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.Continued...