Category Archives: mathematics: general

ArXiv, comments, and “quality control”

Those of you who browsed the arXiv recently may have seen a link to a user survey on top of the page (as of now, apparently no longer online) (update: still available here, until April 26). I ignored it a few times, until a friend brought this particular bit to my attention.


Sure enough, I took the survey. As it turned out, the arXiv was also asking for feedback on what it calls “quality control”: actions such as rejecting “papers that don’t have much scientific value,” flagging papers that have “too much text re-use from an author’s earlier papers” (self-plagiarism) or from papers by other authors (plagiarism), or moderating pingbacks (such as links from blogs or articles) before they appear on the arXiv.

Internet comment sections are in decline everywhere you look. They are mocked, ridiculed, despised. Many websites have closed them already; others have seen their comments become a racist, sexist bog of eternal stench from which any reasonable person is best advised to stay away. I’ve talked about it here at length, with examples and links, and it’s very easy to google up more if you wish.

I’m often told that if a comment section is restricted to “registered” mathematicians posting under their real names, the conversations will be polite and civil, with the rare instances of abuse identified as such and condemned by the community. If that’s what you think, consider that much of what passes for “normal” interactions between mathematicians is viewed as passive-aggressive, if not downright abusive, just about everywhere else. We all know what referee reports can look like, or grant proposal reviews, or MathSciNet blurbs. If you believe that non-anonymity will solve the problem, I could give you many examples of questions from the audience in seminars and conference talks that were at least as problematic as any referee reports I’ve seen.

Women, in particular, get far too many comments questioning our competence, implying that we might not know the basic literature, that we might not really understand our own results, that said results might turn out to be false or trivial if only someone qualified had a look, or some such. We’re also subject to gendered standards of “professionalism” that do not allow us to respond in kind and give as good as we get. But if you tell me that men, too, can get inane, confused, or malicious comments–why, yes, I agree. More reason to refrain from making the arXiv more like YouTube. There’s enough abusive behaviour in mathematics already, on all sides. We should not mandate a form of discourse that has been shown empirically to promote and escalate it. Nor should we mandate having it attached in perpetuity to our formal publication records.

As for “quality control”: there have been well publicized cases where the arXiv moderators might have overstepped in rejecting papers and blacklisting authors. I’m not a fan of flagging papers for “substantial overlap,” either. We often write several consecutive papers in the same area, introducing the same notation each time, stating the same conjectures or prior results for reference, and so on. We might even reuse parts of the same TeX file for such purposes. None of this amounts to plagiarism or self-plagiarism, nor should it trigger red flags.

Now, here’s what all this might mean for the future of the arXiv. Allow me a little bit of speculation here.

The arXiv has become the universally accepted default repository for mathematicians, not only because it provides a service we need, but also because, in not attempting to do more than that, it gives us no reason to not use it. We don’t have to worry that the paper might not “qualify,” that it’s too long or too short, or too expository, or not sufficiently tailored for the “right” audience. We simply post what we think is right. We expect and welcome feedback (I often post papers on the arXiv prior to journal submission, specifically for that purpose), but the site does not allow public abuse or internet flame wars, so no need to worry about that. The bare-bones structure is not a bug, it’s a feature that has been essential for the arXiv’s success.

Currently, the arXiv has little competition. It works well enough for most of us and we have no reasons to look elsewhere. That might change. Discontent breeds business opportunities. The competing site viXra, started by physicists who were dissatisfied with the arXiv’s moderation practices, failed to gain much ground; but if the arXiv were to amp up its “quality control” in ways that test our tolerance, and especially if it were to implement comments and ratings, there just might be a critical mass of scientists willing to try such alternatives. I know I would be looking for them, and I’ve heard from others (including well known mathematicians, and not only women) who feel the same way.

It would be more than ironic if, say, Elsevier or Springer were to set up a competing open access repository where, for a small fee around $100, authors could post their papers on a site guaranteed to be free of comments and ratings. That would obviously discriminate against those unable to pay $100, but there’s nothing stopping anyone from setting up such a site if there is demand, as I assume there would be. Grant holders in many countries are now subject to open access policies that practically mandate the posting of papers on repositories; should we no longer wish to post on the arXiv, we’ll need an alternative. I can’t promise that I wouldn’t switch to a Springer or Elsevier site, in such circumstances. It would be even better if non-profit organizations, such as the AWM for example, were to set up their own preprint archives where the terms of service would reflect the preferences of the membership.

If comments or ratings are allowed retroactively, on papers already posted to the arXiv, then it’s far from clear to me that the arXiv would be able to hold on to such papers. My contract with the arXiv is, essentially, that the arXiv has my permission to distribute my articles on its website and its mirror sites. It does not have my permission to cross-post them on Reddit and Hacker News. By the same token, it does not have my permission to post them on a future site that might continue to use the URL, but would function in substantially different ways. That would have to be renegotiated. Individual mathematicians may have little power in that regard, but if major publishers become involved as per the above, and if they decide to encourage researchers to move their past publications to their servers, then I could think of some interesting ways in which this could develop.

My crystal ball here may well be less than perfect, but I think that some version of this would have to happen. If the arXiv wishes to remain the universal default repository for scientists in the covered areas, the plain vanilla model is the only one that will do that. Quality control is better left to journals, and for those authors who wish to have public discussions about their papers, a wide range of blogs and social media is available. Any changes that alienate a substantial group of users will inevitably lead to the rise of competition, and so within a few years we might well see a variety of arXiv-type sites with different functionalities and user bases.

And that would essentially end the arXiv as we know it.

Update, July 8, 2016: for those coming late to it, I’m also quoted in this Wired article by Sarah Scoles.

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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.

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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.
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Diversity and mathematics

bell curve1

Mother Jones, last year:

According to a new psychology paper, our political passions can even undermine our very basic reasoning skills. More specifically, the study finds that people who are otherwise very good at math may totally flunk a problem that they would otherwise probably be able to solve, simply because giving the right answer goes against their political beliefs.

I was reminded of it while reading the article “Does Diversity Trump Ability? An Example of the Misuse of Mathematics in the Social Sciences” in the Notices of the AMS. The author, Abigail Thompson, takes on a well known and widely cited paper:

“Diversity” has become an important concept in the modern university, affecting admissions, faculty hiring, and administrative appointments. In the paper “Groups of diverse problem solvers can outperform groups of high-ability problem solvers” [1], L. Hong and S. Page claim to prove that “To put it succinctly, diversity trumps ability.” We show that their arguments are fundamentally flawed.

Why should mathematicians care? Mathematicians have a responsibility to ensure that mathematics is not misused. The highly specialized language of mathematics can be used to obscure rather than reveal truth. It is easy to cross the line between analysis and advocacy when strongly held beliefs are in play. Attempts to find a mathematical justification for “diversity” as practiced in universities provide an instructive example of where that line has been crossed.

Thompson proceeds to shred both the “mathematical theorem” and the numerical examples from the Hong-Page paper. The actual paper is available here, and I have satisfied myself that Thompson is not unfair in her mathematical analysis. Her article, however, does not exist in a vacuum. It will be read in mathematics departments, organizations and committees where “diversity” is viewed as a bureaucratic imposition made on them by distant administrators who don’t understand research, even as their few women faculty often find themselves alienated and sidelined. That’s why I would like to add a few things.

First, there are many sound reasons for diversity that have nothing to do the article in question. (I will restrict this post to the benefits of diversity per se, independently of how that diversity was achieved. Affirmative action has its own reasons and will get its own post soon.) It should be common sense, not a mathematical theorem, that there are advantages in having a wider perspective and more than one problem-solving approach. Continue reading

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Filed under academia, feminism, mathematics: general, women in math

Art in the life of mathematicians


This book has been in the works for some years now, and I’m thrilled to finally have a demo copy to show you. The book will be published by the American Mathematical Society. The demo copy has been produced (impressively quickly!) by the Hungarian publisher Ab Ovo. I’m very grateful to Anna Kepes Szemer√©di for envisioning this project in the first place, and for all the hard work she has put into it.

I have contributed an essay on photography. You can download it here, and here is the gallery of the photos I offered to be used in the book. The photo on the cover is also mine. I hope that this will encourage you to purchase the book when it becomes available; I’m only one out of many contributors (see the cover for the list of names), and the book format will add further value through graphic design. If you’re expecting “mathematical art” as exemplified for example by the Bridges conference, I must warn you that this is not what I do. (In the essay, I explain why.) There is some overlap with one of my blog posts from last year: the post was adapted from an earlier version of the essay, and then I used it in writing the final version.

Anna first approached me about this in late 2011. I was much less confident then, both in my photography and in my writing. I have worked on both since then. One thing I wish I’d seen before I submitted my contribution is this classic piece by Linda Nochlin on the absence of great women artists in the history of art. Here’s what she says about “the lady’s accomplishment”:

In contrast to the single-mindedness and commitment demanded of a chef d’ecole, we might set the image of the “lady painter” established by 19th century etiquette books and reinforced in the literature of the times. It is precisely the insistence upon a modest, proficient, self demeaning level of amateurism as a “suitable accomplishment” for the well brought up young woman, who naturally would want to direct her major attention to the welfare of others–family and husband–that militated, and still militates, against any real accomplishment on the part of women. It is this emphasis which transforms serious commitment to frivolous self-indulgence, busy work, or occupational therapy, and today, more than ever, in suburban bastions of the feminine mystique, tends to distort the whole notion of what art is and what kind of social role it plays.

This got me thinking back on what I wrote about photography and wondering for a moment if I might have fallen into the trap of “suitable accomplishment.” In the end, it clarified for me the distinction between the commitment to the process of getting better, and the expectation of achieving a certain level of excellence, and the expectation of gaining public acclaim. I have always been anything but unambitious. Nonetheless, I have never aimed to be a “great artist.” I am not altogether indifferent to success in art, as evidenced by this self-promotional post, but what made me pick up the camera is the pleasure I find in taking photographs. My enjoyment of it is not conditional on finding an audience, receiving public recognition, or on any presumption of greatness. Instead, it comes from trying to get better at it. The pleasure is not in taking the same photographs over and over again, but in expanding my range, improving my technique, seeking out new ideas and solutions. The seriousness of my commitment is in my engagement in the process.

I suppose that this does not make me a lady.

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Would a mathematician pass the Turing test?

You’ve probably seen the recent announcement that a computer passed the Turing test for the first time. The announcement was followed by a healthy round of skepticism and mockery, with transcripts of chats with “Eugene” posted on a number of websites. Among other things, it has been pointed out that introducing “Eugene” as a 13-year-old Ukrainian boy holding a conversation in English would dupe most people into giving him much more slack with regard to awkward language and deficient social graces than they might otherwise.

Well, why Ukrainian boys and not mathematicians? I didn’t get to chat with Eugene. However, here is a conversation that I might or might not have had with an internet user that we’ll call Boris. I’ll let you decide.

IL: Hi Boris. I’m Izabella Laba and I work at UBC.

BORIS: Hi Izabella, it’s nice to meet you. I’m Boris. So what classes do you teach this semester?

IL: Actually, it’s summer and I don’t teach.

BORIS: That’s very interesting. So how many students do you have?

IL: I’ve just told you I don’t teach in the summer. But tell me about your research.

BORIS: As you probably know, I work on modulated gvoorups on questable aussifolds. I have proved that if a modulated gvoorup has a subquestable chain of hyperchenettes, then the aussifold must be oubliettable. This links several areas of mathematics and should have implications for the rapidly developing field of quasialgebraic oubliettability. You can read my papers to learn more about this interesting and exciting area of research.

IL: That sounds fascinating. What is a gvoorup?

BORIS: You can read my papers to learn more about it.

IL: … OK. But can you tell me why you are interested in gvoorups?

BORIS: It is a very interesting and exciting area of research. So what do you work on?

IL: Well, harmonic analysis on fractal sets.

BORIS: It is a very difficult area of research in which it is very difficult to have any new ideas.

IL: How do you know that?

BORIS: This is well known to everyone in the field, even if it has never been published. So who do you work with?

IL: Uhm, I’ve been a full professor for some time now. Are you assuming that I’m a junior researcher because I’m a woman?

BORIS: Women are more interested in teaching than in research. If you’re looking for more women, you should go to a teaching related forum.

IL: I’m pretty sure I’m more interested in research than in teaching.

BORIS: That’s great. So what classes do you teach this semester?

IL: You’re repeating yourself. I’ve answered this already.

BORIS: That’s very interesting. Have you seen my paper on gvoorups from 1995? It may be relevant to your work.

IL: Honestly, I doubt it.

BORIS: It’s been nice meeting you. Please let me know if you have any questions about my paper.

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St. Augustine, Thomas Aquinas, dogma and mathematics

A few weeks ago, I finally got around to reading “Between the Lord and the Priest”, a book-length conversation between Adam Michnik, Jozef Tischner and Jacek Zakowski. I came for the historical content, but stayed in part for certain disputes in Catholic theology in Poland in the 1960s and 70s. It’s not my usual cup of tea; Tischner himself acknowledges that all this was of very limited interest to the general public while trailing well behind contemporary Western European philosophy. It nonetheless describes beautifully some of the disagreements I’ve had with my fellow mathematicians with regard to life in general and social issues in particular.

The framework for the dispute is provided by the long-standing dichotomy between St. Augustine and St. Thomas Aquinas. The way Tischner explains it, Thomism posits eternal, unchangeable truths that must be accepted as dogma and followed in life. It prescribes synthesis, universality, vast generalizations, logical chains of cause and effect all the way back to deity. Augustine, on the other hand, is less sure of himself. Even if the truth, somewhere out there, might be eternal and unchanging, our understanding of it is grounded in history, tradition and experience, and in the end that understanding is all we can ever access. In practice, this is a more bottom-up approach to religion, starting with personal, individual existential questions and then seeking guidance in the Scriptures and the church’s intellectual tradition.

Now, here is where things get interesting. Tischner goes on to say that Thomism, in its methodology and spirit, is actually quite similar to Marxism. Marxists, too, had their axioms of class struggle and dialectical materialism. They presumed to shape human consciousness through class awareness, much as Thomists presumed to shape it through philosophical and religious dogma, with little regard to individual experience and understanding.

That was why Michnik, an atheist and a leftist at odds with communism, tuned into Tischner’s polemics with Thomists. Thomism, like Marxism, represented codified, linear thinking where “one thing always follows from another, and everything is perfectly arranged and therefore very simple.” Tischner found that he could not talk like that to his parishioners – people who’d fought in the war, lived through the horrors of Nazi occupation, made choices that most of us wouldn’t want to think about. Their experience defied the scheme. Michnik, then in his twenties and already a veteran of protests and prisons, trying to graduate from university before his next arrest, had no love for simple explanations of everything, either. He’d rejected Marxism already; he would go on to consider religion, but not if it offered no escape from the same kind of closed-minded thinking, not if it were perfectly arranged with one thing always following from another.

At times, Marxist-Leninist philosophy was almost comical in its straightforwardness. Michnik cites Lenin’s theory of cognitive reflection, asserting that

(1) a world exists “independent” of and “external” to consciousness, and (2)
knowledge consists of approximately faithful “reflections” of that world in consciousness.

The second part of that, understood literally as Lenin indeed intended, is, on a very basic level, at odds with science, and I could say much more along these lines just based on my experience with photography. What’s less funny is the underlying Thomist assumption that there can only be one intellectually correct interpretation and only one right set of conclusions, namely those espoused by the bearer of the dogma, and that any departure from that must be a result of either misinformation or bad faith. When communists censored dissenting opinions, part of it was a genuine conviction that such opinions were obviously nonsensical and therefore there was no reason to disseminate them. When they lost the 1946 referendum in Poland, they blamed it on “confused thinking” and “complete ignorance” among the population. In a similar vein, but centuries earlier, the Catholic Inquisition might first try intellectual arguments, but if the accused were not persuaded, that constituted proof that they were possessed by the devil, because how else could they not agree? Thomists responding to Tischner informed him on a regular basis that he did not really know St. Thomas, because had he known him, he’d love him.

I started drawing my own analogies long before the point where Tischner actually uses the word “mathematical.” Like a good Augustinian, I’ll start with specifics. A couple of weeks ago, in a comment section far away, a mathematician proposed to “solve racism” by generalizing it (to something he never quite defined) so that racism itself would follow easily as a special case. In a different comment section last year, several mathematicians insisted on a purely mechanistic solution to sexism in mathematics. They accepted it as a self-evident axiom that mathematicians were progressive and well-intentioned people who would automatically eliminate sexism from their ranks if it only were pointed out to them. One might of course wonder why it hasn’t worked yet; but one would then be doubting an axiom, an act that’s not only morally reprehensible but, worse, logically inexplicable.

I’m thinking of mathematicians who’ve argued with my blog posts by taking shots at some sentence pulled out of context, the way they might point to an incorrect formula in a math paper. I’m thinking of one person who started a discussion with me, then allowed reluctantly in response to my arguments that he might not be able to change my mind after all, because convincing people is hard in general. Apparently, the possibility of me convincing him had never been on the table. I’m thinking of those who expected I’d stop believing in that gender bias thing if they only could explain it all to me, almost like religious evangelists. Sorry, no. I’ve heard your arguments many times already. I disagree with them, not because I don’t understand them well enough, but because I do. They don’t address my experience, and they never will if you keep starting from your own axioms instead.

It became clear to me over the last couple of years that I’m not, and probably never have been, part of a “mathematical community” of any kind. Sure, some of my best friends are mathematicians. I do my expected share of “service to the community.” But after hours, I’d rather kick off my shoes with people who at least share my logic. I’d rather discuss experiences, not axioms. I’d rather debate someone who’s actually listening to me, not just building his own castles of abstraction.

It’s been claimed by some of those in question that mathematics itself supports Thomist thinking. (As in, “I’m a mathematician and therefore this is how I approach this problem.”) I’m not so clear on that. To some extent, sure: we’re all trained in binary logic and deductive reasoning, as we should be. But in my own research practice, I often work in the Augustinian direction, starting with specific examples and then working towards something more general. Freeman Dyson’s “Birds and Frogs” article comes to mind, except that I’ve met froggy types like me who are incredibly dogmatic on social issues, and birds who are not.

If I were a Thomist, I’d try for a diagnosis, conclusion, and a list of recommendations for my peers. Maybe the problem is when mathematicians Act Like Mathematicians, showing off their smarts where wisdom is called for. Maybe, too, it’s unexamined purchase into the letter of the deductive philosophy of Russell and Bourbaki, without stopping to consider the actual practice of mathematical research; but that argument becomes circular right there. So, instead, I’d rather just leave you with something to think about, and excuse myself from Math Overflow once again.

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