Diversity and mathematics

bell curve1

Mother Jones, last year:

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.)

bell curve2

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.

Author: Izabella Laba

Mathematics professor at UBC. My opinions are, obviously, my own.

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