Progress report

[posic]

by Shinichi Mochizuki -- http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202013-12.pdf

Comments

[posic.livejournal.com]

A proof of the ABC conjecture is a huge result. Half a year is not such a long time at all.

Of course, there is such a theorem. It is called "classification of finite simple groups". They used to say that there isn't (and never was) a single person who understands the whole of it. Even a reasonably meaningful formulation would probably require a semester-long lecture course.

What is being demonstrated is not the limits of human mind. It is the limits of government-run academia.

[leblon.livejournal.com]

A classification theorem is a bit different from a theorem which states that some elementary property holds. When one starts to classify X, there is not guarantee how long the list will be. For finite simple groups the list turned out to be quite long, so the proof is also very long.

I understand that Mochizuki developed a whole theory just for the purpose of proving the ABC conjecture. It is this theory that it takes half a year to learn. Is there another example of such a theorem? The Weil conjectures and the theory of l-adic cohomology perhaps?

[posic.livejournal.com]

Yes, in a sense. People say that the whole of Grothendieck's algebraic geometry was developed with the goal of proving Weil's conjectures in mind.

On the other hand, as the l-adic cohomology is being used far beyond this specific purpose, so one would hope that Mochizuki's theory will prove to be useful for many purposes (and generally, should I say, will expand our understanding of mathematics, or of arithmetic geometry at least). That's what Mochizuki seems to suggest in his report.

In such cases, one can say that while a theory proves validity of a theorem, the theorem in turn proves significance of the theory. This attracts people desiring to learn the theory in order to apply it to other problems.