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.

Various updates

Polynomial configurations in fractal sets: Kevin Henriot, Malabika Pramanik and I have posted a paper where we prove the following result: if a measure μ on a fractal set E in Rⁿ has Fourier decay with some exponent β, and if it also obeys a ball condition with exponent α close enough to n (depending on β and on the constants in both conditions), then it must contain nontrivial configurations given by certain types of systems of matrices with a polynomial term. This is somewhat similar to my earlier paper with Vincent Chan and Malabika Pramanik, on configurations given by systems of linear forms, but there are significant differences. One is of course the polynomial term: we use stationary phase estimates to control the corresponding part of the “counting form” Λ. (Interestingly, while said stationary estimates apply to functions much more general than polynomials, the polynomial form of the nonlinear term is required for the “continuous” estimates which are based on a number-theoretic argument.) Another is that any rate of Fourier decay β>0 suffices, with the caveat that α must be close enough to n, where “close enough” now depends on both the constants and β. This improvement is due to more efficient use of restriction estimates, and extends to the result with Chan and Pramanik as well as my earlier paper with Pramanik on 3-term arithmetic progressions in fractals. A recent result of Pablo Shmerkin shows that the dependence on constants cannot be removed: he proves, for example, that there exists a 1-dimensional (but of Lebesgue measure 0) Salem set on the line that does not contain a nontrivial 3-term arithmetic progression.

Fractal Knapp examples: Kyle Hambrook and I have been asked on various occasions whether our “Knapp example” for fractal sets on the line could be extended to fractals in higher dimensions. In this paper, we combined our construction (with modifications due to Chen) and the classical Knapp example on the sphere to produce fractal Knapp examples of dimension between n-1 and n in Rⁿ.

My profile for Women in Maths: this was published a while ago, in case anyone here is interested.

It’s been a while since I posted any photos here, so here’s one I took today. There will be more on my Google+ page.

A response to Scott Aaronson

Scott Aaronson has been kind enough to respond on his blog to a couple of my tweets. I would like to thank him for his interest and engagement, and encourage everyone to take the time to read his entire post. There is also an excellent discussion in the comments.

Much of the discourse focuses on the use and misuse of jargon in social and physical sciences, and specifically on words such as “privilege,” “delegitimization,” or “disenfranchised.” I’ll address that in a moment, but let me first say that my main reason for objecting to the comment that started this discussion was the phrase “This isn’t quantum field theory” at the end. I understood this, in the context of that comment and the comment to which Aaronson was responding, and in light of the similarity to the well known phrase about rocket science, to imply that social sciences do not have the same complexity as quantum field theory and should not need a multilayered structure where concepts are defined, compared, then used to define further concepts, whereupon the procedure is repeated and iterated, so as to make advanced discourse possible and manageable. Aaronson has now explained that this would be an oversimplification of his position, and I’m glad to stand corrected.

I also would like to speak to some of the other points that he makes about language, feminism, social science, and clarity of writing. I’ll try not to repeat the arguments that his commenters have already made, perhaps better than I could have done it. Still, I have no desire to hide (as some have suggested) behind Twitter’s 140-character limit and avoid making my case at more length. And so, here we are. I will just quote the last two paragraphs from Aaronson’s post, but please do go to his site to read the rest:

Discrete Analysis

You may have seen Tim Gowers’s announcement last week, but if not, here’s the news: we are launching a new arXiv overlay journal called Discrete Analysis. The editorial board consists of Tim Gowers (who will be the managing editor) and Ernie Croot, Ben Green, Gil Kalai, Nets Katz, Bryna Kra, myself, Tom Sanders, Jozsef Solymosi, Terence Tao, Julia Wolf, and Tamar Ziegler. As should be clear from this list of names, the journal will focus on additive combinatorics and related areas such as harmonic analysis, number theory, geometric measure theory, combinatorics, ergodic theory. The temporary journal website is open now, in fact we have already received the first submissions.

“ArXiv overlay” means that we will not be “publishing” papers in the traditional sense. Most of us already typeset our own papers and use the arXiv for quick, reliable, stable worldwide dissemination of our results. It is not clear that mathematical journals can improve much on that; if anything, publication in established journals is currently more likely to impede the dissemination of science through paywalls or embargos than to facilitate it. What we can provide is a refereeing and certification service where we manage the peer review and, when the outcome of the review is positive, attest through publishing the link on the journal website that the paper has been judged to be of suitable quality for publication in Discrete Analysis. Tim’s post has much more information on both the scope of the journal and the technical details of how we expect it to work. If you are finishing an article in one of the covered areas of research, I hope that you will consider Discrete Analysis as a possible publication venue. I’m proud to be on its board.

A few more inside-baseball comments under the cut.

A postscriptum on diversity and learning a language

“The man of the East cannot take Americans seriously because they have never undergone the experiences that teach men how relative their judgments and thinking habits are. Their resultant lack of imagination is appalling. Because they were born and raised in a given social order and in a given system of values, they believe that any other order must be “unnatural,” and that it cannot last because it is incompatible with human nature. But even they may one day know fire, hunger, and the sword.”

— Czesław Miłosz, The Captive Mind

I grew up in Europe, on the other side of the Iron Curtain. I’ve often had to try to explain my country of origin to those born and raised on this side of the Atlantic. Facts can be learned. It’s the lack of imagination that can be the greater problem. It’s disbelief that learning is in fact needed. It’s making assumptions instead of asking questions. It’s demanding a simple picture where the truth is complex. It’s presuming social or political homogeneity where the reality is ripe in conflict and discord. It’s failing, or perhaps not wanting, to understand just how far the circumstances of a different time and place might be from the here and now. and to accept that, were we placed there and then, we would likely behave the same way as those who were in fact so placed.

I’m neither a historian, nor a social scientist, nor willing to accept an unpaid second job. I can only do it in small steps, for my own pleasure. Even just for that, I needed a language that I could use. I needed examples and templates, in English, that I could try to work with. For a long time, I could not find what I wanted. English-language history books, for the most part, neither understood nor cared much about our life down on the ground. At the same time, I had too little in common with those Eastern European writers whose goal in writing was to distance themselves from their own background before witnesses who shared that background and, often, the distancing. That was not the argument I wanted to have. History has already passed judgement on communism and I’m satisfied enough with its verdict. I do, however, want to argue with those who view us with a mixture of pity and condescension, who consider the details of our history unimportant, who dismiss without looking the artistic and intellectual accomplishments of the Eastern Bloc as “couldn’t possibly have been any good,” who bounce the word “communism” here and there like a beach ball but have no idea how that system actually worked.

If you are reading this, you may have already seen my last post on the legacy of Communist and Soviet symbols in Poland:

I learned to give little thought to the walled-off parts of the city. The [Soviet] soldiers were easy to ignore in my daily life: they marched through our streets on their way to or from exercises, but otherwise they and their families stayed within their gated communities. I grew up mocking the unkempt buildings with newspapers in place of window curtains, but also reading children’s books from the Russian bookstore, which was open to the public; as a university student, I returned there for mathematical monographs unavailable in Polish. We resented that the Soviet food stores were well stocked even when ours were empty. Poles, especially children, would sometimes sneak in and shop there: a guard might look the other way, a Russian woman might allow a Polish kid to come in with her. I dreamed of travelling the world, becoming a scientist or an astronaut, but did not know and probably could not imagine what it might be like to live in a city without the Soviet army.

For comparison, here’s an article on how living with Confederate flags and statues in the south of the US was “like having a crazy family member.”

For those of us not born and bred below the Mason-Dixon, it can be really jarring to encounter symbols of the Old South sprinkled all over the place, as though by a casual hand. But given the ubiquity of these symbols, it makes sense that you’d kind of have to let them fade into the background, or you might never leave your house. […]

Everyone deserves to have local pride; it’s just that for a lot of black people in the South, getting to do that means having to swim in the racial messiness that comes with civic life there. The cultures of Southern black folks and Southern white folks have always been defined by a peculiar, complicated familiarity. That might explain why so many black folks have — by necessity — come to look on displays of the Confederate flag with something subtler than apoplexy, why Naima just rolled her eyes at the flags on her campus and moved on. Like a lot of black Southerners, she clearly had a lot more practice holding all of these ideas in her head at once than we Northerners do. The flag matters to her. Of course it matters. It’s just not the only thing that matters.

The weight of dead symbols

The photos in this post are mine, from my visit to Poland in late May and early June. The full set, annotated and viewable as a slideshow, is available here. I have provided English-language links where I could, but much of the information I used is only available in Polish. I marked those links with an asterisk, to save you a click if you do not speak the language.

The Polish-Soviet Friendship (officially, Brothership-in-Arms) Monument in Legnica was built in 1951*. It stands in the Słowiański Square, right in the city center. Two soldiers, one Polish and one Soviet, shake hands while a little girl held by the Pole embraces both of them. Poland, invaded by Germany and the Soviet Union acting in agreement, then devastated in the conflict between them, occupied and plundered by both even as its soldiers fought on every front they could find, finally claimed by Stalin for his Soviet empire, had to be represented as a little girl with no memory, history or trust issues, happy in the care of her saviours. Of course the child had to be a girl. A boy might not project the same naivetė, helplessness or passivity.

The monument is still there. I photographed it just a few weeks ago. The inscription, “To the Soviet Army heroes, from the people of the Legnica region,” had been removed in the 1990s and was never restored. The statue was vandalized repeatedly; past renovations notwithstanding, the neglect is palpable.

From 1952 until the end of the Cold War, Legnica was home to the headquarters of the Soviet forces stationed in Poland*. From 1984 to 1990, it also hosted the central command of the Western Theatre of Strategic Operations of the Warsaw Pact: had the Pact attempted to invade Western Europe, the military directions would have been issued from there. Estimates point* to 80-100K Russian soldiers and civilians stationed in town and at various unmapped bases nearby at any given time. The precise numbers and locations were classified, as was the exact layout of the Soviet-occupied parts of the city, surrounded by walls and guarded by armed soldiers posted at each entrance. The largest one, the “Kwadrat” [Square], measured 39 hectares and was a miniature city within a city, self-sufficient with its own shops, hospitals, cinemas, pools and sport venues. In total, the Soviets occupied about a third of the city’s pre-war area.

I grew up not far from that monument. I often walked past it on my way back from school but rarely thought about it. We acquired the skill of inattention in response to the relentless barrage of words and images that ranged from the hostile to the nonsensical*. “Workers of the world, unite!” “PZPR [the communist party] – the working class’s party, the leadership of the nation!” “We build socialism for people and through people!” Above all, invocations of friendship and brotherly love between Poland and the Soviet Union. Unlike ordinary human friendships that enter quietly and tie little knots here and there, that friendship could not be anything less than eternal, was written into the Polish constitution and had to be pledged and re-pledged every day in the streets of every city. We learned to tune it all out except to mock it. That skill continues to come in handy. Corporate language, often no less Orwellian than Soviet propaganda, washes off me like water off an oiled plate. I can look at ads and zap them off my computer screen without ever engaging with their content. (Sorry, Google and Facebook.)

Gender, conferences, conversations and confrontations

My departmental colleague Greg Martin has posted a paper entitled “Addressing the underrepresentation of women in mathematics conferences.” A comment and a bibliographical reference on page 9 of the text inform us that the paper is intended for publication in the Notices of the AMS. [Update, 3/20: I have been informed by the Notices of the AMS that they did not solicit the paper and will not publish it.] In the acknowledgement at the end, the author thanks “other friends and colleagues, too numerous to list here, for their encouragement and inspiration.” Given that we are employed in the same department, and that I often write here about gender, one might ask whether that large number included me. I would like to make it clear that it did not. Had anyone asked for my opinion, I would have discouraged it and, instead, encouraged the Notices to solicit a very different article.

I would have told them that such an article needs to be grounded in extensive firsthand knowledge of our practices related to conference organizing in mathematics. For that reason, it should be written by someone–better yet, by a group of authors–with broad experience in organizing conferences and an established record of promoting women and minorities in that context. It is not enough to point to the discrepancy between the gender proportions at the bottom and the top of the pyramid, and fall back on studies of gender bias in other fields for an explanation. It is necessary to diagnose the mechanisms that lead to it, addressing directly and specifically our actual practices. That requires experience and access to information including confidential and protected material. If a recommendation is made, it should first be tested in real-life conference organizing, and the results of such attempts should be analyzed. I would also insist that it should be written by a woman or a team of authors including women, and not only because women have direct knowledge of gender bias that men cannot have. Were the Notices to publish an article on the subject, it is likely that this would be suggested as a resource for prospective conference organizers; I know of at least one such attempt before the paper was even submitted. I do not believe that the article can have the necessary moral authority without a woman’s name on it.

Martin starts with, “In the context of mathematics conferences, the subject of gender is somewhat of a taboo. Certainly, bringing up the subject at all during a conference would be deemed outside the norm.” This is not true in my experience. I have organized many conferences. The NSF “broader impact” criteria include “broadening the participation of groups underrepresented in science, mathematics, engineering and technology,” and this carries disproportionate weight in mathematics as other ways of meeting these criteria are rarely available to research mathematicians. Mathematics institutes, in addition to being funded by the NSF and therefore accountable to it, often have their own diversity mandates. The organizers of conferences held under their auspices must report explicitly the number of women speakers and are often asked to increase that number. I have also attended many conferences. I have not found it uncommon, or outside the norm, for the participants to talk about gender-related issues in the space reserved for unstructured interactions. I have had many such conversations myself and have witnessed many more.

It is possible that Greg Martin’s experience has been different. He and I rarely attend the same conferences or talk to the same people. But these sentences point to a deeper issue, and not just with this article: the common belief that the gender problem in mathematics could be fixed if we only talked more about it. I disagree. I have said that I witnessed many conversations on gender at mathematics conferences. I did not say that they were all part of the solution. “Bringing up the subject” can mean complaining about the NSF diversity requirements, pointing out this woman or that one who was clearly only invited because of affirmative action, or explaining how we would all gladly invite more women if only they were a little bit better, even as we reassure everyone within hearing range that we totally believe in gender equality. We sure talk about gender. In terms of pure volume, we may be close to the saturation point already. It is not clear that this is helping.

There follows a long overview of literature on implicit bias and gender discrimination. None of these studies or findings are new to me. I’ve seen them on many feminist blogs and Twitter feeds, have linked to them and written about them here. Still, there is no shortage of people who are less familiar with the subject, and I will be glad if such a reading list is delivered to the mailbox of every mathematician in America and beyond. That is long overdue.

Unfortunately, the original research is problematic. It includes an analysis of the gender make-up of two conferences, the 2014 International Congress of Mathematicians in Seoul and the 2014 Joint Mathematics Meetings of the AMS and the MAA in Baltimore. For both meetings, Martin sets the target benchmark for female participation at 24%, based on the fact that at least 24% of doctoral degrees in mathematics at U.S. institutions were granted to women in each year since 1991.