A People’s History of Mathematics

Like many others, I like to read about the process of making and discovering science and mathematics. I enjoy reading biographies, and especially autobiographies, and especially whatever Feynmann and Atiyah have written about their own work and process of discovery.

Many histories of science and mathematics, however, and many papers in which people argue how others should do science and mathematics, are premised on a view that bothers me a lot: that at any given time there are a lot of people chugging along doing incremental work, and then once in a while a Great Person (usually, a Great Man) comes along and starts a revolution. Then people start chugging along (or “beavering away”, in Atiyah’s wonderful turn of phrase) in the direction dictated by the revolution.

Needless to say, in this view it is the Great Men who are the main principal actors, and the rest are just extras, if not nearly superfluous.

I think that reality is quite different. It is true that our understanding and knowledge of math and science proceeds quite discontinuously. This is quite obvious in mathematical, where a statement can go, in one day, from being an open question for which there seems to be no promising approach to being a solved problem with a new proof that opens up several exciting new directions.

But these discrete advances are not made by geniuses who magically produce an entirely new discovery. Sometimes, a major advance is genuinely ahead of its time (in theoretical computer science, I think that the creation of the modern foundations of cryptography in 1981-82 was such an example). Usually, however, a major advance happens because the time is ripe. Because the work of all the people “beavering away” has completely clarified a number of tools that were previously less well understood, because a large collection of “incremental” results and “observations” has been made, and all the pieces are there, ready to be put together. In fact, major discoveries are usually made independently and around the same time, with a synchronicity that is often stunning.

This is not to detract from the people who actually make great contributions. If finding the right pieces and putting them together was that easy, then we would have revolutions every day. And if it was just dumb luck, then every great discoverer would be a one-hit wonder, while we tend to see the same people making great contributions again and again.

Indeed, realizing that the time is ripe for an advance, and that all the pieces are there, and that they can be put together, is an amazing skill, which is pretty much what people mean by “genius.” Great works of art are not pulled out of nowhere, they are expressions of the sensibility and culture of a given period; seeing a great work of art (be it a movie, a novel, a painting, an installation) makes us think something like “now I better know how I feel.” Similarly, a great leader is not someone who makes other people do what he wants. It is someone who makes people do what they want, who realizes that there is potential for something to be done if only it is articulated in a way that people can identify in.

So, back to mathematics, an extreme version of the Great Man view is something like Lee Smolin’s The Trouble with Physics, in which he proposes a distinction between “seers” and “craftspeople.” Even in the choice of words, it is clear that “seers” are supposed to make their advances by being “visionary,” and it is also clear whose role is more important.

I have seen few discussions on the process of making science and mathematics emphasizing the importance of collective work and the inevitability of major discoveries once the time is right.

(I like very much the philosophy underlying Terry Tao’s essays on how to do mathematics, starting from the opening quotation, and also his essay on what is good mathematics.)

Those thoughts, which I had for a long time, and certainly ever since reading Smolin’s book two years ago, resurfaced when I read about Tim Gowers’s wonderful polymath experiment.

The idea is to try out Massively Collaborative Mathematics, in which an open question and a direction of investigation are posed, and a large number of people make “atomic” contributions, such as asking questions, answering questions, proposing ideas, counterexamples, and any kind of remarks or partial steps towards a proof.

Indeed, if any significant mathematical contribution comes from some kind of combination of pieces all of them well known to some, then the reason why the discovery was not made before was that nobody already knew all the pieces, or at least not from the right perspective. Hence a large part of any mathematical discovery is to find out the lore of some adjacent research area, which is relevant but often not known to the people working on the question itself. If many people join together in this way, however, their collective knowledge is huge, and this process of (re)discovery can be remarkably accelerated.

The “genius” that I defined before as an ability to put together the zeitgeist, could just be in the union of many minds, each doing nothing more than saying what is obvious to them.

Gowers’s inspiration for this project was this post by Michael Nielsen, whose blog makes for very interesting reading. Nielsen’s inspiration, in turn, is open source software, and indeed he talks about open science.

Mathematics, however, is a form of creative endeavor more similar to art than to engineering, and I think that if Massively Collaborative Mathematics works out, its philosophical, and not just practical, consequences (in terms of how we think of creativity, genius, and “paradigm shifts”) would be remarkable.

13 thoughts on “A People’s History of Mathematics

  1. Although it is not particularly related to mathematics, there was a New Yorker article by Malcolm Gladwell last year, which was about precisely your observation that “major discoveries are usually made independently and around the same time, with a synchronicity that is often stunning.”
    In the Air: Who says big ideas are rare?

  2. Pingback: Mathematics, Science, and Blogs « Combinatorics and more

  3. Dear Luca, I think the issue of polarization is important in science and in society. Fortunately, things are overall more pleasant in science and academics compared to art and music. Are blogs, collective efforts, open science, etc. move things towards better science? towards a less polarized science community? We will have to examine these issues carefully and skeptically.

  4. Dear Gil, could you elaborate on what you mean by “polarization” in this context, and in which sense the science community is polarized?

  5. Maybe I used a wrong English word; I meant to say that in art and music the successful great guys (and gals) are having a lot of money and fame while a lot of “ordinary” people in these areas cannot make a living. Academic life and science are better (“less polarized”) in that respect.

    Regarding the assesement of how science proceeds, the role of collective work, the role of few individuals, inevidability of certain developements etc. I think this is an interesting issue and I simply don’t know. (Similar questions are raised regarding political developments. ) Like you am not fond of the term “revolution” in the the context of scientific developements.

    One obvious difference between mathematics and, say, music is that the “audience” is, to a large extent, also the same set of people as the “performers”.

  6. Instead of seers and craftspeople, Terry Tao uses more neutral language and calls them “theory-builders” and “problem solvers”.

    Given the pressures on funding, we currently have a prevalence of the latter, but as Terry points out, this has not always been the case. Not long ago Erdos status as a mathematician was that of a “mere problem solver” while theory builders like Bourbaki were greatly admired. The delta has narrowed greatly since, from both directions.

    CS being such a young field hasn’t had a chance for much theory building. We are still mostly driven by problem solvers and we (rightly) stand in awe of the best of them.

    However there is no need for a dichotomy here. Surely the field is big enough to create room for both types. Other fields within CS have recently started the Hot* workshops to provide a forum for theory building and theory builders.

  7. Alex L-O: The “theory-builders” and “problem-solvers” terminology actually goes back to Tim Gowers’s well known 1999 essay “The two cultures of mathematics” and is not at all equivalent to what is being talked about here. Tim made the case for the “problem solvers”, by arguing that in the end they’re not that different from the “theory builders”, at a time when the case really needed to be made. In particular, he argued (and I agree) that the top problem-solvers such as Erdos can well be thought of as having the “seer” quality.

  8. Thanks for the link. Having read Tim’s piece, it seems to me that his meaning is very much the one we have in mind here. However he takes the argument in a different direction. He argues that in the end problem solvers (at least in combinatorics) are eventually lead to the same grand arching designs as theory builders.

    My claim is that even if they didn’t, there is a need for both types. Both enrich the field and solving a specific problem (Riemann’s hypothesis, P=NP) can be as important as developing a new theory.

    Interestingly enough, the situation in CS is more or less the opposite of math: there are more problem solvers than theory builders and “the acceptable thing to do” is to be a problem solver.

  9. Pingback: Thurston on Understanding and Collective Work in Mathematics « in theory

  10. Izabella Laba cites Gowers’ 1999 essay as the origin of the terms “theory builders” and “problem solvers,” but these terms were already in common use in exactly this context when I was a student in the late 1960’s and early 1970’s, and arose typically in discussions about why Erdos and other Hungarian problem solvers were undervalued in the mathematical community.

  11. The open source community solves a lot of technical problems, whereas the polymath projects takes one at a time. There is no infrastructure in math, and there is a luck of infrastructure in open source. In particular, the resources are going through the standard academic channels, and through the awards. Both the polymath, and the open source (largely) are build by volunteers, who find support from somewhere else. In this respect the award system, and as a particular example, the Clay millennium problems are slowing down the exchange of the information, whereas there is should be mechanism to support polymath (in more general sense) contributors. The current social model support more action (faster and faster actions) than reflection, at least this is the way I see Lee Smolin dichotomy. And this is going to be the case until Open xxx will get a flow of resources built-in. Open source get a boost when big corporations started to move their project into it together with funds.

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