An ABC proof too tough even for mathematicians
In Kyoto, a solitary thinker unveils a solution that’s either a dizzying advance — or a blind alley.
On Aug. 30, a Japanese mathematician named Shinichi Mochizuki posted four papers to his faculty website at Kyoto University. Rumors had been spreading all summer that Mochizuki was onto something big, and in the abstract to the fourth paper Mochizuki explained that, indeed, his project was as grand as people had suspected. Over 512 pages of dense mathematical reasoning, he claimed to have discovered a proof of one of the most legendary unsolved problems in math.
The problem is called the ABC conjecture, a 27-year-old proposition considered so impossible that few mathematicians even dared to take it on. Most people who might have claimed a proof of ABC would have been dismissed as cranks. But Mochizuki was a widely respected mathematician who’d solved hard problems before. His work had to be taken seriously.
Even so, it raised an immediate problem. As a contributor named James Taylor wrote in a post to Math Overflow, a discussion board popular in the tightknit world of higher mathematics, the question amounted to this: Could anyone explain the philosophy behind Mochizuki’s work? The answer was a resounding “no.”
In most fields, including math, researchers move together. They build on one another’s work and cluster around solving big problems, the way physicists did in recent years with the search for the Higgs boson.
Mochizuki was different. Depending on how you calculate it, he’d been working on a proof of ABC entirely by himself for nearly 20 years. During that time, he’d constructed his own mathematical universe and populated it with arcane terms like “inter-universal Teichmüller theory” and “alien arithmetic holomorphic structures.”
Other mathematicians knew he was inventing some exotic and potentially brilliant mathematical machinery, but they had largely ignored his work, deeming it too abstruse and not worth the effort to try and understand.
Now the normally ordered world of higher mathematics is about to do something extremely unusual, plunging into a realm of abstraction and logic that even specialists don’t understand. It’s possible they’ll be stumbling down a colossal blind alley. It’s also possible that the exploration of Mochizuki’s work will change mathematics forever. If Mochizuki is right, he will have done much more than proven the ABC conjecture: This quiet, 43-year-old native of Tokyo will have invented a whole new branch of math and transformed the way we understand numbers.
T he ABC conjecture is a young problem in mathematics, first proposed in 1985 by the mathematicians Joseph Oesterlé and David Masser to describe the relationship between three numbers: a, b, and their sum, c. The conjecture says that if those three numbers don’t have any factors in common apart from 1, then the product of their distinct prime factors (when raised to a power slightly greater than one), is almost always going to be greater than c.
The conjecture intrigues mathematicians because, according to traditional thinking, there shouldn’t be any connection between the prime factors of a and b and the prime factors of their sum. If the ABC conjecture is true, it suggests there’s some hidden property of prime numbers that extends down deeper than we’ve been able to perceive. With his proof, Mochizuki claims to have put his finger on a previously imperceptible thread running through the ordinary operations of addition and multiplication.
“The ABC conjecture is somewhat mysterious,” says Minhyong Kim, a professor at the University of Oxford and the Pohang University of Science and Technology and a longtime acquaintance of Mochizuki. “It is really saying that the process of adding and multiplying ordinary numbers constrains each other in a subtle but precise way. How can there be something new to say about the relation between addition and multiplication? But there seems to be.”
At the time Mochizuki began working intently on ABC, the problem was barely a decade old and little progress had been made, meaning Mochizuki was essentially cutting his own trail. The most notable work on ABC had come from another well-regarded Japanese mathematician, Yoichi Miyaoka, who in 1988 claimed to have proven the conjecture. Miyaoka’s mathematics were considered beautiful and elegant, but ultimately the proof collapsed when other mathematicians checked his work and found serious flaws. Mochizuki’s work amounts to only the third serious attempt to prove the conjecture since then.
In order to understand why Mochizuki’s situation is unusual, it’s useful to compare it to the two biggest discoveries in math in the last 20 years: the British number theorist Sir Andrew Wiles’s proof of Fermat’s Last Theorem (he was knighted for his accomplishment) in 1995 and Russian mathematician Grigori Perelman’s proof of the Poincaré conjecture in 2003. Hailed as singular discoveries, these proofs transformed their authors into the two most famous mathematicians in the world (alongside, perhaps, John Nash of “A Beautiful Mind” fame). But their work built from a base of well-understood mathematics, following routes that others had already speculated could lead to proofs. As a result, the mathematical community was able to verify Wiles’s and Perelman’s proofs in relatively short order.
Not so with Mochizuki’s proof.
Mochizuki made a name for himself in his 20s based on a number of significant contributions to a relatively new, complex subfield of arithmetic geometry known as anabelian geometry. In 1998 he received one of the highest honors in math—an invitation to address the quadrennial International Congress of Mathematicians. He gave his talk in August 1998 in Berlin, then effectively went to ground, disappearing for 14 years to work on ABC.
“I guess he wanted to work on a problem worthy of putting his full powers on,” says Jeffrey Lagarias, a professor of math at the University of Michigan, “and the ABC conjecture fit beautifully.”
Before mathematicians can even start to read the proof, or understand his four papers, they need to wade through 750 pages of Mochizuki’s incredibly complicated foundational work in anabelian geometry. At the moment, there are only about 50 people in the world who know anabelian geometry well enough to understand this preliminary work. Then, the proof itself is written in an entirely different branch of mathematics called “inter-universal geometry” that Mochizuki—who refers to himself as an “inter-universal Geometer”—invented and of which, at least so far, he is the sole practitioner.
“Mathematics is very painful to read, even for mathematicians,” says Kim, explaining why vetting Mochizuki’s proof poses such a formidable task. “Most mathematicians, even people who have the necessary background knowledge in general arithmetic geometry, it’s hard to convince them to put in the energy and time to read the paper.”
Mochizuki is known as an uncommonly clear and poetic writer for a mathematician, but well-established mathematicians don’t have much incentive to put in the years it would take to understand his work: Their research programs are set and unlikely to change dramatically in response. But a handful of up-and-coming mathematicians have seized on Mochizuki’s potential proof as a chance to get in on the ground floor of a possible new field.
Jordan Ellenberg, a professor of mathematics at the University of Wisconsin, is one of them. He’s spent the last two months trying to absorb Mochizuki’s ideas. He’s far from convinced that the proof works, but he’s intrigued by its immense possibility.
“When Wiles proved Fermat,” Ellenberg says, “people were energized to understand his work because they knew he could only have done that if he had understood something true and new about arithmetic. A whole field of mathematics and dozens of people’s careers blossomed out of Wiles’s original paper. That’s the best case scenario with Mochizuki; that’s the hope.”
Vesselin Dimitrov, a graduate student at Yale University, has been concentrating on reading Mochizuki’s preliminary writing as preparation for reading the proof. In a series of e-mails, he explained that he’s drawn by both the challenge of the ABC conjecture and the elegance of Mochizuki’s thinking. “Reading through Mochizuki’s world,” Dimitrov writes, “I am much impressed by the unity and structural coherence that it exhibits. “
Dimitrov stressed that it’s too early to predict whether Mochizuki’s proof will stand up to the intense scrutiny coming its way. In October he and a collaborator, Akshay Venkatesh at Stanford University, sent a letter to Mochizuki about an error they found in the third and fourth papers of the proof. In response, Mochizuki posted a reply to his website acknowledging the error but explaining that it was minor and didn’t affect his conclusions. He is expected to post a corrected version of his proof by January.
Th e math community has reacted to Mochizuki’s proof with equal parts hope and skepticism, though few mathematicians are willing to discuss their doubts on the record out of respect for Mochizuki and a desire not to prejudge the vetting process.
Mathematicians speak of a “brick wall” in mathematical reasoning that has thwarted previous attempts to solve ABC. “Before Mochizuki came along, this problem was viewed as utterly, hopelessly intractable and out of reach, like an out-of-the-solar-system kind of situation,” says Lagarias. In that light, any supposed proof was bound to be greeted with some doubt.
Another source of skepticism is the potential expansiveness of Mochizuki’s accomplishment. It has long been understood by mathematicians that any proof of ABC would have the effect of simultaneously proving four other theorems (the work of Roth, Baker, Faltings, and Wiles) that stand among the most celebrated achievements in math in the last half-century. If Mochizuki has found a way to subsume those monumental results into a single formula, his work would take its place alongside equations like Einstein’s E=mc2 and the inequality behind Heisenberg’s uncertainty principle in terms of its sheer explanatory power. To many, such a discovery seems too good to be possible.
Minhyong Kim thinks that the initial reaction to Mochizuki’s work owes to something else. “Frankly, there are many people who express skepticism because they look at it and they can’t understand what’s going on, and of course when you can’t understand something the most natural initial response is to be skeptical.”
In the coming weeks mathematicians are hoping Mochizuki will provide a kind of “executive summary” of his work—a 20- or 30-page template that traces the pivot points in his logic. Meanwhile, mathematicians like Ellenberg and Dimitrov will continue to poke at Mochizuki’s proof, looking for openings, raising questions, and translating Mochizuki’s ideas into terms that a wider circle of mathematicians can understand.
If Mochizuki’s work makes it through these informal early checks, work groups and conferences will be organized around his ideas. The Clay Mathematics Institute at Oxford has already expressed interest in sponsoring one such workshop. Further down the line—perhaps a year from now, if Mochizuki is able to build up sufficient trust with his peers—Mochizuki will submit his work for journal publication and it will be sent out for peer review.
Mochizuki’s reputation as a gifted mathematician will survive even if his proof turns out to be wrong. But the same cannot be said for his work of the last decade. Mathematicians are tantalized by the possibility of a proof of ABC, but if an error is found early in the vetting process, the math world will likely move on without bothering to explore the rest of the mathematical universe Mochizuki has created. Mochizuki has almost certainly made significant discoveries—but without the allure of a proof, it’s possible no one will take the time to understand them.
Mochizuki, who is known as a shy person and has declined interview requests since publishing the proof in August, will have an important role to play in all of this, answering queries and explaining his work to a math community from which he long ago parted ways.
And this, Kim thinks, might pose the greatest challenge of all. “When you’ve been wrapped up in your own research program for a long time sometimes you lose a sense of what it is that other people don’t understand,” he says. “Other people feel quite mystified as to what he’s doing and part of him, I suspect, doesn’t quite understand why.”
Kevin Hartnett, a freelance writer, lives in Ann Arbor, Mich.
Correction: Because of an editing error, an earlier headline on this story about a mathematics proof gave the wrong city for the scholar who produced it. He works in Kyoto.