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"there isn't new profound number theory in this" I find this claim very surprising.Saying that "there is no new profound number theory" implies a deep understa

"there isn't new profound number theory in this" I find this claim very surprising.Saying that "there is no new profound number theory" implies a deep understanding of the totality of Mochizuki's work, which as far as I know is so complex and far from the current state of the art that nobody really understands it yet. Mochizuki's work may prove to be right, and it may prove to be wrong. But saying there is no new profound number theory at this point seems a little premature.

impendia 14y ago
+1, you ask a very important question, which I can at least sort of answer. I think it's key to understanding why many experts think Mochizuki's work is likely to be wrong. I cannot really give a complete explanation, being unfamiliar with his papers. But to argue by analogy, suppose I had invented a difficult and programming language. How might I persuade you to learn it? One way would be to use it to solve problems that are known to be difficult very easily, elegantly, or extensibly. It is the same with "inter-universal geometry". One should not have to follow Mochizuki's work all the way through his proof of ABC to know whether it is good for anything. What is the quickest route to a surprise? What can Mochizuki now do with less effort than the existing experts? The absence of good answers is behind skepticism of the proof.
- 001sky 14y ago
  As an outsider all this (line of reasoning--not personal) just sounds like is "not invented here" syndrome. That may or may not be fair. But isn't this what tenure is for? You can't get fired for just learning, right? Why is there so much lack of intellectual curiosity and so much self-centered-ness on "academic ambition". Are the two so self-contradictory? How much great work has gotten swept up under the rug by this attitude toward "peer review"?
  - impendia 14y ago
    I am not sure I understand all of your questions, but I assure you that if people believed there was a high chance that the proof was correct, then they would be very eager to read it. Mathematicians, and especially the very best mathematicians, are very eager to learn. I don't know of any great mathematician who demonstrates a "lack of intellectual curiosity". That said, there is much more mathematics available than anyone has time to read. It can easily take an hour a page, or more, to read a dense research paper. Mathematicians, like everyone else, have to be selective about what they choose to learn. It is very rare for great work to be ignored, but here is one example: http://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem http://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem
vhf 14y ago
Perhaps is it not so premature, since this proof relies mainly of this new inter-universal geometry, Take the Fermat-Wiles proof. It's probably the most significant number theory result of the last century, yet it doesn't contain new profound number theory. Its genius is the use of elliptic curves, which is algebraic geometry. I'm not saying Mochizuki's proof doesn't contain new profound number theory, but this is a very possible outcome, we just don't know yet.

impendia 14y ago

+1, you ask a very important question, which I can at least sort of answer. I think it's key to understanding why many experts think Mochizuki's work is likely to be wrong. I cannot really give a complete explanation, being unfamiliar with his papers. But to argue by analogy, suppose I had invented a difficult and programming language. How might I persuade you to learn it? One way would be to use it to solve problems that are known to be difficult very easily, elegantly, or extensibly. It is the same with "inter-universal geometry". One should not have to follow Mochizuki's work all the way through his proof of ABC to know whether it is good for anything. What is the quickest route to a surprise? What can Mochizuki now do with less effort than the existing experts? The absence of good answers is behind skepticism of the proof.

001sky 14y ago
As an outsider all this (line of reasoning--not personal) just sounds like is "not invented here" syndrome. That may or may not be fair. But isn't this what tenure is for? You can't get fired for just learning, right? Why is there so much lack of intellectual curiosity and so much self-centered-ness on "academic ambition". Are the two so self-contradictory? How much great work has gotten swept up under the rug by this attitude toward "peer review"?
- impendia 14y ago
  I am not sure I understand all of your questions, but I assure you that if people believed there was a high chance that the proof was correct, then they would be very eager to read it. Mathematicians, and especially the very best mathematicians, are very eager to learn. I don't know of any great mathematician who demonstrates a "lack of intellectual curiosity". That said, there is much more mathematics available than anyone has time to read. It can easily take an hour a page, or more, to read a dense research paper. Mathematicians, like everyone else, have to be selective about what they choose to learn. It is very rare for great work to be ignored, but here is one example: http://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem http://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem

vhf 14y ago

Perhaps is it not so premature, since this proof relies mainly of this new inter-universal geometry, Take the Fermat-Wiles proof. It's probably the most significant number theory result of the last century, yet it doesn't contain new profound number theory. Its genius is the use of elliptic curves, which is algebraic geometry. I'm not saying Mochizuki's proof doesn't contain new profound number theory, but this is a very possible outcome, we just don't know yet.