This is an article that appeared in the Asahi Shimbun in Japan in the spring of 2020, when the Mochizuki paper was accepted. Since some people seem to be interested in the paid part, I translated the entire article into English.

I think the article is fairly balanced, but I am not sure if it is appropriate to compare the criticism of Mochizuki’s paper to the history of the delay in the evaluation of Galois. The explanation that Perelman’s proof was verified at a faster pace than Mochizuki’s is “because he used familiar mathematical methods” does not seem to be very correct. In any case, it’s a shame that the majority of the article is not being seen by many people since it is a paid publication.

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### ABC Conjecture: “Is the Proof True?” Paper challenged in the U.S. and Europe

Tetsuya Ishikura, May 3, 2020 9:00 am

A paper on the Inter-Universal Teichmüller (IUT) theory published by Shinichi Mochizuki, a 51-year-old professor at Kyoto University’s Research Institute for Mathematical Sciences (RIMS), which claims to have proved the “ABC conjecture,” an extremely difficult mathematical problem, has been the subject of debate, mainly in Europe and the United States. In April this year (translator’s note: 2020), an international journal published by RIMS announced that it had decided to publish the paper after seven and a half years of verification that the proof was correct, but some mathematicians are not convinced.

Immediately after the announcement, the British science magazine New Scientist published an article with such a title: “Baffling 500-page ABC maths proof to be published after eight-year row”. The article introduced a criticism by Professor Peter Scholze of Bonn University in Germany, a young genius who won the Fields Prize at the age of 30, who said, “There are serious and uncorrectable gaps in the paper. Another mathematician in the U.K. said, “I’m changing my view that the proof is flawed. It is a bad situation when it is accepted only by one group and not by others,” he pointed out.

The ABC conjecture is a question about the fundamentals of mathematics, namely addition and multiplication. If proven, it will lead to the solution of a number of unsolved problems and is said to be an achievement worthy of the Fields Prize, the “Nobel Prize of mathematics”.

#### The paper took seven and a half years to verify

However, after Prof. Mochizuki announced that he had proved the ABC conjecture in 2012, it took a series of twists and turns until his paper was verified and published in a mathematical journal. At the end of 2017, PRIMS, the mathematical journal of RIMS, once decided to publish the paper, but doubts erupted from abroad over the method of verification and other issues. On April 3 (note: in 2020), the journal announced that it had finally decided to publish the paper after having several experts check it again. It had been seven and a half years since the announcement.

Even so, there were still many skeptical reports in the US and Europe. The British science magazine Nature said, “The latest announcement seems unlikely to move many researchers over to Mochizuki’s camp”. The U.S. science and technology journal Popular Mechanics said that Future discussions “will revive the debate and bring any flaws to the surface, conclusively, once and for all”.

Scholze told the Asahi Shimbun, “I was surprised by the reports that the paper has been accepted for publication. I stand by my position that the proof is still out there,” although the PRIMS editorial board explained at the press conference that after Professor Mochizuki refuted Scholze’s criticism, there was no second refutation from him. Scholze said, “Professor Mochizuki has not made any substantive rebuttal.”

Why does the criticism not stop? The main reason seems to be that the IUT theory is too difficult and the four papers are too long (646 pages) to be understood by many mathematicians. Definitions of completely new concepts and terms appeared one after another, and it was even said, “I don’t even know what I don’t understand”. The proofs of “Fermat’s Last Theorem” and “Poincaré Conjecture,” which were also extremely difficult problems, were praised with open arms because they were solved using familiar mathematical methods, which is very different from the case of Mochizuki’s paper. Even Prof. Mochizuki’s advisor at Princeton University, the Fields Prize-winning Gerd Faltings, said, “If he insists on the proof, he should make an effort to explain it more clearly. I am also not sure that the ABC conjecture has been proven.” Professor Emeritus Joseph Oesterlé of Sorbonne University in France, who presented the ABC conjecture in 1985, said, “For the proof to be accepted, it needs to be understood by many experts, but we are not in that situation now.”

#### Difficult to understand and “brain-numbing”

Professor Akio Tamagawa of Kyoto University, co-editor-in-chief of the editorial board of PRIMS, also confided, “I have 100% confidence in the peer review process, but the IUT theory is structured like a complex machine, and my brain gets tired of following the logic.” He said, “The more a mathematician is immersed in conventional theories, the more the accumulated knowledge becomes a hindrance. Like a student who is exposed to mathematics for the first time, you have to learn it one by one to keep up.”

There have been cases in the past where innovative theories were not understood for a long time. Teiji Takagi, who is known as the “father of Japanese mathematics,” proposed the “class field theory” in 1920 to describe the relationship between prime numbers, which was not understood at first because of its grandeur. The theory of Évariste Galois, the French mathematician who introduced the concept of “groups” in the 19th century, is now a basic tool in mathematics and physics, but it was not appreciated until after his death. Albert Einstein famously criticized quantum mechanics, saying that “God is not playing at dice.”

On the other hand, Professor Ivan Fesenko of the University of Nottingham, UK, strongly supports the idea. He points out that most of the critics are not experts in Anabelian geometry, which is related to the Mochizuki paper. “They criticized the paper because they did not understand the theory. There are now more than 20 people who understand the theory, and they have all confirmed that there are no mistakes,” he said. The number of young mathematicians who are starting their studies is also increasing, he said. Nobushige Kurokawa, professor emeritus at Tokyo Institute of Technology, has high hopes for the IUT theory, saying, “The IUT theory is not limited to proving the ABC conjecture, but may become a powerful weapon that can influence other difficult problems and fields.”

What does Professor Mochizuki think about this? Professor Mochizuki refused to be interviewed and did not appear at the press conference, but in a commentary for the media, he compared the IUT theory to a computer program and explained, “Each line is simple and easy, but it requires patience to decipher it step by step, just like deciphering a program consisting of tens or hundreds of thousands of lines.”