
An ABC proof too tough even for mathematicians - ot
http://www.bostonglobe.com/ideas/2012/11/03/abc-proof-too-tough-even-for-mathematicians/o9bja4kwPuXhDeDb2Ana2K/story.html
======
dsrguru
The more mathematically-inclined HNers might be interested in Brian Conrad and
Terrence Tao's comments at the bottom of this previous HN article:

[http://quomodocumque.wordpress.com/2012/09/03/mochizuki-
on-a...](http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc)

Edit: Minhyong Kim's initial thoughts seem very interesting as well!

[http://mathoverflow.net/questions/106560/what-is-the-
underly...](http://mathoverflow.net/questions/106560/what-is-the-underlying-
vision-that-mochizuki-pursued-when-trying-to-prove-the-abc/106658#106658)

And for the less mathematically-inclined:

<http://news.ycombinator.com/item?id=4477241>

------
codeulike
If a programmer locked himself away for 14 years and then emerged and
announced he'd written a completely bug free OS, there would be skepticism.
Code needs to be battle tested by other people to find the bugs.

Mathematics is the same, to an extent; one guy working alone for 14 years is
likely to have missed ideas and perspectives that could illuminate flaws in
his reasoning. Maths bugs. If he's produced hundreds of pages of complex
reasoning, on his own, however smart he is I'd say there's a high chance he's
missed something.

Humans need to collaborate in areas of high complexity. With a single brain,
there's too high a chance of bias hiding the problems.

~~~
ekianjo
I do not think it is fair to compare Software and Abstract Mathematics.
Software is mostly an Engineering problem, while Mathematics is mostly Science
and Conjecture.

And even if they locked themselves for 14 years, there is no "single mind at
work" there. Mathematics are built "on the shoulder of giants" through
generations of brain dedicated to it. There is no such thing as pure invention
- new discoveries lead to new theorems, new leads that other people take on.

~~~
georgeorwell
Proofs are programs.

<https://www.google.com/search?q=proofs+are+programs>

~~~
ekianjo
Let me say it in another way.

Abstract Maths is like having a Organic Chemist working on understanding the
transformation of food at the molecular level. A Software Programmer is like a
cook who knows how to mix and prepare food in order to make a delicious dish
out of it.

~~~
fusiongyro
"Programming is one of the most difficult branches of applied mathematics; the
poorer mathematicians had better remain pure mathematicians."– E. W. Dijkstra

~~~
tel
Dijkstra did a kind of programming distant from most of it today. If to him
programming is a difficult branch of applied mathematics, then HN is a forum
talking about how best to calculate your tip after dinner.

~~~
fusiongyro
You guys really are masters of specious analogies. Comparing the best of one
field to the worst of another isn't a legitimate comparison and you know it.

------
dbaupp
Another article with slightly more background on the ABC problem itself (and
possibly slightly less sensationalist). [http://www.nature.com/news/proof-
claimed-for-deep-connection...](http://www.nature.com/news/proof-claimed-for-
deep-connection-between-primes-1.11378)

And the MathOverflow discussion referenced:
[http://mathoverflow.net/questions/106560/what-is-the-
underly...](http://mathoverflow.net/questions/106560/what-is-the-underlying-
vision-that-mochizuki-pursued-when-trying-to-prove-the-abc)

~~~
podperson
I think implying the article is sensationalist is a little harsh. It's
definitely a click-grabbing headline, but overall a decent article on a tough
subject.

~~~
revelation
Seemed extremely hand-wavey to me, and the obscure pictures of 3D wireframe
'A', 'B' and 'C' by the side certainly didn't help.

It tried to explain both the dynamics in math research and the problem at
hand, and in the end explained neither.

~~~
bstpierre
It's the Boston Globe, hand-wavey is all they can really be expected to
publish regarding hard abstract math. I thought they did a decent enough job
for a newspaper.

------
sek
Just read his Wikipedia entry:

> Mochizuki attended Phillips Exeter Academy and graduated in 2 years. He
> entered Princeton University as an undergraduate at age 16 and received a
> Ph.D. under the supervision of Gerd Faltings at age 23.

He is 43 Years old now, I assume he is 100% committed to Mathematics. These
people fascinate me, having a feedback loop that is unbreakable. Especially
for topics where you have a knowledge of something and almost nobody else is
the world is capable of understanding you. It's like Star Trek for the mind.

------
Xcelerate
This article seems to suggets that mathematicians are all too eager to drop
his work at the slightest whiff of any flaw. Could someone more knowledgable
on the subject explain to me why this is?

It is clear that he has already done some very great things in mathematics, so
even if there was a flaw in his proof, I would think his papers would still
have many deep insights that no else had thought of. I mean, it's not like
mathematicians are pressed for time -- if I was one I would certainly dedicate
a lot of time to studying something interesting like this.

~~~
btilly
You clearly do not value the time of mathematicians like they value their own
time.

The estimate that I was told for the average mathematician reading the average
math proof is one page per day.

Thus the average mathematician facing this will see 750 pages just to becomes
of the the 50 people who have mastered the basics of anabelian geometry.
That's 2 years. Then you have to take on some unknown number of years to learn
"inter-universal geometry". Then your reward for doing this is that you are
qualified to read a 512 page proof, which is again going to be a year and a
half. Along the way if you find a mistake in _any_ of it, your reward is to
confirm the immediate guess that most mathematicians have which is that there
is likely a mistake somewhere. (But with this much math, you'll probably find
several "mistakes" that aren't before you find a real one.)

This is years of work, that has nothing to do with anything that you're
already working on. And believe me, a professional mathematician has no
shortage of problems to work on, in areas that they already have the
background for.

If you think that this is unreasonable, well, why don't you volunteer to fix
it? Reading the proof shouldn't take you much longer than it would take to
become a mathematician. And you can learn anabelian geometry during grad
school, so that time is not all wasted.

~~~
Xcelerate
>> The estimate that I was told for the average mathematician reading the
average math proof is one page per day. Thus the average mathematician facing
this will see 750 pages just to becomes of the the 50 people who have mastered
the basics of anabelian geometry. That's 2 years.

Ah, well I did not know this. I appreciate the enlightenment. I see the
trepidation of reading the whole thing then.

------
elliptic
Is this situation similar to that of Louis de Branges & the Riemann Hypothesis
a few years back? I.e, a well-respected mathematician (de Branges had settled
the Bieberbach conjecture in the 80s) releases a proof of an important
unsolved problem using his own poorly understood mathematical technology?

Edit - lest this sound too negative, one should realize that the Bieberbach
proof took a long time to be accepted.

------
sek
A Youtube video with a pretty accessible explanation.
<http://www.youtube.com/watch?v=RkBl7WKzzRw>

------
bnegreve
Would it be possible to use proof assistants like Coq [1] to verify this kind
of proofs ? If not, does anyone know why ?

[1] <http://en.wikipedia.org/wiki/Coq>

~~~
cperciva
Proofs at this level are nowhere near formal enough for existing mechanical
tools to understand. I've seen researchers cite "translated proof X into an
automatically-verifiable form" as a major achievement -- even for very simple
proofs.

~~~
makmanalp
Non-math person here. Wait, so you're saying it's easier for humans to prove
complex theorems than computers? That's nuts.

Doesn't that mean that verifying proofs is then based on a super handwavy
process where a bunch of researchers come to a consensus and say "yes, this is
correct"? Ugh!

What if they all overlook some crucial aspect because of the way they were
trained, or their entrenched / ingrained assumptions?

When you describe a proof with words like you say, aren't there quibbles over
semantic things? But I thought the whole point of a proof was that you leave
nothing to semantics. Everything logically follows from each other.

I thought the main issue with proofs systems was that encoding the concepts
into code was tedious. Could we maybe have a common database of well-known
proofs, axioms, concepts etc, so that you wouldn't have to rewrite common
ideas from scratch? Like a package manager with proof-libraries?

Or is this an issue with the expression power of proofs languages?

~~~
impendia
Math person here.

>Wait, so you're saying it's easier for humans to prove complex theorems than
computers?

Of course! If proving theorems could be readily automated, this would have
been done already. Indeed, automation has succeeded for some kinds of problems
(e.g. google Wilf and Zeilberger's "WZ method" in combinatorics for a
particularly successful example)

>Doesn't that mean that verifying proofs is then based on a super handwavy
process where a bunch of researchers come to a consensus and say "yes, this is
correct"?

Basically, yes.

>What if they all overlook some crucial aspect because of the way they were
trained, or their entrenched / ingrained assumptions?

This happens surprisingly rarely. It helps that we are trained to constantly
question our entrenched assumptions.

>Could we maybe have a common database of well-known proofs, axioms, concepts
etc, so that you wouldn't have to rewrite common ideas from scratch?

Various books fill this purpose. But the problem is that theorems are
difficult to encapsulate. For example, a theorem will say "Assume X, Y, and Z.
Then A and B are true." and you need to use it in some situation where Z isn't
true, but some appropriate variant of Z is, and any expert would know how to
conclude that A and B are true. This is when the handwaviness comes in.

Mistakes do happen. But surprisingly rarely.

~~~
mbell
> Of course! If proving theorems could be readily automated, this would have
> been done already.

It doesn't need to create the proof, just validate it. See P vs NP.

~~~
Someone
To have an automated proof checker validate a proof, someone would have to lay
out every single step in the proof in enough detail for the verifier to do its
work. With current verifiers, that means 'in painstaking detail'. Only highly
educated and motivated people can do that, it takes ages, and it is not very
rewarding (neither financially nor otherwise). Because of that, volunteers are
hard found.

In some sense we won't really know whether a proof is rock-solid until we have
run it through a verifier, using a proven-correct verifier on proven-correct
hardware.

If theorem provers grow to be more easy to use, we may require proofs to be
verified, but we aren't there yet by a wide stretch.

For an idea about how hard this can be, check out
<http://www.cs.ru.nl/~freek/comparison/comparison.pdf: >

_"In 1998, Tom Hales proved the Kepler Conjecture […] with a proof that is in
the same category as the Four Color Theorem proof in that it relies on a large
amount of computer computation. For this reason the referees of the Annals of
Mathematics, where he submitted this proof, did not feel that they could check
his work. And then he decided to formalize his proof to force them to admit
that it was correct. He calculated that this formalization effort would take
around twenty man-years"_

(that effort is ongoing at
<http://code.google.com/p/flyspeck/wiki/FlyspeckFactSheet>)

~~~
xyzzy123
So would it be fair to say then, that these proof checkers are missing the
'libraries' that would be needed to efficiently write verifiable proofs for
modern mathematics?

~~~
cperciva
Exactly. And this is where most of the work on proof checkers is happening --
finding ways that people can input proofs in a less formal way and have
automated code fill in the formal details as a first pass before running the
verification.

------
ph0rque
_...the proof itself is written in an entirely different branch of mathematics
called “inter-universal geometry” that Mochizuki—who refers to himself as an
“inter-universal Geometer”—invented and of which, at least so far, he is the
sole practitioner._

In this universe, at least...

------
genuine
Here is the link to the proof:

[http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20...](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20Teichmuller%20Theory%20IV.pdf)

And the preceding three for context:

[http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20...](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20Teichmuller%20Theory%20I.pdf)

[http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20...](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20Teichmuller%20Theory%20II.pdf)

[http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20...](http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-
universal%20Teichmuller%20Theory%20III.pdf)

------
ArtB
Wouldn't the easiest way to check this proof be to enter it into something
like Coq? That way you'd only have to understand how to translate each step
rather than learn each field.

~~~
Evbn
You would still have to write 100 pages of Coq code, and write every
definition properly, which may be just as hard as learning the field.

------
atas
"Release early release often" applies to Math as well. Wouldn't it be better
for everyone if he hadn't been so secluded and published some of his work in
the meantime?

~~~
taejo
My impression was that he had published some material on IUT before, but
nobody had any reason to read it.

------
pfanner
I'm a physics student. Sometimes I'm thinking if I should completely change my
path to math. I always sucked at it but it seems to be so huge, exciting and
powerful.

------
dbz
Can anyone explain what "inter-universal geometry" is?

~~~
hakaaak
Different universes in mathematics play by different rules and have different
components. Inter-universal Geometry is how these sets of rules and components
can relate and is the bridge to understanding more complex theory.

------
mememememememe
Will the solution(s) to ABC proof be a nightmare to all security protocols
relying on prime number factorization, such as RSA?

~~~
smegel
Apart from cryptography, has prime number research and theorization produced
any other practical applications?

~~~
sold
Finite fields, essential to error-correcting codes, are based on properties of
primes. This forms the basics of efficient data transfer, whether Internet,
CDs or communication with spacecrafts. The book "Number Theory in Science and
Communication" mentions that finite fields were used in verification of the
fourth prediction of general relativity. See that book for more examples.

Here's one from <http://mathoverflow.net/questions/2556>: There is a gamma ray
telescope design using mod p quadratic residues to construct a mask. Gamma
rays cannot be focused, so this design uses a redundant array of detectors
separated from the mask to reconstruct directional information.

Cicidas have a cyclic life spanning a prime number of years. It is supposed
that this is because a predator has harder job aligning with the cycle.
<http://en.wikipedia.org/wiki/Predator_satiation>

I believe that the Möbius function is important in advanced physics:
<http://en.wikipedia.org/wiki/Mobius_function#Physics>

There's a legend that generals used the Chinese remainer theorem to count
soldiers.

Elementary number theory was a motivating factor and foundation for
development of the whole mathematics - algebra, analysis, even logic. Did you
know that proving Godel's incompleteness theorem requires using primes?
[http://mathoverflow.net/questions/19857/has-decidability-
got...](http://mathoverflow.net/questions/19857/has-decidability-got-
something-to-do-with-primes) Transcendence of pi also needs primes.
[http://mathoverflow.net/questions/21367/proof-that-pi-is-
tra...](http://mathoverflow.net/questions/21367/proof-that-pi-is-
transcendental-that-doesnt-use-the-infinitude-of-primes). I did not realize
that earlier. Usage of primes is often invisible. Even if there was no
cryptography or ECCs, it would be silly to demand practical applications from
this subject - it is a foundation for almost everything in mathematics,
sometimes rather concealed.

~~~
smegel
By "practical application" I was also getting at other mathematics/physics
laws/theories that are based on prime theory in some way. I guess you have
highlighted a few.

