An overview of the Polymath projects so far

I've been fascinated by these projects, but I felt that I didn't have sufficient time to really do them justice here. Michael Nielsen has discussed them in several venues so it wasn't clear what I could add. Then I thought about it some more, and I realized that I probably do have different readers than Michael and my view is definitely different than his (plus he nudged me on friendfeed) so here's a discussion for you.

After that rambling preface - you might ask, what's Polymath? It's the name of this project to do massively collaborative mathematics first suggested by Tim Gowers on his blog. He suggested that even a problem that isn't easily segmented might be worked collaboratively online by mathematicians distributed throughout the world. Individual contributions might be like brainstorms - ideas thrown out - not thoroughly completed proofs done offline and uploaded. That having a diversity of people participating would increase creativity and increase the diversity of skills and knowledge brought to bear.   Individual contributions of time and energy would be small, but the actual problem solving would be fairly fast because of the number of collaborators. Some of the concerns about credit and attribution would be addressed by keeping all of the communications public.

This sounds great but the core group has to solve some problems to get this to work. First, they have to find problems that are amenable to this type of work. Second, they have to get participation from knowledgeable mathematicians (these problems aren't something anyone with calc 101 can help with).  Third, they have to find a software tool (or group of tools) that does what they need - that is, once they know what they need.

This is a work in progress, with everyone learning as they go. Learning about collaboration and communication as well as about social software and about math.  Even in this first post linked above, Gowers lays out some policies for the project that, IMHO, provide great value in increasing sociability. He emphasizes being polite, keeping the work online, and keeping focus.

The first project was wildly successful with lots of participation. There are lessons learned on Gowers' blog.

On the plus side, the mathematical result of the project has far exceeded what I thought would be possible in a mere six weeks. ...

Also on the plus side, the project has been genuinely collaborative, and has led, to a remarkable extent, to the kind of efficiency gains that I was hoping for. To give one example, Randall McCutcheon made some very useful comments, but they were in the language of ergodic theory, which I understand only in a very limited way. But Terence Tao is a master at translating concepts back and forth between combinatorics and ergodic theory, so I was able to benefit from Randall's contributions indirectly.

But something I found more striking than the opportunity for specialization of this kind was how often I found myself having thoughts that I would not have had without some chance remark of another contributor. I think it is mainly this that sped up the process so much.

It was also successful in pointing out the shortcomings of doing this work on the original WordPress blog (+wiki and spreadsheet). Some of these shortcomings included:

• it was too fast paced, so some potential contributors gave up
• the question might not have been ideal - required lots of specific background knowledge
• threading vs. no threading
• being in a browser vs. a mailing list, for people who don't live online (intentionally so they aren't distracted)
• no ability to vote contributions as useful or needs work
• long strings of comments - difficult to keep up with when there are more than 50-100 comments
• comments weren't numbered
• comments were a bit narrow so lots of scrolling

Currently the rules are at http://polymathprojects.org/general-polymath-rules/. There's a polymath blog and a polymath wiki. Interestingly, they see publications as being authored by "polymath" - sort of like Bourbaki, I guess.

It's not clear if they have solved the problem of encouraging more people to contribute. It's also sort of ambiguous about the role of moderators and summarizers. People who summarize contributions thus far are highly valued.  Other concerns from the comments include:

• Seems like there is some intimidation (unintentional!) due to the contributions of the Fields Medalists. There is concern that if/when they contribute, the entirety of the project will be credited to them.
• The low participation of women mathematicians (if, indeed, there was more than 1?)
• The conflict if you happen to pick the same problem an individual mathematician is working on for his or her dissertation
• Fear of asking questions that are not even wrong  - that show a misunderstanding of the problem. The answer to this, it seems from the comment thread, is to have a critical mass of stupid questions :)
• Stability and archiving of the threads (hosted on wordpress.com - a free hosting site)

There has been a miniproject since, and they're trying to figure out a good question for the next full "official" polymath project.

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I haven't spent a lot of time studying how mathematicians work. Those at my place of work are applied mathematicians and statisticians, so they publish in SIAM journals, IEEE conferences and journals, and in biomed places (like for syndromic surveillance).  Walsh and Bayma studied the role of CMC in math way back (actual interviews were in like 1991-2). At that time, the specialization in math was an important factor. Even at larger schools, there often would only be one person with that specialization, so there would be isolation.  Some of this was countered by visiting other institutions and going to lots of conferences. A quote from (1996a), page 666:

Respondents noted that it is difficult to grasp the meaning of a piece of research if one relies only on published articles. Respondents argued that transferring mathematical ideas requires face-to-face communication, generally in front of a black board.

Learning math can be an enculturation process, though informal conversation (they cite Sheehan (1990)).

E-mail was widely adopted early on in math and this led to a dramatic increase in co-authored papers.  People might think peripherality is not as big a deal in subjects like math in which you often do not have to buy big expensive equipment. The participants in Walsh and Bayma's study say it's even bigger, because you need access to the big guys to get in early on new research and to share information. Here's a quote from (1996b), page 355

E-mail helps. If you' re sentenced to Podunk, wherever that is, it's not the death sentence it used to be

The publication lag in math at the time was like 19 months, but the in crowd had pre-prints way before. So this created a have and have not situation.

Just like in other studies of CMC, meeting at the beginning in person was really important to building trust (1996a). The use of TeX to transmit mathematical symbols was seen as an important early innovation (do note this,  overlords)

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Given the success so far, and what we knew about CMC in math in 1996, I'm curious to see how polymath will evolve. Will attribution and credit re-surface as an issue for less established contributors?  Does there have to be some separate relationship building exercise if the potential contributors haven't met in meatspace?  Could bloggers use their blogs and what do non-bloggers do? How to get to critical mass in stupid questions (maybe allowing anonymous stupid question submission)? How can these threads be preserved for future use? How can these threads be used in education? Is "official" endorsement of a polymath project a good thing or not? Should polymath projects be centrally managed by a coalition of the willing?

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Walsh, J. P., & Bayma, T. (1996a). Computer networks and scientific work. Social Studies of Science, 26(3), 661-703.

Walsh, J. P., & Bayma, T. (1996b). The virtual college: Computer-mediated communication and scientific work. Information Society, 12(4), 343-363.