Column by Prof. Fumiharu Kato on the “misunderstanding” of the Mochizuki paper

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.

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 (free version)