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Math Trek

A theorem in limbo shows that QED is not the last word in a mathematical proof

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When a top-tier mathematician announced in August that he had proved one of the greatest problems in mathematics, the claim was trumpeted in the New York Times, Nature, Science and the Boston Globe.

But that didn’t make the abc conjecture proven. People often think of mathematics as a solitary pursuit, with a written proof as final product. In fact, it’s an unavoidably social activity, even for mathematicians who prefer to work alone. A theorem isn’t proven until the mathematical community is persuaded that it’s proven. And proofs today are often so complex that that persuasion must happen in person.

Six months after Shinichi Mochizuki of Kyoto University in Japan released his 500-page proof of the abc conjecture, that vetting process has yet to occur. No one has been able to explain the central ideas of the proof. And few people are trying to understand it anymore, with the possible exception of a mathematician or two in Japan.

Whenever major proofs are announced, mathematicians caution that the work might not hold up. Ordinarily it’s a matter of checking for hidden errors, as mathematicians in the field quickly understand the strategy of the proof.

But this time, no one except Mochizuki seems to have any glimmering of how his proof works. It is so peculiar that mathematicians might have dismissed it as the work of a crank, except that Mochizuki is known as a deep thinker with a record of strong results.

Also, they really hope he is right. Though the abcconjecture is only 30 years old, it has become one of the greatest prizes in mathematics, subsuming Fermat’s Last Theorem along with four other major theorems in number theory.

To understand what it says, start with two whole numbers, a and b, that are divisible by small primes raised to large powers. Say a = 210 = 1,024 and b = 34 = 81. Add them together to get c: In this case, c =1,024 + 81 = 1,105. That number happens to be the result when you multiply three other prime numbers raised to small powers (in this case the power is 1): 5 x 13 x 17 = 1,105.

This pattern of two numbers a and b that are divisible by small primes raised to large powers adding up to a number c that is divisible by large primes raised to small powers turns out to be quite common. In the 1980s, mathematicians formulated this observation precisely into the abc conjecture, encapsulating a deep connection between the two most basic mathematical operations, addition and multiplication.

In August, when Mochizuki released the four papers explaining his proof, a number of mathematicians dove into them eagerly. But they couldn’t even understand his vocabulary. Mochizuki had built an entirely new mathematical field, one he named “inter-universal Teichmüller geometry,” and populated it with objects no one had ever heard of: “anabelioids,” “Frobenoids,” “NF-Hodge theaters.” Without any sense of the overarching logic of the proof, his readers bogged down in minutiae.

This isn’t all Mochizuki’s fault: Mathematicians generally don’t necessarily like to read mathematics. “It’s so painful to read someone else’s paper, even if it is a short paper,” says Minhyong Kim of the University of Oxford. The formality and precision necessary for accuracy can interfere with developing intuition, particularly with such unfamiliar mathematics.

Mathematicians began clamoring for Mochizuki to explain the kernel of his ideas, but he refused. Kim explains the refusal this way: “Imagine asking a poet what a poem means: They’d probably say no. What they meant by the poem is what they wrote. I suspect that this is the psychology of the situation for Mochizuki. He said what he wanted to say in the paper.”

Mochizuki has, however, been happy to answer specific questions by e-mail. And recently he released a “panoramic overview,” though many mathematicians find it nearly as impenetrable as the full proof.

Rumors circulated that some of the leading figures in the field had become skeptical of the proof. Out of respect for Mochizuki — and hope that his proof will, in the end, turn out to be right — an effort was made to keep these rumors from circulating on the Internet, and no mathematician would go on the record expressing them. Still, the rumors have further dampened enthusiasm for the hard work of slogging through the four papers. 

Hope rests on a couple of Japanese mathematicians believed to be talking through the proof with Mochizuki, but they are unwilling to talk to the press.

“If something is very, very familiar to you, it’s kind of hard to put yourself in the position of someone who’s never seen it,” says Jordan Ellenberg of the University of Wisconsin–Madison. “What we need is for this stuff to exist in someone else’s brain besides [Mochizuki’s]. We need someone who has just understood it to help us understand it.” 

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