On the MathOverflow Web site, mathematicians from all over the world answer each other's questions.

# The Global Math Commons

For research mathematicians, it’s an all too familiar occurrence. You hit a roadblock, a simple-seeming question that just isn’t in your area of expertise. It feels like the kind of question to which someone out there must already know the answer — but that someone is not you.

In the old days (in other words, more than a year or two ago), your best bet might have been to seek out a relevant expert at math department tea — that is, if you were lucky enough to work in a large department made up of representatives from a wide range of disciplines. If you worked in a small math department, however, you might simply be stuck.

But today math department tea has gone online, in the form of a Web site called MathOverflow. On the site, mathematicians from all over the world ask and answer each other’s questions, chiming in on such topics as stationary non-isotropic spatial stochastic processes and symmetric sequences of blow-ups for a Fulton-MacPherson compactification. Post a question on MathOverflow and the answer typically comes back within an hour or two.

“It has changed the way math is done, more than I thought possible,” says Ravi Vakil, a mathematician at Stanford University who has served in an advisory role to the Berkeley graduate students and postdocs who created the site.

In the year-and-a-half since MathOverflow was launched, more than 4,500 users, ranging from advanced undergraduates to mathematicians at the top of their fields, have posted or answered questions on the site. Thousands more “lurkers” read the posts without adding their voices. Many collaborations and research papers have arisen out of MathOverflow interactions, some of them between mathematicians who had never met before posting on the site.

The core appeal of MathOverflow is its wide variety of interesting and highly advanced questions and answers. At the same time, MathOverflow’s creators have intentionally cultivated a game-like atmosphere — for example, users can garner “reputation points” and “badges” — which can make the site addictive, even for mathematicians whose reputations need no boost.

“I have felt the lure of the reputation points,” acknowledges Fields medalist Timothy Gowers, of Cambridge University. “It’s sort of silly, but nevertheless I do get a nice warm feeling when my reputation goes up.”

The site has attracted a core of loyal users, many of whom would turn up their noses at more frivolous sources of internet addiction, such as Facebook and Twitter.

“MathOverflow has quickly become one of the most important Web sites for professional mathematicians on the internet,” says Greg Kuperberg, a mathematician at the University of California, Davis, who had initially dismissed MathOverflow as “too Twitter-like.” “I used to spend a lot of my time kibitzing on blogs, but not in a way that made me all that happy in the long run, because it was just procrastination,” he says. “Whereas putting a lot of time into MathOverflow is the noblest kind of procrastination.”

## Genesis of the project

MathOverflow was born in one of the places where it is perhaps least needed: the University of California, Berkeley, which boasts more than 250 math faculty and graduate students spanning a wide range of research areas. Yet even there, hitting up against a question outside one’s expertise and not knowing whom to ask isn’t an uncommon experience.

A couple of years ago, Berkeley graduate student Anton Geraschenko starting mulling over the idea of using the internet to “tap into a Borg-like consciousness that houses all mathematical knowledge.” He began talking to other graduate students and postdocs about the idea of using something like a wiki to collect mathematical ideas. Then fellow graduate student David Brown, now an assistant professor at the University of Wisconsin at Madison, showed him Stack Overflow, a popular Web site on which professional computer programmers can post and answer each other’s programming questions.

Stack Overflow seemed to be doing for programmers exactly what Geraschenko wished to do for mathematicians — offer them a way to turn to their community for help with the kinds of specific, answerable questions that someone out there must know the answer to. And the aspects of the programming community that made Stack Overflow work so well were also present in the mathematics community, Geraschenko decided.

“Like programmers, mathematicians like to boil things down to really small problems, if possible,” he says. “They like to get their hands dirty and understand whether something works in a toy version of the problem they’re interested in.”

What’s more, mathematicians tend to enjoy playing with each other’s toy problems. Conversations at math department tea often center around questions of the form “Is there an A that does B but doesn’t have a C?”

“The mathematics community is exactly rigged up to make such a site work,” Geraschenko says.

In a stroke of luck, just a couple of months after Geraschenko started thinking about how to create a version of Stack Overflow for mathematics, the developers of Stack Overflow announced the creation of Stack Exchange, a platform for building “overflow”-type projects. To help their business get off the ground, they offered to host MathOverflow and other early overflow projects free.

MathOverflow went live toward the end of September 2009, and after a few weeks of debugging and refining, its creators announced its existence on October 14 on the Secret Blogging Seminar, a math blog started by some former Berkeley math graduate students. Several other popular math blogs linked to the announcement, and within days the site started getting about 3,500 visits a day. That number has swelled to 10,000 over the past year-and-a-half and continues to grow. “It doesn’t look like it’s slowing down,” Geraschenko says. “Presumably at some point we’ll saturate the market, but we don’t seem to have yet.”

## A relentless focus on research

MathOverflow’s success may be due in large part to its clear mandate: it is for questions that are specific, answerable, and research-level but not so hard that they would constitute a research program in themselves. In other words, it is for tackling the lemmas of the mathematical world.

The creators of MathOverflow have an equally clear vision of what the site is not. It isn’t a social networking site for mathematicians, nor is it a site where they can engage in long discussions about controversial topics in mathematics. “There are already lots of places on the internet where you can go to hang out with technical people – that problem is already solved,” Geraschenko says. The site states clearly that MathOverflow is not a discussion forum, or an encyclopedia, or a homework help site. The site’s moderators have the power to delete questions they feel belong elsewhere.

In the beginning, MathOverflow’s creators encountered some push-back from mathematicians who wanted to broaden the site’s mandate.

“MathOverflow went almost instantly from not existing to having lots of people there,” says Scott Morrison, a Berkeley postdoc who has been a moderator on MathOverflow since its early days. “People thought, ‘This is great; we should open up the site to more things, because there are lots of mathematicians here, and we should talk about all the things mathematicians want to talk about.’

“Then there was this gradual process of reining in some of the extravagances of discussions that really belong on blogs,” he says. “There was a long resistance: we would say something belongs on a blog, and people would very reasonably answer that they don’t have a blog. Eventually the tough solution to that problem won out, but it took a while.”

Part of the moderators’ determination stemmed from their realization that it was important to present MathOverflow as a highly professional site, not a place for “goofing off on the internet,” as Geraschenko puts it. “A lot of mathematicians don’t like the internet, in some sense,” Morrison says. “Of course, they use e-mail and the arXiv, but mathematicians, more than other people, tend to think Facebook and Twitter are an enormous waste of time, and they don’t want anything to do with it. I think MathOverflow’s relentless focus on research only, and no discussion, has really helped get all these fabulous mathematicians to come to the site.”

MathOverflow does offer a separate forum, meta.mathoverflow.net, where users can discuss controversial MathOverflow issues, such as why a question was deleted. The forum gives the site’s moderators a place to analyze tricky issues and reach a consensus. Meta also serves the crucial purpose of diverting flame wars and all the excesses associated with internet discussion forums away from the main site. Any time something unpleasant gets started on the main site — someone insults someone else, say — the discussion is immediately moved to meta, where the rules of discourse are more relaxed. “Meta serves the role of maintaining purity of purpose for the main site,” Geraschenko says.

## Rules and reputations

An essential element of the running of MathOverflow — and a key part of its appeal for some mathematicians — is its system of reputation points.

If, for example, you post a question or answer and one of your peers on MathOverflow likes it enough to “vote it up,” you receive 10 reputation points. If the person who posted the question marks your answer as “accepted,” you get 15 points. If someone votes your posting down (which is allowed but generally frowned upon), you lose two points. Your reputation, the site’s FAQ explains, is “a (very) rough measurement of how much the MathOverflow community trusts you.”

As users garner reputation points, they can win any of various “badges.” For example, more than 3,000 users have earned “Student” badges, meaning that one of their questions received at least one up vote. Ten users have earned the “Famous question” badge, awarded if a question garners 10,000 views.

More important, reputation points earn users privileges on the site. users can’t vote up a question or answer until they have at least 15 reputation points; voting a question down requires at least 100 points (and also costs the user a point, to discourage voting down in all but the most extreme cases). A user with 3,000 points can vote to close a question to further comments, and one with 10,000 points receives access to various moderation tools, such as the ability to delete a closed question.

While the system of points and badges may seem gimmicky, it enables MathOverflow to scale well as it grows. For one thing, the voting system creates an easy, decentralized way for the community to decide for itself which are the truly interesting questions. “Since it’s a really large community where not everyone knows everyone else, the reputation system is very useful for filtering out some of the rubbish,” says Fields medalist Terence Tao, of the University of California, Los Angeles, who uses the site frequently.

What’s more, by creating a hierarchy of users with different privileges, the reputation system puts power into the hands of users who have earned the respect of the community, rather than in a few benign dictators’. “At the high end of this reputation spectrum, there is little difference between users with high reputation and moderators,” the site’s FAQ page states. “That is very much intentional. We don’t run MathOverflow. The community does.”

The reputation system also serves a less exalted function: by tapping mathematicians’ competitive instinct, it turns MathOverflow into a fun and, for many, addictive game. And since whoever is the first to answer a question has a better chance of winning an up vote, the reputation system provides incentive to answer questions promptly.

“There’s definitely a sporting element to it,” says Joel Hamkins, of the College of Staten Island, who currently boasts the distinction of being the user with the most reputation points — more than 38,000. (“I’m the most reputable mathematician in the world,” he jokes to friends.) “Everyone on MathOverflow realizes that these badges are absurd, but nevertheless they’re motivating.”

Hamkins is a self-confessed MathOverflow addict. “My wife threatens sometimes to post a question on MathOverflow asking, ‘Has anyone seen Joel Hamkins?’” he says. “It’s very addictive, but that’s a completely positive thing. I’ve learned a huge amount of mathematics in the last year because of MathOverflow.”

The reputation system, Greg Kuperberg theorizes, appeals to the same type of mathematician who, as a student, enjoyed math contests such as the International Mathematical Olympiad and the Putnam competition. The points and badges are not a good fit for every mathematical temperament, the moderators acknowledge. “There must be hundreds of mathematicians who see these scores next to people’s names and flee in terror,” Morrison says.

While the moderators would like the site to be welcoming to all mathematicians, there is only so much they can do to lure mathematicians who dislike the reputation system or are not big fans of the internet in general, Geraschenko says.

“Some people like the kind of atmosphere MathOverflow provides, and some don’t,” Geraschenko observes. “If people hop aboard, that’s great, but there’s no point in trying to force them.”

## Forging new connections

Enough mathematicians seem comfortable with the MathOverflow format to make the site a very happening place. And for them, MathOverflow offers one of the lowest barriers to entry of any online mathematics sites.

“If you’re going to participate in a blog, for example, either you’re the author and you have to put a lot of effort into a lot of high-quality posts, or you’re a visitor and you don’t get to be on the asking side,” Geraschenko explains. On MathOverflow, a user can simply put a question out there without having to worry about whether it is significant enough to warrant asking mathematicians to spend their precious time and brainpower thinking about it.

“I often get answers to my questions from people I wouldn’t have expected,” Geraschenko says. “And sometimes I get an answer from the ‘obvious suspect,’ but it’s someone I didn’t know well enough to have felt comfortable e-mailing to ask them the question.

“That’s a really nice situation,” he adds. “When someone like that answers my question, I don’t feel as if I’ve imposed on them, since they didn’t have to answer it; but now I’ve made a connection, so I would feel comfortable e-mailing them in the future.”

Some of the connections forged on MathOverflow result in research projects by mathematicians who might not otherwise have realized that they could fruitfully collaborate. Last year, for example, Morrison and one of his research collaborators posted a pair of number theory questions on MathOverflow and quickly found a third mathematician who was interested in their questions; their discussion blossomed into a research paper. “I know no number theory whatsoever, but now I have a number theory paper,” Morrison says. “MathOverflow allowed me to find exactly the people who knew the relevant stuff.”

MathOverflow joins a collection of online tools, such as the arXiv preprint server, that, Ravi Vakil of Stanford says, are gradually broadening mathematics research into an enterprise that welcomes mathematicians who are not necessarily at a few top research institutions. “There are places that, a generation ago, you might say were on the fringes of the mathematics community, but now they are as connected as everyone else,” Vakil says.

MathOverflow gives a voice not just to the experts in a field but also to its more obscure practitioners, and to the graduate students who are its future experts, he adds.

“What’s cool for me is the many names on MathOverflow that I didn’t know but now I look at their papers and am curious when they have a preprint,” Vakil says. “It’s a great way to put one’s finger on the pulse of the mathematics world.”