Monday, December 14, 2009

Massively collaborative mathematics: using blog comments to prove math's toughest theorems

As frequent readers of this blog will know, I'm a huge fan of James Surowiecki's The Wisdom of Crowds. So I was delighted this weekend when I stumbled upon an article titled, Massively Collaborative Mathematics in the NY Times Sunday Magazine. Ironically, for an issue devoted to the "The Ninth Annual Year in Ideas", the online version does not provide easy permalinks or ways to forward to social networks -- so I'll just excerpt the article here:

Massively Collaborative Mathematics

In January, Timothy Gowers, a professor of mathematics at Cambridge and a holder of the Fields Medal, math's highest honor, decided to see if the comment section of his blog could prove a theorem he could not.

In two blog posts — one titled "Is Massively Collaborative Mathematics Possible?" — he proposed an attack on a stubborn math problem called the Density Hales-Jewett Theorem. He encouraged the thousands of readers of his blog to jump in and start proving. Mathematics is a process of generating vast quantities of ideas and rejecting the majority that don't work; maybe, Gowers reasoned, the participation of so many people would speed the sifting.

The resulting comment thread spanned hundreds of thousands of words and drew in dozens of contributors, including Terry Tao, a fellow Fields Medalist, and Jason Dyer, a high-school teacher.

It makes fascinating, if forbiddingly technical, reading. Gowers's goals for the so-called Polymath Project were modest. "I will regard the experiment as a success," he wrote, "if it leads to anything that could count as genuine progress toward an understanding of the problem." Six weeks later, the theorem was proved. The plan is to submit the resulting paper to a top journal, attributed to one D.H.J. Polymath.

By now we're used to the idea that gigantic aggregates of human brains — especially when allowed to communicate nearly instantaneously via the Internet — can carry out fantastically difficult cognitive tasks, like writing an encyclopedia or mapping a social network. But some problems we still jealously guard as the province of individual beautiful minds: writing a novel, choosing a spouse, creating a new mathematical theorem. The Polymath experiment suggests this prejudice may need to be rethought. In the near future, we might talk not only about the wisdom of crowds but also of their genius.

~Jordan Ellenberg, Sunday, Dec. 13, NY Times Sunday Magazine

Massively collaborative mathematics proves Surowiecki's point that:

If you put together a big enough and diverse enough group of people and ask them to make decisions affecting matters of general interest, that group's decisions will, over time, be intellectually superior to the isolated individual, no matter how smart or well-informed he is.

~James Surowiecki, The Wisdom of Crowds

Increasingly, I think the biggest innovations in ALL fields will come from massively collaborative projects -- massively collaborative psychology, massively collaborative economics, massively collaborative astrophysics, etc.

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