
Baffling ABC maths proof now has impenetrable 300-page ‘summary’ - ColinWright
https://www.newscientist.com/article/2146647-baffling-abc-maths-proof-now-has-impenetrable-300-page-summary/
======
dmix
Two paragraphs that highlight the potential source of the problem in the
sublinked article:

> Despite mathematics being a universal language, culture clash could be
> getting in the way, says Kim. “In Japan people are pretty used to long,
> technical discussions by the lecturer that require a lot of concentration,”
> he says. “In America or England we expect much more interaction, pointed
> questions coming from the audience, at least some level of heated debate.”

> There is a growing consensus that Mochizuki has over-engineered his work,
> contributing to the confusion. “Most of the large theories that he builds
> are not essential. He could have written things in a much more streamlined
> way,” says Voloch.

This is a problem in a lot of technical fields but also in writing in general.
It's always easier to write 1000 words to say something than 100 to say it
well.

~~~
KGIII
My verbosity is a shortcoming. Were I able, I'd be more precise and brevity
would be the result. This is actually something I work on improving.

~~~
Declanomous
Not a dig at you, but I tend to use $5 words when a normal word would suffice.
Even though the obscure words usually express my thoughts more precisely than
common words do, they do a worse job of conveying understanding, since people
often do not know the word and infer an incorrect meaning from context.

~~~
KGIII
I try to communicate at a level where the intended audience can easily
understand me. This is also partially out of laziness. I don't like having to
repeat myself.

~~~
ohyes
I'm not sure if this comment thread is a parody of itself.

~~~
Declanomous
That was at least part of my intent.

------
lvh
The article didn't explain the abc conjecture in much detail ("The original
proof is of the long-standing ABC conjecture that explores the deep nature of
numbers, centred on the simple equation a + b = c."). That's unfortunate,
because even though the proof is inscrutable, the conjecture itself isn't very
complicated:

 _If three numbers a, b, and c are coprime and two numbers a and b have large
prime factors, then their sum c generally does not._

To make that a little more formal, it means that the radical (product of
distinct prime factors) of their product abc is usually less than the sum c,
up to some exponent 1+eps:

    
    
        rad(abc)^(1+eps) < c
    

... for some eps. That eps is important! There are infinitely many numbers a,
b, c where this isn't true for some eps (and there are ways to construct
those), but the conjecture states that for any given eps, there are finitely
many.

To find maximal counterexamples (which, because mathematics consistently uses
intuitive terminology, are said to be "high-quality") you typically look for
numbers that are smooth (small prime factors) but not powersmooth (larger
prime-factors-raised-to-a-power). Typically you'll also fix a to be really
small, to limit the search space somewhat. [MM14] describes other methods for
finding triples.

Here's a simple example of a high-quality triple:

    
    
        a = 1
        b = 2*3^7 = 4374
        c = 5^4*7 = 4375
    

Because all factors are unique, rad(abc) = rad(a) rad(b) rad(c).

    
    
        c     > (rad(abc))^(1+eps)
        c     > (rad(a) * rad(b)     * rad(c))^(1+eps)
        5^4*7 > (rad(1) * rad(2*3^7) * rad(5^4*7))^(1+eps)
        5^4*7 > (1      * 2 * 3      * 5 * 7)^(1+eps)
        4375  > (210)^(1+eps)
    

This is a great example, because the exponent (sometimes called q) is about
1.57, which is extraordinarily high. In the first billion c's, there are only
34 with q > 1.4. This is the fifth best such triple we know of. [triples]

Consider a trivial case where all three numbers are prime; a = 2, b = 3, c =
5. Primes are their own radical, so:

    
    
        5 > (2 + 3)^1+eps
    

This is the edge case.

[triples]:
[https://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2](https://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2)

[MM14]:
[https://arxiv.org/pdf/1409.2974.pdf](https://arxiv.org/pdf/1409.2974.pdf)

~~~
lvh
I can no longer edit this comment, but there's a typo at the bottom:

    
    
        5 > (2 + 3)^1+eps
    

... should be

    
    
        5 > (2 * 3 * 5)^1+eps
    

Which is a good example of the "common case", i.e. not a triple of
particularly high quality (this is q=0.4-something).

~~~
legel
Thanks for the edit, I was wondering how that would make sense otherwise. More
generally, thanks for clearly communicating what's really going on! Math can
only be extremely beautiful if it is actually understood.

~~~
dmix
> Math can only be extremely beautiful if it is actually understood.

This is why I've found relearning math very rewarding. Even if I haven't found
as many applications in my day-to-day programming life as I hoped (other than
maybe when writing Haskell-esque languages or when I'm fortunate to get to use
pure FP concepts).

------
math_and_stuff
Some bitter conflict is referenced in footnote 1 of the 300 page summary.

""" The author hears that a mathematician (I. F.), who pretends to understand
inter-universal Teichm¨uller theory, suggests in a literature that the author
began to study inter-universal Teichm¨uller 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. """

I wonder if "I.F." is referring to the same "I.F." quoted in the original
article.

[1]
[http://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/abc_ver6.pd...](http://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/abc_ver6.pdf)

[2]
[https://www.maths.nottingham.ac.uk/personal/ibf/activity.htm...](https://www.maths.nottingham.ac.uk/personal/ibf/activity.html)

~~~
kazinator
I've personally modified thousands of e-mails with the ">" symbol; it's called
replying. That, _per se_ , has nothing to do with fabricating an e-mail; why
mention it?

Of course > symbols are fairly likely to be used in a forged e-mail, possibly
next to fake quoted text; but that then is not simply an alteration of genuine
text by the addition of those symbols.

~~~
mintplant
>I've personally modified thousands of e-mails with the ">" symbol; it's
called replying. That, per se, has nothing to do with fabricating an e-mail;
why mention it?

I read it as "I.F." making it look like something the author wrote was the
author quoting "I.F.", or vice-versa.

For example,

    
    
        From: John Doe <johndoe@gmail.com>
        The meaning of life is 42.
    

becomes

    
    
        From: John Doe <johndoe@gmail.com>
        > The meaning of life is 42.
        Thanks for the help!

------
svisser
The opening sentence of a linked article sums it up quite well: "If nobody
understands a mathematical proof, does it count?"

The abc conjecture may be solved today but only when a sufficient number of
people understand and accept the proof as a proof.

~~~
MikkoFinell
>"If nobody understands a mathematical proof, does it count?"

Just wait until general AI really kicks off, then all of new math will be like
that. It's not that humans are bad at logical thinking, our weakness is
memory. That won't be the case for an artificial agent with instantaneous
perfect recall of everything it has ever seen.

~~~
KGIII
Mathematics is a language. You don't really need to memorize it, you can read
it. Unfortunately, most people aren't exposed to anything higher than
arithmetic.

~~~
posterboy
yeah, well, have fun reading e.g. the stacks project[1] of round about 4000
pages without remember the necessary steps leading up to a corollary.

[1]
[https://stacks.math.columbia.edu/browse](https://stacks.math.columbia.edu/browse)
\- abstract algebra as far as I can tell

~~~
KGIII
You're sure as hell not going to memorize it.

------
ZenoArrow
I'm not a mathematician, so I don't know the significance of the ABC
conjecture, but for those that are laymen like me, here's a quick introduction
to the ABC conjecture:

[https://m.youtube.com/watch?v=RkBl7WKzzRw](https://m.youtube.com/watch?v=RkBl7WKzzRw)

~~~
Joe-Z
Not sure why you're being downvoted. The video actually explains the ABC
conjecture, which the article really doesn't.

Also, the second-highest comment in this thread right now is one explaining
this conjecture.

~~~
ZenoArrow
I don't understand it either, but I've learnt to mostly ignore downvotes.

------
crazygringo
I know there has been a lot of work in automated theorem proving.

I'm curious if anyone here is able to describe the state of the art in that.
Is it really just a toy for basic things right now, or how realistic is it
that we'd ever get to a point where the author of a paper like this could use
a formal language to show that their proof is indeed valid and complete?

~~~
sanxiyn
It is realistic in the sense that it has already been done! But it's still
laborious in the extreme.

Kepler conjecture was stated in 1611 and had been unsolved since. Thomas Hales
started a project to attack it in 1992. After six years of work, he announced
the proof in 1998, in the form of 250 pages of argument and 3 gigabytes of
computer calculation. He submitted it to Annals of Mathematics, one of the
most prestigious math journal, for review. Reviewers and the author tried
valiantly for five years, gave up, and published it in 2003 with a warning
that while reviewers were 99% certain, it couldn't be completely reviewed.

Soon after getting this both rejection and acceptance, Thomas Hales announced
the plan to formalize his proof to remove any uncertainty. It was
enthusiastically received by automated theorem proving community. For a while
Thomas Hales "shopped" for the prover tool to use and basically leaders of
every significant provers tried to "sell" it to him. He decided on HOL Light,
wrote the detailed plan for formalization, and estimated it would take 20
years. He actually carried out this plan, announced the completion in 2014,
wrote the paper on formalization, and submitted the formalization paper, with
21 collaborators, in 2015. The formalization paper was published in 2017.

So there's that. The formal proof of Kepler conjecture is at the moment the
most significant corpus of formalized mathematics in existence.

~~~
chii
> After six years of work, he announced the proof in 1998

then

> plan to formalize his proof ... estimated it would take 20 years

so is it just me, or is the "work" of the normal proof only took 1/5th of the
time it took for the automated/formalized proof?! that seems counter-
productive imho...

~~~
aurelianito
> so is it just me, or is the "work" of the normal proof only took 1/5th of
> the time it took for the automated/formalized proof?! that seems counter-
> productive imho...

It took 5 times to make it not a hunch but a real demonstration. If that's
counterproductive or not remains as a something opinable.

~~~
kazinator
It _looks_ counterproductive when it's not demonstrating informal work to be
_false_.

> _make it not a hunch_

That's not what automation is doing; it doesn't turn conjectures into proofs.
It finds mistakes in proofs; those proofs are not "hunches" but rigorous
efforts.

Something which finds faults demonstrates is value mainly whenever it finds a
fault. (Or at least that is a very easy perception to slide into.)

~~~
sanxiyn
And a fault was found! The original proof, as published, is incorrect. The
error and the fix was published in "A revision of the proof of the Kepler
conjecture" (2009).
[https://arxiv.org/abs/0902.0350](https://arxiv.org/abs/0902.0350)

Note that we now actually know there is no more fault, because formalization
is complete.

------
dmh2000
question: if a reviewer is given a proof to verify, then is it possible to
simply step through each transformation and verify it, without really
understanding the entirety? Or in cases such as this are there steps that are
new or obtuse enough that there isn't a way to verify them without
understanding the whole thing. Or, are there individual steps that require so
much work that it is impractical to verify them?

~~~
rocqua
In practice, a proof contains many steps that are highly non-obvious unless
you have a deep understanding of the subject. This is done not just for the
sake of brevity, but also clarity. In a proof, there are a few key ideas you
want to show. Around that, you need glue to convince the reader that your key
ideas work, and save them from some the tedium and hard work of verifying that
stuff.

A 1/4 page proof, when made as simple as you suggested, could easily bloom to
2 pages. At this point, your reader can no longer find those few key ideas,
and is instead lost in the rigor.

------
nautilus12
This sheds an interesting light on the referee process in that referees are
given an inordinate amount of power in the scientific process.

Suppose I am a referee, then I am a mathematician and I generally want new and
interesting work released having to do with my subject area because I want
more students interested in those topics validating my general area of
research. If I dont plan to personally build on this work (because maybe its
too confusing or doesn't shed light on what I am working on), then in a case
like this where the paper is so confounding, whats to keep me from simply
saying "I understand this and its correct" even if that is not the case? Its
not like if you referee and approve a paper that is later proven wrong
(especially in a case like this) you are run out of the mathematical
community. Do they need to provide detailed explanations as to why they think
its right?

What would stop a referee from simply going along with it and approving this
paper, especially when the community is confounded by it and doesn't want to
do the work itself.

Does the rigor of scientific process ultimately just come down to unrigorous
consensus?

~~~
impendia
Professional mathematician here.

That can occasionally happen with much less important papers; e.g. someone is
asked to referee something, and doesn't really feel like actually reading it
but would still like to see it published. If they are not conscientious they
might only briefly skim the proof and write a hasty referee report.

But ABC is a _major_ claim. The community will not be nearly so quick to reach
a consensus. In particular, from what I can tell now there are a handful of
mathematicians other than Mochizuki vouching for the proof, but even that is
still not enough: overall, the community has yet to come to a consensus that
this is enough.

------
TimTheTinker
The proof apparently rests on an entirely new (not yet vetted) branch of
number theory developed by the proof's author, "Inter-universal Teichmüller
Theory". 45-page overview linked:

[http://www.kurims.kyoto-u.ac.jp/%7Emotizuki/Panoramic%20Over...](http://www.kurims.kyoto-u.ac.jp/%7Emotizuki/Panoramic%20Overview%20of%20Inter-
universal%20Teichmuller%20Theory.pdf)

------
rusanu
> “The language strikes me as substantially more accessible than that of the
> original papers.”

So is not really a summary, is more of a reformulation, an easier to read
rewrite.

------
shad0wca7
This reads like an academic paper created by 'machine learning' to be
unintelligible yet look legitimate at-a-glance...

------
Phosphenes
They should probably break it up into more manageable pieces. Maybe by
starting with Frobenioids?

