MATHEMATICIANS don't have to be brilliant loners. The first analysis of a successful wiki-style mathematical project shows how a large-scale collaboration made up of amateurs and professionals can be just as effective.
The project, called Polymath, was born in January 2009 when University of Cambridge mathematician Timothy Gowers proposed it in a blog post. Another leading mathematician,Terence Tao of the University of California, Los Angeles, flagged the idea on his own blog. The project began in earnest with a wiki for collecting the mathematical contributions made to these and other blogs.
Polymath does more than just bring together pre-existing knowledge, Wikipedia-style. It is an attempt to divvy up and democratise the process of mathematical discovery. Gowers likens it to wiring together multiple mathematical brains to form a “super-brain”. “I'm interested in the question of whether it is possible for lots of people to solve one single problem rather than lots of people to solve one problem each,” he wrote on his blog.
Now computer scientists Justin Cranshaw and Aniket Kittur at Carnegie Mellon University in Pittsburgh, Pennsylvania, have analysed whether his vision has been borne out. As well as identifying the factors that made Polymath a success, they offer a set of design principles for future collective efforts, and highlight strategies for harnessing the talents of mathematically minded individuals who would not otherwise be at the cutting edge of discovery (see “Amateur mathematician: a teacher's tale”).
The pair focused on the first problem tackled by Polymath. Dubbed Polymath1, it was an attempt to find an alternative proof to the density Hales-Jewett (DHJ) theorem. This lies in an area called combinatorics that involves counting and rearranging mathematical objects.
Imagine colouring in squares in a grid. What percentage can you colour before you are forced to make a straight line along a row, column or diagonal? The DHJ theorem says the percentage increases in cubic grids, and in higher-dimensional grids you are able to colour almost all the squares because the extra dimensions give you more “room” to manoeuvre.
Although a proof existed, it was difficult and written in the language of a branch of mathematics called ergodic theory. The challenge was to find a simpler proof based on combinatorics.
Six weeks in, and the 39 contributors to Polymath1 had done it. That's a blistering pace for something so complex. The proof has been submitted for publication under the pseudonym D.H.J.Polymath. “The success of Polymath1 is a proof of concept that you can do math this way,” says Cranshaw.
To reveal how this was achieved, he and Kittur examined the process in detail. It began with Gowers posting an outline of the DHJ problem to his blog, along with some ground rules for working together. Tao and Gowers followed up with posts devoted to different, specific aspects of the proof, to which anyone could post a comment. These comments might consist of a few equations, a suggestion for how to proceed next or proof-checking. After 100 comments, Gowers or Tao would summarise the thread, ensuring loose ideas were turned into mathematical statements. Ideas were also summarised on the wiki, which served as the central location for the effort.
Cranshaw and Kittur mapped out the network of 1228 comments, examining the relationships between them and ranking them in order of importance to the final proof. Commenters ranged from amateur mathematicians to university professors, but the significance of their contributions did not depend on their academic seniority.
While Tao and Gowers were the most prolific commenters, and among the most highly rated, people who made just a few comments also had a large impact on the final proof (see graph). This suggests individuals can help solve a problem without committing a huge amount of time or effort, which was part of Gowers's vision for the project.
Gowers was struck by another benefit provided by the collaboration. “Reading the discussion provides some kind of strange random stimulus that causes your brain to go in to fruitful places where it might not have done otherwise,” he says. “It's a strange effect, but it was a very powerful one.”
The discussion provides a random stimulus that causes your brain to go in to fruitful places
The analysis also identifies the essential role played by Gowers and Tao's leadership. Both have won the Fields medal, mathematics's equivalent of the Nobel prize, and Cranshaw suggests that their status helped to draw in collaborators. “You have grad students who get a chance to work with Terry Tao and Tim Gowers – that just wouldn't happen normally without this project.”
But Tao dismisses any idea that you need a leading mathematician to moderate the discussion. All it takes is someone “willing to invest a fair amount of time and energy into organising the discussion and keeping it coherent”.
Cranshaw and Kittur will present their results next week at the Conference on Human Factors in Computing Systems in Vancouver, Canada.
Cranshaw says there is a “huge” scope to enhance Polymath. He believes the existing structure of consecutive posts is unhelpful, especially for participants who come in late and have to wade through posts in chronological order before contributing.
“In math the relationship between ideas is really important, but I'm not sure that comes across,” Cranshaw says. He also suggests ranking comments by importance as he and Kittur did in their study, or allowing users to vote on comments.
Gowers would like contributions to be arranged on a kind of giant virtual blackboard that would show how different strands of the discussion link together. Translating this vision to a computer screen would be a “major challenge”, he admits.
So is the collaborative Polymath the future of a once-solitary subject? “I view it still as a small-scale experiment,” says Tao. “But I think massively collaborative projects in mathematics will become more common in the future, even if they don't necessarily follow a Polymath-type format.”
Contributing to Polymath, a collaborative approach to solving mathematical problems, requires some mathematical knowledge, but you don't have to be a professional. “I originally didn't think that anything unsolved was at all approachable,” says Jason Dyer, a mathematics teacher in Arizona who participated in Polymath1, and other problems. “Doing this has helped increase my own mathematical confidence.”
Dyer admits he wasn't able to follow all the high-level mathematical arguments, but found he could still make a worthwhile contribution by thinking about a simple kind of logic puzzle known as Fujimura's problem. This involves coins arranged in triangular grids, and asks how many coins can be removed before the triangular structure is lost. “It's not a hard problem,” say Dyer. “It's just nobody had explored it at a real mathematical level.”
As it turns out, Fujimara's problem did not relate to the Polymath1 problem in the way Dyer and others had suspected. But by exploring the puzzle, Dyer was able to contribute significantly to the final proof.
Could Polymath uncover “raw” geniuses outside the traditional community of mathematicians, rather like legendary self-taught mathematician Srinivasa Ramanujan? “The nature of mathematics has changed since Ramanujan's day,” says Terence Tao, a co-founder of Polymath. “In the more mature areas, knowledge of existing mathematics is significantly more important than raw ability.” But, as Dyer shows, non-traditional mathematicians can still contribute.