A few brief comments on the first iteration of the polymath project, Tim Gowers’ ongoing experiment in collaborative mathematics:

- The project is remarkably active, with nearly 300 substantive mathematical comments in just the first week. It shows few signs of slowing down.
- It’s perhaps not (yet) a “massively” collaborative project, but many mathematicians are contributing – a quick pass over the comments suggests that so far 14 or so people have made substantive mathematical contributions, and it seems likely that number will rise further. Unsurprisingly, that number already rises considerably if you include people who have made comments on the collaborative process.
- Regardless of the outcome of the project, I expect that many beginning research students in mathematics will find this a great resource for understanding what research is about. It’s a way of seeing research mathematicians as they work – trying ideas out, making occcasional errors, backtracking, and so on. I suspect many students will find this incredibly enlightening. To pick just one example of why this may be, my experience is that many beginning students assume that the key to research success lies in having great leaps of insight to solve difficult problems. The discussion shows something quite different: you see excellent mathematicians following up every little lead, trying out many different approaches to problems, seeing many, many ideas fail, and gradually aggregating small insights, as a bigger picture only very slowly emerges.
- The discussion so far has been courteous and professional in the highest degree. I suspect such courteous and professional behaviour greatly increases the chances of success in such a collaboration. I’m reminded of the famous Hardy-Littlewood rules for collaboration. Tim Gowers’ rules of collaboration have something of the same flavour.
- One might say that this courtesy and professionalism is only to be expected, given the many professional mathematicians participating. Unfortunately, it’s not difficult to find excellent blogs run by professional scientists where the comment sections are notably less courteous and professional. I’ll omit examples.
- Initially, I wasn’t so sure about the idea of using the linear medium of blog comments to run such a project. It seemed restrictive to use anything less than a multi-threaded forum, if forum software could be found that was geared towards mathematics. (Something like Google Groups would be good, but it doesn’t provide any way to display mathematics, so far as I’m aware.) The linear format has worked much better than I thought it would. Although at times it makes the discussion difficult to follow, the linear format has the benefit of preventing the conversation (and the collaborative community) from fracturing too much. This may be something to think about for future projects.
- Many large-scale collaborative projects make it easy for late entrants to make a contribution. For example, in the Kasparov versus the World chess game, new participants could enter late in the game and come up to speed quickly. This was in part because of the nature of chess (only the current board matters, not past positions), but it was also partially because of the public analysis tree maintained for much of the game by Irina Krush. This acted as a key reference point for World Team decisions, and summarized much of the then-current best thinking about the game. In a similar way, many open source projects encourage late entry, with new participants able to jump in after looking at the existing code base (analogous to the state of the chess board), and the project wiki (analogous to the analysis tree). As the polymath project continues, I hope similar points of entry will enable outsiders to follow what is happening, and to contribute, without necessarily having to follow the entire discussion to that point.

Thank you this fine summary.

Ongoing review of the Polymath Project’s lessons-learned is similarly valuable to the Polymath Project itself—especially to folks (like me) who are contemplating similar enterprises (in simulation-based system engineering).

Just an idea: Mediawiki, the engine behind Wikipedia, has a module for writing Tex-like math symbols.