
The ABC conjecture has still not been proved - thisisit
https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/
======
Sniffnoy
Terry Tao's comment in the comments section is particularly worth pointing
out, I think:

> I do not have the expertise to have an informed first-hand opinion on
> Mochizuki’s work, but on comparing this story with the work of Perelman and
> Yitang Zhang you mentioned that I am much more familiar with, one striking
> difference to me has been the presence of short “proof of concept”
> statements in the latter but not in the former, by which I mean ways in
> which the methods in the papers in question can be used relatively quickly
> to obtain new non-trivial results of interest (or even a new proof of an
> existing non-trivial result) in an existing field. In the case of Perelman’s
> work, already by the fifth page of the first paper Perelman had a novel
> interpretation of Ricci flow as a gradient flow which looked very promising,
> and by the seventh page he had used this interpretation to establish a “no
> breathers” theorem for the Ricci flow that, while being far short of what
> was needed to finish off the Poincare conjecture, was already a new and
> interesting result, and I think was one of the reasons why experts in the
> field were immediately convinced that there was lots of good stuff in these
> papers. Yitang Zhang’s 54 page paper spends more time on material that is
> standard to the experts [...] but about six pages after all the lemmas are
> presented, Yitang has made a non-trivial observation, which is that bounded
> gaps between primes would follow if one could make any improvement to the
> Bombieri-Vinogradov theorem for smooth moduli. [...]

> 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 scepticism 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.

(Comment link:
[https://galoisrepresentations.wordpress.com/2017/12/17/the-a...](https://galoisrepresentations.wordpress.com/2017/12/17/the-
abc-conjecture-has-still-not-been-proved/#comment-4563) )

~~~
ilitirit
Every time I read about Perelman and the Poincare conjecture, I feel compelled
to link this excellent piece from the New Yorker:

[https://www.newyorker.com/magazine/2006/08/28/manifold-
desti...](https://www.newyorker.com/magazine/2006/08/28/manifold-destiny)

It even has a Wikipedia article on it:

[https://en.wikipedia.org/wiki/Manifold_Destiny](https://en.wikipedia.org/wiki/Manifold_Destiny)

~~~
waivek
The piece is inaccurate and most of the mathematicians quoted have distanced
themselves from it. It's journalism at it's worst.

One of the key figures in the article sent this to the New Yorker after it was
published -

[http://www.doctoryau.com/hamiltonletter.pdf](http://www.doctoryau.com/hamiltonletter.pdf)

 _Dear Mr. Cooper_

 _I am very disturbed by the unfair manner in which Yau Shing-Tung has been
portrayed in the New Yorker article. I am providing my thoughts below to set
the record straight. I authorize you to share this letter with the New Yorker
and the public if that will be helpful to Yau._

 _As soon as my first paper on the Ricci Flow on three dimensional mani- folds
with positive Ricci curvature was complete in the early '80's,Yau immedi-
ately recognized it's importance;and although I had proved a result on which
he had been working with minimal surfaces,rather than exhibit any jealosy he
became my strongest supporter.He pointed out to me way back then that the
Ricci Flow would form the neck pinch singularities,undoing the connected sum
decomposition,and that this could lead to a proof of the Poincare conjec-
ture.In 1985 he brought me to UC San Diego together with Rick Schoen and
Gerhard Huisken,and we had a very exciting and productive group in Geomet -
ric Analysis.Huisken was working on the Mean Curvature Flow for
hypersurfaces,which closely parallels the Ricci Flow,being the most natural
flows for intrinsic and extrinsic curvature respectively.Yau repeatedly urged
us to study the blow-up of singularities in these parabolic equations using
techniques parallel to those developed for elliptic equations like the minimal
surface equation,on which Yau and Rick are experts.Without Yau's guidance and
support at this early stage,there would have been no Ricci Flow program for
Perelman to finish. _

_Yau also had some outstanding students at San Diego who had come with him
from Princeton, in particular Cao Huai-Dong,Ben Chow and Shi Wan- Xiong. Yau
encouraged them to work on the Ricci Flow,and all made very important
contributions to the field.Cao proved existence for all time for the
normalized Ricci Flow in the canonical Kaehler case ,and convergence for zero
or negative Chern class.Cao 's results form the basis for Perelman's excit -
ing work on the Kaehler Ricci Flow,where he shows for positive Chern class
that the diameter and scalar curvature are bounded. Ben Chow,in addition to
excellent work on other flows,extended my work on the Ricci Flow on the two
dimensional sphere to the case of curvature of varying sign.Shi Wan- Xiong
pioneered the study of the Ricci Flow on complete noncompact manifolds,and in
addition to many beautiful arguments he proved the local derivative estimates
for the Ricci Flow.The blow-up of singularities usually produces noncompact
solutions,and the proof of convergence to the blow-up limit always depends on
Shi's derivative estimates; so Shi's work is central to all the limit
arguments Perelman and I use. _

_In '82 Yau and Peter Li wrote an exceedingly important paper giving a
pointwise differential inequality for linear heat equations which can be inte-
grated along curves to give classic Harnack inequalities. Yau repeatedly urged
me to study this paper,and based on their approach I was able to prove Har-
nack inequalities for the Ricci Flow and for the Mean Curvature Flow. This
Harnack inequality,generalized from Li-Yau,forms the basis for the analysis of
ancient solutions which I started, and which Perelman completed and uses as
the basic tool in his canonical neighborhood theorem. Cao Huai-Dong proved the
Harnack estimate for the Ricci Flow in the Kahler case,and Ben Chow did the
same for the Yamabe Flow and the Gauss Curvature Flow. _

_But there is more to this story. Perelman 's most important is his noncol-
lapsing result for Ricci Flow,valid in all dimensions,not just three,and thus
one whose importance for the future extends well beyond the Poincare
conjecture,where it is the tool for ruling out cigars,the one part of the
singular- ity classification I could not do. This result has two proofs,one
using an entropy for a backward scalar heat equation,and one using a path
integral.The entropy estimate comes from integrating a Li-Yau type
differential Harnack inequality for the adjoint heat equation,and the other is
the optimal Li-Yau path integral for the same Harnack inequality; as Perelman
acknowledges in 7.4 of his first paper,where he writes "an even closer
reference is [L-Y],where they use "length" associated to a linear parabolic
equation,which is pretty much the same as in our case". _

_Over the years Yau has consistently supported the Ricci Flow and the whole
field of Geometric Flows,which has other important successes as well,such as
the recent proof of the Penrose Conjecture by Huisken and Ilmanen,a very
important result in General Relativity. I cannot think of any other prominent
leader who gave nearly support to our field as Yau has._

 _Yau has built is an assembly of talent,not an empire of power,people
attracted by his energy,his brilliant ideas,and his unflagging support for
first rate mathematics, people whom Yau has brought together to work on the
hard- est problems.Yau and I have spent innumerable hours over many years
work- ing together on the Ricci Flow and other problems,often even late at
night. He has always generously shared his suggestions with me,starting with
the obser- vation of neck pinches,never asking for credit. In fact just last
winter when I finally managed to prove a local version of the Harnack
inequality for the Ricci Flow,a problem we had worked on together for many
years,and I said I ought to add his name to the paper,he modestly declined.It
is unfortunate that his character has been so badly misrepresented.He has
never to my knowledge proposed any percentages of credit,nor that Perelman
should share credit for the Poincare conjecture with anyone but me; which is
reasonable,as indeed no one has been more generous in crediting my work than
Perelman himself.Far from stealing credit for Perelman 's accomplishment,he
has praised Perelman's work and joined me in supporting him for the Fields
Medal.And indeed no one is more responsible than Yau for creating the program
on Ricci Flow which Perelman used to win this prize. _

_Sincerely yours,_

 _Richard S Hamilton_

 _Professor of Mathematics, Columbia University_

~~~
WhitneyLand
If the article was that bad, how come he went through the trouble of hiring
attorneys to draft letters accusing the author of defamation, threaten action,
and never took it any further?

There was never any apology or retraction seeming to remove settlement an a
possible result.

I have no reason to doubt the claims against the article, at first glance
there are a significant number of credible sources supporting the idea the
article was flawed, yet the claims were ineffective legally.

Of course it’s not easy or cheap to pursue a case of defamation, however, I
think it money was the issue he could have rallied some public support. If I
were asked, it’s certainly an issue I would do the due diligence on and
contribute something to if it made sense.

edit: btw why is no one making a movie about this? There are so many
fascinating aspects, the grand challenge, the controversy, the mathematician
who declined a million dollar prize. I’d love to do even a documentary on it,
or watch one if it’s been done.

~~~
qmalzp
Probably because at the end of the day, he doesn't want to fight some legal
battle, he wants to do mathematics.

~~~
hyperpape
In addition (not a lawyer here, but I believe this is well established
territory), defamation is difficult to prove, and suing someone over an
article almost always carries some negative press, regardless of the merits.

~~~
WhitneyLand
That’s the point, I mentioned the difficulty in fact. But he chose to step
across that bridge and then abandon it. What else can we conclude other than,
heat of the moment emotional decision, or that the merits were, if morally
strong enough, not legally strong enough?

------
tontonius
Mochizuki is a badass, elitist rock star of mathematics and he takes his time
to roast the entire field in his papers [1].

P.115 >The adoption of strictly linear evolutionary models of progress in
mathematics of the sort discussed [previously] tends to be highly attractive
to many mathematicians in light of the intoxicating simplicity of such
strictly linear evolutionary models, by comparison to the more complicated
point of view discussed [previously]. This intoxicating simplicity also makes
such strictly linear evolutionary models — together with strictly linear
numerical evaluation devices such as the “number of papers published”, the
“number of citations of published papers”, or other likeminded narrowly
defined data formats that have been concocted for measuring progress in
mathematics — highly enticing to administrators who are charged with the tasks
of evaluating, hiring, or promoting mathematicians.

> Moreover, this state of affairs that regulates the collection of individuals
> who are granted the license and resources necessary to actively engage in
> mathematical research tends to have the effect, over the long term, of
> stifling efforts by young researchers to conduct long-term mathematical
> research in directions that substantially diverge from the strictly linear
> evolutionary models that have been adopted, thus making it exceedingly
> difficult for new “unanticipated” evolutionary branches in the development
> of mathematics to sprout.

I also recommend reading page 114 where he lists some 14 breakthroughs made in
his paper, and how many years earlier they could have been made. The audacity
of this guy!

[1] ALIEN COPIES, GAUSSIANS, & INTER-UNIVERSAL TEICHMULLER THEORY
([http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20...](http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-
universal%20Teichmuller%20Theory.pdf))

~~~
nautilus12
Sounds like a way of justifying the fact that his results are totally silo'd
off from the rest of the mathematical cannon

------
placebo
This got me thinking about human cognition and the meaning of mathematical
truths, or even how does one know that something is true (and I assume that
probably none of this is original).

At which point would Mochizuki’s work be considered proof? If another person
regarded as an authority would claim to fully understand it (but couldn't
explain it better to their peers)? If only 10% of mathematicians specialising
in this field could understand it? If 40%?

What if only a few percent understood the proof, but managed to "prove" to
others that they understood it by using this knowledge to unlock other
difficult problems?

This seems to point that even in mathematics we can never claim to know what
is true. At most, we can hold a belief, an assessment based on some parameters
that increases our degree of belief that something is true.

In fact, there's no need for a controversial proof to show this - it occurs
with the simplest of mathematical proofs - after all, everything we believe in
is based on the ability to detect consistency or lack thereof, which in itself
is based on the ability of a system (in this case, our brains) to have a
memory and notice where something happening now is consistent or inconsistent
with something stored in our memory. Now this is quite problematic in terms of
proof, because how can we trust our memory with a 100% certainty? The answer
is that we can't. We attribute a high probability that what we know is true
based on the consistency of other things we remember, but we can never prove
_anything_ .

~~~
Certhas
"What if only a few percent understood the proof, but managed to "prove" to
others that they understood it by using this knowledge to unlock other
difficult problems?"

Check the Terrence Tao comment below the post (also posted by someone else in
here).

I think if those who claim to understand the proof could use the techniques to
proof other things, or even to provide new proofs of known results, the level
of trust would immediately go up dramatically.

------
kurthr
tl;dr Until someone can explain the proof in something under 100s of hours,
it's not considered proven. It's not useful until someone uses IUT as a tool.

It reminds me of other fields of science (like genetics) where many people
have made a similar "discovery", but one person finally discovers and reports
the results so clearly (e.g. Gregor Mendel) that everyone recognizes them as
the discoverer... even though others came before (e.g. Imre Festetics) with
similar concepts.

~~~
jacobolus
Mendel was a monk and high school teacher whose work (he spent a decade
hybridizing pea plants in the monastery garden) was published in small
regional sources and languished in obscurity for decades. Nobody at the time
understood the importance of his work, and biologists continued under mistaken
models of how inheritance worked. Decades later other botanists did similar
experiments, and then stumbled over Mendel's papers, and credited him for
first developing and experimentally validating the correct model. e.g. Correns
[http://www.esp.org/foundations/genetics/classical/holdings/c...](http://www.esp.org/foundations/genetics/classical/holdings/c/cc-00.pdf)

As far as I know Mendel's model was much simpler and more clearly defined than
anything that came before.

~~~
kurthr
Yes, exactly, he discovered it well enough that people who read his papers
years later didn't need for it to be discovered again. I agree, those at the
time of publishing didn't even consider it to be about inheritance, but those
later did, because his careful work could be replicated... even though Imre
Festetics coined the word genetic and did similar work on sheep in the same
town many years before (he's mostly forgotten).

------
muizelaar
Here's an article on IUT that shows that progress is being made:
[http://inference-review.com/article/fukugen](http://inference-
review.com/article/fukugen)

------
moomin
This. Mathematics is not really the study of what is the case, it’s the study
of why. Even if this was accompanied by a full Coq proof it would be
unsatisfactory because absolutely no insight is being provided.

~~~
ilitirit
Things that provide no insight _now_ may have theoretical or even practical
applications in the future.

[https://mathoverflow.net/questions/116627/useless-math-
that-...](https://mathoverflow.net/questions/116627/useless-math-that-became-
useful/)

I'm not saying that this is the same of Mochizuki's work, but I would not
dismiss it solely on the basis that it's "useles".

~~~
moomin
I don't think anything on that list was bereft of insight, even at the time.
This is a context issue: the problem with Mochizuki's work is that no-one has
managed to point to a single insight other than the abc conjecture. Whilst
it's possible to prove certain results by brute force manipulation, it's not
common. High profile results tend to be accompanied with subsidiary insights.
The applicability of these insights to real world problems is, for the
purposes of this discussion, beside the point.

Anyway, I'm basically repeating what people have already said. So I'll stop.

------
rconti
I posit that there are, in fact, times that one should be doing things other
than closing.

~~~
taneq
"Close" means different things to people in different roles. To a salesman, it
means make the sale. To a mathematician in academia, it means publish. To a
product team, it means ship.

~~~
leoc
"The primary thing when you take a sword in your hands is your intention to
cut the enemy, whatever the means. Whenever you parry, hit, spring, strike or
touch the enemy's cutting sword, you must cut the enemy in the same movement.
It is essential to attain this. If you think only of hitting, springing,
striking or touching the enemy, you will not be able actually to cut him."

[https://en.wikiquote.org/wiki/Miyamoto_Musashi#The_Water_Boo...](https://en.wikiquote.org/wiki/Miyamoto_Musashi#The_Water_Book)

~~~
taneq
"All warfare is based on closing. Hence, when we are able to attack, we must
close; when using our forces, we must close; when we are near, we must close;
when far away, we must close." \- Sun Tsu

------
robinhouston
It’s worth revisiting this very nice article (2013) by Caroline Chen, which
gives a vivid impression of the situation at that time.

[http://projectwordsworth.com/the-paradox-of-the-
proof/](http://projectwordsworth.com/the-paradox-of-the-proof/)

In a way, the only “news” since then is that the mathematical community
_hasn’t_ made any real progress with understanding Mochizuki’s theory.

------
DaggerSpaces
Comments now by Terry Tao and Peter Scholze (PS). That's some heavy fire
power.

~~~
woopwoop
Scholze's comment is particularly interesting.

------
mherrmann
It sounds a little like the opposite of Fermat's famous statement about his
last theorem. "I have a truly marvelous demonstration of this proposition. It
only takes 300 obscure pages to explain."

~~~
vog
I like that story, too, but note that the actual quote was:

"I have discovered a truly marvelous proof of this, which this margin is too
narrow to contain."

[https://upload.wikimedia.org/wikipedia/commons/2/24/Diophant...](https://upload.wikimedia.org/wikipedia/commons/2/24/Diophantus-
II-8.jpg)

[https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Fermat...](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Fermat%27s_conjecture)

~~~
vog
Wow, being downvoted for correcting a misquote. That's HN at its best.

~~~
hardlianotion
The vote was very harsh, but parent was laying out some kind of opposite to
the quote, not misquoting

~~~
vog
Thanks for pointing this out.

BTW, I really wish that short explainations were mandatory for downvotes. This
would have cleared this up early on.

------
adamnemecek
If only we had automated proof assistants that could check the proof quickly.
Oh wait.

~~~
dogecoinbase
This comment does nothing other than demonstrate your own ignorance of
research mathematics.

~~~
adamnemecek
Please keep going.

~~~
qmalzp
I have never seen a theorem prover applied to even basic 100-year-old results
in Algebraic Number Theory. I think you underestimate the difficulty in
translating a mathematical idea into a format a computer program can
understand.

