by Krešo Josić
Today, let's talk massively multiplayer mathematics. The University of Houston's Math Department presents this program about the machines that make our civilization run, and the people whose ingenuity created them.
Mathematics is frequently viewed as a lonely endeavor. Today research teams in many other disciplines consist of hundreds of scientists. Mathematicians often work alone. Think of Andrew Wiles who proved Fermat's Theorem. For 7 years he worked on the problem in his attic every night. He told nobody about what he was doing, until he finally cracked it.
However, recently the Cambridge University mathematician Timothy Gowers 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. Such problems may sound frivolous, but this one 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 gave a far simpler proof of the theorem.
The group discussion on Gowers' blog was organic: Nobody was specifically invited to participate, and there was no leader. Yet the interchange was focused, constructive and polite.
It is important to note what the Polymath project was not. The problem was not broken into tiny bits that could be handled by the average person. Even trained mathematicians will find it difficult 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 initial group of participants was large, only a handful of people persevered — and those that did worked extremely hard to obtain the proof. 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 in their offices and attics. But they can rapidly interchange 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.
I'm Krešo Josić, at the University of Houston, where we're interested in the way inventive minds work.
You can read about massively collaborative mathematics on Timothy Gowers' blog at http://gowers.wordpress.com/2009/01/27/ is-massively-collaborative-mathematics-possible/ You can also find more information about Polymath and the density Hales-Jewett problem when you follow the links.
Here is an understandable introduction to the first project http://numberwarrior.wordpress.com/2009/03/25/ a-gentle-introduction-to-the-polymath-project/.
A more extensive article about massively collaborative mathematics appeared in SIAM Review, vol. 43, Number 3. An interesting article also appeared in Nature http://www.nature.com/nature/journal/v461/n7266/full/461879a.html. It's also mentioned in the New York Times' review of ideas of 2009: http://www.nytimes.com/projects/magazine/ideas/2009/#m. And here are other interesting discussions: http://www.thebigquestions.com/2010/04/08/blogging-tic-tac-toe-and-the-future-of-math/.
A similar discussion of the P-NP problem can be found at http://www.nytimes.com/2010/08/17/science/17proof.html?_r=1&scp=1&sq=P-NP&st=cse.
Above figures: Basic tic-tac-toe figure from MathWorld and the second, from science.org which presents the problem in a high dimensional format.