Massively collaborative mathematics

[bananas]

http://www.nature.com/nature/journal/v461/n7266/full/461879a.html

Nature 461, 879-881 (15 October 2009) | doi:10.1038/461879a; Published online 14 October 2009

Massively collaborative mathematics

Timothy Gowers1 & Michael Nielsen2

The 'Polymath Project' proved that many minds can work together to solve difficult mathematical problems. Timothy Gowers and Michael Nielsen reflect on the lessons learned for open-source science.

ILLUSTRATIONS BY M. HODSON

On 27 January 2009, one of us — Gowers — used his blog to announce an unusual experiment. The Polymath Project had a conventional scientific goal: to attack an unsolved problem in mathematics. But it also had the more ambitious goal of doing mathematical research in a new way. Inspired by open-source enterprises such as Linux and Wikipedia, it used blogs and a wiki to mediate a fully open collaboration. Anyone in the world could follow along and, if they wished, make a contribution. The blogs and wiki functioned as a collective short-term working memory, a conversational commons for the rapid-fire exchange and improvement of ideas.

The collaboration achieved far more than Gowers expected, and showcases what we think will be a powerful force in scientific discovery — the collaboration of many minds through the Internet.

The specific aim of the Polymath Project was to find an elementary proof of a special case of the density Hales–Jewett theorem (DHJ), which is a central result of combinatorics, the branch of mathematics that studies discrete structures (see 'Multidimensional noughts and crosses'). This theorem was already known to be true, but for mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights. There were two reasons to want a new proof of the DHJ theorem. First, it is one of a cluster of important related results, and although almost all the others have multiple proofs, DHJ had just one — a long and complicated proof that relied on heavy mathematical machinery. An elementary proof — one that starts from first principles instead of relying on advanced techniques — would require many new ideas. Second, DHJ implies another famous theorem, called Szemerédi's theorem, novel proofs of which have led to several breakthroughs over the past decade, so there is reason to expect that the same would happen with a new proof of the DHJ theorem.

The project began with Gowers posting a description of the problem, pointers to background materials and a preliminary list of rules for collaboration (see http://go.nature.com/DrCmnC). These rules helped to create a polite, respectful atmosphere, and encouraged people to share a single idea in each comment, even if the idea was not fully developed. This lowered the barrier to contribution and kept the conversation informal.
Building momentum

When the collaborative discussion kicked off on 1 February, it started slowly: more than seven hours passed before Jozsef Solymosi, a mathematician at the University of British Columbia in Vancouver made the first comment. Fifteen minutes later a comment came in from Arizona-based high-school teacher Jason Dyer. Three minutes after that Terence Tao (winner of a Fields Medal, the highest honour in mathematics) at the University of California, Los Angeles, made a comment. Over the next 37 days, 27 people contributed approximately 800 substantive comments, containing 170,000 words. No one was specifically invited to participate: anybody, from graduate student to professional mathematician, could provide input on any aspect. Nielsen set up the wiki to distil notable insights from the blog discussions. The project received commentary on at least 16 blogs, reached the front page of the Slashdot technology-news aggregator, and spawned a closely related project on Tao's blog. Things went smoothly: neither Internet 'trolls' — persistent posters of malicious or purposefully distracting comments — nor well-intentioned but unhelpful comments were significant problems, although spam was an occasional issue on the wiki. Gowers acted as a moderator, but this involved little more than correcting a few typos.

<snip>

Terence Tao's blog is at http://terrytao.wordpress.com/

[phantom power]

1. Cool. Arguably, this is the sort of thing the original Mosaic was created for.

An internet-based tool for science collaboration and sharing content.

[byronius]

2. Sound like the cerebral cortex serving the vast brain that is the Internet.

It's a new thing on Earth. The Democratic Underground is a hardcore information-crunching circuit within that brain, although perhaps not as earth-shakingly powerful as fast scientific cooperation.

Very cool post.

[Fumesucker]

3. I see similar, although not nearly as formal, things happening elsewhere on da tubez..

There's a great deal of collaborative technical brainstorming going on at several forums of which I'm aware.

A new modular brushless axial flux electric motor design for vehicles is being hashed out right now on a forum at endless sphere for instance..

http://endless-sphere.com/forums/index.php