Other studies confirm this result: A diverse team is more innovative on average.

What is going on, Phillips thinks, is that diversity changes the dynamics of a group in a way that makes it more innovative. When we work with others who are like us, we tend to assume they hold similar points of view and share similar information. That makes for easy and comfortable interactions, and it works well when the task at hand is routine. But when a team is trying to do something new that requires knowledge and experience surpassing what any one member can supply, a more challenging social situation leads to better outcomes. When we have to try harder to communicate with collaborators who are different from us, we better articulate our ideas. “Diversity jolts us into cognitive action in ways that homogeneity simply does not,” Phillips writes.

My own problem-solving processes have very little in common with the naive algorithms from the Hong-Page paper. It’s not even close. Phillips, however, strikes a chord. I have noticed, and have mentioned it here, that I often work better when I collaborate: aside from any ideas and expertise that the collaborators might or might not contribute, engagement in collaboration enhances my own mental processes. This is especially pronounced when the collaborators do not share my thinking patterns and force me to articulate them better, in accordance with Phillips’s argument. If you think that it’s only the mathematical and not personal backgrounds that should matter, I will say that the two are not always easily separated, at least not in my experience. I’ll also refer you to the work of James H. Austin on creativity, especially his notion of “Chance IV” (hat tip to Marc Andreessen and his now-deleted blog post “Luck and the entrepreneur”):

[You] have to look carefully to find Chance IV for three reasons.

The first is that when it operates directly, it unfolds in an elliptical, unorthodox manner.

The second is that it often works indirectly.

The third is that some problems it may help solve are uncommonly difficult to understand because they have gone through a process of selection.

We must bear in mind that, by the time Chance IV finally occurs, the easy, more accessible problems will already have been solved earlier by conventional actions, conventional logic, or by the operations of the other forms of chance. What remains late in the game, then, is a tough core of complex, resistant problems. Such problems yield to none but an unusual approach…

[Chance IV involves] a kind of discrete behavioral performance focused in a highly specific manner. [...]

Chance IV favors those with distinctive, if not eccentric hobbies, personal lifestyles, and motor behaviors.

Complex, resistant problems that do not succumb to conventional methods but might sometimes be solved by weird people with eccentric habits? Why, that almost sounds like math research. Before you trivialize my point, no, this does not mean that I choose my collaborators based on the colour of their skin, the more “exotic,” the better. It does mean that I would like to work towards a world where having for example a black collaborator in your own area of mathematics is common and unremarkable, not only because it’s the right thing to do, but also because research progress might be a little bit faster and more exciting in that world.

If “the feminist approach to mathematics” still makes you giggle, you might want to read about the benefits of the feminist approach to computer programming:

What led me to a creative, simple, and extremely fast solution was being part of a feminist community in which people felt comfortable sharing their technical problems, wanted to help each other, and respected each other’s intelligence. Those are all feminist principles, and they make file systems development better.

Please spare me the argument that mathematicians–unlike those horrible computer programmers and gamers–are already respectful, collaborative and open-minded. I happen to agree with Matilde Marcolli that the actual picture is far less rosy than that.

Second, mathematicians are just as likely as social scientists to confirm their political beliefs through incorrect mathematical reasoning. They are especially likely to do so when arguing about issues of gender and diversity. In no particular order:

**50/50 equals 80/20 equals whatever.** According to a study I’ve heard of, when a group has 17% women, this is often perceived as a 50/50 split, and 33% of women is perceived as a majority. This is perfectly consistent with the arguments about “50/50 quota” that I hear from mathematicians all the time. Feminists, supposedly, demand 50/50 gender quota everywhere, even when they are in fact disputing an existing 95/5 or 100/0 split and pointing out that 80/20 or 85/15 might be more desirable. Of course it is possible that, in a specific situation, 80/20 might be unreasonable or unrealistic. It still would not equal 50/50.

**The magical all-purpose bell curve.** We could start with Larry Summers’s theory of “greater male variability”:

It does appear that on many, many different human attributes-height, weight, propensity for criminality, overall IQ, mathematical ability, scientific ability-there is relatively clear evidence that whatever the difference in means-which can be debated-there is a difference in the standard deviation, and variability of a male and a female population. And that is true with respect to attributes that are and are not plausibly, culturally determined. If one supposes, as I think is reasonable, that if one is talking about physicists at a top twenty-five research university, one is not talking about people who are two standard deviations above the mean. And perhaps it’s not even talking about somebody who is three standard deviations above the mean. But it’s talking about people who are three and a half, four standard deviations above the mean in the one in 5,000, one in 10,000 class. Even small differences in the standard deviation will translate into very large differences in the available pool substantially out.

Steven Pinker explains the same theory in much greater detail, complete with graphs and slides. Notably, Pinker makes it very clear that he is talking specifically about the bell curve, not just about some general function whose graph has a bump, and discusses its particular properties such as “a normal distribution falls off according to the negative exponential of the square of the distance from the mean.” He also mentions another version of the bell curve argument where small differences in the mean lead to relatively large differences at the extreme ends. John Allen Paulos, a mathematician, has used that one to explain why, in a corporation employing Koreans and Mexicans, the Koreans might hold all the top posts even if the Mexicans are only a little bit more stupid. Hypothetically, of course. I’ve heard both versions of the argument many times, including in comments here on this blog. Every time, it is assumed that ability, mathematical or otherwise, is clearly described by a bell curve, because of course it is.

I submit that it is not, and here’s a proof. The bell curve is symmetric about its mean value. This means that the two extremes should be about equidistant from the mean. In other words, the average person’s mathematical ability should be exactly halfway between the two extremes, or equivalently, the average person should have one half of the mathematical ability of the very best mathematicians at the level of, say, the Fields Medal or Abel Prize. I don’t believe that there’s any definition or quantification of mathematical ability for which this would be true, even in the roughest approximation. I don’t even believe that the average person has half of the mathematical ability of a typical Ph.D. in mathematics, and then it gets much steeper after that.

Here’s what we do, then: we imagine a bell curve placed so that its maximum more or less coincides with “average” mathematical ability, whatever that may be. On one side, we cut off the far left tail altogether and consider this to be a negligible error. On the other side, though, we look at the tiniest shivers and tremblings of the extreme far right tail, way past the point where we made the cut on the left, and we believe that this will tell us something really deep and meaningful about why there are so few women at the top levels of mathematics. (Click to enlarge.)

Even Summers and Pinker seem to realize that this is way off, because they are always quick to add something about criminality or risk-taking. Because criminality is the negative of math ability? Alrighty, then.

(If you’d like to get more technical: in probability, the bell curve is the limiting distribution of a large number of independent identically distributed variables. On the other hand, the main variables on which high-level mathematical ability might depend–logical thinking, facility with computation, geometrical visualization skills, ability to modularize complex arguments, level of comfort with abstraction, quantitative instincts–are often correlated, and moreover the value of having two or more of these skills seems much higher than the sum of their individual values. Are you already trying to explain it away by coming up with unknown ghost variables and modified bell curves? Chances are that you are only attempting to find a mathematical justification for what you believe already for other reasons.)

**Bayesian priors: now you see them, now you don’t.** A Yale study found that scientists ranked the same job applicant higher, and offered a higher salary, when the candidate’s name was male. In various comment sections around the internet, I saw way too many responses defending such decision-making, based either on bell curves as above or on arguments such as this:

1) The woman on average worked harder to get the same qualification, leaving a man with a greater potential for growth.

As mentioned before, women are more conscientiousness. Across my student years, many just got better marks, because they did homework well and studied more regularly. Even though some got better marks than myself for example, I always felt they were closer to their limits. [...]

2) Women get pregnant. This is a real disadvantage and risk for any project leader. I witnessed myself that a project leader hired a woman with all good intentions, but she got pregnant just after, promised to keep working, but then left. His project was delayed significantly and he said “never again”.

So given the same qualifications, I would rationally go for the man.

This particular comment is anonymous, but I’ve also seen non-anonymous mathematicians making the same arguments on social media and calling them “Bayesian priors.”

Meanwhile, I’ve said a few times that I’m not interested in hanging out on Math Overflow or in publishing my papers in journals that have mandatory comment sections. Part of my rationale is the common sexism of internet comments, from the cesspit of sexism and racism at Hacker News, to situations where two very similar articles generate very different types of comments depending on the gender of the authors, to examples such as this:

[Female] authors are reviewed personally alongside their books, in a way that rarely happens to men. The author Jennifer Weiner tweeted several examples the other day, including “reviews” of herself, Fifty Shades of Grey author E.L. James, and one of my own book: In The New York Times, lead book critic Michiko Kakutani took three paragraphs even to get around to mentioning my book, and on the way there, she quoted — somewhat extensively! — from anonymous comments left on a 2010 essay that I wrote. In a review of, supposedly, my novel.

These are all mainstream sites well frequented by academic audiences, not “obscure gaming sites” or other dark corners of the internet. (There, it gets much worse.) I have already linked to many more such examples, here and and on twitter. I see them every day. And an internet commenting situation, where a post is exposed to many readers and many potential commenters, is certainly a good testing ground for Bayesian priors and the law of large numbers. Yet every time, mathematicians told me that my priors were not valid. These sexist commenters were not mathematicians; or if they were, then it did not happen on a dedicated site for mathematicians only; or if it did, it was a rare exception that I should ignore. And if my general experience with mathematicians has not been great, I should ignore it anyway, because #notallmathematicians.

If nothing else, you should at least acknowledge that if you can have your Bayesian priors, I can have mine.

**The dog ate my logic.** The bell curve arguments, even if interpreted charitably and not obviously wrong on a quantitative level, can only say this: there exists a mathematical model that does not contradict your conclusions. This is at best corroborating evidence, but it’s often treated as a conclusive proof, including by mathematicians. Also along the lines of faulty logic, whenever the subject of female underrepresentation or innate ability comes up, I’m asked whether I really believe that there are no differences between men and women. No, I don’t, why should I? Because “women have to wait longer than men to get promoted and this might indicate unconscious gender discrimination” must necessarily imply “men and women are exactly identical”? Really?

And that’s why I’m writing this post. Because I’m worried that mathematicians will see the Notices article and mistake it for a mathematical “proof” that it should not be necessary to invite women to speak at conferences. A few female graduate students in attendance, yeah sure, but certainly not plenary speakers because diversity is a boogieman.

Please, just for once, prove me wrong.

]]>

This book has been in the works for some years now, and I’m thrilled to finally have a demo copy to show you. The book will be published by the American Mathematical Society. The demo copy has been produced (impressively quickly!) by the Hungarian publisher Ab Ovo. I’m very grateful to Anna Kepes Szemerédi for envisioning this project in the first place, and for all the hard work she has put into it.

I have contributed an essay on photography. You can download it here, and here is the gallery of the photos I offered to be used in the book. The photo on the cover is also mine. I hope that this will encourage you to purchase the book when it becomes available; I’m only one out of many contributors (see the cover for the list of names), and the book format will add further value through graphic design. If you’re expecting “mathematical art” as exemplified for example by the Bridges conference, I must warn you that this is not what I do. (In the essay, I explain why.) There is some overlap with one of my blog posts from last year: the post was adapted from an earlier version of the essay, and then I used it in writing the final version.

Anna first approached me about this in late 2011. I was much less confident then, both in my photography and in my writing. I have worked on both since then. One thing I wish I’d seen before I submitted my contribution is this classic piece by Linda Nochlin on the absence of great women artists in the history of art. Here’s what she says about “the lady’s accomplishment”:

In contrast to the single-mindedness and commitment demanded of a chef d’ecole, we might set the image of the “lady painter” established by 19th century etiquette books and reinforced in the literature of the times. It is precisely the insistence upon a modest, proficient, self demeaning level of amateurism as a “suitable accomplishment” for the well brought up young woman, who naturally would want to direct her major attention to the welfare of others–family and husband–that militated, and still militates, against any real accomplishment on the part of women. It is this emphasis which transforms serious commitment to frivolous self-indulgence, busy work, or occupational therapy, and today, more than ever, in suburban bastions of the feminine mystique, tends to distort the whole notion of what art is and what kind of social role it plays.

This got me thinking back on what I wrote about photography and wondering for a moment if I might have fallen into the trap of “suitable accomplishment.” In the end, it clarified for me the distinction between the commitment to the process of getting better, and the expectation of achieving a certain level of excellence, and the expectation of gaining public acclaim. I have always been anything but unambitious. Nonetheless, I have never aimed to be a “great artist.” I am not altogether indifferent to success in art, as evidenced by this self-promotional post, but what made me pick up the camera is the pleasure I find in taking photographs. My enjoyment of it is not conditional on finding an audience, receiving public recognition, or on any presumption of greatness. Instead, it comes from trying to get better at it. The pleasure is not in taking the same photographs over and over again, but in expanding my range, improving my technique, seeking out new ideas and solutions. The seriousness of my commitment is in my engagement in the process.

I suppose that this does not make me a lady.

]]>

]]>

The official press release on Mirzakhani’s research is available, as are the citations for the other Fields medalists. I’d like to speak to what the selection of a female Fields medalist means to me as a woman and a mathematician. In that, I would like to paraphrase something that Melissa Harris-Perry has said about the election of President Obama. Mirzakhani’s selection does exactly nothing to convince me that women are capable of doing mathematical research at the same level as men. I have never had any doubt about that in the first place, and I have said so here many times. What I take from it instead is that we as a society, men and women alike, are becoming better at encouraging and nurturing mathematical talent in women, and more capable of recognizing excellence in women’s work. I’ve said here before that the highest level of achievement within the age limit set for the Fields medals requires a confluence of both exceptional talent and favourable circumstances. Talent must be recognized, nourished, directed in productive ways, accomplishment must be acknowledged and promoted. Among the setbacks I experience every day and hear about from other women, Mirzakhani’s award offers a reason for guarded optimism, a point of evidence that sufficient dents have been made in the many layers of glass ceilings that a woman could break through all of them to the highest level.

]]>

The problem with science is that, for all its wonders, it lacks narrative and story-line. Science (and maths) is about facts, and the laboratory testing of elements.

It is not primarily about people.Women – broadly speaking – are drawn to the human factor: to story, biography, psychology and language.

This self-proclaimed people specialist keeps referring to women as “females,” the noun more often than the adjective. For instance: “Biology and nature, he suggested, will generally nudge females away from [science and engineering].” Here’s to biology, I guess. And to consistency.

Here’s one good rebuttal, with further links. This essay in particular matches a great deal of my own experience. But I also want to question the “science is not about people” line from a different angle–the one that scientists adapt enthusiastically and unquestioningly in every funding application, from individual grants with a training and/or collaborative component, to conference funding, to large institute grants. For example:

The mandate of PIMS [Pacific Institute for Mathematical Sciences] is to:

- promote research in and applications of the mathematical sciences of the highest international caliber
- facilitate the training of highly-qualified personnel at the graduate and postdoctoral level
- enrich public awareness of mathematics through outreach
- enhance the mathematical training of teachers and students in K-12
- create mathematical partnerships with similar organizations in other countries, with a particular focus on Latin America and the Pacific Rim.

NSERC pays 1.15M per year for this, and that amount does not include provincial funding or support from participating institutions. I suppose one might argue about the precise meaning of “primarily,” but the “human factor” does not exactly seem unimportant. You could also look at the webpages of individual institute programs:

The purpose of this programme is to bring together researchers in these diverse areas of mathematics, to encourage more interaction between these fields, and to provide an opportunity for UK mathematicians to engage with an important part of the mathematical computer science community.

This is very standard language. Every conference, workshop and institute program aims to bring together researchers, encourage interactions, promote the exchange of ideas, contribute to training, engage the community. Every conference proposal and grant application emphasizes it. Every funding agency demands it. Every mathematics institute derives its very existence from this notion.

And how do women score here? In light of their natural, biologically determined talents and inclinations, surely we should be looking for women scientists in particular to manage all those human interactions, or at least to participate in them significantly? PIMS has never had a female director or deputy director. Among the more than 120 participants in the program I linked above, there are 3 that I recognize as women. There are many more such examples, more that I could ever have the time to list. Women are often underrepresented at conferences (read the comment section for testimonials), both as speakers and as organizers, and when they are represented proportionally or better, this is often framed as an affirmative action gimmick rather than genuine appreciation of their contributions.

We sing the importance of communication, interaction and connection-making at the bean counters, then ignore it in our own deliberations. We take pride in choosing conference speakers based on “scientific merit,” defined as a best paper contest with an all-male jury, even when good arguments can be made that the “human factor” should in fact count towards scientific merit. And heavens help anyone who might raise the idea of inviting more women to conferences based on their alleged skills in interpersonal communication. And I don’t see women being overrepresented among institute directors, deputy directors, or other high profile research facilitators, all positions for which women should be particularly well qualified by the virtue of biology and nature.

Consistency, indeed.

]]>

Well, why Ukrainian boys and not mathematicians? I didn’t get to chat with Eugene. However, here is a conversation that I might or might not have had with an internet user that we’ll call Boris. I’ll let you decide.

IL: Hi Boris. I’m Izabella Laba and I work at UBC.

BORIS: Hi Izabella, it’s nice to meet you. I’m Boris. So what classes do you teach this semester?

IL: Actually, it’s summer and I don’t teach.

BORIS: That’s very interesting. So how many students do you have?

IL: I’ve just told you I don’t teach in the summer. But tell me about your research.

BORIS: As you probably know, I work on modulated gvoorups on questable aussifolds. I have proved that if a modulated gvoorup has a subquestable chain of hyperchenettes, then the aussifold must be oubliettable. This links several areas of mathematics and should have implications for the rapidly developing field of quasialgebraic oubliettability. You can read my papers to learn more about this interesting and exciting area of research.

IL: That sounds fascinating. What is a gvoorup?

BORIS: You can read my papers to learn more about it.

IL: … OK. But can you tell me why you are interested in gvoorups?

BORIS: It is a very interesting and exciting area of research. So what do you work on?

IL: Well, harmonic analysis on fractal sets.

BORIS: It is a very difficult area of research in which it is very difficult to have any new ideas.

IL: How do you know that?

BORIS: This is well known to everyone in the field, even if it has never been published. So who do you work with?

IL: Uhm, I’ve been a full professor for some time now. Are you assuming that I’m a junior researcher because I’m a woman?

BORIS: Women are more interested in teaching than in research. If you’re looking for more women, you should go to a teaching related forum.

IL: I’m pretty sure I’m more interested in research than in teaching.

BORIS: That’s great. So what classes do you teach this semester?

IL: You’re repeating yourself. I’ve answered this already.

BORIS: That’s very interesting. Have you seen my paper on gvoorups from 1995? It may be relevant to your work.

IL: Honestly, I doubt it.

BORIS: It’s been nice meeting you. Please let me know if you have any questions about my paper.

]]>

]]>

The connection between Fourier-analytic properties of measures and geometric characteristics of their supports has long been a major theme in Euclidean harmonic analysis. This includes classic estimates on singular and oscillatory integrals associated with surface measures on manifolds, with ranges of exponents depending on geometric issues such as dimension, smoothness and curvature.

In the last few years, much of my research has focused on developing a similar theory for fractal measures supported on sets of possibly non-integer dimension. This includes the case of ambient dimension 1, where there are no non-trivial lower-dimensional submanifolds but many interesting fractal sets. The common thread running through this work is that, from the point of view of harmonic analysis, “randomness” for fractals is often a useful analogue of curvature for manifolds. Thus, “random” fractals (constructed through partially randomized procedures) tend to behave like curved manifolds such as spheres, whereas fractals exhibiting arithmetic structure (for instance, the middle-thirds Cantor set) behave like flat surfaces. There is a clear connection, at least on the level of ideas if not specific results, to additive combinatorics, where various notions of “randomness” and “arithmetic structure” in sets of integers play a key role.

The paper discusses three specific questions that I have worked on: restriction estimates, differentiation estimates, and Szemeredi-type results. I’ve also mentioned some open problems. At this point, I feel like we’re only started to scratch the surface here; there is much more left to do, for example optimizing the exponents in some of the estimates I’ve mentioned and, perhaps more importantly, figuring out what properties of fractal measures determine such exponents.

]]>

If you haven’t yet read this classic essay by Linda Nochlin on the question of why there have been no great women artists, I recommend it very highly. The essay is from 1971, but Nochlin’s points remain very much relevant to today’s arguments about why there have been so few great women philosophers, or mathematicians, or whatever.

Nochlin starts out by questioning the common notion of a “great artist” as a singularity that exists independently of society and history. The truth is, it takes at least a village. Great artists are enabled by the society they live in, draw on its artistic traditions, engage in a dialogue with other practitioners. Indeed, if artistic greatness depended only on innate talent, it would be very difficult to explain what Nochlin calls “conditions *generally* productive of great art,” such as must have existed, for instance, in the 15th century Florence and Rome, or in France in the second half of the 19th century. (We’ll note here that much of the same can be said of mathematics.)

The society also establishes standards for what qualifies as “great art,” and what does not. In the pre-impressionist Europe, historical painting– understood broadly so as to include biblical scenes, Greek and Roman mythology– was considered the highest and most prestigious form of art. Landscapes, still-lifes, portraits, and other suchlike were deemed less worthy. To wit:

Until the 20th century, Mona Lisa was one among many and not the “most famous painting” in the world as it is termed today. Among works in the Louvre, in 1852 its market value was 90,000 francs compared to works by Raphael valued at up to 600,000 francs.

“Great art,” going back to ancient Greece and Rome and then again starting with Renaissance, more often than not depicted naked and partially naked human bodies. Think Michelangelo, Raphael, Titian, Botticelli, Rubens. Even when the figures are clothed, the paintings still display a thorough knowledge of human anatomy. Such knowledge was usually gained through extensive study of the nude model, a practice that continues to be a mainstay of art programs. And yet, as Nochlin explains in detail, nude models (both male and female) were forbidden to women painters before the end of the 19th century. That right there explains completely why there has been no female Michelangelo or Raphael.

Nochlin cites many other ways in which the society refused to enable women artists: the apprenticeship system, access to academic educational institutions such as the Ecole des Beaux-Arts, opportunities to establish suitable relationships with art patrons, and more.

But the part I want to highlight here is the prevailing attitude to “the lady’s accomplishment”:

In contrast to the single-mindedness and commitment demanded of a chef d’ecole, we might set the image of the “lady painter” established by 19th century etiquette books and reinforced in the literature of the times. It is precisely the insistence upon a modest, proficient, self demeaning level of amateurism as a “suitable accomplishment” for the well brought up young woman, who naturally would want to direct her major attention to the welfare of others–family and husband–that militated, and still militates, against any real accomplishment on the part of women. It is this emphasis which transforms serious commitment to frivolous self-indulgence, busy work, or occupational therapy, and today, more than ever, in suburban bastions of the feminine mystique, tends to distort the whole notion of what art is and what kind of social role it plays.

In Mrs. Ellis’s widely read The Family Monitor and Domestic Guide published before the middle of the 19th century, a book of advice popular both in the United States and in England, women were warned against the snare of trying too hard to excel in any one thing:

“It must not be supposed that the writer is one who would advocate, as essential to woman, any very extraordinary degree of intellectual attainment, especially if confined to one particular branch of study. ‘I should like to excel in something’ is a frequent and, to some extent, laudable expression; but in what does it originate, and to what does it tend? To be able to do a great many things tolerably well, is of infinitely more value to a woman, than to be able to excel in any one. By the former, she may render herself generally useful; by the latter she may dazzle for an hour. By being apt, and tolerably well skilled in everything, she may fall into any situation in life with dignity and ease–by devoting her time to excellence in one, she may remain incapable of every other.”

Which of course brings to mind this famous quote from G.H. Hardy:

… it is undeniable that a gift for mathematics is one of the most specialized talents, and that mathematicians as a class are not particularly distinguished for general ability or versatility. If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity or age. [...]

It is very hard to find an instance of a first-rate mathematician who has abandoned mathematics and attained first-rate distinction in any other field. There may have been young men who would have been first-rate mathematician if they had stuck in mathematics, but I have never heard of a really plausible example. And all this is fully borne out by my very own limited experience. Every young mathematician of real talent whom I have known has been faithful to mathematics, and not from lack of ambition but from abundance of it; they have all recognized that there, if anywhere, lay the road to a life of any distinction.

And Paul Halmos:

What does it take to be [a mathematician]? I think I know the answer: you have to be born right, you must continually strive to become perfect, you must love mathematics more than anything else, you must work at it hard and without stop, and you must never give up.

I’d like to show these quotes, together, to everyone who professes the theory of “greater male variability”:

It does appear that on many, many different human attributes-height, weight, propensity for criminality, overall IQ, mathematical ability, scientific ability-there is relatively clear evidence that whatever the difference in means-which can be debated-there is a difference in the standard deviation, and variability of a male and a female population. And that is true with respect to attributes that are and are not plausibly, culturally determined. If one supposes, as I think is reasonable, that if one is talking about physicists at a top twenty-five research university, one is not talking about people who are two standard deviations above the mean. And perhaps it’s not even talking about somebody who is three standard deviations above the mean. But it’s talking about people who are three and a half, four standard deviations above the mean in the one in 5,000, one in 10,000 class. Even small differences in the standard deviation will translate into very large differences in the available pool substantially out.

From my professional point of view as a mathematician, here’s how I see this argument. Take a fluid, complex, multidimensional quality such as “math skills.” (Or such as “propensity for criminality,” for that matter.) Assume that this quality can be uniquely quantified, on some scale that covers all types and ranges of “math ability.” Assume further that the resulting distribution is described by a bell curve, because why not. Condition on events of probability practically zero, assume that the same generic, first-approximation model is still accurate on a scale and in a range where it was never meant to be applied, and draw your conclusions about women faculty at R1 universities. It’s not even clear to me that there’s anything here that could be defended. Should you nonetheless feel like reading another rebuttal, this one cites a few studies that refute the hypothesis.

But G.H. Hardy and Mrs. Ellis should also be a part of this discourse, as their writing goes a long way towards explaining why we find the Greater Male Variability theory so intuitive and appealing. The theory “looks right” and “makes sense” to us for the reason that it matches deeply ingrained behaviour standards and stereotypes, the same ones that Hardy and Mrs. Ellis articulate so well. We’ve lived for centuries in a culture that has discouraged women from focused achievement–and by “discouraged” I mean “actively prevented”–directing them towards unassuming mediocrity instead. We’ve lived in a culture that has propagated the stereotype of a woman as an all-round dilettante, while encouraging men possessed of any discernible talent to pursue it to distinction.

These cliches are not even close to dead. Excessive achievement is still often considered “unladylike.” The single-minded focus described by Hardy is still discouraged in women, and rarely available to them in any case.* Our society still disparages the actual existing achievements of women so as to better fit the narrative. More insidiously, we’re confusing nurture for nature. If Mrs. Ellis could have her way with bell curves for women, I assume we’d be talking about very small deviations indeed. The apparent smaller female variability is very easily explained when one considers that those women whose class and financial status might have allowed opportunities for intellectual or artistic achievement, were also the most prone to being brought up according to Mrs. Ellis’s strict standards. Instead, we’re ascribing it to genetics, evolution and essence.

To paraphrase Neil DeGrasse Tyson: if we lived in some other world where G.H. Hardy and Mrs. Ellis have never existed, I’d be very happy to have a conversation about genetic differences and greater male variability. But not here.

———–

*It’s not just family obligations. It’s also that our jobs have extra components that are not required of men, from special feminine negotiating skills, to learning to manage the masculine traits of ambition and competence and compensating for them with “feminine” qualities, to the special attention we must pay to our appearance. Take your pick.

]]>