In the mid-20th century the encyclopedic works of French mathematician Nicolas Bourbaki traced every mathematical concept back to the subject’s foundations in the theory of sets—the stuff of Venn diagrams—and changed the face of his field. Like many of his notions, Bourbaki existed only in the abstract: he was the pseudonym for a tight-knit group of young Parisian researchers. The Internet-age version could be D.H.J. Polymath, another collective pseudonym who could define a new style of mathematics.
Polymath began life on the blog of Timothy Gowers, a University of Cambridge winner of the Fields Medal, mathematics’ most coveted prize. In a blog post in January 2009, Gowers asked whether spontaneous online collaborations could crack hard mathematical problems—and if they could do so in the open, laying the creative process out for the world to see. Web-based scientific collaborations and even “crowdsourcing” are now common, but this one would be different. In typical online collaborations, scientists each perform a small amount of research that contributes to a larger project, Gowers pointed out. In some cases, citizen-scientists such as bird-watchers or amateur astronomers collectively can make significant contributions. “What about the solving of a problem that does not naturally split up into a vast number of subtasks?” he asked. Could such a problem be solved by the readers of his blog—simply by posting comments?
For a first experiment, Gowers chose the so-called density Hales-Jewett theorem. This problem, Gowers says, is akin to “playing a sort of solitaire tic-tac-toe and trying to lose.” The theorem states that if your tic-tac-toe board is multidimensional and has sufficiently many dimensions, after a short while it is impossible to avoid arranging X’s into a line—you cannot avoid winning no matter how hard you try. Mathematicians have known since 1991 that the theorem was true, but the existing proof used sophisticated tools from other branches of math. Gowers challenged his blog’s readers to help him find a more elementary proof, a problem generally considered quite hard.
The project took off a lot faster than Gowers expected. Within six weeks, he announced a solution. Turning the proof into a conventional paper took longer, especially because the argument was scattered across hundreds of comments. But last October the group posted a paper on the online repository arxiv.org under the name of D.H.J. Polymath, where the initials are a reference to the problem itself.
In another way, however, the project was a bit of a disappointment. Just six people—all professional mathematicians and “usual suspects” in the field—did most of the work. Among them was another Fields medalist and prolific blogger, Terence Tao of the University of California, Los Angeles.
Pooling talent has its advantages, Gowers says. When trying to solve a problem, mathematicians usually make many failed attempts, in which they try lines of reasoning that can turn out to be “blind alleys,” after weeks or months of work. Often those lines of reasoning that seem promising to one expert look obviously fruitless to another. So when every attempt is exposed to public feedback, the process can become much faster.
Tao describes the experience as “chaotic” but a lot of fun and “more addictive than traditional research.” Gowers has since kicked off a few more online collaboration projects, and so has Tao—and nonprofessionals have begun to contribute in ways that are “genuinely useful,” Gowers says. These high-brow amateurs included a teacher, a priest and a math Ph.D. who now works in computing. But how widely the approach will be adopted is unclear. A number of hard problems may be suitable, Tao says, such as devising an algorithm for playing chess that is not based on the brute-force calculation of possible future moves. Famous mathematical conjectures may not be as amenable, because those problems tend to have a long history—and experts already know all the blind alleys.
Rafael Núñez, a cognitive scientist at the University of California, San Diego, who has studied the mental and social process of doing mathematics, points out that problem solving is just another human activity. When mathematicians work together in front of a blackboard, they communicate in subtle ways with their voice and body language, clues that will be lost in online collaborations. But mathematicians will adjust to the new medium, just like people have adjusted to doing all kinds of other things in a connected world, Núñez notes: “Anything we do online is different, not just mathematics.”
In the end, the open nature of the project may have been its most important feature. As Gowers wrote on his blog, Polymath may be “the first fully documented account of how a serious [math] research problem was solved, complete with false starts, dead ends, etcetera.” Or, as Tao puts it, the project was valuable because it showed “an example of how the sausage is made.”