When one is an expert in a niche area of mathematics, it becomes difficult to judge their workThomas T, Unsplash

Mathematics is often seen as the purest of all sciences and perhaps the closest humans can get to absolute truth. While there have always been debates about whether mathematics was invented or discovered, and whether mathematical objects can be said to exist, the idea of proof is what makes mathematics different to experimental sciences. A mathematical proof is a chain of reasoning that starts from just a few basic assumptions (for example, in geometry, you might assume that you can always draw a straight line between any two points), with logical steps using these assumptions, eventually leading to the desired result.

Attempted proofs that turned out to contain mistakes have been published, the most famous example being Wiles’ proof of Fermat’s last theorem (which was fixed the following year), but otherwise, the proof is accepted by the mathematical community and considered true, becoming a part of mathematics (just like Pythagoras’ theorem or the quadratic formula). Either a proof is correct, or it is not.

Well, usually.

The story of the proof – or claimed proof, or failed proof, depending on who you ask – of the abc conjecture is a situation almost without precedent in mathematics. For the past decade, one mathematician claims to have proved an important unsolved problem – but the few people who understand it are split on whether it actually works.

“Is it enough to know that a statement is true, or is part of mathematics being able to explain why?”

The abc conjecture dates back to 1985, proposed by David Masser and Joseph Oesterle (in mathematics, a conjecture is a result believed to be true but not yet proven). The conjecture is based on the simple equation a + b = c; simply put, it suggests that if and b are divisible by large powers of prime numbers, then will not be. If true, this would have many implications for the field of number theory, including a simpler proof of Fermat’s last theorem and the resolution of many other problems.

In 2012, the Japanese mathematician Shinichi Mochizuki posted four papers on his website, claiming to prove the conjecture. At the time, Mochizuki was a highly respected mathematician, and the announcement was greeted with excitement. The papers together are 500 pages long and contain many new ideas; the mathematician Jordan Ellenberg wrote on his blog comparing his initial impressions of the work to “reading a paper from the future, or from outer space”.

Several years later, despite conferences intended to explain Mochizuki’s work to other mathematicians, only a handful of people claimed to understand the proof. It was incredibly long, difficult to read, and heavily dependent on other work Mochizuki had developed over the previous twenty years. Mochizuki was also accused of not doing enough to try to communicate his ideas.

In 2017, the mathematician Frank Calegari wrote a blog post titled “The abc conjecture has (still) not been proved”, calling the situation was “a complete disaster” for number theory. He argued that there were three possibilities: that someone would clearly explain the proof, that someone would find an error in it, or that neither would happen and that “the mathematical community [would move] on” until someone independently proved the result.

“His initial impressions compared the work to reading a paper from the future, or from outer space”

Calegari also argued that whether or not Mochizuki’s work is correct is irrelevant if it is not understandable, comparing it to a hypothetical proof written in an undeciphered language. This is reminiscent of the debate on computer-assisted proofs in mathematics. Essentially, some results in mathematics have been proved not using methodological arguments, but instead using computer programs that tested thousands of possible cases, sparking a debate about the purpose of proof; is it enough to know that a statement is true, or is part of mathematics being able to explain why? However, the difference between computer-assisted proofs and Mochizuki’s proof is that most people believe in the correctness of the former. For Mochizuki’s work, few mathematicians understood enough of the proof to consider themselves able to comment on its accuracy – that is, until 2018.

That year, two leading German mathematicians, Peter Scholze and Jakob Stix, published a report arguing that a specific part of Mochizuki’s work, central to the proof, was wrong. This was based not only on their detailed study of the work but also on a meeting with Mochizuki. However (and unusually), Mochizuki did not agree with the criticism; neither side could convince the other.


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Probability puzzles and problems

In 2020, it was announced that the proof would be published in the Japanese journal RIMS (where Mochizuki, controversially, is the chief editor). Most believed that this would change a few minds. Peter Woit, a mathematical physicist, wrote that “abc is still a conjecture”, calling the mathematicians who believe in it “a small circle of die-hards” and criticising Mochizuki’s response to the criticism for focusing on personal attacks.

That is where the situation stands today; a few mathematicians who think the proof is valid, more who think it is not, and the rest of the community, who have chosen not to put in the huge effort required to try to understand it. I make no claim to be able to comment on its validity, but if most mathematicians are not convinced, then neither am I.