x + y

CHAPTER 1

INTRODUCTION

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.

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 as a whole, which is still dominated by men—not in sheer number, as in the mathematical world, but in concentration of power.

When I was young I didn’t have any female mathematician role models. There were hardly any female mathematicians in the first place, but also I felt no particular affinity with those I did encounter, and no special desire to be like them. I did grow up with strong, high-achieving women around me at all levels, though: my mother, my piano teacher, the headmistress of my school, the UK’s prime minister, the Queen.

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?

WHAT IS MATHEMATICS?

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. I have written about this extensively before. 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.

HOW DOES MATHEMATICS DO THINGS?

Mathematician Sir Tim Gowers writes of the “two cultures of mathematics,” which he characterizes as problem-solving and theory-building.

It seems to me that in the wider world, mathematics is seen as all about problem-solving, and that theory-building is mostly unheard of or otherwise neglected. Of course, the two are not cleanly separate, and in the end I think mathematics is most satisfying when the two come together: where building a theory also solves some problems. But what does it mean to “build a theory”?

Mathematical theories are descriptive, not prescriptive. They describe something that we see happening at the root of a situation, but they are not only there to predict how the situation will unfold. More broadly, they are there to help us think differently about the situation, to illuminate it, to help us understand how certain aspects of it are functioning. The theory is abstract—we have disregarded some of the surface details in order to see what is happening inside.

In mathematics a theory often starts with an idea or a possibility, and is built up as a collection of conclusions about that idea, or consequences of that idea, or properties of that idea. Sometimes it’s really just a reframing of an existing idea, but a slight reframing can lead to a monumentally different theory. If you stand at the top of the building formerly called the Hancock Center in Chicago, you can look across the city and see a vast jungle of “civilization” sprawling out to the horizon. But if you just shift your angle a tiny bit so that you’re looking straight down Michigan Avenue, the jungle becomes a grid system that will suddenly spring up before your eyes in near-perfect alignment. Sometimes in life all it takes is a slight shift in perspective, and this is often how new mathematical theories arise too.

I am going to propose a theory, or possibly just a reframing, to solve a problem.

WHAT IS THE PROBLEM?

The problem I am going to address in this book is the divisiveness of arguments around gender equality. Not all arguments are like this, but too many of them are. Sometimes the arguments push people in opposite directions because of the way we talk about gender, and sometimes arguments are futile because there is little clarity about what exactly is being discussed. Take the word “feminism,” for example. One problem right from the start is that there are so many different definitions of feminism—it means completely different things to different people. As a result, there is a certain amount of talking at cross-purposes. Some people use the narrowest possible definition so that they can obviously justify denigrating it. For example:

Other people use the broadest possible definition in order to persuade everyone to support it, or even to convince themselves that they are working toward its causes no matter what they’re doing. For example:

With some people’s definition of feminism being highly restrictive and some other people’s encompassing almost everything, it is a divisive concept even before it has even really gotten past the definitions.

Aside from these arguments about definitions, there are various reasons some people are anxious not to associate with the word “feminism” at all. Some people stereotype feminists as angry, man-hating, anti-family women, and this is an off-putting image for potential feminists of any gender. This highlights another source of the divisiveness of feminism: it inherently seems to separate women from men even as it is trying to overcome that separation. It’s hard to argue about men and women without acknowledging that they are different; if they weren’t different, we wouldn’t refer to them as “men” and “women.” And then we get distracted and absorbed by arguments about the ways in which men and women are and aren’t different. That argument is a distraction and a detraction.

The problem with this lack of clarity is that it draws us into a meta-argument—an argument about what we should be arguing about. There is an argument that the only experiences of oppression shared by all women are by definition those experienced by the most privileged of women: white women (more specifically, rich white straight cisgender women). There are many problems other than, and possibly worse than, the problems of those women, such as racism, homophobia, wealth inequality. We can thus also be distracted by a competition about which issue is the most important and pressing, instead of addressing them all.

Who benefits from these meta-arguments? I suggest it is the people who currently hold power and who want to keep it that way. While women fight each other about what feminism should mean, and about which intersectional branch of feminism should be given the loudest voice, and about who is the most oppressed, and while men oppressed on the grounds of race, sexual orientation, wealth, status, upbringing, education, gender expression, physical strength, or prowess in sports fight to be heard too, while all this disagreement rages between people who feel disadvantaged in any way, the people currently holding power can rub their hands in glee and consolidate that power.

The problem of definitions and meta-arguments is something that mathematics is very good at sorting out.

A MATHEMATICAL APPROACH

In mathematics we propose a theory by making a basic definition. We illuminate the definition by exploring some key examples. We justify the theory by demonstrating in what way it is ubiquitous and helpful. It might be helpful because it enables us to solve particular problems. It might be helpful because it enables us to think more clearly. I will argue that the theory I propose in this book does both.

In math, things need to be ubiquitous enough and helpful enough. They don’t need to be helpful in all possible situations; even being helpful in one situation is enough, although the helpfulness does need to be weighed against any sense in which it might also be obstructive. This is different from evidence-based arguments. A mathematical theory isn’t judged by the statistics of what it achieves in a large randomized sample. When working with evidence we typically test something on a large sample and see if it makes a statistically significant difference. Is it better than doing nothing, on average? In the case of a new pharmaceutical drug, is it better than placebo, on average? In the case of differences between men and women, are there differences between male behavior and female behavior, on average?

Mathematical deductions happen somewhat differently. First of all, they proceed according to logic, not evidence. Second, they are more likely to ask: Under what circumstances does this make a difference? Or even: Under what circumstances might this make a difference? This is very different from asking whether something makes a difference on average.

It is important to remember that averages—whether mean, mode, or median—do not apply to individuals. “The average person” is not a real person. Saying that “the average person” does something-or-other certainly doesn’t mean that most people exhibit that behavior, and it does not tell us anything concrete about an individual in front of us. This includes the amount of sleep people need (some people really don’t need much), the amount of calories needed to maintain a healthy weight (some people need to eat a lot less than the official daily recommended amounts), whether doing exercise makes you feel better (according to research averages it does, but it doesn’t work on everyone), whether or not you can win the lottery (it is so unlikely as to be impossible, and yet people do win almost every week).

It is often pointed out that individual experiences do not generalize to large groups, but the reverse is also true: the average experiences of a large group do not apply to individuals. So how are we to think about people?

Mathematical deduction often operates by something more like case study. Something worked in this one situation; what made it work? And can we figure out how to replicate it or get it to work more widely? This is the sort of thought process I will be applying.

Importantly, this is a sense in which talking about personal experience is valid, as long as you’re not claiming that your personal experience is necessarily universal, statistically significant, or typical. It’s a case study, and so one can ask, as mathematicians do: What made this work, and how can we build on this idea?

I will use many case studies throughout this book, highlighting what I find meaningful about some of the brilliant people who have achieved things in the world. Many of these will be women who achieved things in their own way, not by emulating the heroism, bravery, record-breaking, fearlessness, or athleticism of men. Sometimes this means they are undervalued in society, and I’ll talk about that too. I’ll talk about the work of mathematician Emmy Noether, scientists Jocelyn Bell Burnell and Rosalind Franklin, the activist Susan Burton and her advocacy for a justice system that is more humane rather than punitive, the more famous and less famous achievements of Florence Nightingale, the business principles of Dame Stephanie Shirley and Mary Portas, the career trajectory of Michelle Obama, the experiences of Professor Deirdre McCloskey in her “crossing” (as she describes it) from male to female.

As is manifestly evident, I am not a biologist, psychologist, philosopher, historian, sociologist, gender theorist, biographer, neuroscientist. I am a mathematician. So I am going to write this as a mathematician, theorize as a mathematician, and explore as a mathematician.

I am going to propose a reframing of the entire discussion around gender. This consists of focusing on relevant character traits instead of gender, and has at its center some new terminology to help us get started with this new dimension. I will show a vast array of ways in which this can help us move forward, whether we are women or not. Like many mathematical reframings, it is in a way only a very small step further than what many brilliant people have written about and understood before me. But somehow everything seems to stop just short of proposing a new dimension and new vocabulary. Much has been written about the problem without proposing a solution.

My research field, category theory, also began with an apparently small step involving a new dimension and new vocabulary. And yet this had an effect on the entire way of thinking of contemporary mathematics. I think something similar is possible for how we think about gender. What I am proposing is not a mathematics of gender, but a mathematical approach to gender. I am going to use mathematical thought processes to rethink our whole approach to gender.

THE PROCESS OF MATH

Math is not just about calculating answers. In fact, the theory-building part of math isn’t really about calculating answers at all. On the right is a sketch of some of the phases involved when mathematicians build a theory. Often a first step is spotting patterns. At a basic level the patterns might be patterns in numbers, such as the fact that all multiples of 10 end in 0. They might be patterns in shapes; for example, the following grid of triangles has patterns of bigger triangles of various sizes, built up from the small ones.

Humans have used patterns throughout the ages and across many (or all) cultures as a way of building up large ideas based on small ones. Nature also uses patterns to build complex structures such as petals on flowers or the spirals of a pineapple. At root, patterns are about connections between different things, or between different parts of the same thing. We can then think of patterns more abstractly to include phenomena that aren’t necessarily visual, such as patterns in behavior, which are essentially similarities between one person’s behaviors at different times, between different people’s behaviors, or more broadly between the behaviors of different animals or different communities.

Math is about making increasingly abstract connections between things. The abstraction consists of forgetting particular details about a situation in order to see some deep similarity. This is how making analogies works, but math takes it a step further than just declaring that a similarity exists; it moves up one level in abstraction by examining what the similarity really consists of, and treating that as a new concept.

For example, the equation

is analogous to the equation

But in math we don’t just leave it at that—we say

for any numbers a and b. This abstraction makes our point simultaneously less ambiguous and also more open to generalization, in the sense of broadening to include more examples.

Similarly, if we think about a pattern of women not speaking up in meetings, and female students not asking questions in class, the similarity at an abstract level could be summed up as “women not speaking up in mixed-gender environments.” At this point we have not measured how prevalent this pattern is, we haven’t found what causes it, and we haven’t found out how to change it. But in identifying the pattern we have made a start, by shaving off extraneous details and homing in on what is really important. This is a crucial step in building a good theory.

The abstraction then helps as we wonder why a pattern might be arising. Math is all about asking why, and delving deeper and deeper for more and more fundamental answers. A superficial answer to why women are likely to speak up less in mixed groups is “because they’re women.” We could test that theory, and what we are likely to find is that there are women who actually do speak up, and men who don’t. We might find a statistical link between being a woman and not speaking up. If the link is only statistical, however, then causation isn’t fully determined. In abstract math we seek to go a level deeper. Can we look deeper to find a causation? Why does being a woman cause this phenomenon statistically? We might then find that rather than it being about someone being a woman per se, it’s about relationships between people, and about how different people relate to each other. The idea of studying relationships rather than intrinsic characteristics matches a major advance in modern mathematics: the development of my research field, category theory.

THE IDEA OF CATEGORY THEORY

Category theory is thought of as a “foundational” branch of math because it looks at how math itself works. Before category theory there was set theory, which has a different ideology and thus very different technicalities. Set theory is based on the idea that the fundamental starting point of math is membership—that is, whether or not a given thing is a member of a particular set. This is a bit like parts of society being all about what “clubs” you’re a member of, whether that is literal clubs like the old-fashioned and deliberately exclusive gentlemen’s clubs, or more abstract clubs, like political “tribes” of people who believe the same thing more because of their perceived membership in that tribe than anything else.

Category theory takes a different starting point: relationships. It is built on the idea that we can understand a lot about something or someone by looking at their relationships with those around them.

Here are some pictures of how those two points of view differ. A set can be depicted as a collection of objects enclosed within a boundary, where inclusion and exclusion are determined by some intrinsic characteristics:

By contrast, a category can be depicted as a collection of objects with a network of arrows showing how they’re related, a bit like in a family tree. In this case, instead of showing parent-child relationships we can show which numbers are related to each other as factors:

In a way this is only a slight change in perspective, but it took modern mathematics in a whole new direction. For our theory about gender, we can think of the set-theoretic approach as defining the set of women on earth according to some intrinsic characteristics. Sometimes this set is said to be determined by chromosomes, sometimes by reproductive organs, sometimes by hormones. In fact, none of these definitions is as clear-cut as some people assume, and furthermore the definitions do not match each other.

I will instead take an approach based in the ideas of category theory. This consists of thinking about how people relate to one another rather than what their biological descriptions are. Across history there have been attempts to define masculinity and femininity in terms of behavior, and how men and women relate to other people, and there are also definitions of gender as a social construct, as opposed to sex as a biological description. But in this book I will instead propose a theory that only looks at how people relate to one another, without trying to impose genders on those behaviors. These behaviors may or may not be statistically or biologically related to gender, but investigating that would be a different kind of study and is not really the point. Questions of whether certain behaviors are based in biology or society are a distraction; in a way, it doesn’t matter, because all that matters is how people actually do relate to one another.

The idea of focusing on relationships instead of on intrinsic characteristics is an important part of the idea of thinking “categorically.” I mean this in the mathematical sense of thinking according to the ideas of category theory, rather than in the non-technical sense of thinking in a fixed, clear-cut, immovable way. It’s perhaps an unfortunate reuse of the word because, as I will explain, one of the important aspects of mathematical categorical thinking is flexibility.

The flexibility of category theory comes from thinking about relationships and freeing ourselves from thinking about intrinsic characteristics. This means that we have the potential to apply the ideas much more widely, to situations involving very different types of people, or even to zoom in and out and think about smaller-scale things like games and toys or about larger-scale things like schools, companies, and society more broadly. An example I wrote about in my previous book The Art of Logic involved thinking about power structures giving rise to structural privilege in society, such as being rich, or white, or male, or different combinations of them. The relationships we then consider are those between rich people and non-rich people, or between white people and non-white people, or between male people and non-male people. All those relationships combine into this diagram showing the relationships between people with different combinations of those particular types of privilege:

The arrows just depict hypothetical loss of one type of privilege, without making any claims about the causes or consequences.

As we are only thinking about the relationships between people who do and don’t have those types of privilege, we can then apply the ideas to any other type of privilege among any other type of people, such as restricting our attention to women and thinking about the privilege of being rich, white, or cisgender.