tar0log

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.”

In the April 3 issue of the Japanese business magazine Weekly Diamond, Professor Fumiharu Kato of the Tokyo Institute of Technology wrote about the reasons for the “misunderstanding” surrounding IUT.

Prof. Kato published a book on IUT theory titled “Mathematics That Bridges Universes” in 2019, which became a best-selling scientific book in Japan. He was one of the researchers who held seminars and collaborated with Prof. Mochizuki while he was building the IUT theory, and is known as an evangelist of the IUT in Japan.

Kato himself wrote about the article on twitter, “I wrote something important about the IUT theory. Why do leading experts make elementary mistakes when they see the IUT? Please have a look. I hope you enjoy it.”

I’m not sure why he wrote such an “important” article in a domestic business magazine, but anyway, I translated it into English so that people overseas can read it. However, since this is a column for a general magazine, it does not include technical explanations. I used DeepL for the translation.

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The contents of Prof. Mochizuki’s “new report” published following the publication of his paper on the ABC conjecture

Prof. Fumiharu Kato, Tokyo Institute of Technology

For the first time in a long time, bright news in mathematics has hit the world.

On March 5, it was reported that a series of papers on the Inter-Universal Teichmüller (IUT) theory, which solves the ABC conjecture, were published in a special electronic edition of the international journal PRIMS. The papers, written by Prof. Shinichi Mochizuki of Research Institute for Mathematical Sciences, Kyoto University, were accepted for publication in the journal in February 2020 after seven and a half years of peer review since their submission in August 2012.

I have explained in this column twice (May 30, 2020 and July 18, 2020) what the ABC conjecture is, why it is the most difficult problem in modern mathematics, and how Professor Mochizuki confronted this difficult problem.

A Prescription for Misconceptions

The background of the ABC conjecture is the complex intertwining of the “addition” and “multiplication” of numbers. Therefore, in order to disentangle addition and multiplication, IUT theory considers multiple mathematical “universes” (i.e., a set of environments in which mathematics can be performed).

Thus, the ABC conjecture was solved with a new flexibility that was not available in conventional mathematics.

In conjunction with the publication of this paper, Professor Mochizuki published a new report on his website titled “On the Essential Logical Structure of Inter-universal Teichmuller Theory in Terms of Logical AND “∧”/ Logical OR “∨” Relations“. In the report, Prof. Mochizuki carefully explains the “logical structure” on which the entire IUT theory is based.

As I have already mentioned in part in my book “Mathematics that Bridges Universes” (KADOKAWA, 2019), the IUT theory is unfortunately not correctly understood by all researchers even now due to its novelty, and there are many people who have fundamental misunderstandings about it.

In 2018, researchers critical of the IUT theory and Prof. Mochizuki and his colleagues started a debate on the mathematical correctness of the IUT theory, but it stopped after a few months (due to the unilateral departure of the critics). Although the history of the debate is available to anyone in the form of reports from both sides, it is very regrettable that, regardless of the mathematical content of the debate, only the rumor that “the debate ended in failure” remains.

Even now, various rumors and speculations about the merits of the IUT theory are flying around (even among professional researchers) in a dimension other than the important mathematical content.

The most important role of this report by Prof. Mochizuki is to diagnose the structure of “misunderstandings” that have been revealed in various ways since the 2018 debate and to clarify prescriptions for them in the dimension of a proper mathematical discussion, which is a different dimension from these “rumors”.

According to Professor Mochizuki, most of the misunderstandings about IUT theory are misunderstandings at the level of “logic”. Moreover, it is a misunderstanding at the “elementary and simple” level. Professor Mochizuki explains the structure of this misunderstanding from the very basic level of discussion of “logical product ∧” and “logical sum ∨”, that is, “AND” and “OR”.

In fact, many of the controversial points in the IUT theory paper were so obvious to those who understood IUT that they did not even understand why additional explanation was necessary.

What is the background to such “elementary” misunderstandings? This report can be said to be a useful “first aid” for all those who want to understand IUT theory by taking a scalpel to the background of the misunderstanding.

One aspect of the explanation is this: IUT theory requires us to consider separately two dimensions that are inextricably linked in the very nature of “number”, such as addition and multiplication.

For this reason, we set up multiple mathematical stages, set up loose “links” between them, and then perform the calculations “all at once” in a “multiradially defined algorithm”. What is important in doing so is that we need to prepare several “mutually different copies of the same thing” between the different stages, and carefully distinguish and manage them.

Proper management of copies

For example, in a coordinate plane with an x-axis and a y-axis, both the x-axis and the y-axis are copies of the “number line”. So they are “the same thing”.

However, in the coordinate plane, they must be properly distinguished (labeled as x and y) and treated separately (otherwise, the plane would not be a plane).

Thus, in mathematics, it is necessary to carefully distinguish between “different” copies of the “same” object, or conversely, to equate them. This distinction and identification can be used in different ways depending on the situation, but the way to do so is not always explicit.

It is done almost unconsciously because of “familiarity” or “training” since middle school or high school mathematics, or in the paradigm of modern mathematics.

In IUT theory, however, in order to disentangle addition and multiplication, the structure of numbers must be pushed down to a basic layer, and the management of copies of the same single object must be properly performed at a more fundamental level.

Of course, the management of copies of the same single object itself is not much different from what conventional mathematics has done, but many people are not accustomed to distinguishing between copies of an object at such a basic level.

That is why the IUT theory is very detailed and explicit about how to do this. Still, it is hard to get used to doing consciously what we have been doing unconsciously. And if you don’t manage even one copy correctly, you will make an elementary mistake.

However, this is also different for each person, and for those who can do it right from the start, it is difficult to understand what is so difficult about it. This is the basic structure of the typical misunderstanding of IUT theory.

In the first place, it may be hard to believe that a rigorous science like mathematics can have a “correctness debate”.

Historically, however, mathematics has passed through many controversies. For example, the debate on the foundations of mathematics in the 19th and 20th centuries. It may be time for modern mathematics to go through a major controversy and be reborn.

Translated with www.DeepL.com/Translator (free version)