| Each week Quanta Magazine explains one of the most important ideas driving modern research. This week, senior math writer Jordana Cepelewicz describes how math is becoming an increasingly collaborative discipline, and why. | | Collaboration Is Changing Math for the Better By JORDANA CEPELEWICZ | | The archetype of the lone genius toiling in isolation still looms large over how we imagine mathematics. That's not without good reason: Until a few decades ago, single-author papers were the norm in the subject. But most papers now have two or three authors.
As Hugo Duminil-Copin, a winner of the 2022 Fields Medal, told me, finding a good collaborator is like "when you meet your soulmate." (Some mathematicians embrace that sentiment more literally.) So why did it take so long for mathematicians to figure out how to work together?
With each passing year, there is a lot more math to know. To make progress, mathematicians need to be familiar with a staggering amount of material, often in very different fields. That might involve not just building bridges between one area of math and another, but also gaining insights from computer science, physics and other subjects. It's more than any one person can get their head around.
But collaboration isn't only about finding the perfect partner. In the last few decades, mathematicians have gained many new ways to collaborate: email, Zoom, blogs, wikis, the arxiv.org preprint server, databases like the On-Line Encyclopedia of Integer Sequences, and even 4chan. Technology is making it possible to do research not just with one or two close collaborators, but with dozens or hundreds.
What's New and Noteworthy
In 2009, the mathematician Tim Gowers posted a challenge on his blog: "Is massively collaborative mathematics possible?" He encouraged anyone who wanted to participate to do so: to discuss, and hopefully solve, a math problem of his choosing in the comments section of the blog. "If a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well," he wrote.
It worked. One and a half months, 27 contributors and more than 800 comments later, the project yielded not one but two new proofs, as well as progress on related problems.
Since then, Gowers and others have streamlined the process and developed it into the Polymath project, which encourages mathematicians to put forth half-baked ideas, examine problems through different lenses and engage in lively discussion. Their crowdsourcing approach has made it possible for mathematicians — in just a few weeks or months — to solve or make progress on problems that might have otherwise taken years. It has the added benefit of leaving a record of how proofs come together: What worked and what didn't? Who contributed which ideas?
Take one of the biggest open problems in number theory — the twin primes conjecture, which asserts that there are infinitely many primes that differ by only 2 (like 11 and 13, 17 and 19, or 1,049 and 1,051). In 2013, to the shock of the math community, Yitang Zhang, then an unknown mathematician, announced a major advance: By himself, he had proved that infinitely many primes differ by at most 70 million. Though that number was huge — far greater than 2 — it was a lot smaller than infinity. And thanks to a Polymath project involving dozens of mathematicians, as well as another breakthrough by the mathematician James Maynard, this number has since been driven down to 246.
Mathematicians are now engaging in a second type of large-scale collaboration enabled by the rise of "proof assistants" — software that helps mathematicians verify that a proof is ironclad, without any gaps in logic, by prompting them to write it out systematically in a computer language. Since 2017, for instance, mathematicians have been building a digital mathematical library by translating axioms, definitions, theorems and proofs into the software program Lean. More recently, they've started using Lean to check the accuracy of new proofs, to automate certain steps in mathematical arguments, and more.
In traditional collaborations, each mathematician has to verify their collaborators' work before trusting that their proof is correct. Now, Lean can do that instead. And writing a proof in Lean makes it much easier to coordinate: Mathematicians can break up a proof into little chunks and work on each one separately.
That technical innovation might one day allow collaborations that involve hundreds, even thousands of people — something more common in subjects like physics, but unheard of in math.
As the mathematician Terry Tao said on Quanta's The Joy of Why podcast last month, "The way we do mathematics is definitely changing." | | | The Joint Mathematics Meetings channel on YouTube has a lecture by Tao about the transformational power of computers in mathematics — including Lean's role in making new kinds of collaborations possible. | | | Nature published an opinion piece by Tim Gowers and Michael Nielsen about the origins and hopes of the Polymath project. "Who would have guessed that the working record of a mathematical project would read like a thriller?" they wrote. | | | The Atlantic published a feature in 1967 about the mathematician G.H. Hardy, whose collaboration with John Littlewood has been described as "the most remarkable and successful partnership in mathematical history." (As biographers have noted, Hardy and Littlewood even delineated axioms to guide their work together.) | | | | | | |
