Being a woman means many things.

Many of those things really have nothing at all to do with being a woman—they are contrived, invented, imposed, conditioned, unnecessary, obstructive, damaging, and the effects are felt by everyone, not just women.

How can mathematical thinking help?

As I am a woman in the male-dominated field of mathematics, I am often asked about issues of gender: what it’s like being so outnumbered, what I think of supposed gender differences in ability, what I think we should do about gender imbalances, how we can find more role models.

Excerpt adapted from *x + y: A Mathematician’s Manifesto for Rethinking Gender*, by Eugenia Cheng. Buy on Amazon.

However, for a long time I wasn’t interested in these questions. While I was making my way up through the academic hierarchy, what interested me was ways of thinking and ways of interacting.

When I finally did start thinking about being a woman, the aspect that struck me was: Why had I not felt any need to think about it before? And how can we get to a place where nobody else needs to think about it either? I dream of a time when we can all think about character instead of gender, have role models based on character instead of gender, and think about the character types in different fields and walks of life instead of the gender balance.

This is rooted in my personal experience as a mathematician, but it extends beyond that to all of my experiences, in the workplace beyond mathematics, in general social interactions, and in the world itself, which is still dominated by men, not in sheer number as in the mathematical world, but in concentration of power.

I worked hard to be successful, but that “success” was one that was deﬁned by society. It was about grades, prestigious universities, tenure. I tried to be successful according to existing structures and a blueprint handed down to me by previous generations of academics.

I was, in a sense, successful: I looked successful. I was, in another sense, not successful: I didn’t *feel* successful. I realized that the values marking my apparent “success” as deﬁned by others were not really my values. So I shifted to finding a way to achieve the things I wanted to achieve according to my values of helping others and contributing to society, rather than according to externally imposed markers of excellence.

In the process I learned things about being a woman, and things about being a *human*, that I had steadfastly ignored before. Things about how we humans are holding ourselves back, individually, interpersonally, structurally, systemically, in the way we think about gender issues.

And the question that always taxes me is: What can I, as a mathematician, contribute? What can I contribute, not just from my experience of life as a mathematician, but from mathematics itself?

Most writing about gender is from the point of view of sociology, anthropology, biology, psychology or just outright feminist theory (or anti-feminism). Statistics are often involved, for better or for worse: statistics of gender ratios in different situations, statistics of supposed gender differences (or a lack thereof) in randomized tests, statistics of different levels of achievement in different cultures.

Where does pure mathematics come into these discussions?

Mathematics is not just about numbers and equations. Mathematics does *start* with numbers and equations, both historically and in most education systems. But it expands to encompass much more than that, including the study of shapes, patterns, structures, interactions, relationships.

At the heart of all that, pumping the lifeblood of mathematics, is the part of the subject that is a framework for making arguments. This is what holds it all up.

That framework consists of the dual disciplines of abstraction and logic. Abstraction is the process of seeing past surface details in a situation to find its core. Abstraction is a starting point for building logical arguments, as those must work at the level of the core rather than at the level of surface details.

Mathematics uses these dual disciplines to do many things beyond calculating answers and solving problems. It also illuminates deep structures built by ideas and often hidden in their complexity. It is this aspect of mathematics that I believe can make a contribution to addressing the thorny questions around gender, which are really a complex and nebulous set of ideas hiding many things.

A Case Study

We can use mathematical thinking to help us unravel arguments about gender differences, and evaluate questions like: Are men and women innately different in some way? And if so, is it justifiable to treat them differently? To examine and refute these, it helps to first show the weakness of arguments that suggest gender imbalance is “just the way of things.” (But in the end, rather than just refuting these arguments, we need to reframe the entire debate so that we can stop thinking about gender differences where they aren’t relevant and stop getting involved in arguments that mainly serve the people who currently hold power in society.)

Why do we persist in thinking about gender differences? I think it’s telling to think about who benefits, when we think about why this research is even being done. Why is anyone trying to prove that there are innate differences between men and women in intelligence, scientific ability, competitiveness, or any other traits that seem to confer high status in society?

One general reason to cling to the idea of innate ability is to give ourselves an excuse for not being good at something. If I claim that I just have no natural aptitude for sports, that gives me an excuse for being very, very bad at sports. Conversely, when people declare that I am very talented at the piano, that negates the thousands of hours of practice I have put in. People can declare themselves to be a right-brained, “creative” person, and use that as an excuse for being disorganized. They can boast of being a left-brained, “logical” person, and use that as an excuse for being insensitive. (This is in spite of the fact that the left-/right-brain theory has been largely debunked.)

The more invidious reason to claim that people are born with certain traits is to avoid having to help people do any better. This is a way of not having to address our prejudices. If we can somehow argue that women are innately less intelligent than men, then we won’t have to address issues of inequality in education, science, business, politics, and every echelon of power. If “innate” biological differences are found, they become fodder for people who seek a pseudo-rational basis to maintain structures that discriminate against women.

If the arguments are about biology, what can math do for us here? Math gives us a framework for making justifications and also for evaluating them, providing a way of assessing the value of any particular opinion. This is why math can be relevant to all sorts of things that don’t appear to be obviously “mathematical.” Mathematics is too often seen as being all about numbers and equations, in which case anything that does not involve numbers or equations appears to be not “mathematical.” But I think anything that involves some sort of justification can be examined mathematically.

A mathematical justification is called a **proof**. It is like a kind of journey. It has a starting point, a destination, and a way of getting from the starting point to the destination using logical deductions. And so we evaluate it by thinking about the starting point, and thinking about the logical deductions.

I am going to use this approach to evaluate some existing arguments about gender differences, and then make a theory of how these arguments are flawed. But since these existing arguments are not stated quite like mathematical proofs, the first thing to do is to find the (attempted) logical structure of the argument and express it a bit more like a mathematical proof by reducing it to its bare bones. This process of stripping away outer layers is an important step in the mathematical process. The outer layers often obscure what the real structure of the argument is, a bit like sleight of hand, and so stripping away those layers often exposes the flaws in the argument. This is one of the reasons that math uses very precise language and abstractions, to leave less possibility for that sort of misdirection. It’s a bit like the fact that it would be hard to carry a concealed weapon on a nude beach.

Step 1: Identify the Logical Structure

Here is one much-discussed argument about the gender imbalance in science and math, involves the idea of assessing people according to “systemizing” and “empathizing”: The claim is that men’s brains tend to be stronger in systemizing than empathizing, and systemizing is important in mathematics, so it is to be expected that there are more men than women mathematicians.

This looks a bit like a simple string of implications:

**1. Being a man implies being better at systemizing.**

**2. Being better at systemizing implies being better at math.**

**3. Therefore, being a man implies being better at math.**

Now, if these were valid logical implications of the sort used in mathematical proofs, then the conclusion would be correct. This is because in pure logic if we know “X implies Y” and also “ Y implies Z,” then it is logically valid to conclude “X implies Z.”

However, in the situation I’ve described here, they’re not really logical implications. They are something more complex and difficult. The first step is a statistical observation, not a logical implication. It has been observed that men, on average, tend to be better at systemizing than empathizing, according to some proposed definition of these things.

The next step, the idea that systemizing is important in mathematics, is somewhere between an assumption and an observation. The idea that it is important in mathematics sounds logical, but that makes some assumptions about what “systemizing” really means and what skills are really important for research mathematicians (as opposed to people who are very good at mental arithmetic or math exams). There are some observational studies backing up this idea, but in that case the result goes back to being an observed statistical correlation.

The fact that these are statistical observations then raises the question of whether the effect is something innate about men or something cultural. A more honest chain of argument would go like this:

**1. Men have been observed to be statistically more likely to be stronger at systemizing than empathizing, for some very specific definitions of these words.**

**2. A correlation has been found between this notion of systemizing and becoming a mathematician.**

**3. Therefore, we might expect more men than women to become mathematicians.**

This is a rather weaker conclusion, reflecting how weak the steps in the argument actually are. It does not reveal anything about whether it is fair or biologically inevitable that the gender imbalance persists.

Carefully dissecting an argument in this way enables us to uncover its shortcomings. It often turns out that there are many small weaknesses compounding each other, and this can be more confusing than an argument with one large and obvious flaw. However if we see the same pattern of multiple small weaknesses in a diverse range of situations, understanding the general pattern can help us understand each individual case.

Step 2: Develop a General Theory Through Abstraction

One important step in the mathematical process is to make a general theory that can then shed light on more than just one situation. Mathematicians often do this with the help of abstraction, stripping away some external details to show the bare bones of a situation, which can then be seen to be the bare bones structure of other situations. This was the point of my introducing the letters X, Y, and Z in place of some parts of the argument used earlier—to focus on the logical structure of the argument that did not really depend on the details of what X, Y, and Z actually represented in this particular case. Having done that to show what a sound logical argument would look like, we can contrast it with what the weak, un-sound argument looks like, which is something like this:

**1. Men are observed to have quality Y on average, under some select circumstances.**

**2. Quality Y is believed to be good for activity Z without any very strong basis.**

**3. “Therefore,” men are naturally better (or worse) at Z.**

**4. “Therefore,” we don’t need to do anything about the imbalances in favor of men in activity Z.**

It’s worth noting that this general argument form is applicable very widely to many situations other than gender where arguments about imbalance are raging, including disagreements about race, wealth, educational background, sexual orientation, and so on. One advantage of abstraction is that it helps us to see connections between a broad range of situations beyond the matter directly under consideration.

Anyway, the weak argument gets subtly but invalidly morphed into one that seems much stronger via a series of sneaky slides, as in the preceding example. “Men are statistically more likely to be better at systemizing than empathizing” turned into “Being a man implies being better at systemizing,” involving some unsound deductions about statistics.

The abstract version of this slide is something like this:

**men have quality Y → on average men have quality Y**

There’s another slide that turns “Men are observed to be better at systemizing” into “Men are by nature better at systemizing,” assuming the observed quality to be a result of nature, not nurture. This is the sort of deceptive argument that enables some people to assert that gender differences are biological and therefore gender imbalances in the world are not the fault of discrimination. The abstract version is like this:

**men are observed to have quality Y→ men naturally have quality Y**

And then there’s the slide that turns “Men are better at systemizing” into “Men are better at math,” where the thing that has (supposedly) been measured is taken as a proxy for something much harder to measure. The abstract version is like this:

**men have quality Y**

**↓**

**men are better at Z**

where Y has been casually swapped for Z without much justification or fanfare. These three surreptitious slides can be combined to make arguments dramatically weaker through these less noticeable increments. This means that although we start at the top of the following diagram, we can sneakily claim that we are anywhere further down it by sliding down the arrows, but each time we move along an arrow the argument becomes more flawed.

Step 3: Test the Theory

The generality of this theory means that it can be applied to a wide range of examples where gender imbalance is found. In mathematics a theory is judged by the breadth of examples it unifies and the amount of light it sheds on those examples, so after making a mathematical theory we typically test it by trying it out on some more examples. For example, this one could be applied to another type of argument that has been used to justify gender imbalances in academia, this time in physics:

**1. Men have more academic citations than women in physics.**

**2. Citations are a measure of how good you are at physics.**

**3. Therefore, men are better than women at physics.**

**4. Therefore, it is fair that there are more men than women in physics.**

The first point is fairly well documented, but the second assertion involves less of a slide and more of an enormous leap of faith. The conclusion “Men are better than women at physics” may well be true statistically if we take a snapshot in time right now and take “better at physics” to mean more successful at making progress in advancing theories, but concluding that this is a fair situation is another giant unjustified leap: men might be more successful because the world favors them unfairly.

This method of thinking has revealed multiple flaws in some existing arguments around gender imbalance. We have seen examples where those arguments involve sneaky shifts replacing one statement by another which sounds similar superficially, but on closer inspection is only equivalent on the basis of a large and unproved assumption. As a result, the conclusions about gender differences have those unproven assumptions embedded in them.

There is a strong perception of differences between men and women, and understandably so—there are some fairly obvious general differences between men and women physically. But there are flaws in taking those differences too seriously or concluding too much about those differences. Instead of asking whether gender differences are innate, it is more productive to ask in what sense they are innate, to what extent they are innate, and what the point is of basing our world on those differences.

*Excerpted from x + y: A Mathematician’s Manifesto for Rethinking Gender by Eugenia Cheng. Copyright © 2020. Available from Basic Books, an imprint of Hachette Book Group, Inc.*

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