In April, an eBay auction offered math and science aficionados a rare opportunity: to link their names, albeit through 5 degrees of separation, with one of the most famous mathematicians of the 20th century. Trumpeting the title “Decrease your Erdös number!” the auction presented bidders with the chance to collaborate on a research project with the seller, who had collaborated with someone who had collaborated with someone who had collaborated with someone who had collaborated with Paul Erdös, the Hungarian mathematical prodigy who died in 1996.

Erdös was an eccentric, legendary figure with no fixed address. He worked with mathematicians all over the globe, coauthoring papers with more than 500 other researchers during his lifetime. Just as some film buffs calculate movie actors’ fame by measuring their degrees of separation from Kevin Bacon, mathematicians calculate their “Erdös numbers.” A mathematician who has published a paper with Erdös has an Erdös number of 1. A mathematician who has published a paper with someone who has published a paper with Erdös has an Erdös number of 2, and so on.

The eBay seller was William Tozier, a scientific consultant in Ann Arbor, Mich., whose Erdös number is 4. He launched the auction as a joke, in his words, “one morning before I’d had enough coffee.” Quickly, however, the auction took on a life of its own, sparking a vigorous debate both about the limited research opportunities available to amateur mathematicians and about the ethics of selling an Erdös number.

**Exclusion principle**

Although Tozier started the auction on a whim, his offer was serious: 40 hours of his time to collaborate on a project of the winning bidder’s choosing in one of Tozier’s areas of expertise, which include machine learning and complex systems. If the collaboration resulted in a published research paper, which Tozier said was likely but not guaranteed, the auction winner would earn an Erdös number of 5.

Tozier, who is interested in social networks, started the auction as an informal experiment about how news spreads through social and professional circles. He was curious to see how quickly gossip about the auction would spread and whether anyone would actually take him up on his unusual offer.

“It’s hard to get real data about social networks without infringing on people’s privacy, so I thought it would be interesting to set this out there as something between a joke and an experiment,” he says. “I like to approach things in this ‘let’s see what happens’ mode.”

Tozier told only four friends about the auction, asking them to pass on the news to their own friends. He also posted a link to the auction on his Weblog (http://williamtozier.com/slurry/2004/04/index.html). From there, Tozier says, news about the auction made its way quickly through the “blogosphere,” and soon, the mathematics, computer science, and other technical communities were buzzing with the story. Mathematician Jerry Grossman of Oakland University in Rochester, Mich., who maintains the official Web site for the Erdös number project, told Tozier that visits to the site skyrocketed after the auction started.

When Tozier set up the auction, he had no idea who, if anyone, would respond. To his surprise, his offer of collaboration struck a chord in a wide range of individuals, including engineers, teachers, and businesspeople, who had ideas for research projects but felt excluded from the mathematical community. During the 10 days of the auction, Tozier heard from more than 100 would-be researchers, whose frustration echoed complaints Tozier had been hearing for more than a decade.

“Through my years as a researcher and corporate consultant, I’ve met lots of extraordinarily smart people who love math, read about it in books, try things out, and explore, but because they’re not members of the [mathematics] community, they can’t put their observations back into the pool of collective knowledge,” he says. “They do what is arguably research but can’t share it with people who would take it further or put it in perspective.”

As a result, he adds, “most of these people cannot do anything at all about their ideas, and they cannot apply their skills.”

It’s hard for people outside of academia to share their ideas with mathematicians, agrees Ronald Graham, a mathematician at the University of California, San Diego who has an Erdös number of 1. “I’ve run into people who are serious researchers in other areas but have some math idea and don’t know what to do,” he adds.

Mathematicians get so many letters from crackpots claiming to have proved amazing theorems, Graham says, that it would take too much time to separate the wheat from the chaff.

Spurred by the emotional impact the auction was creating, Tozier decided to devote some of its proceeds to launch a free online community to promote scientific collaboration among laymen and mathematically trained professionals outside of academia. The idea, he says, is to create a forum in which individuals can post research proposals, find collaborators, review each other’s work, and publish research findings. He is now talking to nonprofit and governmental groups about the best way to put together such a community.

**A mockery of the system**

Although his auction officially ended on April 30, Tozier has no money in hand, because there was a saboteur among the bidders. In the auction’s final seconds, an anonymous bidder placed a winning bid of $1,031, then announced on a Weblog that had been chronicling the auction (http://www.zephoria.org/thoughts/archives/

2004/04/21/decrease_your_erdos_number.html) that he had no intention of paying the money or collaborating with Tozier.

The winner—who later identified himself as Jose Burillo, a mathematician at the Polytechnic University of Barcelona in Spain with an Erdös number of 3—wrote that he had placed the winning bid “to stop the mockery this person is doing of the paper/journal system” and called the auction a “travesty.”

At first, Burillo says, he found the idea of the auction merely funny. But as the auction progressed and individuals appeared to be placing serious and substantial bids, Burillo started to fear that the prospect of buying an Erdös number might be an unhealthy temptation for a mathematician trying to climb the academic ladder. “Someone could be desperate enough for an Erdös number of 5 to pay for it,” he says.

An Erdös number of 5 isn’t that big of a deal: More than 66,000 mathematicians can claim the honor. Yet in some parts of the world, Burillo says, Erdös numbers are taken very seriously, and mathematicians post their number prominently on their résumés. Even where that is not the case, he says, the number of papers a mathematician or scientist publishes can play a critical role in his or her advancement.

“I don’t want to see the system by which I am going to be evaluated tarnished by the possibility that people can buy or sell their coauthorships,” Burillo says. “Joint papers have to be worked and earned, not sold, auctioned, or bought.”

Tozier retorts that the current system is far from perfect. It’s common in many sciences, he says, for an eminent researcher to be listed as the coauthor of a paper researched and written by, say, a graduate student working in the researcher’s lab. Sometimes, he says, the researcher doesn’t even bother to read the paper. By contrast, Tozier says, he was offering a genuine collaboration to the winner of the auction.

While Burillo is motivated by idealism, Tozier says, his defense of the status quo “summarizes everything self-centered and exclusionary about the academic life.”

Burillo isn’t the first to fear that Erdös numbers could be exploited. Graham says that when Erdös died, mathematicians worried that researchers might start frivolously adding Erdös’ name to papers he had nothing to do with, thereby netting themselves an Erdös number of 1.

So far, those fears appear to have been unfounded, Graham says. Erdös’ name is still appearing on new papers, but they are generally written by mathematicians who collaborated with Erdös years ago and are only now getting around to writing about their work.

“Erdös has slowed down a bit, but death didn’t completely stop him,” Graham says.

Erdös would have been amused by the fuss over the eBay auction, Graham guesses. “He thought the whole idea of Erdös numbers was kind of goofy.”

Meanwhile, Burillo’s maneuver may have succeeded in dealing only a temporary setback to the Erdös number auction. Tozier is considering re-running the auction on eBay. He also intends to follow through on his idea of creating a collaborative community of amateur mathematicians, which would be open to the unsuccessful bidders. “I very seriously intend to bring those people together,” he says.