
Mochizuki's proof of the ABC conjecture accepted for publication - fovc
https://www.nature.com/articles/d41586-020-00998-2
======
gautamcgoel
Math student here. This whole process reflects very poorly on Mochizuki, IMO.
The proper thing to do when claiming to have solved a major open problem is to
make yourself available to the mathematical community, for example by giving
lectures on your work. Releasing a preprint on your website and expecting
other mathematicians to drop what they're doing and devote years of their time
to understand your obscure paper just screams of arrogance. Also, publishing
your work in a journal where you are the editor isn't a good look; it gives
the impression of a conflict of interest. The critique by Scholze et al is the
final nail in the coffin as far as I'm concerned.

~~~
wolfgke
The whole situation is in my opinion much more complicated.

Here are some texts from the internet concerning this:

David Michael Roberts - A Crisis of Identification: [https://inference-
review.com/article/a-crisis-of-identificat...](https://inference-
review.com/article/a-crisis-of-identification)

Two Quora posts:

[https://www.quora.com/Did-Peter-Scholze-and-Jakob-Stix-
reall...](https://www.quora.com/Did-Peter-Scholze-and-Jakob-Stix-really-find-
a-serious-flaw-in-Shinichi-Mochizukis-proof-of-ABC-conjecture/answer/Xiaowei-
Xu-1)

[https://www.quora.com/What-do-you-think-about-Stix-and-
Schol...](https://www.quora.com/What-do-you-think-about-Stix-and-Scholze-
coming-out-and-saying-Mochizukis-proof-of-the-abc-conjecture-is-
wrong/answer/Xiaowei-Xu-1)

Also relevant:

[https://thehighergeometer.wordpress.com/2019/01/18/taylor-
du...](https://thehighergeometer.wordpress.com/2019/01/18/taylor-dupuy-on-
mochizukis-iutt-infamous-corollary-3-12/)

~~~
hyperpape
Thanks for this. Tao’s comment made quite an impression on me, but important
for outsiders like me to remember that the facts on the ground can change as
time goes on.

~~~
Koshkin
To quote, "To take an extreme example, if Mochizuki had carved his argument on
slate in Linear A and then dropped it into the Mariana Trench, then there
would be little doubt that asking about the veracity of the argument would be
beside the point. The reality, however, is that this description is not so far
from the truth."

~~~
chx
That's Frank Calegari's writing at
[https://www.galoisrepresentations.com/2017/12/17/the-abc-
con...](https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-
has-still-not-been-proved/) not Tao's. Tao left a comment there:
[https://www.galoisrepresentations.com/2017/12/17/the-abc-
con...](https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-
has-still-not-been-proved/#comment-630)

------
gizmondo
> Acceptance of the work in Publications of the Research Institute for
> Mathematical Sciences (RIMS) — a journal of which Mochizuki is chief editor,
> published by the institute where he works at Kyoto University

That's exceptionally bad optics.

~~~
battery_cowboy
From TFA:

> Mathematicians often publish papers in journals where they are editors. As
> long as the authors recuse themselves from the peer-review process “such a
> case is not a violation of any rule, and is common”, says Hiraku Nakajima, a
> mathematician at the Kavli Institute for the Physics and Mathematics of the
> Universe in Tokyo formerly part of Publications of RIMS’s editorial board.
> Mehrmann confirms that this would not violate EMS guidelines.

> Kashiwara said that Mochizuki had recused himself from the review process,
> and had not attended any of the editorial board meetings about the paper.
> The journal has previously published papers from other members of the
> journals’ editorial board, he said.

~~~
lisper
Speaking as someone who used to make a living by publishing (though not in
mathematics): the problem with this is that even though you may have recused
yourself, everyone who is still in the room knows that you will be passing
judgement on their papers at some point in the future. So recusal does not
entirely remove the potential for conflict of interest.

~~~
jessriedel
The claim isn't that there was no bias, the claim is that this degree of bias
is not unusual for the field.

------
moomin
I mean, I’m enjoying the show, but this changes nothing. Very few people claim
to understand the proof and some of the few that do think it’s wrong.

Contrast with Wiles, where they _did_ understand it, they _did_ find a gap in
his proof and he fixed it to everyone’s satisfaction.

~~~
yodsanklai
> Very few people claim to understand the proof

I don't have any idea what this proof looks like and I can assume it's very
complex. But, could it be modular enough that you can check the proof without
understanding it?

For instance, imagine the proof is made of 50 lemmas. One could check the main
theorem derives from the 50 lemmas. And checking each individual lemma could
be left to other mathematicians.

~~~
Someone
[https://www.quantamagazine.org/titans-of-mathematics-
clash-o...](https://www.quantamagazine.org/titans-of-mathematics-clash-over-
epic-proof-of-abc-conjecture-20180920/):

 _Despite multiple conferences dedicated to explicating Mochizuki’s proof,
number theorists have struggled to come to grips with its underlying ideas.
His series of papers, which total more than 500 pages, are written in an
impenetrable style, and refer back to a further 500 pages or so of previous
work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford
University, has called “a sense of infinite regress.”_

It’s not possible to “check the main theorem derives from the 50 lemmas” if
each of the lemmas uses terminology one doesn’t understand or even has never
heard of.

~~~
throwaway_pdp09
I would expect the base axioms to be specified by the author of the paper.
That is, by the guy who claims to understand it.

After that the steps of the proof should be just turning the handle, and
perhaps the distance between steps to be bridged automatically. Maybe you
wouldn't understand what was going on and likely not understand the end
product any better but at least by symbol manipulation alone you should be
able to make a path and verify that _start_ via _steps_ arrives at
_conclusion_

Of course I am not a mathematician and haven't a clue but in principle it
seems viable and if it isn't I'd really love to know why.

~~~
ttz
The problem is that the base "axioms" (really, definitions) he's working off
of are basically alien language, and almost nobody understands why they are
relevant. Add to that he is not saying "turn the crank like this", but is
effectively saying "trust me when I say the cranks turn and out pops the
answer, but if you dig into the machinery yourselves you'll see what I mean".

On the other hand, there was a similar case with a German mathematician
beforehand, and he turned out to be correct.

~~~
throwaway_pdp09
I don't buy that the base axioms are alien _and it doesn 't matter anyway._

I can give you an alien language of logical axioms, and a series of steps,
none of which a random person wouldn't be able to understand, but could still
lead them through a series of steps, showing that the outcome could be derived
from the input via the axioms.

Now, if nobody could understand the axioms and steps of this but they still
got a consistent outcome, mathematicians would still be interested if for no
other reason than it was self-consistent. That alone would be a result.

But if they had that, I suspect that would also give them a framework for
understanding it.

> "trust me when I say the cranks turn and out pops the answer, but if you dig
> into the machinery yourselves you'll see what I mean".

automate that and there'll be no need to trust him.

~~~
Someone
This (purported) proof is 500+ pages. For a mathematical proof, that’s
enormous. Sampling a few of the issues of the journal in question, its longest
papers seems to be about 40 pages
([http://www.kurims.kyoto-u.ac.jp/~prims/pdf/43-1/contents-43-...](http://www.kurims.kyoto-u.ac.jp/~prims/pdf/43-1/contents-43-1.pdf),
[http://www.kurims.kyoto-u.ac.jp/~prims/pdf/42-3/contents-42-...](http://www.kurims.kyoto-u.ac.jp/~prims/pdf/42-3/contents-42-3.pdf))

And those are papers that an expert can skim to get an idea about the flow of
the proofs. This one, if you jump in, says you things like _“every fooable
bazz is a bar”_ , where _foo_ , _bar_ and _baz_ are new terms no mathematician
has any intuition for that, likely, got defined in terms of other hitherto
unknown terms _qux_ , _quux_ and _quuz_ , making it impossible to judge
whether that statement has merit, or how it leads to proving the abc
conjecture.

And yes, an automated proof checker would be nice to have, but we aren’t there
yet, by a wide margin. Getting there would make this proof a lot longer.

(For a loose analogue: imagine that the published solution to a “mate in 2”
problem in chess wouldn’t just be “Queen g1, check, King h8, rook h2, mate”,
but “whites queen is on field d1, by rule r queen can move horizontally across
empty fields, there are no empty fields between d1 and g1, that doesn’t bring
white’s king in check (blacks pawn on e4 can’t take it because…, blacks knight
on c6 can only move to… because of rules…, so white can move their queen to
g1. That brings black in check because of rules…, etc)

------
hn_throwaway_99
> But one mathematician who prefers to be quoted anonymously says that editors
> and referees handling these papers might have been in a nearly impossible
> situation. “If the best mathematicians spend time trying to work out what’s
> going on and fail, how can one referee on his own have any chance?”

I totally disagree. When someone is making extraordinary claims, _the burden
is on them_ to go the extra mile to explain their reasoning, none of which
Mochizuki has done. The easy decision in this case should be to leave things
at the current "default" state unless a higher burden of proof is met.

I think it reflects especially poorly that when confronted with criticisms
that Mochizuki just waved the criticisms away with what was basically a "you
mere mortals misunderstood my greatness", without taking the effort to engage
and explain himself. I'm not in the math community so could be
misunderstanding, but that's certainly the sense I got reading this article.

~~~
pixiemaster
> the burden is on them to go the extra mile to explain their reasoning

that’s not how extraordinary math worked in the past

~~~
QuesnayJr
They go the extra mile to explain it to _other experts_. When Perelman
produced his proof of the Poincare conjecture, he traveled all over the place
explaining his result.

------
confuseshrink
I'm not a mathematician but coming from the software world if one guy wrote a
massive program (I'm assuming 600 pages is massive) in "an impenetrable,
idiosyncratic style" you could virtually guarantee it would not be correct.

~~~
wolfgke
Surely not the best example, but Terry A. Davis' (RIP) TempleOS was written in
a very idiosyncratic style (he even invented his own programming language
(HolyC) for this).

~~~
voldacar
TempleOS is awesome

Being able to mix text, images, 3d models, etc in a terminal session is
something that modern OSes still haven't gotten around to doing. Terry may
have been crazy but he was right about a lot of things and unlike many, he had
the skills to implement his ideas

------
avip
The first comment in
[https://news.ycombinator.com/item?id=15971802](https://news.ycombinator.com/item?id=15971802)
(one of _many_ HN posts discussing this ongoing situation) is intriguing.

~~~
westoncb
From Terry Tao's comment:

> 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.

This seems like the crux of the controversy: what is the true value in the
300+ "pages of set up"? Clearly he's providing a new framing for the problem.
If that new framing ends up being applicable to other problems (which was
presumably the author's intent), then the "pages of set up" are the true value
here, not the proof of the abc conjecture itself.

When Tao says, "no smaller fragment of this setup having any non-trivial
external consequence whatsoever"—this is a comment about the state of things
_so far_ —but whether that will change is unknown.

Maybe all of that "set up" ends up having no more general utility whatsoever,
or maybe we shouldn't even expect to have found external consequences yet:
very fundamental re-framings may be very disconnected from applications. Maybe
Mochizuki has found one path from the new framing back into an area of
contemporary mathematical interest, and maybe further exploration will yield
an abundance of new paths—maybe some highways.

Anyway just playing devil's advocate since the prevailing stance on this seems
to be against Mochizuki in a way that feels odd to me.

------
dang
Previous threads on this:

[https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...](https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=comments%3E1%20abc%20conjecture&sort=byDate&type=story)

[https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...](https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=comments%3E1%20mochizuki&sort=byDate&type=story)

~~~
vosper
If anyone on iOS Safari gets a blank white page following these links: you may
have to turn off your content blocker(s) (in my case Firefox Focus)

------
doall
This is Mochizuki's informal blog post in Japanese about the current
situation.

[https://plaza.rakuten.co.jp/shinichi0329/diary/202001050000/](https://plaza.rakuten.co.jp/shinichi0329/diary/202001050000/)

Although he had lived in the US for more than a decade and has no problem with
the English language, he seem to have a kind of "western culture allergy" that
is written in detail in the post below:

[https://plaza.rakuten.co.jp/shinichi0329/diary/201711210000/](https://plaza.rakuten.co.jp/shinichi0329/diary/201711210000/)

I think the "allergy thing" is the reason he doesn't want to follow the
ordinary "western approved way" and do a tour in the US.

Also I have read somewhere that he is open to mathematical discussions via
online or if you visit him in Japan.

------
akvadrako
For a decent take in support of the publication, see this recent philosophy of
mathematics paper:

[https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf](https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf)

------
mellosouls
TL;DR: obscure, long "proof" of major conjecture by respected mathematician to
be published in a journal he is closely associated with despite several years
of scepticism by the wider maths community his attempt is successful.

Complicated somewhat by possible language and cultural barriers, and his
perceived reluctance to fully engage with his critics or the maths world
outside his home country.

It's an interesting and odd story that has been rumbling on for the last few
years.

~~~
throwlaplace
There's no language barrier Mochizuki is completely fluent in English.

~~~
mellosouls
He seems to insist on only communicating his ideas in Japanese though.

~~~
throwlaplace
The paper is written in English and scholz for example traveled to Japan and
communicated with him in English

~~~
mellosouls
Although if you look at his rather colourful webpage, the strong, almost
defensive, warning that Japanese will be the language used (at a Japanese
institution of course) doesn't detract from the anecdotal suggestions that
have dogged the saga of a reluctance to engage more fully with the world
beyond his local region.

Scholze and Stix had to go to him.

[http://www.kurims.kyoto-u.ac.jp/~motizuki/students-
english.h...](http://www.kurims.kyoto-u.ac.jp/~motizuki/students-english.html)

~~~
knolax
I would be pissed too if I were a professor and some undergrads insisted on
speaking about mathematics to me in Japanese simply because they view it as a
"lingua franca". The argument that "English is the lingua franca" is akin to
"let's use Electron because web is the most cross-platform".

~~~
olodus
That is true. It is sometimes annoying that English speakers so often never
tries to learn a new language and just expects everyone else to use theirs.

However, one could view it as the only common language between you and someone
else and use it for that reason. In my mind the one that know more languages
are able to more easily communicate with more people more accurately, and as
such have a leg up on teaching and learning from others. These ideas might be
able to travel faster since they can travel by English and Japanese. Though
maybe that is just wishful thinking from my side.

~~~
knolax
In my experience monolingual people underestimate the effort it takes to use
your non-dominant language(s). For example, I could theoretically communicate
in Spanish, but the effort it takes to deal with romance language conjugation
is a legitimately unpleasant experience for me. Mochizuki might know English,
but it might be the case that using it is an unpleasant experience to him. In
addition, just as there are English proficiency requirements for international
students in the US, I feel it would be unreasonable to pursue higher education
in Japan without even knowing enough Japanese to talk about math.

------
peter_d_sherman
Excerpt:

"The saga began when Mochizuki, a respected number theorist quietly posted his
preprints on 30 August 2012 — not on arXiv.org, mathematicians’ preferred
repository, but on his own webpage at RIMS. Written in an impenetrable,
idiosyncratic style, the papers seemed to entirely consist of mathematical
concepts that were completely unfamiliar to the rest of the community —

 _“like you might be reading a paper from the future, or from outer space”_ ,

wrote Jordan Ellenberg, a number theorist at the University of
Wisconsin–Madison, on his blog soon after the papers appeared."

...Which makes it all the more a subject of curiousity -- and worth looking
at...

------
credit_guy
The top 2 math journals are, in many people's minds, Annals and Inventiones
(i.e. Annals of Mathematics and Inventiones Mathematicae). The fact that this
work, which is supposed to be of utmost importance, was not published in one
of these two, is not confidence inspiring. That it was published in a journal
where the author is chief editor is downright scandalous.

------
haecceity
He should have written the proof in Lean or something.

~~~
JulianWasTaken
As far (though little) as I know, that's way beyond feasible still -- lean has
a ton I think, but is nowhere near covering the "seam" of mathematics -- many
fundamental objects are not even defined themselves yet, so you'd have to do
lots of extra work there, not to mention all the work to define your own new
things.

Lean is amazing (and I played around with it a bit, should do so more) but I
don't suspect it's realistic to expect new theorems of this magnitude to be
written in it at this point yet, doing so is a huge huge effort on top of the
life-altering effort that the theorem itself requires.

------
jesuslop
Had we a slick method of formalizing proofs for computer verification, no
polemics would be possible.

~~~
Koshkin
Good point; was it not possible in this case?

------
rkagerer
Maybe try publishing it first, _then_ see if it's newsworthy.

This isn't something you hype like a iPhone.

~~~
abnry
Add to it that it has been known for a long time that this paper would be
published in this journal.

------
ur-whale
Once again, this is a rather perfect example of why there is a great need in
Math for a universal formal proof language that can be verified by computers.

If this existed, the burden of proof would be on Mochizuki to present his
proof in a language that can actually be understood by others and by machines.

~~~
QuesnayJr
The problem is that producing a formal proof that can be verified by a
computer is incredibly boring. Since it's incredibly boring, nobody is going
to do it. It's not a matter of a universal formal proof language existing
(there are lots of frameworks that are powerful enough), it's matter of one
being easy enough to use that people will willingly use it.

~~~
gravypod
What would need to exist to make such a proof specification utility less
boring? I know many engineers who never thought they'd use unit tests until I
was able to demonstrate immediate workflow benefit that made their job far
easier than before. If you have experience in higher math, what would need to
exist to make such specification useful?

~~~
QuesnayJr
The challenge is making a tool that's sufficiently ergonomic. It's the UI
problem from hell. The problem is that people are much smarter than computers,
so you can just specify the broad strokes and your audience can fill in the
details as needed.

------
cronocr
It seems mathematicians are not aware of code obfuscation. Mochizuki objective
is to keep this knowledge in Japan and protect this technological advantage.

~~~
papeda
If his objective is to "keep this knowledge in Japan", circulating the
preprint and then publishing it in the journal he edits is a weird strategy.

~~~
cronocr
Weird, why? Since he can't keep this information from leaking around the
world, the best strategy is to obfuscate it and show his trusted partners the
concepts required to understand it. By gaining notoriety around the world, he
is also attracting more partners, and he can filter them out (ie. he knows
english, but will only do lectures in japanese, inside the country). If at
some point the information gets fully decoded, he will be recognized as having
solved the problem anyway. In the meanwhile there is a technological advantage
to keep.

~~~
knolax
What is the technological advantage of proving the ABC conjecture? Seems like
any application for it would comparatively take much more time to develop than
whatever time advantage can be gained through obsfucation.

