
Titans of Mathematics Clash Over Epic Proof of ABC Conjecture - digital55
https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/
======
Moodles
I have a MSc in mathematics so I am by no means an expert in mathematics
proofs but I get the gist of it. To me a proof is literally a logical argument
that you can follow to "believe" that a theorem is true. I do worry how many
people actually understand or verify mathematics proofs. How many people have
actually read and verified Perelman's or Wiles' proofs (I'm only singling them
out because they're famous, not because they're particularly sketchy). We seem
to think that once a proof is published in a reputable journal then it is
definitely true, but really we should only think "a small number of qualified
people have read it and think it is fine". The foundations actually seem a bit
shakier to me than people seem to appreciate. Perhaps machine-verified proofs
will be the answer one day.

~~~
barbecue_sauce
You have an MSc in mathematics but don't consider yourself an expert?

~~~
Smaug123
Where high-school mathematics was analogous to learning to form letters with a
pencil, undergraduate BSc mathematics is analogous to learning to form words
and short sentences.

By the end of your BSc in the analogy, you can look at a sentence and
recognise at least what sort of genre it might be part of, and you can write
down many of the most common words as well as a number of rehearsed useful
sentences.

An MSc is like reading some examples of long sentences and short paragraphs,
and being shown some long paragraphs or even chapters of books (which you have
no hope of understanding, but you try, and the experience is salutary). If
you're lucky, you did a research project in which you perhaps rewrote a
certain very specific short paragraph in your own words.

The job of a research mathematician is analogously then to read and write
books.

~~~
shoo
> If you're lucky, you did a research project in which you perhaps rewrote a
> certain very specific short paragraph in your own words.

Well stated. this pretty accurately characterises my experience of
undergraduate studies & an honours year in mathematics

------
lacker
I remember an instructor explaining during the math olympiad training program:
"A proof doesn't have to be in any particular format. A proof is an argument
that can convince other mathematicians."

At this point, it seems clear that we do not have a proof of the ABC
conjecture. Perhaps someone will be able to improve this to make a real proof
- after all, there were some errors in the first Wiles proof of Fermat's Last
Theorem, but they ended up being minor errors that were fixed up once they
were found. But what we have is not a proof.

~~~
westoncb
It's interesting that this article doesn't touch on this aspect of the
situation at all, but when reading about Mochizuki's proof in the past it was
presented as clear that proving the ABC conjecture was one application of a
new general theory—not that all of Mochizuki's innovations are tied
specifically to that problem.

I was given to understand that his work was more like Grothendieck's, where
he's developing this entirely new, super general system. But you can imagine
if Grothendieck for example had done all of his work for years and years in
isolation and then presented Schemes, Topoi, and Motives to everyone all at
once, using them in combination to prove something familiar, and expecting
other mathematicians to just learn and understand them all in order to verify
a single proof.

I think in that case, folks would not have been immediately so interested in
the new constructions Grothendieck had come up, viewing it initially as
mountains of unnecessarily alien concepts built up just to give other
mathematicians a hard time ;)

(Not to identify the two mathematicians overly—I'm not sure how appropriate
that really is—but I remember that being my impression when first digging into
Mochizuki's work, and I haven't seen it mentioned here yet.)

~~~
Sniffnoy
This is where Terry Tao's comment from December seems relevant:
[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)

------
jim777
> When he told colleagues the nature of Scholze and Stix’s objections, he
> wrote, his descriptions “were met with a remarkably unanimous response of
> utter astonishment and even disbelief (at times accompanied by bouts of
> laughter!) that such manifestly erroneous misunderstandings could have
> occurred.”

Is this normal in advanced Math or is this guy kind of a jerk?

~~~
a-nikolaev
Math academic community is actually very healthy overall, in my experience.

~~~
WilliamEdward
see: monty hall problem

But anyway, mathematics is not any more unhealthy than any other field, but it
does have toxicity like all others do as well.

~~~
ssmmww
My understanding is that the Monty Hall problem was just a cute puzzle
published in a pop-sci magazine that confused a few professors who perhaps
ought not to have been confused by it. I don't think it says anything at all
about the health of the mathematical community.

~~~
fjsolwmv
It was also ambiguously worded, which is the bane of probability problems
posed by people who think they are clever (like vos Savant).

~~~
mannykannot
I will repeat what I wrote about this point a couple of weeks ago: According
to vos Savant, in most of the cases where the correspondent actually had a
correction to offer for her alleged error, they were assuming the intended
interpretation of the puzzle.

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

Terence Tao and Peter Scholze (and others) join the discussion in the comments
section.

~~~
avip
Tao's commentary is especially interesting for me as a non-mathematician:

>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.

Likely such is the understatement-ish way of a mathematician to say "This is
clearly wrong, but I have not the time to dig in and find where exactly"

~~~
neuronexmachina
I think it might be more akin to a programmer complaining about a
monolithic/black-box chunk of code that can't easily be split into
understandable and/or unit-testable functions.

~~~
sorokod
For the analogy, I'd replace 'unit-testable' with reusable

~~~
shoo
I'd argue for something stronger: both reusable and useful (outside of the
initial application) and novel.

It's possible to write structurally reusable components that are useless, i.e.
they do a thing that no one wants or needs.

------
westoncb
> “The abc conjecture is a very elementary statement about multiplication and
> addition,” said Minhyong Kim of the University of Oxford. It’s the kind of
> statement, he said, where “you feel like you’re revealing some kind of very
> fundamental structure about number systems in general that you hadn’t seen
> before.”

I can kind of see that, but there's something in it that gives me the feeling
of arbitrariness.

It's looking at the relationship between prime factors of A, B, C in A + B =
C, where no prime factors may be shared between A, B, or C. The specific
relationship in question is between the magnitude of C and the unique prime
factors of A, B, and C multiplied together.

To take an example from the article, 5 + 16 = 21 meets the basic requirements
since the prime decomposition looks like (5) + (2 * 2 * 2 * 2) = (7 * 3) —no
factors are shared. But, the quantity we're supposed to relate to C is (5 * 2
* 7 * 3), since it is the product only of the unique primes.

Does that not feel arbitrary, to drop the repetitions? So it ends up being
that the sought out smaller products arise because they had large numbers of
repetitions, so we removed more when forming the product.

But I suppose there are probably just some deep number theory mysteries behind
it which give justification, so it only feels arbitrary to someone like myself
who is basically ignorant on the subject :)

~~~
edflsafoiewq
Another way to think about it is to consider what needs to happen for rad(abc)
to be much less than c. This can only happen if the "dropping" you mention
drops a whole lot of factors. In other words, it can only happen if a, b, and
c all contain mostly large powers of primes.

The prime factorization of a number is essentially "random". That makes
numbers that are large powers of primes rare; it's like rolling a die and
continuously getting the same face. Now if you add two numbers, the prime
factorization of the sum does not appear to be related to the prime
factorization of the summands, so adding two very rare numbers and getting yet
another very rare number seems unlikely. And that seems to be roughly what the
abc conjecture says: it is not likely that all of a, b, and c are rare.

~~~
westoncb
Great explanation—thanks. I think I got the upshot, though I am unclear on
what function rad() is.

~~~
rocqua
rad is the radical of a number [1]. In short rad(x) is the product of all
unique prime factors of x.

So rad(8) = 2, rad(24) = 6, rad(75) = 15

[1]
[https://en.wikipedia.org/wiki/Radical_of_an_integer](https://en.wikipedia.org/wiki/Radical_of_an_integer)

------
Obi_Juan_Kenobi
Slightly OT: it was very nice to read an article that made such wonderful use
of links. Just about every case where material could be directly linked, it
was done.

I particularly enjoyed the link to Dr. Calegari's blog post, as it was
interesting to read the comments in 'real time' and compare that with the
author's synthesis. Very good article!

~~~
timb07
Even further OT: I was good friends with Frank Calegari's older brother (also
a mathematician) at school, and used to play bridge with Frank at university.
I'm not a mathematician but I do sometimes read articles when they're posted
on HN, so it was a surprise to see someone I know mentioned. :p

------
drenvuk
I think this would count as esoteric if you ignore the drama portion of it.
Just to get to the point of understanding what they're talking about would
probably take months of dedicated uninterrupted work. Neat. I have no idea
where to even start.

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

~~~
xxpor
I think what makes these number theory arguments so interesting to me is how
simple the theorems are, but how impenetrable the proofs are. For example,
Fermat's last theorem is so easy to understand, but there's no chance in hell
I'll ever understand Wiles' paper proving it.

~~~
dagw
You might not understand all the details of the paper, but you can definitely
follow the thrust of the argument and see the elegance in it without advanced
math skills.

The really hard part (and main part of Wile's contribution to the proof) was
proving something called the Taniyama–Shimura–Weil conjecture. However if you
skip that bit and just accept that the conjecture is true the rest of the
steps of the proof are both elegant and relatively easy to follow (if you
gloss over some of the details) for anybody with a decent grasp of basic math.

The book Fermat's Last Theorem by Simon Singh is a pretty great read and does
a good job of outlining the basic structure of the proof for anybody with a
decent grasp of high school mat.

------
S4M
Coincidentally I recently finished a MOOC about prime numbers where the ABC
conjecture was introduced, I believe in order to show how mysterious prime
numbers still were. The MOOC is here if someone is interested:
[https://courses.edx.org/courses/course-v1:KyotoUx+011x+2T201...](https://courses.edx.org/courses/course-v1:KyotoUx+011x+2T2018/b46e533438fc4cb98eeef569abbd837a/)

Apart from that, the article makes me curious about the state of automated
mathematical proof. I remember having read an article posted here about a
mathematician (I think he was a Field medalist) who was claiming that
automated proof were the future of mathematics as they would enable
mathematicians to collaborate much more easily, removing problems of trust in
others' proof.

------
vertline3
Peter Scholze is too valuable to be lost to the hall of mirrors that is
Mochizuki's conjecture. :P He is young and at the peak of his powers. Even
I've heard of him.

------
hedvig
So Mochizuki has had years to revise and restate things right? Knowing that it
hasn't been well received wouldn't one want to clarify things? Did he move on
to a different project, or is he so arrogant that once he submits a paper he
walks away from it?

------
auggierose
A few years ago I asked Mochizuki via email if he would be interested in
organising with me an effort to mechanise his proof. At the time I was
developing a collaborative interactive theorem prover, for which this would be
an ideal application. He politely declined, and I am kind of happy about this
now, because I need a few more years anyway before my theorem prover could be
used for something like this without being a pain for its users :-D

------
spydum
I do have to wonder, if the proof is so complex, and yet doesn't offer
insights to any other proof or theories.. does it matter?

Older threads for reference:

    
    
        https://news.ycombinator.com/item?id=15971802
        https://news.ycombinator.com/item?id=4502856
        https://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture

------
ur-whale
To me, this article highlights two things: \- the sore need for a way of
automatically verifying proofs. \- the sore need for a formal, agreed-upon
language in which math proofs can be written.

The language of math has always been much more mushy than mathematicians are
willing to concede, and the chickens are coming to roost: 21s century math has
become so complex and sophisticated that very few people can actually even
read the content of proofs, much less understand them.

Shinozuki's case is an extreme example of that: after almost ten years, even
he other experts in the field aren't sure of what he's saying.

There is a clear need for formalizing the language of mathematics in a way
that allows machine to verify he validity of a proof.

------
matthewbauer
I wonder if language is a factor in this at all? I know mathematics is written
entirely in English but most likely discussions still happen in local
languages. Could part of the proof be lost in translation?

~~~
doall
I think Mochizuki won't have any problems with English, since he spent his
school days, up to PhD in the US.

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

About Hoshi, I have seen him in a conference video before, and have an
impression that he isn't good at English, which can make a bit difficult in
discussion.

I don't know about Scholze and Stix.

~~~
laingc
Scholze has a strong accent, but speaks flawless, sophisticated English.

------
graycat
The math community can't accept a new result is true without a good proof. As
part of good, the proof has to be readable by peer reviewers. Net, if the
writing is tough to read by some of the best qualified peers, then the writer
has more work to do, in particular, to learn to write better.

We have lots of examples of good writing in math from Bourbaki, W. Rudin, P.
Halmos, H. Royden, and more.

Sorry 'bout that: No one wants math to miss out on a great, new result, but
the writer has to do their job first, in particular, do good math writing.

------
peq
Mathematicians should do more computer verifiable proofs to avoid discussions
like this.

> Definitions went on for pages, followed by theorems whose statements were
> similarly long, but whose proofs only said, essentially, “this follows
> immediately from the definitions.”

This sounds perfect for machine checked proofs, but I guess the proofs are
actually a lot more involved than they are presented.

~~~
cycrutchfield
Per Peter Scholze (from
[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-4619)):

>One final point: I get very annoyed by all references to computer-
verification (that came up not on this blog, but elsewhere on the internet in
discussions of Mochizuki’s work). The computer will not be able to make sense
of this step either. The comparison to the Kepler conjecture, say, is entirely
misguided: In that case, the general strategy was clear, but it was unclear
whether every single case had been taken care of. Here, there is no case at
all, just the claim “And now the result follows”.

~~~
MaxBarraclough
> The computer will not be able to make sense of this step either.

What does that mean, then? That a leap is being made that depends on the
reader's fuzzy intuitions, rather than the established axioms? If that's the
case, it's not a formal proof at all, no?

Or am I way off here?

~~~
konschubert
Maybe he says that the computer won't help understand the proof since the
proof is missing a step.

He may be saying the equivalent of "A computer won't help you to prove that
1+1=3“

~~~
MaxBarraclough
> He may be saying the equivalent of "A computer won't help you to prove that
> 1+1=3“

But it can. You need to have the computer 'comprehend' the relevant axioms, of
course.

If your 'proof' is in fact just arguing the case for new axioms, that isn't a
proof at all, it's a misunderstanding of what 'axiom' means. (They're
definitions, not profound universal truths.)

------
onemoresoop
[http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-
discussions-...](http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-
discussions-2018-03.html)

~~~
dvdkhlng
Thanks for the link. As that seems very relevant to the discussion, let me add
some context: This is Mochizuki's answer to the recent criticism titled
"Report On Discussions, Held During The Period March 15 – 20, 2018, Concerning
Inter-Universal Teichmuller Theory (IUTC)"

For TL;DR see around page 40 in that PDF:

> Indeed, at numerous points in the March discussions, I was often tempted to
> issue a response of the following form to various assertions of SS (but
> typically refrained from doing so!): __Yes! Yes! Of course, I completely
> agree that the theory that you are discussing is completely absurd and
> meaningless, but that theory is com- pletely different from IUTch! __

> Nevertheless, the March discussions were productive in the sense that they
> yielded a valuable first glimpse at the mathematical content of the
> misunderstandings that underlie criticism of IUTch (cf. the discussion of §
> 3). In the present report, we considered various possible causes for these
> misunderstandings , namely:

> (PCM1) lack of sufficient time to reflect deeply on the mathematics under
> discussion (cf. the discussion in the final portions of § 2, § 10);

> (PCM2) communication issues and related procedural irregularities (cf. (T6),
> (T7), (T8));

> (PCM3) a deep sense of discomfort ,or unfamiliarity ,with new ways of
> thinking about familiar mathematical objects (cf. the discussion of § 16;
> [Rpt2014], (T2); [Fsk], § 3.3).

> On the other hand, the March discussions were, unfortunately, by no means
> sufficient to yield a complete elucidation of the logical structure of the
> causes underlying the misunderstandings summarized in § 17.

------
zokier
So what would this imply over the larger body of Mochizukis work? As far as I
have understood the situation, the abc proof is more of a sideproduct of him
developing this completely new system(?) for maths rather than the main thrust
of the works. Is it all falling down like house of cards, or is there still
workable and useful bits there?

~~~
contravariant
Somewhere in this comment section someone quoted the following line by
Terrence Tao:

>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.

so it's not so much that the main result is called into question but there are
some useful and workable bits left over. Rather the main result is called into
question _because_ there are no other useful and workable bits.

------
swerveonem
Will others fork his proof to fill in the gaps? Is there a github for maths
proofs?

------
notoriousjpg
We need Yasantha on this

------
CodeSheikh
My God! This Titans' website needs urgent attention:
[http://www.kurims.kyoto-u.ac.jp/~motizuki/top-
english.html](http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html)

~~~
dan-robertson
Why? It might be worse than a <p> and a <ul> with links to papers but it loads
ok and it’s basically a point of pride amongst academics to have an ancient or
amateurish looking handmade website. Perhaps it is a signal that time is spent
on academic work or a that the site was made by the academic (and so has
useful materials for academics) and not by some publicist who does not provide
useful academic materials.

