One, Two, Many:
Individuality and Collectivity in Mathematics

MELVYN B. NATHANSQN

“Fermat's last theorem” is famous because it is old and easily understood, but it is not particularly interesting. Many, perhaps most, mathematicians would agree with this statement, though they might add that it is, nonetheless, important because of the new mathematics created in the attempt to solve the problem. By solving Fermat, Andrew Wiles became one of the world's best known mathematicians, along with John Nash, who achieved fame by being crazy, and Theodore Kaczynski, the Una- bomber, by killing people.

Wiles is known not only because of the problem he solved, but also because of how he solved it. He was not part of a corporate team. He did not work over coffee, by mail, or via the Internet with a group of collaborators. Instead, for many years, he worked alone in an attic study and did not talk to anyone about his ideas. This is the classical model of the artist, laboring in obscurity. (Not real obscurity, of course, since Wiles was, after all, a Princeton professor.) What made the solution of Fermat's last theorem so powerful in the public and scientific imagination was the fact that the story comported with the romantic myth: solitary genius, great accomplishment.

This is a compelling myth in science. We have the image of the young Newton, who watched a falling apple and discovered gravity as he sat, alone, in an orchard in Lincolnshire while Cambridge was closed because of an epidemic. We recall Galois, working desperately through the night to write down before his duel the next morning all of the mathematics he had discovered alone. There was Abel, isolated in Norway, his discovery of the unsolvability of the quintic ignored by the mathematical elite. And Einstein, exiled to a Swiss patent office, where he analyzed Brownian motion, explained the photoelectric effect, and discovered relativity. In a speech in 1933, Einstein said that being a lighthouse keeper would be a good occupation for a physicist. These are the kind of stories that give Eric Temple Bell's Men of Mathematics its hypnotic power and inspire many young students to do research.

Wiles did not follow the script perfectly. His initial manuscript contained a gap that was eventually filled by Wiles and his former student Richard Taylor. Within epsilon, Wiles solved Fermat in the best possible way. Intense solitary thought produces the best mathematics.

Gel'fand's List

Some of the greatest twentieth-century mathematicians, such as Andre Weil and Atle Selberg, had few joint papers. Others, like Paul Erdo's and I. M. Gel'fand, had many. Erdo's was a master collaborator, with hundreds of co-authors. (Full disclosure: I am one of them.) Reviewing Erdo's's number theory papers, I find that in his early years, from his first published work in 1929 through 1945, most (60 percent) of his 112 papers were singly authored, and that most of his stunningly original papers in number theory were papers that he wrote by himself.

In 1972–73 I was in Moscow as a postdoc studying with Gel'fand. In a conversation one day he told me there were only ten people in the world who really understood representation theory, and he proceeded to name them. It was an interesting list, with some unusual inclusions and some striking exclusions. (“Why is X not on the list,” I asked, mentioning the name of a really famous representation theorist. “He's just an engineer,” was Gel'fand's disparaging reply.) But the tenth name on the list was not a name, but a description: “Somewhere in China,” said Gel'fand, “there is a young student, working alone, who understands representation theory.”

Bers Mafia

A traditional form of mathematical collaboration is to join a school. Analogous to the political question, “Who's your rabbi?” (meaning “Who's your boss? Who is the guy whom you support and who helps you in return?”), there is the mathematical question, “Who's your mafia?” The mafia is the group of scholars with whom you share research interests, with whom you socialize, whom you support, and who support you. In the NewYork area, for example, there is the self-described “Ahlfors-Bers mafia,” beautifully described in a series of articles about Lipman Bers that were published in a memorial issue of the Notices of the American Mathematical Society in 1995.

Bers was an impressive and charismatic mathematician at NYU and Columbia who created a community of graduate students, postdocs, and senior scientists who shared common research interests. Being a member of the Bers mafia was valuable both scientifically and professionally. As students of the master, members spoke a common language and pursued common research goals with similar mathematical tools. Members could easily read, understand, and appreciate one another's papers, and their own work fed into and complemented the research of others. Notwithstanding sometimes intense internal group rivalries, members would write recommendations for one another's job applications, review their papers and books, referee their grant proposals, and nominate and promote each other for prizes and invited lectures. Being part of a school made life easy. This is the strength and the weakness of the collective. Members of a mafia, protected and protecting, competing with other mafias, are better situated than those who work alone. Membership guarantees moderate success but makes it unlikely to create really original mathematics.

The Riemann Hypothesis

The American Institute of Mathematics organized its first conference, “In Celebration of the Centenary of the Proof of the Prime Number Theorem: A Symposium on the Riemann Hypothesis,” at the University of Washington on August 12—15, 1996. According to its website, “the American Institute of Mathematics, a nonprofit organization, was founded in 1994 by Silicon Valley businessmen John Fry and Steve So- renson, longtime supporters of mathematical research.” The story circulating at the meeting was that the businessmen funding AIM believed that the way to prove the Riemann hypothesis was the corporate model: To solve a problem, put together the right team of “experts” and they will quickly find a solution.

At the AIM meeting, various experts (including Berry, Connes, Gold- feld, Heath-Brown, Iwaniec, Kurokawa, Montgomery, Odlyzko, Sar- nak, and Selberg) described ideas for solving the Riemann hypothesis. I asked one of the organizers why the celebrated number theorist Z was not giving a lecture. The answer: Z had been invited, but declined to speak. Z had said that if he had an idea that he thought would solve the Riemann hypothesis, he certainly would not tell anyone because he wanted to solve it alone. This is a simple and basic human desire: Keep the glory for yourself.

Thus, the AIM conference was really a series of lectures on “How not to solve the Riemann hypothesis.” It was a meeting of distinguished mathematicians describing methods that had failed, and the importance of the lectures was to learn what not to waste time on.

The Polymath Project

The preceding examples are prologue to a discussion of a new, widely publicized Internet-based effort to achieve massive mathematical collaboration. Tim Gowers began this experiment on January 27, 2009, with the post “Is massively collaborative mathematics possible?” on his Weblog http://gowers.wordpress.com. He wrote, “Different people have different characteristics when it comes to research. Some like to throw out ideas, others to criticize them, others to work out details, others to re-explain ideas in a different language, others to formulate different but related problems, others to step back from a big muddle of ideas and fashion some more coherent picture out of them, and so on. A hugely collaborative project would make it possible for people to specialize.…In short, if a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well.” This is the fundamental idea, which he restated explicitly as follows: “Suppose one had a forum…for the online discussion of a particular problem.…The ideal outcome would be a solution of the problem with no single individual having to think all that hard. The hard thought would be done by a sort of super-mathematician whose brain is distributed among bits of the brains of lots of interlinked people.”

What makes Gowers's polymath project noteworthy is its promise to produce extraordinary results—new theorems, methods, and ideas— that could not come from the ordinary collaboration of even a large number of first-rate scientists. Polymath succeeds if it produces a super- brain. Otherwise, it's boring.

In appropriately pseudo-scientific form, I would restate the “Gowers hypothesis” as follows: Let qual(w) denote the quality of the mathematical paper w, and let Qual(M) denote the quality of the mathematical papers written by the mathematician M. If w is a paper produced by the massive collaboration of a set of mathematicians, then

A reading of the many published articles and comments on massive collaboration suggests that its enthusiastic proponents believe the following much stronger statement:

Superficially, at least, this might seem plausible, especially when proposed by one Fields Medalist (Gowers) and enthusiastically supported by another (Terry Tao).

I assert that (1) and (2) are wrong, and that the opposite inequality is true:

First, some background. Massive mathematical collaboration is one of several recent experiments in scientific social networking. One of the best known is the DARPA Network Challenge. On December 5, 2009, the Defense Advanced Research Projects Agency (DARPA) tethered ten red weather balloons at undisclosed but readily accessible locations across the United States, each balloon visible from a nearby highway, and offered a $40,000 prize to the first individual or team that could correctly give the latitude and longitude of each of the ten balloons. In a press release, DARPA wrote that it had “announced the Network Challenge…to explore how broad-scope problems can be tackled using social networking tools. The Challenge explores basic research issues such as mobilization, collaboration, and trust in diverse social networking constructs and could serve to fuel innovation across a wide spectrum of applications.”

In less than nine hours, the MIT Red Balloon Challenge Team won the prize. According to the DARPA final project report, “The geolocation of ten balloons in the United States by conventional intelligence methods is considered by many to be intractable; one senior analyst at the National Geospatial Intelligence Agency characterized the problem as impossible. A distributed human sensor approach built around social networks was recognized as a promising, nonconventional method of solving the problem, and the Network Challenge was designed to explore how quickly and effectively social networks could mobilize to solve the geolocation problem. The speed with which the Network Challenge was solved provides a quantitative measure for the effectiveness of emerging new forms of social media in mobilizing teams to solve an important problem.”

The DARPA Network Challenge shows that, in certain situations, scientific networking can be extraordinarily effective, but there is a fundamental difference between the DARPA Network Challenge and massive mathematical collaboration. The difference is the difference between stupidity and creativity. The participants in the DARPA Network Challenge had a stupid task to perform: Look for a big red balloon and, if you see one, report it. No intelligence required. Just do it. The widely disbursed members of the MIT team, like a colony of social ants, worked cooperatively and productively for the greater good, but didn't create anything. Mathematics, however, requires intense thought. Individual mathematicians do have “to think all that hard.” Individual mathematicians create.

In a recent magazine article (“Massively collaborative mathematics,” Nature, October 15, 2009), Gowers and Michael Nielsen boasted, “The collaboration achieved far more than Gowers expected, and showcases what we think will be a powerful force in scientific discovery—the collaboration of many minds through the Internet.” They are wrong. Massive mathematical collaboration has so far failed to achieve its ambitious goal.

Consider what massive mathematical collaboration has produced, and who produced it. Gowers proposed the problem of finding an elementary proof of the density version of the Hales-Jewitt theorem, which is a fundamental result in combinatorial number theory and Ramsay theory. In a very short time, the blog team came up with a proof, chose a nom de plume (“D.H.J. Polymath”), wrote a paper, uploaded it to arXiv, and submitted it for publication. The paper is: D.H.J. Polymath, “A new proof of the density Hales-Jewett theorem,” arXiv:0910.3926.

The abstract describes it clearly: “The Hales-Jewett theorem asserts that for every r and every k there exists n such that every r-colouring of the n-dimensional grid {1,…,k}ⁿ contains a combinatorial line.…The Hales-Jewett theorem has a density version as well, proved by Fur- stenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemeredi's theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large n needs to be.”

A second, related paper by D.H.J. Polymath, “A new proof of the density Hales-Jewett theorem,” arXiv:0910.3926, has also been posted on arXiv.

These papers are good, but obviously not Fields Medal quality, so Nathanson's inequality (3) is satisfied. A better experiment might be massive collaboration without the participation of mathematicians in the Fields Medal class. This would reduce the upper bound in Gowers' inequality (1), and give it a better chance to hold. It is possible, however, that Internet collaboration can succeed only when controlled by a very small number of extremely smart people. Certainly, the leadership of Gowers and Tao is a strong inducement for a mathematician to play the massive participation game, since, inter alia, it allows one to claim joint authorship with Fields Medalists.

After writing the first paper, Gowers blogged, “Let me say that for me personally this has been one of the most exciting six weeks of my mathematical life.…There seemed to be such a lot of interest in the whole idea that I thought that there would be dozens of contributors, but instead the number settled down to a handful, all of whom I knew personally.” In other words, this became an ordinary, not massive, collaboration.

This was exactly how it was reported in Scientific American. On March 17, 2010, Davide Castelvecchi wrote, “In another way, however, the project was a bit of a disappointment. Just six people—all professional mathematicians and usual suspects in the field—did most of the work. Among them was another Fields Medalist and prolific blogger, Terence Tao of the University of California, Los Angeles.”

Human beings are social animals. We enjoy working together, through conversation, letter writing, and e-mail. (More full disclosure: I've written many joint papers. One paper even has five authors. Collaboration can be fun.) But massive collaboration is supposed to achieve much more than ordinary collaboration. Its goal, as Gowers wrote, is the creation of a super-brain, and that won't happen.

Mathematicians, like other scientists, rejoice in unexpected new discoveries and delight when new ideas produce new methods to solve old problems and create new ones. We usually don't care how the breakthroughs are achieved. Still, I prefer one person working alone to two or three working collaboratively, and I find the notion of massive collaboration aesthetically appalling. Better a discovery by an individual than the same discovery by a group.

I would guess that even in the already interactive twentieth century, most of the new ideas in mathematics originated in papers written by a single author. A glance at MathSciNet shows that only two of Tim Gow- ers's 42 papers have a co-author. (Terry Tao responded to this observation by noting that half of his many papers are collaborative.)

In a contribution to a “New Ideas” issue of The New York Times Magazine on December 13, 2009, Jordan Ellenberg described massive mathematical collaboration with journalistic hyperbole: “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.”

It is always good to rethink old prejudices, but sometimes the re- evaluation confirms the truth of the original prejudice. Massive collaboration will produce useful results, but it will not meet the standard that Gowers set: No mathematical super-brain will evolve on the Internet and create new theories that will yield brilliant solutions to important unsolved problems. It won't happen. Gowers cited the classification of the finite simple groups as a kind of massive collaboration, and this is a perfect example. It was a useful result. Ignoring the contentious question of whether the proof was or is finally correct, which is an inherent problem of massive collaboration, the work is definitely boring. As far as I know, neither brilliant insights nor new techniques have come out of the proof and been applied to create new areas of mathematics or solve old problems in unrelated fields. It is more engineering than art. Recalling Mark Kac's famous division of mathematical geniuses into two classes, ordinary geniuses and magicians, one can imagine that massive collaboration will produce ordinary work and, possibly, in the future, even work of ordinary genius, but not magic. Work of ordinary genius is not a minor accomplishment, but magic is better.