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The Polymath project: scope of participation

by Michael Nielsen on March 20, 2009

As I’ve mentioned before, over the past seven weeks mathematician Tim Gowers has been running a remarkable experiment in how mathematics is done, a project he dubbed the Polymath1 project. Using principles similar to those employed in open source programming projects, he used blogs and a wiki to organize an open mathematical collaboration attempting to find a new proof of an important mathematical theorem known as the density Hales-Jewett (DHJ) theorem. The problem was a tough one. Gowers, an accomplished professional mathematician, initially thought that the chances of success were “substantially less than 100%”, even adopting a quite modest criterion for success.

Last week, Gowers announced that the problem was “probably solved”. In fact, if the results hold up, the project has exceeded expectations. The original goal was to find a new proof of an important special case of the DHJ theorem using a particular approach Gowers suggested (or explain why that approach failed). This goal broadened over time, and the project appears to have found a new proof of the full theorem, using an approach different to that Gowers originally proposed. A writeup is in progress. Inspired by the work of Polymath1, mathematician Tim Austin has released a preprint claiming another proof of DHJ, and citing Polymath1 as crucial to his work.

The scope of participation in the project is remarkable. More than 1000 mathematical comments have been written on Gowers’ blog, and the blog of Terry Tao, another mathematician who has taken a leading role in the project. The Polymath wiki has approximately 59 content pages, with 11 registered contributors, and more anonymous contributors. It’s already a remarkable resource on the density Hales-Jewett theorem and related topics. The project timeline shows notable mathematical contributions being made by 23 contributors to date. This was accomplished in seven weeks.

The original hope was that the project would be a “massive collaboration”. Let’s suppose we take the number above (23) as representative of the number of people who made notable mathematical contributions, bearing in mind that there are obviously substantial limitations to using the timeline in this way. (The timeline contains some pointers to notable general comments, which I have not included in this count.) It’s certainly true that 23 people is a very large number for a mathematical collaboration – a few days into the project, Tim Gowers remarked that “this process is to normal research as driving is to pushing a car” – but it also falls well short of mass collaborations such as Linux and Wikipedia. Gowers has remarked that “I thought that there would be dozens of contributors, but instead the number settled down to a handful, all of whom I knew personally”.

These numbers take on a different flavour, however, when you note that the number of people involved compares favourably even to very successful open source collaborations, at the seven-week mark. 7 weeks after the inception of Wikipedia it had approximately 150 articles. This is considerably more than the Polymath1 wiki, but keep in mind that (a) the Polymath1 wiki is only a small part of the overall project; and (b) I doubt anyone would disagree that the average quality on the Polymath1 wiki is considerably higher. Similarly, while Linux has now received contributions from several thousand people, it took years to build that community. 6 months after Linus Torvalds first announced Linux there were 196 people on the Linux activists mailing list, but most were merely watching. Many had not even installed Linux (at the time, a tricky operation), much less contributed substantively. I’m willing to bet more than 196 people were following Polymath1.

A great deal can be said about scaling up future projects. I believe this can be done, and that there are potentially substantial further benefits. For now, I’ll just make one remark. Long-lived open-source collaborations sometimes start with narrowly focused goals, but they typically broaden over time, and become more open-ended, allowing the community of contributors to continue to grow over time. That’s certainly true of Linux, whose goal – the construction of an operating system kernel – is extremely broad. At the same time, that broad goal naturally gives rise to many focused and to some extent independent problems, which can be tackled in parallel by the development community. It may be possible to broaden Polymath1’s goals in a natural way at this point, but it seems like an interesting challenge to at the same time retain the sharp problem-oriented focused that characterized the collaboration.