The mathematics of wheel reinvention

The first talk I attended at this year’s JMM in Seattle was Tim Gowers’s lecture on how the internet and mass communication might change mathematics. Tim started out by listing some of the more dysfunctional features of how we do mathematics today, then suggesting how they might be improved. I very much agree with that part, and I would like to mention a few points from it here.

Our basic and most important unit of discourse is a research article. This is a fairly large unit: effectively, we are required to have a new, interesting and significant research result before we are allowed to contribute anything at all. Any smaller contributions must be bundled and packaged into units of acceptable size, or else they go unacknowledged. A comparison that came to my mind was having to conduct all transactions in twenty-dollar bills. Whatever your product is, you would have to sell it for $20 or else give it away for free, with nothing in between. It should not be difficult to see why this would not be am ideal environment for doing business. We should have smaller bills in circulation. It should be possible to make smaller contributions–on a scale of, say, a substantive blog comment–and still have them count towards our professional standing.

Our culture is extremely competitive. We value beating others more than we value helping them. All that matters is getting “there” first and scooping everyone else on our way. Intermediate results are worth far less. Additionally, this prioritizes one specific type of contributions over all others, even in those cases where a different order of priorities might be more reasonable. A good expository paper might have more impact on its area of mathematics than a middling research article; and yet, expository work is rarely, if ever, taken seriously by funding agencies and tenure committees.

We spend a great amount of time and energy on reinventing the wheel. A mathematician working on a problem might start with relatively small reductions, observations and lemmas that, by themselves, do not qualify as journal-publishable units; if that effort is not successful, these smaller contributions are lost and the next person working on the same problem has to reprove them all over again. Moreover, information such as “this method didn’t work, and here’s why” might be very useful to that next person. If nothing else, a great deal of time might be saved that would otherwise be spent on trying out unsuccessful approaches. Yet, there is currently no system in place to circulate such information and reward those who provide it.

Consider also how we work and collaborate. We are all gifted in different ways: some are better at imagining new ideas, some at asking questions, some at turning informal sketches into rigorous proofs, some are walking encyclopedias of the relevant literature. Yet, we have decided that each of us has to be self-sufficient and do all of these things equally, instead of allowing people to focus on what they do best and forming collaborations based on complementary skills. (I’d add that such collaborations obviously exist, including in my own experience. We just pretend, at least in official paperwork, that this does not happen.)

I agree with all of this, and I’d love to see us abandon the old ways and adapt new ones. We are far too invested in forcing everyone to fit the same mould. In a profession we like to call creative, I’d love to see more diverse and varied career paths and modes of expression. I’d love to see the flow of information a little bit less hampered by our ambition and competitive instincts. Think of all the theorems we could prove if we allowed more people into the field and, instead of hampering their intellectual power, harnessed it to the full.

I do not believe, however, that such changes are inevitable, and I have very little faith that they will be forced by the internet and other means of mass communication. It takes more than technology to change the culture. The early evidence is not encouraging. The basic discourse unit is still the research paper, except that we now post these on the arXiv. Other types of research contributions are still not being counted towards career progress, even as the subject comes up in discussions over and over again. We are as competitive and territorial as ever. The Polymath projects came and went; one was successful, another one was somewhat productive, others fizzled out. They did attract more participants than conventional math collaborations, but they never became truly “massive” as originally envisioned. People still ask questions on Math Overflow, and sometimes they get useful answers, but it never became the universal communication and collaboration platform that some of its early enthusiasts seemed to imagine. Other, smaller discussion boards went mostly unnoticed. There’s not much actual research that gets done on public blogs or social networks.

At the end of the talk, someone raised the diversity point in a question. The participants in Polymaths, Math Overflow and other similar projects are even less diverse than the general population of research mathematicians. Is there a reason why women and minorities tend to stay away from such venues? What can mathematicians do to ensure that all of us feel welcome to participate? I do not feel that Tim really answered that. He said (and I hope that I’m summarizing it fairly) that all those changes are just going to happen, like it or not, because they bring a more efficient way of doing mathematics and nobody will want to give up on that. It is an unfortunate fact that some people feel less comfortable on the internet, but in the end we will all just have to get over it.

I would like to suggest a different answer.

Later that day in Seattle, I attended an AWM panel on mathematics research conferences for women. I had been skeptical. I had not had much help from other women when I was coming up. But I have also heard from women who had, more recently, had such help and considered it pivotal for their careers, and I wanted to learn more about how that can happen. And indeed, there were many women in that room, both among the panelists and in the audience, who had organized such conferences and participated in them, and who had a great deal of experience and practical knowledge to share. They did not just talk about how nice it would be to have more collaborations between women and how good it would be for them, then wait for said women to show up. Instead, they went out to look for candidates and offered them specific, concrete reasons to participate. They helped in setting up collaborative groups, taking into account individual strengths and trying to match people with complementary skills and areas of expertise. There is an actual record of research that gets done at such conferences and in the follow-up work, far more substantial than anything that Math Overflow (for example) could document. Their grant applications do well enough in merit-based competitions.

Closer to my own experience: I once wrote, in a 2011 post that still gets a fair number of views, that I was not active in online communities in general. That has changed. For the last few years, I have been spending less time on non-mandatory professional activities and more on non-mathematical matters. I did get involved in online communities, some public, some less so. I’ve followed others. I did not always feel comfortable there, but I nonetheless found some communities where the benefits of participation did justify the time investment. The point that’s relevant here is that these communities know a great deal about participation, moderation, and group dynamics on the internet. There are no simple solutions or guarantees. No merit point system or software-based fix can compete with humans bent on having their way. There is only a constant stream of complex, human situations that has to be moderated by actual live people who have experience in such matters. Neutrality and impartiality is a myth and a cop-out. The moderators have to be clear on where they stand, what they will allow, and what they wish for the group to accomplish.

Some of the younger Math Overflow enthusiasts might tell me, as they have done in the past, that mathematics is always objective and that mathematical statements are verifiably true or false. As long as the discussion is limited to mathematics, there should be no room for emotions or opinions. To which I would respond, have I mentioned that mathematicians are extremely competitive and that this is not always best for the development of the field? And that I wasn’t the only one to observe that, either? We do not check our ambition at the gate when we open a browser, or our drive to be the first or the most important. I’ve been to enough committee meetings. I’ve seen enough passive-aggressive referee reports and grant proposal reviews. I’ve seen enough young mathematicians complain about the awful ways of the older generation, then follow in their exact footsteps. The internet does not fix such problems; it magnifies them instead.

We need to recognize that there is a wide variety of contributions that can be made to mathematical research, and learn to appreciate and credit properly all of them, not just those we’ve been focusing on. Guess who else has had that problem? There is a vast body of feminist writing on that exact subject, on how the types of contributions that are usually made by women, for social, cultural or any other reasons, also tend to be less valued. There are many studies documenting, for example, sharp salary drops in those professions that become feminized. There are many books and articles analyzing the mechanism of how this happens, the failed attempts to correct it, the circumstances in which a change can take place. I’ve linked to one example in an earlier post: Linda Nochlin’s classic essay on why there have been no great women artists. The essay was originally published in 1971:

Basic to the question are many naïve, distorted, uncritical assumptions about the making of art in general, as well as the making of great art. These assumptions, conscious or unconscious, link together such unlikely superstars as Michelangelo and van Gogh, Raphael and Jackson Pollock under the rubric of “Great”—an honorific attested to by the number of scholarly monographs devoted to the artist in question—and the Great Artist is, of course, conceived of as one who has “Genius”; Genius, in turn, is thought of as an atemporal and mysterious power somehow embedded in the person of the Great Artist.

Does this sound like a problem we might have in mathematics? But keep reading, and you’ll learn more. “Great art” meant, for a long time, large paintings and sculptures with biblical or mythological themes; women were excluded from making such art for the simple and basic reason that they were not allowed to work with nude models. They could still paint landscapes, portraits or still lifes, but those were considered as lesser art. This changed only when a seemingly unrelated social change took place: the ranks of European art collectors, previously limited to aristocracy and high-ranking clergy, were augmented by the emerging middle class looking for art to decorate their rather less palatial dwellings. Without that, not only might Berthe Morisot have never been recognized as a great painter, but we might have never heard of Claude Monet, either.

I’m not asking you to agree with my politics. I’m not even asking you to discuss it with me at all. I’m not saying that Polymaths will be more successful when we remodel our houses, either, so don’t take it too literally. But in the spirit of not reinventing the wheel, I’m telling you that there are plenty of people out there who have studied very similar problems, who have written about them and have tried to address them. Look at the similarities, the differences, take note of what worked and what didn’t, just like we do in mathematics. I’m also suggesting that it’s completely possible to learn from those you might otherwise disagree with. I have been doing that all my life, through the entirety of my mathematical career. I have never had any other choice.

We’re not Luddites, scared of the internet. We’re ahead of you. We’ve likely spent more time on the internet than you did. The feminists, radical leftists, and social justice warriors have long been busy building the kinds of communities that mathematicians, for the most part, only wish for. When we object to unmoderated comment sections, it’s not because we don’t know how they work. It’s because we do. The decline of comment sections and their eventual exit from much of the internet were not at all unexpected from where I was.

Mathematicians can try to learn from that. Or they can reject it all wholesale, refuse to listen, refuse to read the relevant literature because surely they can figure it all out by themselves, then reassure us that Fermat’s theorem shouldn’t really be that hard to prove if we only try to use prime factorization of integers.

Your choice.