Comment by yarvin9 | Hacker News Reader

Comment by yarvin9 | Hacker News Reader

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It somehow fits that this is halfway up the front page, and there are no comments on it.

A good metric for the vitality of an academic field is the level of genuine interest researchers take in the substance of their peers' work. A good proxy for this metric is the use of preprint servers. Math and physics dominate, obviously. Interestingly, these are also the fields which the Soviet scientific system screwed up the least.

Even in math, as the case of Mochizuki (and de Branges) shows, there are limits.

What's the actual tangible reward for investing a year or two of your career in learning Mochizuki's world? Suppose you showed that inter-universal geometry was really, truly, a new major subfield of mathematics? Would math departments all over the world hire the faculty needed to make this subfield a reality? Making you, the reviewer, who didn't do original work but only checked someone else's work, a big shot? Not #1 in IUG, but maybe #2 or #3?

I'm not a math guy, but it's hard to see. And yet, 40 years ago, this might well have been the outcome. Conclusion: maybe we really do live on Trantor.

There is a lot to read and it has only been posted here for around three hours. What kind of substantial comments are you expecting in such a short amount of time? The linked article itself is a decent read and the papers (conveniently not directly linked to in the article) themselves are quite long.

>What's the actual tangible reward for investing a year or two of your career in learning Mochizuki's world?

I'd guess that would greatly depend on what Mochizuki's world allows one to do.

> I'd guess that would greatly depend on what Mochizuki's world allows one to do.

Um, it's abstract math, it's not going to help you build a flying car. Isn't proving a major conjecture enough?

Most of mathematics predates the modern American academic system. This immense body of work wasn't developed by people with a careful eye on the best way to get a good tenure-track job. It was developed by people who did math for one main reason: they were fascinated by math.

I am not a mathematician, but I haven't seen anyone suggest that Mochizuki's world is boring. It's difficult for me to imagine 19th-century mathematicians resisting the opportunity to become students again in a new, promising, and unfamiliar world.

The 21st-century reaction seems to be: I already got that degree, you want me to start over? WTF? Why? What's in it for me? How do I make a name for myself by studying someone else's theory, which might not even be true? And it's a pretty sensible reaction, given the institutions we have.

A nitpick: More mathematics has been created/discovered/whatever since 1950 than before. It's not true that most of math predates the American academic system. The modern academic system, for all its flaws, has fostered an explosion of new ideas.

> Isn't proving a major conjecture enough?

Not necessarily. And not because of a lack of tangible rewards, but because of a lack of mathematical ones.

Mathematicians are not just after true statements, or proof, but understanding, and new theories that open up new areas of study. For that reason, a new proof of an old theorem is often quite interesting.

I seem to recall reading that Wiles' proof of Fermat's last theorem was sensational, but ultimately less exciting than it could be--the proof did not create a new way of viewing or systematizing things so much as apply very specialized machinery to a specific problem. Take that with a grain of salt--I don't have a hundredth of the mathematical background to judge it myself.

Your general statements are correct but your example of Wiles is quite wrong. Wiles ideas and their developments in the general area of "modularity of Galois representations" (which this comment box is too small to explain) led to a huge explosion of results. Taylor's work on the Artin conjecture and the Sato-Tate conjecture, Khare-Winterberger proof of the Serre conjecture,...

Thanks for the correction. My memory is hazy, and I must have read that about some other result, or merged different comments into something no one said.

Sure, but most things we take for granted today started as abstract math. We wouldn't be communicating with each other right now if it wasn't for the abstract math developed by many others.

I guess now that you were being sarcastic in your first post. There may be an argument that the reaction may be sensible in relation to the institutions that exist, but those are not the only possible institutions.

Fwiw, he has been posted before:

https://hn.algolia.com/?query=Mochizuki&sort=byPopularity&pr...

I doubt any pre-tenure mathematician would spend the necessary time to check the proof, as it would be such an immense career risk. Post-tenure mathematicians are not really worried about being hired anywhere.

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