The wonderful story of a math proof is true only in Japan

As a scientific journalist, there are some stories that you have returned forever. There is a climate change, of course – the biggest story of our time. Then there are big questions, such as “what is meditatures?” Or “Are we alone in the universe?”, etc. But one of the stories I could not stop a math argument that flows over a decade. I call it “The proof is only true in Japan”.

Let me start. In 2012, mathematical Shinichi Mohizici In Kyoto University, Japan, published an amazing set of paper, across 500 pages, detail what he called the theory of inter-universal (IUT) theory. It is a framework for stretching and deforming our ordinary concepts of mathematical items, such as numbers and adding, transferring them to new “universities” replacing new insights. “He literally takes the custom things in terrible ways and rebuild them with new universities,” A mathematician who told me the time.

That’s a pretty cool sound, but what makes it a bigger claim that MoChizuki claims to use ABC consecture, a deep problem with ABC’s current 40th. Unlike iut, the statement of the assumption can be easily understood, because it only begins with equation A + b = c, which each letter for a whole number, or integer.

You also need to know that every integer can be divided by its main reasons, the main numbers that serve as building blocks for all numbers. For example, the main causes of 21 are 3 and 7, while they were 12, 2 and 3. Of 12, 2 with self-defined species at once.

What does ABC consecture suggest that if you multiply different Prime reasons to A, B and C together, the result is usually greater than c. For example, if we have 12 + 21 = 33. Our main main reasons are 2, 3, 7 and 11 (11 and 11 (11 and 11 and the larger restrictions on the abundance of the increase.

Besides telling us something deep about the nature of numbers, verifying the ABC conjecture, in turn, opening a thorough mathematical results that depends on the bad proof Fermat’s last theorem – So MoChizuki’s claim is a big deal. There is only one problem: nothing can understand it. During publication in 2012, mathematicians compared Mochizuki’s work on a paper from “outer space“, So the alien is his way.

Not unusual for mathematicians to process a while in the process and fully understand a great proof, but the next occurring may not be like mathematical history. A few years after the initial post of Mochizuki on his website, his epic proof has not been accepted for a formal publication of a journal. By the end of 2014, He posted an updateThe other mathematicians were criticized not to share his job, teaching that researchers spent the time to study under Kyoto University for many months understood it.

This impasse went on for years, with Conferences devoted to IUT understanding and even Publishing a 300-page “summary” of task. By 2017, it is said that a dozen people understand Mochizuki’s work – they all submit to his intense robbery. But Mochizuki’s refusal to leave Japan to attend international conferences or engage in the wider world – he never responds to a request for the interview New ScientistFor example – mean most of the mathematicians do not want to spend time learning a theory that may not pay.

But then, two mathematical Germans seem to carry a resolution to this saga. ATTEND Peter Scholze at the University of Bonn and Jakob Stix At Goethe University Frankfurt, two heavyweights in the field. They joined a week working with Mochizuki in Tokyo in March 2018, and in September announced they found a Durable Disease in his proof.

The error concerned a part of the proof called Corollary 3.12, seen as a vital part of MoChizuki’s efforts to solve the ABC conjecture, which scholze and stix claimed suffered from an unjustified leap of logic. “We agree with no evidence,” writes the pair, without answering a request to comment for this article.

But in an extraordinary twist, Mochizuki and his acylets refused to accept the suggestion of any mistake. In 2020, the IUT Papers accepted for publication In the journal checked by peer Publications in the research institute for math sciences. While this is a respected journal, detractors target the potential for the conflict of interest Mochizuki (and is) the editor-of-chief without publishing. The papers reflect a 2021 issue in the journal.

So, we have: a proof that most mathematicians believe that the mistake, but that is a little japanese insunist devotees. Since the formal publication, Mochizuki supporters have set the inter-universal geometry center to improve the IUT and even launched a $ 1 million prize for anyone who can show that it is false – Something scholze and stix seem to be done. Another prize is also given each year to people who have committed an essential development of the IUT study – the first such prize, with an award $ 100,000, went to Mochizuki and his colleagues.

The outcome is definitely strange, and yet the story does not end there because another challenge has entered the arena. In recent years, Kirti Joshi At the University of Arizona repeatedly published at work claiming both Mochizuki and Scholze / Scholze / Schox error, but he has a solution Resolve the crisis. He admitted to the error founded in Corollary 3.12, but the scholze disagree. Meanwhile, Mochizuki’s response was not very modified, refused to join Joshi’s questions directly and instead Post a criticism Calling Joshi’s work “deep without knowledge” and lack “any meaningful mathematical content of anything”.

In May, Joshi published a paper titled Final Report of Mochizuki-Scholze-Stix Controversy showing attempt to draw a line under the proceedings. “I checked claims of (Mochizuki, scholze and stix) in respectful detail, and given a canonical formation of Mochizuki’s infinite work. Joshi declined a request for commentary.

Is there anything else we can have a satisfactory answer to who is right? In the purity of maths seeming to come down to the human name – call, we may need to call the unmatched adjudicator. In surprise, such thing is in the form of formal proofer checkers. The idea is that you interpret every part of a math proof of a machine-read form, and then a checked computer that is correct each step.

If it is easy sound, it is not – the process of “formalization” is complicated and difficult, and some evidence is evaluated in this way. When I first wrote about the idea of ​​meeting Mochizuki’s job with a proof checker in 2015Mathematicians have told me that may be more difficult than the original proof. The goods moved since then, as artificial intelligence has Starts used in the formalizationBut I think a long time, long way from Mochizuki’s job is true outside Japan.