Mathematics is frequently viewed as a lonely endeavor. Research teams in other disciplines can number in the thousands - the "Initial Sequencing and Analysis of the Human Genome" (Nature, vol. 412, p. 565) has about 2900 authors. In contrast, mathematicians still frequently work alone. Think of Andrew Wiles who worked alone and in near secrecy for 7 years to prove Fermat’s Theorem.

Recently Timothy Gowers, a Cambridge University mathematician and Fields medalist, asked whether massively collaborative math is possible. Can a large group of mathematicians “efficiently connect their brains” to attack difficult problems? As an experiment Gowers asked for a simple proof of the density Hales-Jewett Theorem. The theorem describes certain aspects of high dimensional tic-tac-toe games. Suppose you are given an H dimensional tic-tac-toe board with n cells to a side, and that you can choose to color each cell choosing from a palette of c different colors. The theorem states that, if H is sufficiently large, one column, row, or diagonal must contain cells of the same color. The problem is not of purely theoretical interest - it has important applications in computer science.

When Gowers proposed the problem, the only known proof of the theorem was exceedingly complicated. Gowers asked the mathematics community to collectively devise a simpler proof. And the mathematics community responded enthusiastically. The resulting Polymath Project advanced through an interchange of ideas on Gowers’ website. In a relatively brief 7 weeks the participants came up with a far simpler proof.

One could argue that this type of exchange is not really new. After all, Usenet has been around for many years. Many of us participated in useful discussions before the newsgroups got buried under a torrent of spam (not of the "edel" variety). Excellent forums for the discussion of mathematical ideas still exist - for instance Math Overflow.

However, the Polymath project is different. In my experience, answers to questions posed in mathematics forums are typically generated by one person, and report on what is known. The group discussion on Gowers’ blog was organic: While nobody was specifically invited to participate, and there was no leader, the solution was a result of an intense interchange which was focused, constructive and polite.

It is also important to note what the Polymath project was not: The problem was not broken into tiny bits that could be handled by the average calculus student. Even experts may find it challenging to follow the flow of ideas recorded on Gowers’ website. It is also not true that thousands of people from across the globe contributed ideas from which the solution spontaneously emerged. While the a initial group of participants was large, only a handful of people persevered – and those that did worked extremely hard to obtain the proof. Can this effort still be called crowdsourcing?

Perhaps it is obvious, but it is still important: The success of the Polymath project had nothing to do with machine intelligence. All ideas were contributed, evaluated and developed by humans. Modern technology provided a very efficient means for the interchange of ideas. But machines had no role in creating them.

The rapidity of modern communication is changing the way we do things. Throughout human history we have solved scientific and engineering problems collectively. However, past technology limited the speed at which ideas flowed between us. The Polymath project demonstrated what can happen when such barriers are lifted. Mathematicians will always spend time thinking alone. But they can now rapidly exchange the resulting ideas with colleagues across the world. This has the potential to profoundly change the practice of mathematics – a practice that has remained largely unchanged through millennia.

To learn more about the Hales-Jewett Theorem you can, of course read the informative and collectively edited entry on Wikipedia. More about the Polymath project can be found here.

Krešimir Josić