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.Continued...