An attempt to fix problems with a controversial mathematical proof has itself become mired in controversy, in the latest twist in a saga that has been running for over a decade and has seen mathematicians trading unusually pointed barbs.
The story began in 2012, when at Kyoto University, Japan, published a 500-page proof of a problem called the ABC conjecture. The conjecture concerns prime numbers involved in solutions to the equation a + b = c, and despite its seemingly simple form, it provides deep insights into the nature of numbers. Mochizuki published a series of papers claiming to have proved ABC using new mathematical tools he collectively called Inter-universal Teichm眉ller (IUT) theory, but many mathematicians found the initial proof baffling and incomprehensible.
While a small number of mathematicians have since accepted that Mochizuki鈥檚 papers prove the conjecture, other researchers say there are holes in his argument and it needs further work, dividing the mathematical community in two and prompting a prize of up to $1 million for a resolution to the quandary.
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Now, at the University of Arizona has published a proposed proof that he says fixes the problems with IUT and proves the ABC conjecture. But Mochizuki and his supporters, as well as mathematicians who critiqued Mochizuki鈥檚 original papers, remain unconvinced, with Mochizuki declaring that Joshi鈥檚 proposal doesn鈥檛 contain 鈥渁ny meaningful mathematical content whatsoever鈥�.
Central to Joshi鈥檚 work is an apparent problem, previously identified by at the University of Bonn, Germany, and at Goethe University Frankfurt, Germany, with a part of Mochizuki鈥檚 proof called Conjecture 3.12. The conjecture involves comparing two mathematical objects, which Scholze and Stix say Mochizuki did incorrectly. Joshi claims to have found a more satisfactory way to make the comparison.
Joshi also says that his theory goes beyond Mochizuki鈥檚 and establishes a 鈥渘ew and radical way of thinking about arithmetic of number fields鈥�. The paper, which hasn鈥檛 been peer-reviewed, is the culmination of several smaller papers on ABC that Joshi has published over several years, describing them as a 鈥淩osetta Stone鈥� for understanding Mochizuki鈥檚 impenetrable maths.
Neither Joshi nor Mochizuki responded to a request for comment on this article, and, indeed, the two seem reluctant to communicate directly with each other. In his paper, Joshi says Mochizuki hasn鈥檛 responded to his emails, calling the situation 鈥渢ruly unfortunate鈥�. And yet, several days after the paper was posted online, Mochizuki , saying that Joshi鈥檚 work was 鈥渕athematically meaningless鈥� and that it reminded him of 鈥渉allucinations produced by artificial intelligence algorithms, such as ChatGPT鈥�.
Mathematicians who support Mochizuki鈥檚 original proof express a similar sentiment. 鈥淭here is nothing to talk about, since his [Joshi鈥檚] proof is totally flawed,鈥� says at Westlake University in China. 鈥淗e has no expertise in IUT whatsoever. No experts in IUT, and the number is in two digits, takes his preprints seriously,鈥� he says. 鈥淚t won鈥檛 pass peer review.鈥�
And Mochizuki鈥檚 critics also disagree with Joshi. 鈥淯nfortunately, this paper and its predecessors does not introduce any powerful mathematical technology, and falls far short of giving a proof of ABC,鈥� says Scholze, who has emailed Joshi to discuss the work further. For now, the saga continues.
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