Controversial "Alien's Math" Has A New Translator. Can He Reform Its Reputation?

If aliens really are out there, interacting with them is going to be difficult. It’s not that they’ll just speak a different language – it’s more that the very building blocks of communication themselves will probably be different. The sounds; the relationships between ideas; heck, the conceptual basis of words themselves – without a real-life babel fish, we’d basically be screwed.

Now imagine what their math would look like. 

That’s basically how most mathematicians have regarded IUT, or “Inter-Universal Teichmüller Theory” – the weird, seemingly impenetrable system invented by Japanese mathematician Shinichi Mochizuki in 2012. It’s known as the “alien language” of math, so weird and complex that even the best minds around struggle to fathom it.

But now, it has a new translator – and its fans are excited.

What is Inter-Universal Teichmüller Theory?

IUT is famous for its sci-fi-esque reputation – but its story began long ago. Almost 400 years ago, in fact – with a now infamous statement, hastily scribbled into a margin too small to contain its proof.

Fermat’s Last Theorem – the statement that “the equation xⁿ + yⁿ = zⁿ has no integer solutions for n ≥ 3” – would, despite its creator’s claims of having “discovered a truly marvelous proof” already, bamboozle the world of math for centuries. It wasn’t until 1995 that it was finally vanquished, when Andrew Wiles published the result of his life’s work on the theorem and effectively dropped the mic on every number theorist working on it until that point.

But 10 years before that, a pair of mathematicians were batting around a different approach to solving the then-open problem: a proposition now known as the “abc conjecture”, which, if proven true, would immediately and pretty obviously imply Fermat’s Last Theorem.

Like the Theorem that inspired it, the abc conjecture looks simple enough: it just says that, given three coprime positive integers a, b, and c such that a + b = c, then the product of the distinct prime factors of abc is almost always larger than c. For example, you can take, say, a = 9, b = 4, and c = 13: the product of the distinct primes is then 2 × 3 × 13 = 78, which (we hope you will agree) is indeed larger than 13.

Sounds easy, right? But as you may have already guessed from the way we keep calling it the abc conjecture rather than the abc theorem, proving the statement turned out to be pretty difficult. The first major attempt, from French mathematician Lucien Szpiro in 2007, was swiftly retracted after problems were found within the argument, and the second… well. The second attempt gave us the IUT.

“A few months ago […] Shinichi Mochizuki claimed he had a proof of the ABC Conjecture,” wrote mathematician and author Cathy O’Neil back in November 2012. But, while “I’m not saying Mochizuki will never prove the ABC Conjecture,” she declared, “he hasn’t yet.”

Now, Mochizuki was a respected mathematician with a formidable resume, and a reputation among his peers for being a smart guy – but with this new proof, he had raised more than a few eyebrows. The proof was “sprawling,” O’Neil wrote, spanning thousands of pages in total, and it relied on a huge collection of supporting literature, some of which wasn’t even peer-reviewed itself. It included “mathematical objects that nobody has ever heard of,” O’Neil pointed out; “he introduces them in his tiny Mochizuki universe with one inhabitant.”

The whole thing had been built within the weird new framework of IUT, and it was dense, idiosyncratic, and frankly impenetrable. “In order for it to be a proof, someone, preferably the entire community of experts who try, should understand it, and he should be the one explaining it,” explained O’Neil. “So far he hasn’t even been able to explain what the new idea is.”

“The issue […] is that nobody understands what he’s talking about, even people who really care and are trying,” she wrote, “and his write-ups don’t help.”

An “alien’s language”

If mathematicians were frustrated by Mochizuki – well, the feeling was mutual. As increasing numbers of experts gave up on the “proof”, its creator became ever more crotchety about what he saw as a simple refusal to think outside the box and accept his ideas. He published multiple documents online trying to elucidate his ideas, often slighting respected peers along the way – and rarely convincing onlookers as to the legitimacy of his proof.

“Essentially the claim Mochizuki is making […] is that the most accomplished and talented young mathematician in his field is an ignorant incompetent, and that everyone Mochizuki has consulted about this agrees with him,” concluded Peter Woit, a mathematician at Columbia University, in 2021. 

“It’s hard to imagine a more effective way to destroy one’s own credibility and to convince people not to bother to try and make sense of the [proof],” he wrote.

Still, Mochizuki’s proof didn’t go away. As unverified and unaccepted by the mathematical community as it was, it had two things going for it: firstly, a handful of fervent and otherwise generally well-respected supporters, and secondly, the fact that it was literally so impenetrable that nobody could figure out what was wrong with it.

“Mathematicians are often very careful,” Frank Calegari, an algebraic number theorist at the University of Chicago, wrote in 2017. “We are usually trained as mathematicians to consider an inability to understand an argument as a failure on our part.”

“Mathematicians are very loath to claim that there is a problem with Mochizuki’s argument because they can’t point to any definitive error,” he explained. “So they tend to be very circumspect (reasonably enough) about making any claims to the contrary.” 

And so, for more than a decade now, Mochizuki’s IUT has more or less languished in a sort of mathematical limbo. It’s not accepted as true by any stretch, but unable to be formally rejected – at least, until somebody (or a group of somebodies) translates it into something more explicable to a general audience.

And that’s where Zhong-Peng Zhou comes in.

To infinity… and beyond

Aged 28, and with an aborted PhD in graph theory behind him – an area of math closer to computer science than number theory – Zhou’s experience with pure math is almost entirely homegrown. Literally: his new paper, in which he describes several refinements and novel applications of IUT, is reportedly the result of a lone obsession with the topic, worked out during his off hours from a full-time job as a software engineer.

His work, he said in a social media post, “has only made some minor innovations and explorations” – but it nevertheless was enough to impress Ivan Fesenko, a distinguished professor of mathematics at Westlake University, China, and ardent long-time supporter of IUT. Spurred by his paper, Fesenko invited Zhou to re-enter academia as a visiting student under Fesenko’s supervision – and according to Fesenko, it’s going extremely well.

“His results are infinitely stronger than Wiles[’s],” Fesenko told the South China Morning Post last week – indeed, if confirmed, Zhou’s methods could prove Fermat’s Last Theorem in just one page, rather than Wiles’ 130. And that’s just one potential benefit of proving the abc conjecture – in fact, a solution would immediately prove dozens of open problems, as well as having applications in cryptography, quantum computing, and physics. 

It’s a tantalizing prospect. But until Zhou’s “translations” of the “alien” language are published, it’s likely that many mathematicians will (understandably) retain a healthy skepticism of IUT. 

“Proof is a social construct: it does not constitute a proof if I’ve convinced only myself that something is true,” wrote O’Neil. “It only constitutes a proof if I can readily convince my audience, i.e. other mathematicians, that something is true.”

So far, Mochizuki has failed to do that. Should Zhou succeed, it would mark a vibe shift in math of a magnitude rarely seen – and a boost to Zhou’s reputation that would match it. After all, O’Neil concluded, “whoever ends up interpreting this to the world will be responsible for the actual proof and should be given credit along with Mochizuki.” 

“It’s only fair,” she wrote. “And it’s also the only thing that I can imagine would incentivize someone to do such a colossal task.”

Zhou's paper has been posted as an unreviewed preprint to the arXiv.