No One Can Tell If We’ve Solved One Of Math’s Greatest Problems

man and math formulas
“Genius after all ain’t ennything more than elegant kommon sense.” —Josh Billings, “Nosegays”

In A Nutshell

In August 2012, mathematician Shinichi Mochizuki quietly posted four papers on his website that would guarantee his place in the history books. Totaling 500 pages, they solved the famous ABC conjecture, a longstanding pure maths problem first proposed in the 1980s. It was cause for celebration.

At least, it should have been. Unfortunately, Mochizuki’s proof was so advanced and so complex that no other mathematician alive can understand it. As a result, no one actually knows whether he solved the ABC conjecture or is simply a deluded madman.

The Whole Bushel

In the world of pure maths, Shinichi Mochizuki is a name frequently revered. A reclusive mathematician based at the Kyoto University’s Research Institute for Mathematical Sciences (RIMS) in Japan, his work is widely respected. So when he claimed in 2012 to have found a proof for the infamous ABC conjecture, the world prepared to celebrate.

Then they actually read the proof, and the champagne was quickly stowed away again.

The ABC conjecture is something no one has ever come close to solving. A simple-seeming equation (a+b=c), it raises profound questions about the true nature of numbers. It’s not hyperbole to say solving it would be the mathematics achievement of the century. So it was always going to require a complex proof. The trouble was, Mochizuki’s proof was perhaps the most complex paper ever written.

It’s not that his work was necessarily wrong, just that nobody on Earth could understand it. To reach his conclusion, Mochizuki invented an entirely new branch of extremely abstract mathematics known as “Inter-universal Teichmuller Theory” and then used it so densely that even those who could figure out its basics became lost. Number theorist Jordan Ellenberg said, “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.”

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It doesn’t help that Mochizuki himself has been no help at all. Famously reclusive, his social skills make Sheldon Cooper look like a big fat party animal. He has so far refused to lecture on his theories, leave Japan, or give any help to anyone beyond a few testy remarks about the “need for researchers to deactivate the thought patterns that they have installed in their brains and taken for granted for so many years.” While he’s offered to teach a handful of other mathematicians his new branch of math, this still means there’s no one truly independent who can review his work.

As of October 2015, only four people in the world have managed to read Mochizuki’s proof all the way through. Three years on, we still don’t know if he’s a genius or a raving lunatic (or both). As for the ABC conjecture, it could haunt mathematics yet for a very long time.

Show Me The Proof

Nature: The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof
New Scientist: Mathematician’s anger over his unread 500-page proof

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