Hacker News Reader

Comment by Xcelerate | Hacker News Reader

This article seems to suggets that mathematicians are all too eager to drop his work at the slightest whiff of any flaw. Could someone more knowledgable on the subject explain to me why this is?

It is clear that he has already done some very great things in mathematics, so even if there was a flaw in his proof, I would think his papers would still have many deep insights that no else had thought of. I mean, it's not like mathematicians are pressed for time -- if I was one I would certainly dedicate a lot of time to studying something interesting like this.

View on HN

You clearly do not value the time of mathematicians like they value their own time.

The estimate that I was told for the average mathematician reading the average math proof is one page per day.

Thus the average mathematician facing this will see 750 pages just to becomes of the the 50 people who have mastered the basics of anabelian geometry. That's 2 years. Then you have to take on some unknown number of years to learn "inter-universal geometry". Then your reward for doing this is that you are qualified to read a 512 page proof, which is again going to be a year and a half. Along the way if you find a mistake in any of it, your reward is to confirm the immediate guess that most mathematicians have which is that there is likely a mistake somewhere. (But with this much math, you'll probably find several "mistakes" that aren't before you find a real one.)

This is years of work, that has nothing to do with anything that you're already working on. And believe me, a professional mathematician has no shortage of problems to work on, in areas that they already have the background for.

If you think that this is unreasonable, well, why don't you volunteer to fix it? Reading the proof shouldn't take you much longer than it would take to become a mathematician. And you can learn anabelian geometry during grad school, so that time is not all wasted.

>> The estimate that I was told for the average mathematician reading the average math proof is one page per day. Thus the average mathematician facing this will see 750 pages just to becomes of the the 50 people who have mastered the basics of anabelian geometry. That's 2 years.

Ah, well I did not know this. I appreciate the enlightenment. I see the trepidation of reading the whole thing then.

Even more importantly, you might take all that time to review it and the miss a mistake that's actually there, thus wasting all of your time (for some definition of waste).

Right now we are waiting for his corrected paper, as the current one contradicts some known number theoretic results. Mochizuki put uP an amazing amount of scaffolding and invented a new area with new abstractions. People who I trust claim that there isn't new profound number theory in this, so then it may become a matter of whether to study an learn it for its own sake, which carries less incentive if there is no immediate application.

Plus: mathematicians really don't have spare time to learn every area that comes along. Allegedly Poincare was the last true polymath. Given how math has been developing for centuries and you never really need to throw things away it continually becomes harder to get up to speed in any particular field. There's a famous quote about how the Princeton undergraduate math program is so rigorous that it brings you up to speed, to the point of early twentieth century mathematics. In this way math is incomparable to CS.

Nice to know, thanks!

"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.

+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.

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"?

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

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.

Seems quite the opposite. The article says most mathematicians have well-established lines of research and are disinclined to devote years to understanding a different specialization. The mathematicians who are trying to understand the proof are described as young, up-and-coming mathematicians with little to lose and a lot to prove (and an opportunity to potentially get in on the ground floor of a new theoretical framework -- "inter-universal geometry")

Mathematics - at least some kinds of it - is largely a game of symbols. You define something and then see if you can prove anything about it. Most of the time, it turns out to reduce to group theory (e.g. Galois theory basically consists of reducing the study of fields to the study of groups). Sometimes you come up with a novel structure about which you can say something. Occasionally, this gives you a result that feeds back into another branch of mathematics, or even into number theory - as with Wiles' proof of Fermat's Last Theorem. But most of the time, this is just some funny structure and you say "well, that's nice" and move on.

So Mochizuki has almost certainly defined some new objects and proven some things about them. But mathematicians do that all the time (admittedly on a smaller scale). If you just want to look at something, there's plenty of "new territory" around - stuff that doesn't require learning a new notation, and hasn't had Mochizuki looking at it for ten years already. The only reason some arbitrary new structure is interesting enough to spend your time learning about is if you can tie it back to number theory.

> Most of the time, it turns out to reduce to group theory

I'd be really interested if you could justify this! Sure, group theory has a huge number of applications across mathematics, but to say that it reduces to group theory "most of the time" seems completely implausible.

What I mean is: if you try, starting from nothing, to make up the definition of some new kind of object, and then start proving things about it, most of the time you find that you've defined a group or something that reduces to a group. I don't have a quantitative measure of this (I can't really define a probability distribution for "random objects you think up"), it's just personal experience. Put another way: I find it quite hard to think up a new object that doesn't turn out to just be a group.

Have you tried thinking up new mathematical objects? What do you tend to come up with? (Not snarky, I'm genuinely interested - maybe some people's creativity works differently to mine)

funny. yesterday i was trying to explain to my friend, a physics grad student like myself, why i wanted to study algebraic geometry. i said, it's because physicists only know about groups, because everything seems to be groups. but rings have more structure than groups and fields have more structure than rings and that's what algebraic geometry is about. this comment is probably too late. you are right that most things turn out to be a group, but not "just a group".

> Have you tried thinking up new mathematical objects?

Yes, I have. Off the top of my head, the objects I'm interested in are often monoidal, but usually lack inverses.

Strings don't have any natural group structure (though as free monoids there is an obvious way to extend them into free groups), nor metric spaces or context-free grammars or finite state machines.

Of course groups are ubiquitous, but I think your claims of universality are vastly exaggerated.

That wasn't the takeaway I got from the article at all -- and the original proof of Fermat's Theorem by Wiles had flaws that took some work to patch over, but in no way diminished his accomplishment.

I think you've just misinterpreted some of the language used.

"Mochizuki has almost certainly made significant discoveries—but without the allure of a proof, it’s possible no one will take the time to understand them."

"if Mochizuki is able to build up sufficient trust with his peers"

"if an error is found early in the vetting process, the math world will likely move on without bothering to explore the rest of the mathematical universe Mochizuki has created"

I was basing my post on these quotes.

I guess the reason those didn't stick in my mind is that (I think?) they were all interjected by the author -- all the quotes from actual mathematicians said otherwise.

Consider the incentives involved in studying his work carefully for years, and those of being in academia.

I don't think it's even just the incentives of academia, but of life/interest as well. Tenured professors could choose to devote their next few years to learning inter-universal geometry. But they have to guess: is this likely to be a more interesting/useful thing to learn than something else they could learn in those same years? It seems so far that most are heuristically guessing no.

I do not think that is the case. Their is the skeptism resulting from the huge amount of work between accepted math and his results that give mathmaticians concern. It is almost definite that his work contains some flaws, what is not definite is wheather it contains fatal flaws. Many are not willing to invest the time to learn something that might turn out to be useless.

Some relevant background > http://news.ycombinator.com/item?id=4781825

Reply on news.ycombinator.com