Shinichi Mochizuki (Photo source: Kyoto University)
One day in 2012, Shinichi Mochizuki, a Japanese mathematician, posted four papers on his website. The papers, which total more than 600 pages, claim that he has solved the ABC conjecture, one of the biggest problems in mathematics today. However, after the paper was published, for a long time, very few mathematicians could read the paper. Shinichi Mochizuki's proof has also become an unsolved case in the mathematical community.
On April 3 of the same year, Shinichi Mochizuki's proof of the ABC conjecture was published in the Journal of the Institute of Mathematical Analysis (PRIMS,—— as the name of the journal suggests, the sponsor of this academic journal was none other than Kyoto University, where Shinichi Mochizuki worked, and the editor-in-chief of the journal was none other than Shinichi Mochizuki himself.
Despite the controversy, the acceptance of the paper means that this long (both time and length) ABC conjecture has finally entered a new phase. Next, Shinichi Mochizuki's proof will be judged by more peers.
In 2017, a lecture by Japanese mathematician Fumimoto Kato, author of "The Language of Mathematics: Shinichi Mochizuki's IUT Theory" and close friend of Shinichi Mochizuki, sparked heated discussions and led to the publication of the book.
Although this book is a popular reading book to interpret Shinichi Mochizuki's "Teichmüller Theory (IUT Theory)", it is not difficult to read, and by showing Shinichi Mochizuki's mathematical research process, it mainly conveys questions such as what mathematicians do, how mathematicians think, and the underlying logic and deep thinking methods of mathematics. It is suitable for many people who are interested in mathematics and want to learn mathematical thinking.
撰文 | 达维德·卡斯泰尔韦基（Davide Castelvecchi）编译 | 吴非来源 | 环球科学
More than 30 years of conjecture
The ABC conjecture that Shinichi Mochizuki is trying to solve is a number theory puzzle with a history of more than 30 years. This conjecture reveals a deep connection between integer addition and multiplication. Each positive integer can be uniquely expressed as the product of prime numbers. Now, a, b, c are 3 positive integers, where a + b = c. If a and b can be decomposed into many small prime numbers, then c can only be decomposed into a few large prime numbers.
The idea was first proposed by the French mathematician Joseph Oesterlé in a lecture in 1985. Östler made it casually at the time, but David Masser, a Swiss mathematician who was present, realized that the conjecture might be important and began promoting it as a form of generalization. For this reason, the ABC conjecture is also known as the Östler-Mather conjecture.
A few years later, Noam Elkies, a mathematician at Harvard University, realized that if the ABC conjecture was correct, it would have a profound impact on the field of integer equation research—the proof of the ABC conjecture would solve a long list of famous unsolved diphantom equations at once.
How to understand the importance of the ABC conjecture? Let's first look at the most important breakthrough in the history of Diophantine equation research. The American mathematician Louis Mordell proposed a conjecture in 1922 that the vast majority of Diophantine equations either have no solution or only a finite number of solutions, and Modell's conjecture was proved in 1983 by the German mathematician Gerd Faltings, for which he was awarded the Fields Medal three years later. And if the ABC conjecture is correct, you know not only how many solutions there are to the equation, but also a list of them all.
If confirmed, this would be one of the most astonishing mathematical achievements of the century, and a revolution in the study of integer equations. Based on the ABC conjecture, a completely new method for proving Fermat's theorem may also emerge.
Ten years of hard fighting
Shinichi Mochizuki was born in Tokyo in 1969 and was brought to the United States by his family at an early age. At just 16 years old, the gifted Shinichi Mochizuki became an undergraduate student in the Department of Mathematics at Princeton University. Soon, he went straight to his Ph.D. studies, and his supervisor was none other than Faltins, who had proved Modell's conjecture.
In 1994, Shinichi Mochizuki, who received his Ph.D. from Princeton University, returned to Japan to work at the Institute of Mathematical Analysis. In the years that followed, he gained international fame for his work. But at the beginning of the 21st century, Shinichi Mochizuki "closed" himself in the institute, no longer attending international conferences, and rarely even leaving Kyoto. Shinichi Mochizuki did, however, keep in touch with other number theorists, knowing that Shinichi Mochizuki's ultimate goal was the ABC conjecture. He had few competitors: the vast majority of mathematicians explicitly avoided the problem, believing it to be too tricky.
On August 30, 2012, Shinichi Mochizuki published his proof online. However, instead of preprinting the site, he quietly posted the paper on the website of the Institute of Mathematical Sciences. Not only were the four papers staggeringly long, but they were also a headache for other mathematicians – some of which were completely unfamiliar to him. Shinichi Mochizuki uses astonishing, almost messianic language when describing new theories, and he even refers to the realm he created as "intercosmic geometry."
Jordan Ellenberg, a number theorist at the University of Wisconsin-Madison, said of Shinichi Mochizuki's paper: "It's like reading a paper from the future or outer space." His mentor, Faltins, also said: "I tried to read part of it, and then at some stage, I gave up. I don't understand what he's doing" and criticized him for "not articulating his thoughts more clearly".
The first to claim to understand these proofs was Ivan Fesenko of the University of Nottingham in the United Kingdom. Feshchenko carefully studied Shinichi Mochizuki's work, and in 2014 visited Shinichi Mochizuki to claim that he had verified the correctness of the proof. The overarching theme of intercosmic geometry, as Feshchenko explains, is that we must look at integers from a different perspective—adding aside, and multiplication as some kind of structure that can extend variation. Standard multiplication will be a special case in a class of structures, just as a circle is a special case for an ellipse. But for the vast majority of mathematicians, these papers are still "books from heaven", and Shinichi Mochizuki's conjecture has reached an impasse for a time.
Defects can't be repaired?
In December 2017, Japan's Asahi Shimbun reported that Shinichi Mochizuki's papers had been officially accepted by the magazine, and it was reported at the time that it was PRIMS, which Shinichi Mochizuki was the editor-in-chief of, who received these papers, but the editor of PRIMS denied this rumor at the time.
A few months later, Shinichi Mochizuki's situation was even worse. Two respected mathematicians, Peter Scholze of the University of Bonn and Jacob Stix of the Goethe University, winners of the 2018 Fields Medal, refuted Shinichi Mochizuki's proof and pointed out that one specific key passage was "seriously flawed beyond repair". Scholzer, an authority in the field of number theory, said in an interview: "I think the ABC conjecture is still open, and anyone has a chance to prove it." ”
On April 3, 2020, two of Shinichi Mochizuki's colleagues at the Institute of Mathematical Analysis held a press conference to announce that Shinichi Mochizuki's proof of the ABC conjecture had finally been accepted by academic journals and would soon be officially published, and the journal that accepted these papers was PRIMS.
However, the news of the acceptance of the paper does not seem to have caused more mathematicians to move to Shinichi Mochizuki's camp. "I can safely say that attitudes in academia haven't changed much after 2018," said Kiran Kedlaya, a mathematician at the University of California, San Diego. He had spent years trying to prove the ABC conjecture.
It should be pointed out that it is not uncommon for mathematicians to publish papers in academic journals where they are editors. Hiraku Nakajima, a mathematician at the University of Tokyo's Covili Institute of Physics and Mathematics, said that as long as authors avoid the peer review process, "it doesn't break any rules and it's very common." ”
In the mathematical community, acceptance of a paper is usually not the end of peer review, and only after a peer consensus can a conclusion be truly established. This process can take years.
For Shinichi Mochizuki's work, "the result will not be either exactly right or undesirable." Ehrenberg said. Even if the proof of the ABC conjecture is not realized, his methods and ideas can slowly penetrate the entire mathematical community, and researchers may find them useful in other ways. "Based on what I've learned about Shinichi Mochizuki, I do think it's very likely that these documents contain interesting or important math. Ehrenberg said.
But there's also a risk that things will go in the other direction, adding that "I think it would be too bad if we forgot about it." ”
Reference Links:
https://www.scientificamerican.com/article/mathematical-proof-that-rocked-number-theory-will-be-published/
"Mathematical Proof to Revolutionize the Universe?" (Global Science, December 2015)
"Looking at the Universe in the Language of Mathematics: Shinichi Mochizuki's IUT Theory" Author: [Japanese] Fumimoto KatoPublisher: People's Posts and Telecommunications PressPublication time: 2024-02This book is a popular reading book to interpret Shinichi Mochizuki's "Teichmüller Theory (IUT Theory)". The author translated Mochizuki's essays and ideas into a language that could be understood by the general reader, and created this handbook for the interpretation of the "IUT Theory". The book focuses on the interpretation of the thinking context of the "IUT theory" and its great significance to the modern mathematical system, and also shows the thinking methods of mathematicians. This book is suitable as a primer for mathematics researchers and mathematics enthusiasts to understand the "IUT theory", and also as a reference book for students to understand mathematical thinking methods.