Following my last two posts on women in mathematics and the internet, I was challenged to turn my crystal ball sideways and look at it again. I have talked about what I oppose (comments on the arXiv). I have talked about initiatives that are successful but labour-intensive and difficult to pull off (research conferences for women). Are these the only choices we have? Must the internet disadvantage women in math?
The fact is, the positive impact of the internet on my own career would be hard to overestimate. I had long-distance collaborations by email that kept me going when I was isolated at my institutions of employment. I made new mathematical contacts over the internet. I do not need the departmental coffee room to keep track of research developments or professional opportunities. I get my news from blogs, social media postings and online discussions.
It might be too much to claim that, without the internet, my isolation would have killed my research career. Remote communication existed long before computers, even if it was less efficient. It is also possible that, in other circumstances, I might have made different career choices. Yet, the particular career I did have was largely shaped by the internet, and, given that women are especially likely to be isolated within their institutions, it should be safe enough to say that my experience was not unique. It is easy to overlook this kind of impact when it’s all around us, uncontroversial and taken for granted. Still, it’s there, a vital lifeline to those of us who might otherwise have been left stranded with no way back in.
We should not forget career advice. Perhaps you’re negotiating a job offer. Articles and blog posts can tell you about the process: the timeline, the framing and manner of speech, the range of what might be expected. You can ask about your specific case in a trusted discussion forum. But when I first went on the market, I did not even know that one was supposed to negotiate at all. Somehow, I’m still here. I’m not always sure how that even happened. The withholding of information has always been a means of control, and the internet is the best antidote to it that we have.
We can, and should, go much further. In recent years, I have been making a conscious effort to avoid those environments that I consider suboptimal for me, and to spend more time instead in feminist spaces, many of them online, with people who share deeper ties with me than mere geography and profession. As my commitment and involvement there increased, as I learned and grew in these spaces, as I began to pay more attention to how they were optimized for growth and learning, I found that this also affected the ways I approach mathematics and especially mathematical collaborations. I found the advantage that has been missing from my mathematical career all along.
Mathematics is increasingly collaborative. The percentage of single-authored articles decreased from 91% in the 1940s to 54% in the 1990s; I could not find the current number, but my estimate would be under 30%. Even when the authorship is not shared, our thinking often develops in interactions with others. Anecdotal evidence of this is abundant. Psychological theories provide corroborating backstories. Tradition tells us to leave the office from time to time and talk to the colleague down the hall.
The composition of collaborative groups matters. Some teams never go past collating individual contributions and stapling them together. Some actively suppress the work of some of their members. And there are others that work like magic: the coworkers pick up each other’s ideas, run away with them, refine them through dialogue and cross-examination.
There are no unambiguous criteria for what makes a great collaborative group, nor is there a simple way to create one. It’s not enough to put people in one room and instruct them to work together, or provide financial incentives, or sign institutional collaboration agreements. True creativity does not come to heel and does not respond to whistles from self-appointed leaders. There is, however, research on the subject, for instance at Google:
Team A may be filled with smart people, all optimized for peak individual efficiency. But the group’s norms discourage equal speaking; there are few exchanges of the kind of personal information that lets teammates pick up on what people are feeling or leaving unsaid. There’s a good chance the members of Team A will continue to act like individuals once they come together, and there is little to suggest that, as a group, they will become more collectively intelligent.
In contrast, on Team B, people may speak over one another, go on tangents and socialize instead of remaining focused on the agenda. The team may seem inefficient to a casual observer. But all the team members speak as much as they need to. They are sensitive to one another’s moods and share personal stories and emotions. While Team B might not contain as many individual stars, the sum will be greater than its parts.
Within psychology, researchers sometimes colloquially refer to traits like “conversational turn-taking” and “average social sensitivity” as aspects of what’s known as psychological safety–a group culture that the Harvard Business School professor Amy Edmondson defines as a “shared belief held by members of a team that the team is safe for interpersonal risk-taking.” Psychological safety is “a sense of confidence that the team will not embarrass, reject or punish someone for speaking up,” Edmondson wrote in a study published in 1999. “It describes a team climate characterized by interpersonal trust and mutual respect in which people are comfortable being themselves.”
Psychological safety. That’s right. You could even call Team B a “safe space.” Let me remind you here that the research described in the article was done at Google, not in your local feminist circle or student activist group, and that the direct goal of that research was increasing productivity rather than promoting social justice. “Safe spaces” are not about coddling or allowing no disagreements. They create conditions where certain types of distractions and obstacles are temporarily removed so that everyone can focus on the actual task at hand. Instead of mocking things we don’t understand, we might do better to acknowledge that people work better when they feel supported, disengage when they are attacked or disrespected, and have less mental capacity left for research when a good chunk of it is diverted towards watching their back. That goes for all of us, not just minorities or marginalized groups.
But the matter of social justice cannot be avoided, for minorities and the marginalized people are much less likely to feel supported and psychologically safe in their everyday working environments. That colleague down the hall might be a sexist and racist jerk. The committee evaluating conference proposals might be more inclined to see “leadership potential” in men. Women are less likely to be heard, and more likely to be interrupted or harassed. We face constant low-level negativity, nitpicking, comments implying that perhaps we do not understand our own research, that our results are trivial or false or some such. This has been my experience all along.
Imagine attending a harmonic analysis research seminar. The speaker tries to talk about the Fourier restriction problem for the sphere, but she can’t really get started because the guy in the front row does not believe in Plancherel’s theorem and demands to see a proof of it before the talk can proceed. The two postdocs next to him are carrying a conversation about the merits and demerits of different ways of normalizing the Fourier transform. The guy behind them has noticed that the speaker, in an expository blog post on another subject several months earlier, used the phrase “divide the set {1,2,…,N} into three sets of size N/3”; perhaps she does not understand that not all numbers are divisible by 3? The dude in the corner would like to know how to define the Fourier transform on quaternions and whether Plancherel’s theorem would be true in that setting. Several people in the back start grumbling: all this talk about quaternions and normalization is so boring, we should cancel the harmonic analysis lecture and have a seminar on algebraic number theory instead.
It’s not necessarily that the quaternion dude is a horrible person and should be banned from seminars. He could be a valuable contributor in a different group. He could also be genuinely interested in learning abstract harmonic analysis, in which case he should read a couple of books. It’s not about whether questions should be allowed in seminars, or whether they are being asked in good faith, or whether a harmonic analyst should be able to reproduce the proof of Plancherel’s theorem on the spot. It’s not relevant whether we “can deal with criticism,” “can stand the heat,” or “have thick enough skin.” (If we couldn’t or didn’t, we wouldn’t be here.) It’s not even relevant whether the expert on divisibility by 3 is sexist, or whether he just does it to everyone, as if that somehow made it better. All of these things can be discussed at length elsewhere, some other time. The point here is, nobody in that room is learning anything about the restriction problem for the sphere. Nobody is going to leave it with new insights on the subject. The seminar does not accomplish its intended purpose.
I’m exaggerating here, but not by much. I have witnessed real-life equivalents of everything I just described and I’ve had such things happen to me many times, although not all of them in the same seminar. I have also had discussions with mathematicians about feminism that looked exactly like that.
We like to say that mathematics is 100% objective. Statements are either true or false, their truth value being absolute and not subject to opinions. Any fact must be proved before it can be accepted. Any proof, if correct, must stand up to any amount of scrutiny and interrogation. Any dispute can be resolved by appealing to mathematical principles. Power structures and relations are not relevant: if an undergraduate student points out an error made by a Fields medallist, the truth of it is decided based on mathematics rather than seniority. All this may be true of our final product, at least in theory. (In practice, it is very easy to find published research papers where a lemma is missing an assumption or the constants do not add up.) But that’s not how we work at the creative stages of research and discovery.
There, we deal in conjectures and speculation. We build castles in the sand. We design tentative proof schemes where, initially, every part is incomplete or outright false. We must think past the nearest obstacle and towards the eventual goal. We draw general conclusions from examples that might not be representative. We make simplifying assumptions that are mutually contradictory. We don’t bother to cross all t’s and dot all i’s until the very end. At the same time, we also have to interrogate our ideas, test them against known facts and examples, look for gaps and errors. It’s a balancing act. When we strategize, we want to move forward, but also to keep it realistic. When we question our arguments, it is to focus, refine, or redirect them, not to shut down the process.
Collaborations, at their best, offer an advantage in that regard. The dreamer and the devil’s advocate can be played by different people, whereas the lone researcher has to do two things simultaneously that are at odds with each other. Having to explain your vision to someone else forces you to clarify it. A collaborator can push you in a new direction when you’re running in circles, pull you back when you get sidetracked, step in when you are discouraged.
By the same token, we are exposed and vulnerable in that process. It’s very easy to attack our ideas for being insufficient or false as stated. Of course they are, at that stage. It’s easy to dismiss someone else’s contributions just because we do not understand them at the time. It’s easy to accuse us of not engaging in good faith when we honestly do not have the information we need. With the defences of mathematical formality set aside, everything we say can be questioned, nitpicked and torn apart at will. Those coming with a hammer and eager to swing it will always find an abundance of nails. There is ample room for abuse of power: senior versus junior, well connected versus isolated. Few people speak out about this, but when they do, it’s not pretty.
We find, as the Google researchers did, that there is no simple recipe for a good collaboration. Some groups thrive on loud, animated discussions. Some go out for beer every time after work. Some talk quietly and disband promptly at 5 pm. Descriptions like “supportive,” “helpful,” “polite,” or “too aggressive,” can mean very different things to different people. It helps to have “high ‘average social sensitivity’ — a fancy way of saying they were skilled at intuiting how others felt based on their tone of voice, their expressions and other nonverbal cues”. It can also help to have codes of conduct. Tim Gowers’s Polymath rules include the following:
3. When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly, stupid comments are welcome here. (In the sense in which I am using “stupid”, it means something completely different from “unintelligent”. It just means not fully thought through.)
4. If you can see why somebody else’s comment is stupid, point it out in a polite way. And if someone points out that your comment is stupid, do not take offence: better to have had five stupid ideas than no ideas at all. And if somebody wrongly points out that your idea is stupid, it is even more important not to take offence: just explain gently why their dismissal of your idea is itself stupid.
5. Don’t actually use the word “stupid”, except perhaps of yourself.
That’s about as far as mathematicians are willing to go in allowing others to regulate their behaviour. We kick and scream when someone tries to tell us what we can say or do. When faced with the argument that some regulation is necessary, for example because some of us do not feel welcome in many professional mathematical environments, we rationalize and explain away the conflict. We don’t need rules because we are nice people who will not behave badly. In the unlikely case that someone should abuse the privilege, such actions will be noted by the community and the offender will be put in place. What do you mean, not true? Are you sure that it happened? Perhaps you just misunderstood? And wasn’t it an isolated incident anyway? And if we had such rules, how would we know that they wouldn’t misfire? Purely hypothetical situations invented for the sake of argument carry more weight than testimony and proof of real-life abuse.
Given the vehemence and universality of such attitudes, I have to assume that there must be more to it than a garden-variety knee jerk reaction. I would conjecture that when we oppose having constraints imposed on our behaviour, we might be doing so in the belief that this would hamper our creativity. It’s likely that the further we are from “being able to be completely and utterly ourselves all the time at work”, the higher price we pay in terms of stress, burnout and loss of productivity. Even our sexist and racist biases might be a means of saving mental energy. Academia does not teach us to worry about the possible cost to others.
Women rarely get to be completely and utterly their selves in a mathematics department. If we try, there can be consequences. Yet, all those years of restraining ourselves, pretending that we didn’t hear that joke, smiling politely when we are told that of course we are being taken very seriously but other priorities are more important – that, too, takes a toll, in the excitement that we no longer feel, in the research that does not get done. Even if we try to “just ignore” or “just forget,” as we are often counselled to do, we might not be doing ourselves any favours.
And as we defend ourselves from the negativity, as we fight to stay on our feet, we don’t have the time to ponder the hypothetical benefits of the psychological safety that we might never have. We don’t know how to create good, functional groups, if we’ve never been in one that worked for us. We don’t know what it’s like to be supported, or how to support others. We don’t know how to challenge and be challenged in the right measure. We know how to deal with criticism, grow thick skin and withstand the heat. We don’t always know how to reach out for more than that.
It’s liberating to be able to speak freely and to hear others do so. That was my experience under communist censorship, and that is my experience now. And as a lifelong mathematician who has spent decades working on abstract problems and has had ample opportunity to observe what circumstances are or are not conductive to such research, I can testify that the release is not only emotional. It’s a brain reboot, a power boost for our intellectual enterprise. Whether it’s talking openly about sexism, collaborating on a project without having to adopt the proper feminine manner of behaviour, or stretching my mind with like-minded people in directions entirely unrelated to my profession, it’s not only my well-being that benefits from it, but also, each and every time, my mathematics.
For all these reasons, I believe that we need to have our own working spaces. They don’t have to be exclusive to women or any other specified groups, or to follow any particular model, but they need to be spaces where we set the rules and control the membership. There, we can speak in our normal voices and smile only when we want to. When our faculties are not preoccupied with choosing the right level of deference, or with designing tactics to deflect those who would speak over us, we have the mental room to think about what we want to say. When we don’t have to defend what we said already from the same attacks many times over, we can think about what to say next. Then we can go ahead and say it. We stop running in circles and start moving forward.
I want to be clear on which spaces I’m talking about. I’m not interested in institutional “equity committees” that are in the same relation to women and minorities as HR departments to company employees. I’m not interested in organizational structures where feminism is a path to administrative advancement and where those who discriminated against me can demand and gain access and leadership. I also want to distinguish between political and intellectual spaces. In politics, it is necessary to make alliances, tailor the message to the audience, argue Gender 101 again and again as needed. In intellectual truth-seeking, it is necessary to see political compromise for what it is and to seek what lies beyond it. I’m a scientist, not a politician. I made my choices a long time ago. I’m far from dismissing the importance of politics. I can’t afford to do so. At the same time, the means are pointless without the end.
We need the kind of spaces that grow quietly and organically when like-minded people find each other. We need them for companionship, comfort, solidarity, but also for intellectual development unhampered by the constraints imposed on us elsewhere in the profession. The numbers are still against us, as are our geographical and professional realities. But we have the internet. We can create our own places there, render the physical distances irrelevant, and emphasize common interests and personal compatibility. Membership is based on commitment, not on entitlement. Arguments are common, but there is sufficient agreement on basic principles to allow grounds for a conversation. These are not the only places that should exist, or the only ones I will ever frequent, but I have found that these are the best environments to study, grow, explore and create.
There is no shortage of templates. If nerds writing fan fiction about an obscure character from an obscure film can meet up online and work on their writing together until they become successful published authors, then various groups of women in mathematics should be able to do something similar. But it was in feminist and other similar circles that I learned more about cooperation and collaboration than anywhere else. I learned how such groups work, not only by watching and participating, but also through explicit discussion and analysis. They never just “go with the flow,” assuming correctly that the absence of structure does not translate magically into equality and happiness. They articulate the goals, spell out the rules, teach their members to follow them, and make a conscious effort to ensure everyone’s psychological safety. Yes, this can mean trigger warnings, low tolerance for certain types of jokes, or using terminology that might sound alien to those not used to it. These are mutually agreed tools to create conditions where contentious, difficult and sensitive matters can be discussed. We, in mathematics, could use more of that. I’m tired of the mathematical communities where nothing is ever spelled out, everything is assumed, and the best time to define expectations is always after we fall short of them.
This post is a work in progress. I’m not pretending to have a definitive treatise on Women And Mathematical Collaboration. I’m just trying to keep track of what I’m learning, as many bloggers do. The question of women’s spaces is one that comes up often and in many contexts, and is asked especially often by young women. I feel that it’s crucial for us to have such spaces and, at the same time, that various formal women’s organizations or mentoring networks are only one small part of what we need, that we have to go beyond them into a different territory that we haven’t mapped out yet. This is my attempt to account for my own experience. I had little guidance on this when I entered the profession. You shouldn’t have to start from the same place.
The Accidental Mathematician 