Not mentioned: Kedlaya was Mochizuki's student and then later his teaching assistant back in '92-'94 (for a tough course[1]). Mochizuki hasn't had that many students; Kedlaya might be one of a few to have an advantage in understanding his approach.
The bad news is that the new participants seem to have made no progress on the IUT papers themselves. But the good news is that some substantial progress was made on the background material, and that there are some folks still interested in continuing.
I don't understand how something, that anyone understands, can be this cryptic. Someone some time ago mentioned that all of Einstein's major papers on special and general relativitiy can be covered in a semester or two.
Isn't the hard part discovering this stuff?
How can anyone discover mathematical proofs but it is so impossible to communicate them that nobody in the world can follow your train of thought? I mean people are really, really good at communicating. It's kind of what we do.
I don't understand how anyone can publish proofs that don't succumb easily to understanding by experts who concentrate on them for a long time, perhaps with tiny hints by the author.
It's not as though we're looking at fragments written in a long-lost language we have no access to, from a culture nobody has access to, on unknown subjects, and with the authors dead hundreds of years ago! The author is right there. The subject isn't even art (like poetry criticism or something), it's fully rigorous.
Understanding someone's proof (for devoted experts) should be easy, right? What gives?
> I mean people are really, really good at communicating.
On my experience, people are usually really bad at communicating. It requires additional effort to imagine what information might be required for somebody else to follow your train of thought. Think of code documentation: What the original author percieves as obvious is cryptic to somebody else because context has been omitted for the sake of brevity. The author needs to invest a lot of time to actually dump his entire mindstate into documentation. Even if he does make an effort, not everybody is a good writer. If he doesn't care about others understanding his stuff (which also seems to be the case with Mochizuki) it gets even worse.
Also remember that Mochizuki basically worked alone on his stuff for 20 odd years. He built his own world to work in and it's just as hard for collegues to catch up to his level in that world as it is for e.g. non-programmers to understand programming (which takes years).
> context has been omitted for the sake of brevity
Such omissions should be forbidden. Writing down the context in comments and in any other ways available should be a minimum requirement for any new code written. I do enjoy a good puzzle once in a while, but I don't want to reverse engineer someone's brain every time I read code. We should really focus more on helping other people understand the code we write. Some tools can help, but if the original author keeps most of the information to himself it's going to be really hard to follow the code.
The papers on Inter-universal Teichmueller theory span over 500 pages alone, and they build on material developed earlier that needs to be understood too. Those papers were published a little over 3 years ago, so claiming that they can't be understood “by experts who concentrate on them for a long time” might be a bit premature.
but the article doesn't make it sound like it's not "fully understood." the article makes it sound like not even the summary is known - for example, what parts of the other papers are used, what isn't. As though the experts didn't even known what parts of theories were being referred to. very bizarre.
Did you even read our article, jsprogrammer? That's what I'm reacting to. It says:
>A consensus emerged that the highlight of the workshop was a lecture on 9 December by Kiran Kedlaya, an arithmetic geometer from the University of California, San Diego. He zeroed in on a result from a 2008 paper by Mochizuki that linked the statement of the abc conjecture to another branch of maths called topology. The link was immediately recognised as a crucial step in Mochizuki’s grand strategy.
This is why I'm making the reference to ancient lost texts. I mean, if the result of a 2008 paper is crucial, you'd think the guy would have just mentioned this fact. It's as though they're investigating some long dead guy's hidden thoughts. This is a direct quote: "There is still no clear answer to lingering questions about how things are ultimately going to fit together." What? Wouldn't the guy just tell you how they're ultimately going to fit together? I'm utterly perplexed at the process the article describes, when the author is available and can give hints.
What I understood from http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-works... (posted elsewhere here), is that the problem stems from differing motivations. On one hand, Mochizuki wants to make his proof fullyt complete and correct (being both fully general and covering all special cases), but the community wants broad strokes intuitions. From Mochizuki's perspective, he has told everyone how things fit together, but fit together means something different for others. The workshop attendees want to know what things they can gloss over to get the big picture, but as of yet, nobody who understands the proof has succeeded in communicating that information, probably because they don't share the same motivation.
I did read the article. It wasn't clear that you were referring to that specific paragraph.
> I mean, if the result of a 2008 paper is crucial, you'd think the guy would have just mentioned this fact.
Maybe he did? Or maybe it is just not crucial.∎
The article just states "The link was immediately recognised as a crucial step in Mochizuki's grand strategy."...that is just the writer's opinion.
Linking these papers to topology is trivial. The word "topological" alone occurs 111 times in the first paper. A lot of what I'm reading in this article, and others, is people complaining about not understanding something that they (probably/apparently) have not even read.
Personally I find this more mind boggling than that it can't be communicated easily. Clearly, a proof of 500 pages will be extremely difficult to verify or even understand.
However... That a concept taking just a paragraph to explain* for people who know math requires a proof of over 500 pages. THAT is what I find mind boggling and I find hard to understand. That it can just balloon in size like that, especially given that mathematics is a language of logic.
The halting problem proves that mathematics is hard. There is no algorithm that can prove/disprove any decidable statement. Not even a a superslow algorithm.
While people make source code to be as clear as possible, mathematicians go the other way.
One idea would be to invert the proof, going top-down, instead of bottom up (in the explanation), then clearly dividing the subproofs instead of one big "wall of text" (yes, I know, there are numbered proofs/lemmas, but still it is not very well organized)
Edit: looking at the paper above it seems the author really tries to be helpful, the paper seems very organized (but it is long)
It is essentially a communications problem. Brian Conrad summed it up:
For every subject I have ever understood in mathematics, there are instructive basic examples and concise arguments to illustrate what is the point to generally educated mathematicians. There is no reason that IUT should be any different, especially for the audience that was present at Oxford. Let me illustrate this with a short story. During one of the tea breaks I was chatting with a postdoc who works in analysis, and I mentioned sheaf theory as an example of a notion which may initially look like pointless abstract nonsense but actually allows for very efficient consideration of useful ideas which are rather cumbersome (or impossible) to contemplate in more concrete terms. Since that postdoc knew nothing about what can be done with sheaf theory, I told him about the use of sheaf cohomology to systematize and analyze the deRham theorem and topological obstructions to construction problems in complex analysis; within 20 minutes he understood the point and wanted to learn more. Nobody expects to grasp the main points of IUT within 20 minutes, but if someone says they understand a theory and does not provide instructive visibly relevant examples and concise arguments that clearly illustrate what is the point then they are not trying hard enough. Many are willing to work hard to understand what must be very deep and powerful ideas, but they need a clearer sense of the landscape before beginning their journey.
I remember at university how, when attending the few multi-disciplinary meetings or get-togethers, I would marvel that these people - on the same campus! - didn't get together more often in order to share and stimulate growth or share strategies, which inevitably occurred during these meetings.
I wonder why a handful of qualified mathematicians couldn't get together and go to Japan to sort this proof out...
Having worked in a university mathematics department I noticed that the pure mathematicians tended to keep themselves to themselves, for instance not attending talks by visiting mathematicians unless it was of direct relevance to their research. In contrast, we theoretical physicists would listen to any visiting speaker just for fun - even on occasion experimental particle physicists, applied mathematicians or computing specialists.
Perhaps pure maths is just more deeply specialised and leaves no possibility of such dilettanteism.
I think it's probably because it's very difficult to understand a pure math talk if you didn't already have significant understanding of the subject before the talk.
I don't work on pure mathematics, but I work in a field of computer science where there is both theoretical and empirical/applied research, and I do, and enjoy doing, both.
However, when it comes to attending talks, I do go to a lot of talks on empirical subjects just for fun, but I hardly ever go to a highly theoretical talk for fun. I only do that if the talk is highly related to the specific work I do; if you see me in a theoretical talk otherwise, it's probably just for social compromise.
The reason is that I can perfectly understand the gist of an empirical paper in 20 or 30 minutes, where there is no way I will understand a piece of theoretical research by listening to a talk if I haven't gone through the paper carefully before (and if you have done that, there is often no point in going to the talk anyway). Honestly I think the talk format doesn't lend itself too well to highly theoretical work. I have learned a lot about empirical research and obtained many useful ideas from talks, but in the theoretical field, the useful ideas and insights I got from talks are few and far between, and I would probably have obtained them more efficiently from reading papers anyway...
Yes, part of what I was trying to say is why not get that handful of folks together and go to Japan, to the source. Logistics? Ego? Funding? Perhaps a trip is in the cards...
This was just one workshop that happened to be at Oxford. Probably it occurred the other way around to you imagine - "let's host a workshop here [at Oxford], what shall be the topic?"
Article says there is another one planned in Japan that will no doubt be attended by a (proper, less those put off!) subset of those at Oxford.
Money and time. I'm sure you could straight-up pay some of the qualified mathematicians to figure it out, but you'd need to make it worth their while. Offer to buy out their salary for a couple of years, plus hire them a few postdocs and grad students to further their own projects, and I'm sure a few would be happy to help. But it would cost the better part of $1M for each guy.
There was an earlier workshop in Japan. From wiki:
"A workshop on IUT was held at RIMS [Japan] in March 2015 and in Beijing in July 2015. A CMI workshop on IUT theory of Mochizuki was held in December 2015 in Oxford, IUT-Oxford (2015). A workshop on IUT will be held in July 2016 at RIMS."
I view some of these math researchers like startup founders who are absolutely brilliant technically, but who lack almost all ability to interact with customers, investors, etc. As a solo founder, they will fail. However, if they attract a co-founder who complements their skills, they can be wildly successful. It sounds like Mochizuki would have benefited greatly from such a co-author, in order to truly prove his result to the community.
Math is objective. The proof is either correct or not. Human interaction doesn't matter, except for perhaps marketing the importance of the results. In this case, the history of the problem itself has done the marketing, so any valid solution would be a wild success regardless of how socially eccentric the researcher is. Another recent example of such a mathematician is Grigori Perelman.
If the proof cannot be understood, it might very well be the case that another researcher more concerned with advancing knowledge for us all, will be credited. There are many roads to Rome.
Proofs that cannot be grasped are like unproven truths. Worth very little... They only stand through the credibility of the inventor. It downgrades the mathematician to a politician.
Mathematics, being a human endeavor, is fundamentally about human interaction. There's no point to doing math if there is no communicating it to other people.
It's the only pure way to describe existence. That isn't meaningful? Mathematics is a language, first and foremost. To argue otherwise is to fundamentally misunderstand its goals, methods, and form.
You can draw all the squiggly lines on paper that you want, but it won't mean anything until someone else interpets it. You cannot prove logic that only exists in your own head, because you cannot prove yourself rational. Your very existence does not even make sense without other people perceiving you.
Mathematics as done by humans is not just objective. There's a very social aspect to it; if you dress up the most amazing result in a drab and ununderstandable manner then nobody will notice it. You have to take the effort to make your result digestible and relatable, even for experts.
The proof can be completely fine, but if no one can understand it then it won't be accepted.
Mochizuki would have benefited greatly from such a co-author, in order to truly prove his result to the community.
Except that, from reading the article, it sounds like he doesn't much care about proving his result to the community. Don't make the mistake of assigning the motives and values of a founder to a late-career tenured mathematician. According to the article, he doesn't care for travel. It's not a stretch to conclude that he doesn't care to communicate his proof to other mathematicians.
I'm guessing he found an argument that was sufficient to convince him (and only him), put it online as a slight concession to his colleagues, and then kicked his feet back.
Surely everyone else would benefit from his having a co-author, but it sounds like he can't be bothered, and doesn't care if no one else understands his ideas.
> True to form, Mochizuki himself did not attend, although he did answer participants’ questions through Skype
Why not present his papers through skype? If the problem is just a dislike of travelling. It could also help with any issues related to public speaking or that type of thing.
I guess we can now ask a lot of simple questions in math for whose there can be answer, but the solution itself is untenably long to be produced or understood by mere mortal, not a computer or computer-like individual.
Same way as we can't really reason about inner workings of machine learning model that yields useful results.
Just because his proof is long, convoluted, and cryptic doesn't mean that there won't be a simpler proof down the road. Maybe some of the tools just haven't been developed yet. Look for example at how the Greeks and Romans struggled with mathematical problems many of which are today easily solvable in high school because we have advanced techniques and a positional notation for numbers.
As far as I understand it, a lot of the problem stems from the fact that Mochizuki is quite a hermit. And that's a problem if you want others to understand what you have done for years.
This workshop was crucial that it showed several people that trying to understand the proof could be a worthwhile task.
> As far as I understand it, a lot of the problem stems from the fact that Mochizuki is quite a hermit. And that's a problem if you want others to understand what you have done for years.
Interesting. Reinforces the idea that you should attend conferences and talk to others in your field, to ensure that you speak the same language and don't miss out on helpful advice or new tools that may (for instance) make a proof easier to follow.
I'm not mathematician but I guess if there is really only one possible proof for a given question (Gödel might have proven that this is impossible but I don't know for sure), then the question isn't actually an interesting one. Because this would mean that the question touches only very few topics.
But the abc conjecture is interesting because it has lots of links with deep questions in number theory.
So far, we don't know if there are any proofs that are beyond the human intellect. I don't think that we currently believe such a thing to exist theoretically.
Every provable conjecture should be understandable in principle by a generic human mind.
Now, there for certain systems that are so complex that proofs within those systems are beyond us. Not because we couldn't understand them but because learning and applying all the rules of that system takes longer than a human mind is able to live (currently).
But that's just a practical limitation, not a theoretical one.
I agree. More generally I have often wondered if pure mathematics (and theoretical physics [1]) will reach such a level of abstraction that it becomes impossible to digest all the established ideas in order to build on it. The mathematicians attending this workshop have an ability to manipulate extremely abstract concepts that is superior to the vast majority of humans, and even they can't seem to handle this proof.
the reason they fail to understand the proof seems the same everybody else fails at math. funny even the greatest mathematicians can have this problem.
"A consensus emerged that the highlight of the workshop was a lecture on 9 December by Kiran Kedlaya, an arithmetic geometer from the University of California, San Diego. He zeroed in on a result from a 2008 paper by Mochizuki5 that linked the statement of the abc conjecture to another branch of maths called topology. The link was immediately recognised as a crucial step in Mochizuki’s grand strategy. "
I'm not a mathematician at all but does anyone know if this may have implications in the field of cryptography, because it involves prime numbers? Also, the word "cryptic" in the title. ;)
I just learned about Math 55 and would trade plenty to teach that course.
The problem here is that Mochizuki uses too many German Nazi references.
Those people had no academic integrity whatsoever; and their works are exclusively stolen, through war crimes and the Holocaust.
After you do the right thing, and translate Mochizuki's nomenclature into French and Slavic; the proof is no longer valid, and the problem is negligible.
For the record, Oswald Teichmüller is a fraudulent Nazi war criminal who robbed Felix Hausdorff during the Holocaust.
The undeniable fact that Oswald Teichmüller is solely published in a journal of racial propaganda makes Mochizuki's choice of terminology questionable at the very least.
The fact that I am being suppressed for this demonstrates unambiguously that Mochizuki's proof is nothing more than media hype and puffery.
I can agree with this being extremely problematic in a political sense - it's horribly wrong to rip someone off, even more so by robbing them during wartime. Quoting such stolen material is a kind of passive support and should not be tolerated.
BUT - how does that affect it from a purely scientific perspective? Is the quoted stolen material also incorrect? Because it would be equally bad, IMHO, to ignore an important and correct scientific finding just because it was produced in an unacceptable way.
If I exaggerate a bit, it's similar to someone experimenting on babies to discover a cure for AIDS - sure, they're a terrible person and should not receive praise/compensation for their discovery, their actions should be condemned. But should we also throw away the discovered cure?
Interesting how Teichmueller has a bunch of stuff named after him, yet there's very little information about him to be had. There are some references on Google Scholar: https://scholar.google.de/scholar?q=%22o+teichmueller%22&btn.... He is certainly published in more than a single journal, however.
On the other hand, Felix Hausdorff, who also appears to have worked in the same field (though topology is probably a far broader field than I can understand) has plenty of information readily available. This is conjecture, but the mere lack of solid information on Teichmueller could lend credence to some of what you say.
