Thursday, October 29, 2015

When is a proof a proof?

The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof 

I don't often post about mathematics, but Peter Woit's blog had a link to this article in Nature about a lengthy proof that hardly anyone in the field can understand:
But almost everyone who tackled Mochizuki's proof found themselves floored. Some were bemused by the sweeping — almost messianic — language with which Mochizuki described some of his new theoretical instructions: he even called the field that he had created 'inter-universal geometry'. “Generally, mathematicians are very humble, not claiming that what they are doing is a revolution of the whole Universe,” says OesterlĂ©, at the Pierre and Marie Curie University in Paris, who made little headway in checking the proof.

The reason is that Mochizuki's work is so far removed from anything that had gone before. He is attempting to reform mathematics from the ground up, starting from its foundations in the theory of sets (familiar to many as Venn diagrams). And most mathematicians have been reluctant to invest the time necessary to understand the work because they see no clear reward: it is not obvious how the theoretical machinery that Mochizuki has invented could be used to do calculations. “I tried to read some of them and then, at some stage, I gave up. I don't understand what he's doing,” says Faltings.

Fesenko has studied Mochizuki's work in detail over the past year, visited him at RIMS again in the autumn of 2014 and says that he has now verified the proof. (The other three  mathematicians who say they have corroborated it have also spent considerable time working alongside Mochizuki in Japan.) The overarching theme of inter-universal geometry, as Fesenko describes it, is that one must look at whole numbers in a different
light — leaving addition aside and seeing the multiplication structure as something malleable and deformable. Standard multiplication would then be just one particular case of a family of structures, just as a circle is a special case of an ellipse. Fesenko says that Mochizuki compares himself to the mathematical giant Grothendieck — and it is no
immodest claim. “We had mathematics before Mochizuki's work — and now we have mathematics after Mochizuki's work,” Fesenko says.

But so far, the few who have understood the work have struggled to explain it to anyone else. “Everybody who I'm aware of who's come close to this stuff is quite reasonable, but afterwards they become incapable of communicating it,” says one mathematician who did not want his name to be mentioned. The situation, he says, reminds him of the Monty Python skit about a writer who jots down the world's funniest joke. Anyone who reads it dies from laughing and can never relate it to anyone else.
 All rather odd.

1 comment:

Not Trampis said...

all rather odd or even!