What Is Research?

March 24, 2009

Concluding notes on the polymath project — and a challenge

Filed under: polymath,Wikis — vipulnaik @ 4:27 pm

In this previous blog post, I gave a quick summary of the polymath project, as of February 20, 2009. The project, which began around February 2, 2009, has now been declared successful by Gowers. While the original aim was to find a new proof of a special case of an already proved theorem, the project seems to have managed to find a new proof of the general case. There’s still discussion on how to clean up, prepare the draft of the final paper, and wrap up various loose ends.

In a subsequent blog post, Gowers gave his own summary of the project as well as what he thinks about the future potential of open collaborative mathematics. Michael Nielsen, who hosted the Polymath1 wiki where much of the collaboration occurred, also weighed in with this blog post.

In Gowers’ assessment, the project didn’t have the same kind of massive participation as he had hoped for. People like Gowers and Terence Tao participated quite a bit, and there were also a number of other people who made important contributions (my own estimate of this is around eight or nine, based on the comment threads, with an additional three or four who made a few crucial comments but did not participate extensively). But it still wasn’t “massive” the way that Gowers had envisaged it. Nielsen felt that, for a project just taking off, it did pretty well. He compared it to the early days of Wikipedia and the early days of Linux, and argued that the polymath project did pretty well compared to these other two projects, even though those projects probably had a lot larger appeal.

Good start, but does it scale?

Before the polymath project began (or rather, before I became aware of it), I wrote this blog post, where my main point was that while forums, blogs and “activity” sound a lot appealing, the greater value creation lies in having reliable online reference material that people can go to.

Does that make me a critic of polymath projects?

Well, yes and no. I had little idea at the time (just like everybody else) about whether the particular polymath project started by Gowers would be a success. Moreover, because Ramsey theory is pretty far from the kind of math I have a strong feel for, I had no idea how hard the problem would be. Nonetheless, a solution within a month for any nontrivial problem does seem very impressive. More important than the success in the project, what Gowers and the many others working on it should be congratulated for is the willingness to invest a huge amount of time into this somewhat new approach to doing math. Only through experimentation with new approaches can we get a feel for whether they work, and Gowers has possibly kickstarted a new mode of collaboration.

The “no” part, though, comes from my strong suspicion that this kind of thing doesn’t scale. As I mentioned in the earlier post, there are by definition very few blogs and forums that can be hubs of activity. Gowers and Tao are good mathematicians, but more importantly, they’re famous, well-respected, and their blogs are closely followed by many others. When an experiment is started by Gowers and promoted by Tao on his blog, that naturally attracts a lot of people to go look at it, and a small fraction of those people might comment. I doubt that an equally well-considered problem-solving attempt would be successful if started by a less famous and well-followed mathematician.

This is an important distinction, because while Gowers’ experiment shows that it is possible to choose a new mode of collaboration when there are sufficiently big names that promote it, it has not yet been demonstrated that it is possible to achieve a new scale of collaboration. In other words, it is not clear whether this will just create yet another “research institute” for problem-solving to add to the plethora of institutes that already exist, or it will enable the explosion of a large number of research institutes. If we just get a couple of “research institutes” — namely online collaborative environments created by a few famous mathematicians, then this in itself is a major step forward, but it isn’t revolutionary.

I’m not saying it will not scale. It just seems unlikely, as of now, that it will — given all that I’ve seen of the general tendency to have a few cluster points of famous people and famous websites.

Open source versus open content versus open science?

How strong are the parallels between the open source movement(s), the many crazes for open content, and open science, of the kind that Gowers, Nielsen, Tao, and many others are trying. I suspect that they are significantly weaker than Nielsen claims. As I described in my previous post, the appeal of a content bag such as Wikipedia is the fact that everybody knows exactly where to get “knowledge” and where to contribute it — a combination of extensive modularization, canonical naming, and strong internal linking make it very easy to locate the Wikipedia entry. This creates a positive feedback loop encouraging people to direct their efforts towards editing the Wikipedia entry.

Open source is a little different from open content. First, the barrier to entry is high — fewer people can write code that runs than the number of people who can write content in a natural language. Second, code doesn’t have the same kind of pinpoint feel as a Wikipedia entry. Thus, there isn’t that much of a mass appeal to writing code. On the other hand, programs and code get used by a much larger number of people than those who’re writing it — so there is a sense of social utility and power that comes from writing good code, or contributing to a larger piece of code. Further, I suspect that there are a lot more people who code compared to the number of people who do mathematical research — which might mean that in comparative terms, mass collaboration for coding is easier.

With open science, the participating group is fairly limited, and the audience is not much larger. Moreover, if open science is designed around ad hoc problem solving, it doesn’t have the Wikipedia-like feature of people knowing where to find and where to contribute ideas. These are all major drawbacks. They are partially overcome when people with relatively higher profiles take up the cause, but can they be completely overcome? I don’t really know, but probably not.

A challenge

It’s exciting to see new efforts, and I’m eager to see what further steps polymath projects take. As I said earlier, we don’t know what works unless somebody tries it, and the more varied the range of things that is tried, the more we get to learn.

My personal focus is on creating a nicely navigable sea of knowledge, information, and ideas, rather than on enhancing participatory forms. The nicely navigable sea should follow the pinpoint principle: easy to locate specific nuggets of facts, determine interesting relationships between them, and explore further. It’s true that there is some sort of complementarity between participation and knowledge creation, but setting the focus on the knowledge that gets created and its degree of accessibility strikes me as qualitatively different from setting the focus on the extent of activity created.

One of the things I’ve been experimenting with is subject wikis, the most developed of them being the group properties wiki. If the measure of success here is participation, then these have not (yet) been successful, though the every-optimist in me feels that they will. If usage is a success parameter, then again, they haven’t been successful yet, but they haven’t entirely been unsuccessful — as I discussed here, the group theory wiki, since it moved to its new location less than a year ago, has recorded more than 75,000 human pageviews (and a lot more views by search engines and bots) — not a large number, but not negligible either. It currently grosses around 8000 human pageviews a month.

What does this have to do with polymath? Not much. But if my basic insight about information availability being more important in the long run than participation, then I could claim that:

A subject wiki on Ramsey theory would be more useful to future potential problem solvers of questions related to polymath1, compared to the polymath1 wiki itself.

That’s a hard hypothesis to test objectively, but there can at least be a kind of rough judgment. I’m interested in taking on the challenge. I’ll get a Ramsey theory subject wiki started. My knowledge of Ramsey theory here is severely limited, so collaboration is gladly welcomed — but the claim I’m making is that in any event, it will soon turn into a more valuable resource than polymath1. Does the claim hold? That’s the challenge here.

As I’ve said before, it is only by trying things that we figure out what works and what doesn’t work, that we subject our convictions to rigorous testing and find out which of them hold water.

UPDATE: It has been six months since I wrote the post and I haven’t gotten around to a subject wiki on Ramsey theory. This refutes my conjecture that it would be esasy to start. My other conjecture, that it could be built to a level where it would be more useful than polymath, remains open.

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7 Comments »

  1. [...] Ramsey theory subject wiki: This is a challenge I took up in this blog post, and I plan to work on it, hopefully reaching an interesting stage by August [...]

    Pingback by Success parameters for the subject wikis – The Subject Wikis Blog — March 24, 2009 @ 5:50 pm

  2. I am wondering what superiority a general Ramsey theory resource would have over what already exists (Wikipedia pages on the subject, Mathworld, etc.).

    If I wanted to contribute to the world’s general understanding of Ramsey theory, I would much rather edit Wikipedia than some isolated resource.

    Comment by Jason Dyer — March 24, 2009 @ 8:46 pm

  3. Hi Jason,

    Your point is well-taken. The whole lure of editing Wikipedia, as I’ve pointed out numerous times, is that quite a few people go there — both to read and to edit. So in order to have something more useful than Wikipedia, I need to offer something that is so much better that the initial advantage that Wikipedia has is overcome.

    Can this be done? I don’t know if I’ll be able to do this for Ramsey theory, but I have built quite an extensive wiki resource for group theory. If you’ve not had a look at it, I urge you to do so, see how it is structured. Even if you don’t have much knowledge of group theory, looking at the structure and organization of the wiki will probably give you a picture of whether this is, or could be, better than what you see at Wikipedia. And I’d like to hear your specific views. (There are also wikis in topology, commutative algebra, and other topics, though they’re not as well-developed).

    Finally, let me mention that Wikipedia is an encyclopedia. The kind of subject-specific knowledge structuring that any deep mathematical subject such as group theory or Ramsey theory, or even a subject such as Euclidean geometry, needs, is neither encouraged nor tolerated at Wikipedia. Before starting the subject wikis, I did contribute some of my knowledge of group theory to Wikipedia, relics of which still remain — however, as I came to understand Wikipedia’s goals and policies, and interacted more with Wikipedia editors, I figured out that this is not the place to contribute such knowledge (Note: This happened much before I became a “Wikipedia critic”. Even in its most idealistic form, Wikipedia does not enable this kind of knowledge structuring).

    Thanks once again for your comment.

    Vipul

    Comment by vipulnaik — March 25, 2009 @ 12:45 pm

  4. I cannot believe this will work!

    Comment by Martin-Edelgard — July 30, 2009 @ 12:25 am

  5. [...] @ 12:51 am It’s been some time since I last wrote about the polymath project (see this, this for past coverage), and an even longer time since I wrote an extremely lengthy blog post about [...]

    Pingback by Collaborative mathematics, etc. « What Is Research? — August 9, 2009 @ 12:51 am

  6. [...] Polymath again Filed under: Culture and society of research — vipulnaik @ 10:42 pm Tags: polymath Timothy Gowers and Michael Nielsen have written an article for Nature magazine about the polymath project (I blogged about this here and here). [...]

    Pingback by Polymath again « What Is Research? — October 26, 2009 @ 10:43 pm

  7. I think the idea is an excellent one, maybe what is needed is bringing it to the attention of more mathematicians… maybe an article in some of the major blogs and also circuilating the web pages to many college and universities….also toall the engineering blogs… and even article to the general public through major newspapers or even on blogs with major readership… sell the idea.

    Comment by rudy — January 7, 2013 @ 3:54 am


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