Mathematics—Who's She?

“Mathematics is not built from rigorous logical necessity.” - Lucy Spouncer, a 23 year old PhD student at Edinburgh University explores how mathematics is constructed as a reflection of the minds, culture, priorities and passions of the established practicing community of mathematicians.

Mathematics as a discipline gets its authority from its extolled virtues of objectivity and universality. Therein lies the problem. It’s a well-known psychological phenomenon, those who present as unbiased, consistently practice bias to a higher degree. What does this mean for the ‘unbiased discipline’? It means that if you are not represented in the mathematical community, mathematics does not care about what is important to you.

Question the audience.

Imagine you’re at a conference, maybe you’re at the Max Planck Institute in Bonn. Imagine you’re engrossed in a talk about your current favourite research topic, maybe it’s about Alexei Kitaev’s conjectures around topological quantum field theories (TQFTs) and categorical spectra. Imagine you start to get a bit lost towards the second half of the lecture. You look around and wonder whether to ask the man to your left for clarification. Maybe it’s Kitaev himself, the reason 100 researchers are attending this two week conference. Or you could ask the man in the row behind you, or the man to your right, or the man to his right. Maybe you look around and notice the man sat in every other seat in the lecture theatre, except yours. Imagine you’re the only woman in an audience of 50 men; maybe you’re me.

And in that moment, all I wanted was to know what it’s like to sit at a conference in a room full of women and talk about maths. I want to know what that maths looks like, what it feels like, and I want to see those voices shape the future of mathematics in my lifetime.

In practice.

Locally, mathematics is about truth, but consider how mathematicians actually work. We spend 90% of our time struggling with mathematics we do not understand. Once we’ve understood a proof, technique, paper, textbook, we move on, to something else we don’t yet understand. The conclusion is that the practice of mathematics requires sustained faith in a direction. Mathematics requires trust, both in ourselves and our community, that our shared struggles with complexity, and our confusion will lead to something meaningful. We depend upon validation that our work is important.

So, how does our work become important? Well, the 100 people at that conference in Bonn weren’t there because of logical necessity. They were there because someone convinced them this particular mathematical story was career-worthy. The narratives we perpetuate literally determine which mathematical directions get pursued and which get abandoned.

Our voices.

Many people tell stories, so what determines whose voices get heard?

Asking questions in talks gets you noticed. Introducing your work to people gets you noticed. Speaking up regularly in group discussions, answering the questions of others, and being genuinely interested in others’ work gets you noticed. However, speaking at a conference gives you authority, and having your work referenced by speakers validates your competence. Your work, publicly contextualised in an active area of research, by someone with authority, that gives you influence.

There are two clear measures for whose voices have influence and authority: the first is who has the conversational floor, the second is who gets interrupted, or who doesn’t. Interruption is a double-whammy; it’s often a direct message that what the speaker has to say is not valued, it’s also often a sign that the interrupter believes that what they have to say is more important. So, who speaks more in group discussions? Who is socialised to interrupt more often? And who gets interrupted?

The data.

Collected by the Women Interrupted app, conversational data shows that women get interrupted on average 1.67 times a minute in the UK, 1.43 times a minute in the US, and as an extreme example 8.28 times a minute in Pakistan. Though the interruptions come considerably more from men than women, the bias is present in all genders. Additionally, the interruptions from women are far more likely to be encouraging, to support or expand upon the current viewpoint, rather than shut it down.

A 2014 study showed that women are interrupted on average 33% more than men [1]. From casual conversations to female justices at the Supreme Court [2], the trends are the same. That might be okay though, if men and women got equal airtime. But they don’t. One prime example is a 2017 study of 250 academic seminars in 10 countries [3], which saw men asking 2.5 times as many questions as women. What about when women actually speak their (gruellingly fought for) half of the time? Naturally, they are perceived as dominating the conversation [4].

The pattern reveals a maddening catch-22: women are interrupted more, speak less, yet are still seen as talking too much. For a woman, taking a central role in conversation is as comfortable as blasting death metal in a yoga class.

Hidden bias.

A word of caution, we should always be wary when pronouncing our own communities as inclusive, since bias, when it becomes invisible, is much harder to call out. Recent data from a reflective survey on a research retreat for Edinburgh’s Algebra Geometry and Quantum Fields Centre for Doctoral Training reveals a telling paradox. Some 64% of people often held back from contributing to mathematical discussions, due to imposter syndrome, fearing insufficient knowledge, or preferring to listen first; when they actually did contribute, almost all felt valued. Even in inclusive research environments, where the barrier isn’t actually hostile reception, it is anticipatory anxiety that silences. The interruption patterns and authority gaps create psychological barriers that prevent contribution before it even begins.

Imagine you’re a category theorist in a room of mathematical physicists and someone asks what an adjunction is. You’ve spent the last nine months computing adjoints in a grid of 12 compatible adjunctions to understand and generalise deformation quantisation for operads, also as an adjunction. Just as you open your mouth to answer, a couple of mathphys guys fall over each other, loudly (and incoherently), to give their answers. The first attempt misses the mark and they get frustrated at each other. Even though you want to interrupt and correct them, not only would it feel rude, but you just could not compete with their volume and confidence. Even if you started speaking at the same time it’s likely that they, and everybody else, would expect you to stop first.

All the while we interpret confidence as competence, all the while we let systemic unconscious bias dictate who adopts a speak-first or listen-first policy, competent marginalised voices will not shape the language and future of mathematics.

Inside or out.

In the community researching, say, Anabelian geometry, researchers at Kyoto dominate the field. Outside of the St. Petersburg school of research within geometry and mathematical physics, others on the same topics are unlikely to make progress. In higher category theory, especially in more abstract research, if your supervisor isn’t spearheading the movement, you’re likely to be left behind. There are distinct cultural and social barriers to entry. This influenced my decisions when preparing my PhD applications. Exploring the collaborators and students of potential supervisors, I noticed the representation of the LGBTQ+ community jump up whenever higher category theory or homotopy theory was mentioned. No doubt this has something to do with Emily Riehl trailblazing with homotopy type theory, recognising the role of queer identity in her research, and Mike Hill with the organisation Spectra. It’s quite clear that each group is drawn to the language, philosophy and culture of the mathematics they end up in, and wherever they see themselves reflected in it.

Defining interpretations.

Diversity in authority in mathematics is not just about fairness, it’s about the quality and validity of mathematical knowledge. When I claim that “mathematics is social,” I’m not just talking about workplace dynamics, or getting to see your friends at conferences, I’m not trying to sell you the idea that ‘collaboration is good’ in disguise. I’m talking about the fundamental epistemological fact that mathematics itself is produced through social practices. Closing the authority gap opens up the possibilities of what mathematical objects can mean.

Mathematical objects don’t have fixed meanings. When I describe a manifold, for example, I’m not just conveying information, I’m performing interpretive acts that were learned through my training in particular mathematical communities. This hermeneutical argument skims the surface of the ongoing research of Irida Altman on ‘Textuality and mathematics’[5]. First I might write a symbolic presentation using charts, then draw a 3-dimensional blob projected onto a chalkboard. Eventually, I would move on to precise verbal analogies of holes in higher dimensions, and locality principles. I might throw in the statement that manifolds are the ‘right’ model for spacetime.

The ‘obvious’ ways to understand mathematical concepts are actually culturally specific conventions. They seem universal only because they’re shared within our community. Despite the written literature, whole areas of mathematics and physics have had to be rederived when the last of those who spoke it were gone; a proof is only a proof if enough people understand and believe it. Maths is a living language, form is content, and our language is inherently human. Maths is social.

The observer.

The only way that mathematics retains objectivity and universality is by acknowledging that truth means the subjective truth of the human experience.

I recall a comment from Graeme Segal at the Categorical Symmetries conference at the International Centre for Mathematical Sciences in Edinburgh last summer, when he was questioned about locality as an inherent property of a physical system. He said that locality does not live in some separate Platonic realm, our laws of physics represent our human experience of the world. Locality is our observed and lived reality, and we cannot separate the observer from the phenomenon, nor from our models. In mathematics, it’s even harder to get recognition of this idea.

Amplify quiet truths.

Inclusive mathematics means recognising the gap between mathematical truth and mathematical authority, in its language and culture, in its epistemological strength, its direction, and whose voices have the authority to shape it. So, how do we address this authority gap?

At a table at the Bonn conference dinner, an associate professor and speaker at the conference, Nils Carqueville mentioned why someone in his field might care about my project. Assuming that I go on to formulate the poisson and deformation quantisation operads in spectra, we could expect to use them for studying deformation theory of TQFTs and lattice systems in the spectral setting. Kitaev’s spectrum provides a natural starting point, however, even more interesting deformations likely lie beyond the characterisation of semisimple theories, in topological twists of supersymmetric theories or supersymmetric gauge theories for example. What a vision that would be! Oh, to be able to connect spectra, lattice systems, TQFTs, and deformation theory through my project.

To everyone else at the table, my project’s value was literally constructed in conversation with Carqueville. When he connected my work to the broader conference vision, it wasn’t just intellectual validation, it was breaking through the authority gap that might otherwise have kept my mathematical insights marginalised.

Those with authority, recognise it, and use it to amplify the work of others. Endeavour to understand, who has to work for the authority that others get without question, and why?

Authority in action.

The undisputable authority on The Authority Gap, Mary Ann Sieghart, presents several key strategies to combat this bias in her Guardian ‘best politics book of the year’[6].

1. Choose a woman to ask the first question. When chairing meetings or seminars, - this sets the tone for balanced participation throughout the discussion.
2. Acknowledge underrepresented collaborators at the start, at conferences and workshops, begin both talks and discussions with their names. Before attributing yet another theorem to the established titans of your field, pause to consider if other quieter voices may have gotten there first.
3. Amplify and attribute in the moment. When a female colleague makes a valuable point, repeat it while giving her credit.
4. Plaster department walls with posters of female, ethnically diverse, and queer mathematicians [7], and don’t just advertise their persecution, they didn’t pursue mathematics to talk about how hard it was. They came to do maths—advertise their actual research.

If you’re reading this and my accounts feel alien to you, you could just take my word for it. If that’s not sufficient, I wonder whose authority is.

And if this post resonates with you—intervene and amplify. □

References

[1] Influence of Communication Partner’s Gender on Language
[2] Female Supreme Court Justices Are Interrupted More by Male Justices and Advocates
[3] Women’s visibility in academic seminars: women ask fewer questions than men
[4] Speaker sex and perceived apportionment of talk | Applied Psycholinguistics | Cambridge Core
[5] Textuality and Mathematics: A Hermeneutics
[6] The Authority gap: Why Women Are Still Taken Less Seriously Than Men
[7] Successful female leaders empower women’s behavior in leadership tasks - ScienceDirect