Flawed solution to famed math problem spurs cyber soap opera.

Oh my God, did you hear about Navier-Stokes? ©iStockphoto.com/Dmitry Bezkorovayny

Mathematics is a field rarely associated with human drama, but earlier this month, a soap opera-worthy narrative unfolded in the wooly world of conjectures, theorems and lemmata. 

It all started when a mathematician tackled one of math’s most enduring open problems—one that happened to be worth $1 million—in a paper she posted online. The journal Nature quickly published a story on its web site; news of a great mathematical breakthrough began to spread.

But less than two weeks after she posted the paper, the author learned that she had made an error and withdrew her work. In another era—as recently as, say, 10 years ago—that would have been the end of the story.

“Before blogs, here’s the way this normally would have shaken down.  [She] would have put the paper out, people would have heard about it and downloaded it,” said Peter Woit, a physicist and mathematician at Columbia University. “Other people see it over a few days or weeks. There would have been a much slower exchange taking place, privately.”

Oh, but times have changed. In the world of instant communication and public access to sophisticated research, this small story blossomed into a veritable cyber-drama. The narrative at “Not Even Wrong,” Woit’s blog, escalated quickly. Within a week, it had become a revealing chronicle of scientific hope, human disappointment, and the perils of undertaking the often messy enterprises of science and math in the age of the blog.

It began on Oct. 5, when Woit started a new thread titled “Navier-Stokes Equations Progress?” A Lehigh University mathematician, Penny Smith, had posted a paper online that tackled a famous unsolved problem. In the abstract of her paper, Smith claimed to have proven the existence of a solution to the Navier-Stokes equations. (Mathematicians are often just as concerned with proving the existence of a solution, as they are with finding the solution itself.) The equations describe the behavior of liquids and gases in motion, and a solution would give scientists insight into the chaotic natural phenomenon of turbulence.

The Navier-Stokes problem, which has lingered unsolved for more than 150 years, was designated a Millennium Prize problem by the Clay Mathematics Institute in Cambridge, Mass; anyone who solves it will win $1 million from the Clay Institute.

Smith posted her paper at arXiv.org, an online preprint server owned by Cornell University where physicists and mathematicians often place their work before it has gone through peer review or been published in a journal. The arXiv (pronounced “archive”) serves as an online billboard where scientists can share ideas and research, but lately it seems to have taken on a new importance. 

In 2002 and 2003, the reclusive Russian mathematician Grigory Perelman used the arXiv to post three ground-breaking papers that contained a proof of another million-dollar Millennium Prize problem: the Poincaré Conjecture. Even though Perelman’s work had not appeared in a refereed journal, he was named a winner of this year’s Fields Medal, math’s highest honor. (Perelman declined to accept the award.)

People flocked to Woit’s blog, eager to discuss Smith’s paper. Giddy admirers congratulated Smith while cautioning each other against getting too excited until the work had been verified.

“Since [Perelman’s] proof came out of nowhere, maybe people were thinking, ‘oh it’s going to happen again,” Woit said of Smith’s proof. “I was amazed at how much attention it got.”

But doubts about Smith’s work began to surface; uncertainty swelled until the morning of Oct. 8, when Smith withdrew the paper from arXiv because it contained a “serious flaw.”

The drama, however, didn’t stop. Readers of Woit’s blog, some of them physicists and mathematicians themselves, proved to have much more to say, especially to Smith, who declined to comment for this article. Some reminded her that advancement in mathematics requires risks and failures, while others criticized her work. Some lobbed personal insults.

Though the flare-up was not an isolated incident—flawed proofs show up regularly on arXiv—the blog discussion it ignited suggests that the protocols of mathematical research may have to change in the face of growing technology. It might also give mathematicians of the future a strong incentive to be hyper-meticulous about their work.

“Penny did not do anything wrong by posting her paper on arXiv.org; I just wish she had been more careful about assessing the correctness of her work before she posted the paper,” said Deane Yang, a mathematician at Brooklyn, New York’s Polytechnic University, via email. “But in the end, no matter how hard we try to catch our own mistakes, we’re all capable of doing the same thing she did.”

Perhaps mathematicians will have to reconsider posting their work on arXiv. On the one hand, the site can disseminate information quickly within a wide mathematical audience. This worked to Perelman’s advantage; news of his breakthrough spread quickly through cyberspace, and any interested mathematician—or layperson—could download history in the making. Of course, at the time, his papers were new and subject to scrutiny, and for three years mathematicians scoured his equations for signs of errors.

On the other hand, however, this same public scrutiny can subject a researcher who submits a flawed paper to public grief. As scholars often spend years preparing a single paper or result, the swift appearance of a grievous fault can be devastating.

“This incident serves as part of our learning process on how to use the Internet and blogs to communicate and discuss ideas more effectively,” Yang said. “Blogs have made these discussions much more public, drawing both appropriate and inappropriate comments from a broad range of people. This makes a lot of mathematicians very uncomfortable.”

Smith’s own posts to Woit’s blog reveal that she began to suffer under the weight of the experience: “ArXiv is supposed to be a preprint file—not a journal or a newspaper,” she wrote. “This has hurt me a lot.”

With the discussion showing few signs of quieting down after nearly a week, Woit decided to shut down the forum on Smith’s proof and send everyone home. “This endless discussion has become both unpleasant and tedious,” he wrote. “No one seems to have anything else to say about mathematics.”

Originally published October 26, 2006

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