Sinichi Mochizuki And The Impenetrable Proof

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To complete the proof, Mochizuki had invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” number theorist Jordan Ellenberg, of the University of Wisconsin–Madison, wrote on his blog a few days after the paper appeared.

Here's a really important point:

For Mochizuki's work, “it's not all or nothing”, Ellenberg says. Even if the proof of the abc conjecture does not work out, his methods and ideas could still slowly percolate through the mathematical community, and researchers might find them useful for other purposes. “I do think, based on my knowledge of Mochizuki, that the likelihood that there's interesting or important math in those documents is pretty high,” Ellenberg says.

Sounds like it would be foolish to bet that Mochizuki's proof works. Seems like the smart bet is that there's at least one significant error in it somewhere. But it also sounds like there might be something even more valuable in there than a proof of the abc conjecture: sounds like he might possibly be onto new methods of proof. An old prof of mine used to say--taking Cantor as an example--that, as interesting as it was to discover that there are more reals than natural numbers, what was really important there was the discover/invention of the method of diagonalization... And Peirce says that every major advancement in science is an advancement in methods.

   Anyway, it'll be interesting to hear what number theorists are saying about this in a couple of years.