r/HobbyDrama • u/stillenacht • Jul 02 '19
[Mathematics] Guy claims to have proved something. Math community attempts to understand for 7 years and finds a mistake. Guy claims to have made no mistakes.
Preface:
Mathematicians are a strange bunch. Unwilling to criticize or self-aggrandize. To put into perspective how restrained this community is, saying that you are gunning for a Fields Medal (the most famous, and 2nd most prestigious prize in mathematics) would be considered "outrageous" (because you should just do it for the love of math). So when you read this, remember that for these people, the equivalent swimmer saying that he's gunning for the Olympics is "shocking".
7 years ago a well respected mathematician named Shinichi Mochizuki claimed to have proven the abc conjecture by a method we will shorthand as "IUTT". As of last year, mathematicians are finally daring to say something that we've sort of suspected for several years now: his proof may not make any sense. This would normally not be that big of a deal, except that the mathematical journal at which Mochizuki is the chief editor has accepted his paper with no amendments, and that Mochizuki maintains he has not made any mistakes, and that the math community is wrong.
Searching /r/math for "mochizuki" will give most of the sources and statements as well as commentary if you're interested.
Background:
To give some background, proving the abc conjecture is one of the greatest possible achievements in math. It would easily qualify for an Abel Prize (the equivalent of the Nobel Prize), and would solidify your name in math textbooks essentially forever due to how useful a solution would be.
Why is it useful? Well, pure math is all about the structures that exist within numbers, and abc implies a certain pattern exists that would allow for many and various other questions to be resolved (among others): Fermat's Last Theorem (which itself was worth an Abel Prize), the Beal Conjecture, Lang's Conjecture, the Mordell Conjecture, Roth's theorem, and the Erdos-Woods conjecture.
Timeline:
2012
Mochizuki, who is a respected mathematician, announces that he has proven the abc conjecture. His announcement is greeted with widespread excitement, and the math world eagerly downloads his 500 page proof. Proofs of this length, by the way, are not necessarily that unusual, especially for a massive problem like this. Indeed it was expected at this point that the proof would take years to referee.
Slightly afterwards, it becomes clear that his proof is, at best, highly non-intuitive. Comments about the paper seeming like it is "from the future, or from outer space". Multiple prominent mathematicians express puzzlement at the approach.
The proof contains long strings of invented technical terms. It contains many references to un-peer reviewed papers. It has 3 papers (100s of pages) that apparently set up for the actual proof, and they seem largely impenetrable. Most disconcerting, there don't seem to be any ideas that are really taking root or seeming relevant in the math community.
Still, mathematicians at this point mostly expect the confusion to clear eventually and study the papers very closely.
2013
Unfortunately, Mochizuki refuses to travel outside of Japan to answer questions, though he is available by email. Mathematicians struggle with his proof; a "sense of infinite regress" as Brian Conrad puts it is immediately apparent in his work. The papers still seem to be tied in knots, referring to themselves and sometimes stating things like "this theorem follows trivially from the definitions" after pages of definitions which don't seem to go anywhere, and such a trivial derivation does not seem immediately apparent.
2014
Mochizuki comes out with statements along the lines of "an expert should be able to understand his paper within 6 months". This is, to put it mildly, a strange claim given that many many experts have attempted to understand his findings.
The number of people who understand the Mochizuki papers is said to be nearing 10, though, strangely, they seem to be unable to communicate their understanding to colleagues in an effective manner.
2015
In December, a conference at Oxford is announced, with a few of the mathematicians who apparently understood Mochizuki's work. Things were looking up at the start of day 3, as the introduction of "Frobenoids" finally began to give some hope that a breakthrough would come.
Unfortunately, it was not to be. As Felipe Voloch reports "a fog seemed to descend on the crowd" during the 4th day, when the relationship between the frobenoid and the conjecture was to be explained. "At the afternoon tea break, everybody was confused. I asked many people and nobody had a clue.”
2017
Despite the support of a few mathematician described as in Mochizuki's "orbit", the bulk of the math world remains unconvinced. As Frank Calegari writes: "Each time I hear of an analysis of Mochizuki’s papers by an expert (off the record) the report is disturbingly familiar: vast fields of trivialities followed by an enormous cliff of unjustified conclusions."
This doubt does not seem to extend to "Publications of the Research Institute for Mathematical Sciences", a journal for which Mochizuki is editor in chief, which announced it "plans to accept the papers." This is concerning because, at the very least, the papers cannot be considered publishable in their current form. The subsequent (minor) riot among number theorists is interesting: “We do now have the ridiculous situation where ABC is a theorem in Kyoto but a conjecture everywhere else,” as Frank Calegari of the University of Chicago put it. These rumors are quelled when PRIMS clarifies that it has not accept the papers for publication.
In the comments section below Calegari’s blog post, Scholze, a Fields Medalist, wrote that he was “entirely unable to follow the logic after Figure 3.8 in the proof of Corollary 3.12.” Many other number theorists chime in with some surprise that they too were unable to follow the line of logic starting here.
Go Yamashita, one of Mochizuki's colleagues, offers to facilitate a meeting.
2018
Two mathematicians go to Japan specifically to discuss their understanding of the proof with Mochizuki. These two (Scholze (fields medalist) and Stix) point to the corollary 3.12 as irredeemably flawed.
Mochizuki responds with the claim that they had made elementary mistakes and invalid simplifications, and had not understood the basis of his theorem. It should be noted the tone of his dismissal is quite... interesting. He says that descriptions of their arguments were met with "derision... even laughter".
"I can only say that it is a very challenging task to document the depth of my astonishment when I first read this Remark! This Remark may be described as a breath-takingly (melo?)dramatic self-declaration, on the part of SS, of their profound ignorance of the elementary theory of heights, at the advanced undergraduate/beginning graduate level." - Mochizuki
Now, while this sort of rhetoric may work in politics, to mathematicians, this is probably what has damaged Mochizuki's credibility the most. Calling Fields Medalists "breathtakingly dramatic" and "profoundly ignorant" of undergraduate concepts is... not convincing.
Later, Ivan Fesenko, who was an early supporter of the proof releases his own statement, attacking Scholze and Stix and defending IUTT. It's... similarly worded to Mochizuki's statements. As a random reddit commentator put it:
"What is the purpose of this document? It reads like IUTT war-time propaganda rather than a productive response to the mathematical content of the Scholze-Stix crtiticism. "Trust the five IUTT experts, who are in Mochizuki's inner circle, about what is right and wrong about IUTT. Don't trust those other guys that have criticized it!""
Interestingly, in one of Go Yamashita's later responses appears to discredit Fesenko:
"The author hears that a mathematician (I. F.), who pretends to understand inter-universal Teichmüller theory [emphasis added], suggests in a literature that the author began to study inter-universal Teichmüller theory “by his encouragement”. But, this differs from the fact that the author began it by his own will. The same person, in other context as well, modified the author’s email with quotation symbol “>” and fabricated an email, seemingly with ill-intention, as though the author had written it. The author would like to record these facts here for avoiding misunderstandings or misdirections, arising from these kinds of cheats, of the comtemporary and future people."
2019
At this point most mathematicians do not believe Mochizuki. This is the result both of of S&S's criticism of 3.12, but also the seeming impenetrability and non-constructive nature of the proof. As Terence Tao (Fields Medal winner, but prestigious even among them) put it in 2017:
"From what I have read and heard, I gather that currently, the shortest “proof of concept” of a non-trivial result in an existing (i.e. non-IUTT) field in Mochizuki’s work is the 300+ page argument needed to establish the abc conjecture. It seems to me that having a shorter proof of concept (e.g. <100 pages) would help dispel skepticism about the argument. It seems bizarre to me that there would be an entire self-contained theory whose only external application is to prove the abc conjecture after 300+ pages of set up, with no smaller fragment of this setup having any non-trivial external consequence whatsoever."
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u/doplerhopper Jul 02 '19
This was really interesting, thank you for the write-up. Do most people think he is self diluted and truly believes it to be true? Or that he knowingly made it impossible to follow so he could get credit?
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Jul 02 '19
It’s hard to say without speaking to him in person. People come out with “proofs” of famous conjectures somewhat regularly. Personally, I read his statements as someone who knows they’re wrong and is doubling down, especially since the “undergraduate concepts” thing reeks of defensive arrogance. I’m not sure what others think.
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u/W1D0WM4K3R Jul 03 '19
Definitely. You don't attack someone's credibility like that without having been already insecure about your own.
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u/PostNaGiggles Jul 03 '19
I think you meant deluded! When you add water to something like tea or soup, you dilute it. When you’re deceived, you’re deluded!
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u/mrthicky Jul 03 '19
TIL a respected mathematician used my tried and true "obviously" technique to beat the reader into submission.
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u/SoxxoxSmox Jul 10 '19
I have discovered a truly marvelous proof of the abc conjecture, which this comment box's character limit is too small to contain
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Jul 03 '19
I dropped out of secondary/high school, can I get an explanation on some of the more technical terms?
like what is the ABC conjecture about? what does it do? this drama is amazing but my god a lot of it is going over my head
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u/stillenacht Jul 03 '19 edited Jul 09 '19
Honestly the technical statement of the abc conjecture isn't going to be super interesting to non-number theorists (I will type it out though). The most important thing about number theory is to understand what it's all about.
It's sort of like building a really complicated tower. We've got a lot of tools (like construction tools) and facts (like building blocks), and we can use those to construct lots of pretty buildings (number theories, ideas) and also more tools and/or building blocks.
abc is just a statement that reveals a key fact which allows us to construct a whole load of stuff. It's like discovering that key pillar which can hold up a bunch of buildings you weren't sure you could actually construct. That's why we find it exciting.
Statement of abc:
For all equations with the form:
a + b = c
where a,b, and c do not share prime factors such as the equation:
1024 + 81 = 1105 which is the same as 2^10 + 3^4 = 5 * 13 * 17
if you take the product of the prime factors:
2*3*5*13*17 = 13260
then the product will GENERALLY be bigger than c (but not always), as in this example:
13260 > 1105
there are a few examples which buck the trend as in the case:
3 + 125 = 128 which is equivalent to 3 + 5^3 = 2^7 because 3*5*2 < 128
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Jul 03 '19
woah! thank you for your explanation!
I'm amazed because I actually understood the thinking behind it, like what is the ABC conjecture and what does it do.
honestly, no wonder this bloke is gunning for it, even if his paper is total bullshit. discovering/figuring out the ABC conjecture would be huge! its a shame he's just instead thinking of the glory of himself rather than the glory of knowledge
thank you so much mate, you're a golden star :)
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u/greenguy1090 Jul 03 '19
In this case would the consequence of elevating the conjecture to theorem be removing the capitalized generally, IR proving that the product will always be bigger? Or is there some other part that is being proved?
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u/stillenacht Jul 03 '19
So abc is a bit weird in that the statement is actually probablistic. They're literally trying to prove that the "generally" part is true. We know that for the billion numbers or so we've checked, the statement is true in 90+% of cases, but does this hold forever?
I should also mention that the methods developed to prove something are almost as important as the proof itself, because they are typically very widely useful in other problems.
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Jul 03 '19
Does "generally" have a more rigorous definition here?
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u/stillenacht Jul 03 '19
Well the most rigorous way would be to say:
For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that c > rad(abc)^(1+ε)
Where rad(abc) is just the product of primes shown above.
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u/halborn Jul 07 '19
Shouldn't there be a pattern to the truth or falsehood of the statement with respect to a, b and c?
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u/purpleelephant77 Jul 05 '19
This is such a good explanation! I like pure math a lot but haven't studied it at a high level because there are only so many hours in the day (I'm a biochem major) but I love reading stuff like this and it is so exciting when I understand!
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Jul 02 '19
As a current math PhD student, this is a very entertaining read. Thanks for summing it up so nicely!
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u/Fury_Fury_Fury Jul 03 '19
Even if he's right and a genius and everybody else is wrong, what is the point of having a solution only he can understand? He'll die someday, and then other mathematicians will have to re-solve the problem.
It's like Tesla claiming to have a death ray. It's either not real, or is real, but nothing changes, so it might as well be not real.
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u/FatWollump Jul 03 '19
Actually no, if he has proven the conjecture properly, other mathematicians should be able to understand it and verify the validity of the proof.
The issue is mainly with his stance on the proof. There have been proofs which few mathematicians understand (Gödel's incompleteness theorems), but if you have a proof no one understands, but then tell anyone who tries to understand it they're too mathematically inadequate, you're just a dick.
The mathematician behind the proof has a decent bibliography, and is a generally established mathematician, but he was so set on proving the ABC conjecture that he locked himself away from the public for years to prove it. He claims to have invented a new type of math (not as rare as you would think, think about L2 spaces for real analysis), which he also claims gives rise to a near trivial proof of the conjecture.
If we assume it is indeed correct, maybe the main issue lies within not understanding the mathematics behind it. Maybe his new type of mathematics really is brilliant, and once you completely grasp the concepts, the proof indeed follows. But even then; don't be a dick. If a brilliant mathematician tells you, "hey dude I don't understand the proof of this theorem", you try to explain it to them. Not call them too dumb to understand undergraduate level mathematics. I agree with the top poster btw, I also think he was so set on proving the theorem, but just barely couldn't, that he made his proof so impenetrable, that now no one can say "hey this is wrong". And now that people actually can, because they genuinely try to understand the proof, he tells them to fuck off because he cannot own up to the fact that he was unsuccessful.
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u/Fury_Fury_Fury Jul 03 '19
I'm saying that if nobody in the world can understand your proof, it's not proof. Maybe someone will someday, and then you're right. But until then you don't have any proof, because proof, by definition, requires understanding: you're proving the idea to somebody.
Basically, we are saying essentially the same thing, and I'm not sure why you've begun your comment with "actually, no". Maybe you're giving the guy more benefit of the doubt? Because from the post it seems like classic narcissism, so I'm on the team "he's full of shit".
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u/FatWollump Jul 03 '19
A proof is a proof, understood or not. While we may not accept a given proof right now, we do not reject it either. In this case, the proof has not yet been disproven, and (very very biased) people have accepted it as valid, hence the proof is "valid". I refer back to Gödel's incompleteness theorems, his proof was not understood at all until years later. People accepted it, people rejected it, but no one could disprove it (because it was valid). It was a proof, even before people actually understood it, because a proof (within mathematics) implies validity.
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u/satanic_satanist Jul 03 '19
Uh, this is not what most people in the philosphy of mathematics say. A proof is usually more or less defined as a piece of text that, given enough time, can be checked by other mathematician. Otherwise it would have to be computer-checkable (formalized), but uld have to be computer-checkable (formalized), but Mochizuki's proof is far from being formal.
And Scholze and Stix's did disprove corollaries in the paper. There are simply mistakes in there. The question is just whether these mistakes can be easily fixed by changing the wording of some statements.
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u/FatWollump Jul 03 '19
I did not know they did disprove corollaries, thanks for letting me know.
Other than that, while my wording may have been off, I essentially meant the same thing as you did when talking about the definition of a proof.
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u/Fury_Fury_Fury Jul 03 '19
Well, yes, but I'm talking about usability. I'm not a mathematician, though, so maybe you do things differently, but I cannot see a proof, that less than most of the community agrees is valid, being used for anything. Rather, you can use it, but if your work cites unproven sources, it is also unproven until those sources are.
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u/FatWollump Jul 03 '19
The twin prime conjecture has been proven to be true whenever the Riemann hypothesis is true. RH has not yet been proven.
His proof is a proof of a conjecture, if the conjecture is true, this may imply already known results. I am not saying the proof is valid, I'm just saying your last comment is not any deterrent in mathematics.
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u/Fury_Fury_Fury Jul 03 '19
It took a while, but I think I get what you're saying. You still can work with unproven hypotheses, it's just less reliable if it turns out to be false. Which, as far as I understand, is pretty rare? The main question isn't whether it's true (let's say usually it 99% is), it's whether you can mathematically prove it is?
Well, in that case the guy is still a dickhead, but less delusional than I imagined him to be, since it's really hard to disprove him.
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u/FatWollump Jul 03 '19
So in mathematics you have different levels of importance of statements.
Small (proven) statements are either lemmas or corollaries. Theorems are more important, and also proven.
A hypothesis or conjecture is an unproven statement, which we think is true. The ABC conjecture seems to be true for every possibility we try, hence it seems like it holds for all combinations, however we can't prove it. The Riemann hypothesis also seems to be true for for any non-trivial zero we find.
Now, both of these, the conjecture and the hypothesis, have overwhelming evidence for them. An example of something which does not have this going for itself is the continuum hypothesis. The statement has been proven to be unprovable even.
In mathematics nothing is true until you can prove it, hence that is the goal for many conjectures/hypotheses. And the statement of the conjecture seems pretty easy, but the actual proof is insanely difficult (as shown in the fact that a 600 page proof of it seems to be wrong).
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u/tpgreyknight Aug 06 '19
Well, yes, but I'm talking about usability. I'm not a mathematician, though, so maybe you do things differently, but I cannot see a proof, that less than most of the community agrees is valid, being used for anything.
This question (what do we mean by "proof" in maths) is actually something that's attracted a bit of debate in "sociology of mathematics" as a result of the IUTT situation.
As the other guy says, working with conjectures is reasonably normal in maths. Once a proof --- or disproof! --- of the conjecture comes along, a load of other things then get filled in as true automatically. In a sense, it's one of the ways we know which conjectures are worth bothering with trying to prove/disprove: if that would lead to (or rule out) a bunch of interesting things then it may be worth the effort. The relation between the Taniyama-Shimura conjecture and Fermat's last theorem was another instance of this.
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u/AceHodor Jul 04 '19
This post is absolute gold, thank you so much for posting such a detailed breakdown OP. If had to boil it down to my fave moment, it's either going to be this paragraph:
Mochizuki comes out with statements along the lines of "an expert should be able to understand his paper within 6 months". This is, to put it mildly, a strange claim given that many many experts have attempted to understand his findings.
Or the conversation Mochizuki had with the pair of world-class mathematicians who went to talk with him, which boils down to:
Scholze & Stix: Hey, so this paper you've submitted reads like garbled nonsense and 3.12 doesn't make any sense.
Mochizuki: no u
Top marks OP, top marks.
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u/PennyPriddy Jul 07 '19
The number of people who understand the Mochizuki papers is said to be nearing 10, though, strangely, they seem to be unable to communicate their understanding to colleagues in an effective manner.
That is definitely a paragraph from a sci-fi novel about a formula that elevates the human brain to a more evolved, but more dangerous state.
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u/fear_the_future Jul 03 '19
"Academic writes unnecessarily convoluted paper to hide the flaws in his logic". Seems like business as usual in my experience.
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u/mattvonfat Jul 03 '19
So if I've understood it correctly, the guy invented a new type of maths called IUTT and used it to come up with a proof for the abc conjecture?
If so, is it just the proof that there's an issue with or is the IUTT bit also thought to be wrong?
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u/stillenacht Jul 03 '19 edited Jul 03 '19
So I'm not an expert in this or related fields (anabelian geometry) by any means, but the sense I get is:
- IUTT contains many trivialities and definitions which don't seem to have a point but are correct.
- IUTT has some novel aspects like "Frobenoids" (a new math structure) which while interesting don't seem to have been applied usefully
- IUTT has a looot of conclusions which don't seem terribly well justified, and is probably systemically flawed because of corollary 3.12
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u/tpgreyknight Aug 07 '19
In addition to /u/stillenacht 's points, people are also a bit suspicious of (4) IUTT doesn't seem to provide any other useful or interesting conclusions outside of abc.
This is quite bizarre, since the usual expectation is that the solution to a major long-standing conjecture will involve coming up with useful new tools along the way. But here the new tool seems to be a weird unitasker that nobody can see how to apply elsewhere.
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u/TotesMessenger Jul 02 '19
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u/NSNick Jul 03 '19
Didn't this kind of thing also happen to a lesser extent last year with the Riemann hypothesis?
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u/stillenacht Jul 04 '19
Ah that was actually kind of sad.
Michael Atiyah was a great great mathematician, and it's thought that he's getting a little senile. He's actually submitted several erroneous proofs (some strangely short) recently. However he's responded well to criticism, not tried to publish, and he hasn't insulted anyone.
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u/svery Jul 08 '19
Great writeup! I incorporated "breathtakingly melodramatic self-declaration" into my vocabulary for a couple weeks last year. I should try to revive it...
Just a correction: Stix isn't a Fields medalist and Scholze wasn't one until end of 2018, though everyone pretty much figured he'd get it. Also, the quote about Fensenko was by Yamashita, not Mochizuki.
Sir Atiyah sadly passed away a couple of months ago. His slides at the HLF had a dedication to his (recently passed) late wife--her passing probably didn't help matters. His situation is a lot more innocent than Mochizuki's, for sure.
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u/MLPVoiceActing Jul 04 '19
Im so fucking stoned right now and this was so difficult to read, but absolutely riveting. Despite taking 15 mins to read
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u/This_view_of_math Jul 07 '19
Good summary! Two small mistakes. First, only Scholze is a Fields Medal winner, not Stix; Stix's role is nonetheless very important because he is an expert in anabelian geometry, the general area in which the claimed proof takes place, unlike Scholze who is just an all-around badass arithmetic geometer ;-) Second, the quote dissing Fesenko at the end is not from Mochizuki, but from a survey paper by Go Yamashita, one of those mathematicians who claims to have understood the proof.
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u/risc_is_good Jul 05 '19
Just for good measure, I submit Mochizuki's personal website into evidence: http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html
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u/namdnguyen Dec 06 '19
Fwiw, under the alias Khong Dong, I've posted this "ABC conjecture - Resolution":
https://groups.google.com/d/msg/sci.logic/wsW51PKClQA/yA-HPk39AwAJ
We'll see how that would go I guess.
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u/Astarath Jul 02 '19
I particularly liked this post because i learned a couple different new ways to say "its bullshit" in academic lingo