Progress report

[posic]

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

Comments

[chaource.livejournal.com]

Amazing! Mochizuki proposes that the way to understand his theory is to read several thousand pages, going one by one through all definitions and theorems...

I just wonder whether we are not witnessing a demonstration of the limits of the human mind. A mathematical result requires many logical steps to be understood; but surely a lot depends on how many steps are needed (thousand, ten thousand, a million, ...). After a certain number of steps, it becomes impossible to comprehend a result during a single discussion session or a single seminar, and one needs two or three seminars. Beyond a yet greater number of steps, one needs a series of seminars, or a one-semester lecture course. During this time, the brain will slowly develop more structures that will allow more logical steps to be skipped and still retain full understanding. But maybe there are some mathematical theories that require more than one year of such study.

So Mochizuki claims, after some empirical experience, that a competent and suitably prepared mathematician is able to build the necessary brain structures within a few months.

It is also interesting that each of the two mathematicians who went through the procedure was compelled to give many hundreds of technical comments. It is obviously a sign that the exposition of the theory had to be adapted, each time, so that their brains could build the necessary structures. Perhaps several more such iterations are required until the exposition can be more widely understood?

[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.