
Biggest mystery in mathematics in limbo after cryptic meeting - signa11
http://www.nature.com/news/biggest-mystery-in-mathematics-in-limbo-after-cryptic-meeting-1.19035
======
JonnieCache
Here's a detailed report on the situation from one Brian Conrad, who attended
the meeting: [http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-
works...](http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-
brian-conrad/)

------
aamar
Not mentioned: Kedlaya was Mochizuki's student and then later his teaching
assistant back in '92-'94 (for a tough course[1]). Mochizuki hasn't had that
many students; Kedlaya might be one of a few to have an advantage in
understanding his approach.

The bad news is that the new participants seem to have made no progress on the
IUT papers themselves. But the good news is that some substantial progress was
made on the background material, and that there are some folks still
interested in continuing.

[1]
[https://en.wikipedia.org/wiki/Math_55](https://en.wikipedia.org/wiki/Math_55)

------
logicallee
I don't understand how something, that anyone understands, can be this
cryptic. Someone some time ago mentioned that all of Einstein's major papers
on special and general relativitiy can be covered in a semester or two.

Isn't the hard part discovering this stuff?

How can anyone discover mathematical proofs but it is so impossible to
communicate them that nobody in the world can follow your train of thought? I
mean people are really, really good at communicating. It's kind of what we do.

I don't understand how anyone can publish proofs that don't succumb easily to
understanding by experts who concentrate on them for a long time, perhaps with
tiny hints by the author.

It's not as though we're looking at fragments written in a long-lost language
we have no access to, from a culture nobody has access to, on unknown
subjects, and with the authors dead hundreds of years ago! The author is right
there. The subject isn't even art (like poetry criticism or something), it's
fully rigorous.

Understanding someone's proof (for devoted experts) should be easy, right?
What gives?

~~~
mmarx
The papers on Inter-universal Teichmueller theory span over 500 pages alone,
and they build on material developed earlier that needs to be understood too.
Those papers were published a little over 3 years ago, so claiming that they
can't be understood “by experts who concentrate on them for a long time” might
be a bit premature.

~~~
logicallee
but the article doesn't make it sound like it's not "fully understood." the
article makes it sound like not even the summary is known - for example, what
parts of the other papers are used, what isn't. As though the experts didn't
even known what parts of theories were being referred to. very bizarre.

~~~
jsprogrammer
The summary is known. Have you read the abstracts?

~~~
logicallee
Did you even read our article, jsprogrammer? That's what I'm reacting to. It
says:

>A consensus emerged that the highlight of the workshop was a lecture on 9
December by Kiran Kedlaya, an arithmetic geometer from the University of
California, San Diego. He zeroed in on a result from a 2008 paper by Mochizuki
that linked the statement of the abc conjecture to another branch of maths
called topology. The link was immediately recognised as a crucial step in
Mochizuki’s grand strategy.

This is why I'm making the reference to ancient lost texts. I mean, if the
result of a 2008 paper is crucial, you'd think the guy would have just
mentioned this fact. It's as though they're investigating some long dead guy's
hidden thoughts. This is a direct quote: "There is still no clear answer to
lingering questions about how things are ultimately going to fit together."
What? Wouldn't the guy just _tell_ you how they're ultimately going to fit
together? I'm utterly perplexed at the process the article describes, when the
author is available and can give hints.

~~~
shmageggy
What I understood from [http://mathbabe.org/2015/12/15/notes-on-the-oxford-
iut-works...](http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-
by-brian-conrad/) (posted elsewhere here), is that the problem stems from
differing motivations. On one hand, Mochizuki wants to make his proof fullyt
complete and correct (being both fully general and covering all special
cases), but the community wants broad strokes intuitions. From Mochizuki's
perspective, he _has_ told everyone how things fit together, but _fit
together_ means something different for others. The workshop attendees want to
know what things they can gloss over to get the big picture, but as of yet,
nobody who understands the proof has succeeded in communicating that
information, probably because they don't share the same motivation.

------
lelf
If you don't follow how it came mathematicians have so much trouble with some
papers:

This is part I — [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)

~~~
IshKebab
Jesus. This surely requires a computational proof.

~~~
ilitirit
It is essentially a communications problem. Brian Conrad summed it up:

 _For every subject I have ever understood in mathematics, there are
instructive basic examples and concise arguments to illustrate what is the
point to generally educated mathematicians. There is no reason that IUT should
be any different, especially for the audience that was present at Oxford. Let
me illustrate this with a short story. During one of the tea breaks I was
chatting with a postdoc who works in analysis, and I mentioned sheaf theory as
an example of a notion which may initially look like pointless abstract
nonsense but actually allows for very efficient consideration of useful ideas
which are rather cumbersome (or impossible) to contemplate in more concrete
terms. Since that postdoc knew nothing about what can be done with sheaf
theory, I told him about the use of sheaf cohomology to systematize and
analyze the deRham theorem and topological obstructions to construction
problems in complex analysis; within 20 minutes he understood the point and
wanted to learn more. Nobody expects to grasp the main points of IUT within 20
minutes, but if someone says they understand a theory and does not provide
instructive visibly relevant examples and concise arguments that clearly
illustrate what is the point then they are not trying hard enough. Many are
willing to work hard to understand what must be very deep and powerful ideas,
but they need a clearer sense of the landscape before beginning their
journey._

[http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-
works...](http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-
brian-conrad/)

------
mipapage
I remember at university how, when attending the few multi-disciplinary
meetings or get-togethers, I would marvel that these people - on the same
campus! - didn't get together more often in order to share and stimulate
growth or share strategies, which inevitably occurred during these meetings.

I wonder why a handful of qualified mathematicians couldn't get together and
go to Japan to sort this proof out...

~~~
zbyszek
Having worked in a university mathematics department I noticed that the pure
mathematicians tended to keep themselves to themselves, for instance not
attending talks by visiting mathematicians unless it was of direct relevance
to their research. In contrast, we theoretical physicists would listen to any
visiting speaker just for fun - even on occasion experimental particle
physicists, applied mathematicians or computing specialists. Perhaps pure
maths is just more deeply specialised and leaves no possibility of such
dilettanteism.

~~~
Al-Khwarizmi
I think it's probably because it's very difficult to understand a pure math
talk if you didn't already have significant understanding of the subject
before the talk.

I don't work on pure mathematics, but I work in a field of computer science
where there is both theoretical and empirical/applied research, and I do, and
enjoy doing, both.

However, when it comes to attending talks, I do go to a lot of talks on
empirical subjects just for fun, but I hardly ever go to a highly theoretical
talk for fun. I only do that if the talk is highly related to the specific
work I do; if you see me in a theoretical talk otherwise, it's probably just
for social compromise.

The reason is that I can perfectly understand the gist of an empirical paper
in 20 or 30 minutes, where there is no way I will understand a piece of
theoretical research by listening to a talk if I haven't gone through the
paper carefully before (and if you have done that, there is often no point in
going to the talk anyway). Honestly I think the talk format doesn't lend
itself too well to highly theoretical work. I have learned _a lot_ about
empirical research and obtained many useful ideas from talks, but in the
theoretical field, the useful ideas and insights I got from talks are few and
far between, and I would probably have obtained them more efficiently from
reading papers anyway...

------
ryporter
I view some of these math researchers like startup founders who are absolutely
brilliant technically, but who lack almost all ability to interact with
customers, investors, etc. As a solo founder, they will fail. However, if they
attract a co-founder who complements their skills, they can be wildly
successful. It sounds like Mochizuki would have benefited greatly from such a
co-author, in order to truly prove his result to the community.

~~~
enewc
Math is objective. The proof is either correct or not. Human interaction
doesn't matter, except for perhaps marketing the importance of the results. In
this case, the history of the problem itself has done the marketing, so any
valid solution would be a wild success regardless of how socially eccentric
the researcher is. Another recent example of such a mathematician is Grigori
Perelman.

~~~
moron4hire
Mathematics, being a human endeavor, is fundamentally _about_ human
interaction. There's no point to doing math if there is no communicating it to
other people.

~~~
webkike
You can argue that everything can be reduced to human interaction, but
mathematics is certainly not about human interaction in any meaningful sense

~~~
moron4hire
It's the only pure way to describe existence. That isn't meaningful?
Mathematics is a language, first and foremost. To argue otherwise is to
fundamentally misunderstand its goals, methods, and form.

You can draw all the squiggly lines on paper that you want, but it won't mean
anything until someone else interpets it. You cannot prove logic that only
exists in your own head, because you cannot prove yourself rational. Your very
existence does not even make sense without other people perceiving you.

~~~
webkike
Languages do not require human interaction. I do not argue that there is no
mathematical language.

~~~
moron4hire
Language without human interaction is an undefined operation, like raising
zero to the zeroth power, or clapping with one hand.

~~~
drdeca
Are you including interacting with oneself?

------
serge2k
> True to form, Mochizuki himself did not attend, although he did answer
> participants’ questions through Skype

Why not present his papers through skype? If the problem is just a dislike of
travelling. It could also help with any issues related to public speaking or
that type of thing.

------
ilitirit
Here's a comic that makes light of the situation:

[http://www.sandraandwoo.com/comics/2014-11-24-0636-hodge-
the...](http://www.sandraandwoo.com/comics/2014-11-24-0636-hodge-theater.png)

From:

[http://www.sandraandwoo.com/2014/11/24/0636-hodge-
theater/](http://www.sandraandwoo.com/2014/11/24/0636-hodge-theater/)

------
guard-of-terra
I guess we can now ask a lot of simple questions in math for whose there can
be answer, but the solution itself is untenably long to be produced or
understood by mere mortal, not a computer or computer-like individual.

Same way as we can't really reason about inner workings of machine learning
model that yields useful results.

Talk about transhumanism.

~~~
bhaak
Just because his proof is long, convoluted, and cryptic doesn't mean that
there won't be a simpler proof down the road. Maybe some of the tools just
haven't been developed yet. Look for example at how the Greeks and Romans
struggled with mathematical problems many of which are today easily solvable
in high school because we have advanced techniques and a positional notation
for numbers.

As far as I understand it, a lot of the problem stems from the fact that
Mochizuki is quite a hermit. And that's a problem if you want others to
understand what you have done for years.

This workshop was crucial that it showed several people that trying to
understand the proof could be a worthwhile task.

~~~
guard-of-terra
What if there won't be? What if, for many of questions, the only proof is as
decipherable as MD5 of question?

~~~
bhaak
That's probably highly improbable. Look at how many different proofs the are
for the pythagorean theorem: [http://www.cut-the-
knot.org/pythagoras/](http://www.cut-the-knot.org/pythagoras/)

I'm not mathematician but I guess if there is really only one possible proof
for a given question (Gödel might have proven that this is impossible but I
don't know for sure), then the question isn't actually an interesting one.
Because this would mean that the question touches only very few topics.

But the abc conjecture is interesting because it has lots of links with deep
questions in number theory.

~~~
guard-of-terra
Okay. What if there's a lot of proofs, but the shortest of those is still not
within reach of human intellect?

~~~
bhaak
So far, we don't know if there are any proofs that are beyond the human
intellect. I don't think that we currently believe such a thing to exist
theoretically.

Every provable conjecture should be understandable in principle by a generic
human mind.

Now, there for certain systems that are so complex that proofs within those
systems are beyond us. Not because we couldn't understand them but because
learning and applying all the rules of that system takes longer than a human
mind is able to live (currently).

But that's just a practical limitation, not a theoretical one.

~~~
guard-of-terra
I'm talking about practical limitations. It turns out human exploration
becomes impractical :(

~~~
bhaak
Practical limitations can be overcome.

At first it was artificial light sources, then glasses, then machines speeding
up calculation and now machines that do calculations themselves.

There is certainly a practical limit of what a human mind can grasp but that
limit is far away and with tools, it is even much farther.

We won't run out on difficult problems to solve anytime soon.

------
timwaagh
the reason they fail to understand the proof seems the same everybody else
fails at math. funny even the greatest mathematicians can have this problem.

------
espeed
"A consensus emerged that the highlight of the workshop was a lecture on 9
December by Kiran Kedlaya, an arithmetic geometer from the University of
California, San Diego. He zeroed in on a result from a 2008 paper by
Mochizuki5 that linked the statement of the abc conjecture to another branch
of maths called _topology_. The link was immediately recognised as a crucial
step in Mochizuki’s grand strategy. "

------
PaulWillis
I'm not a mathematician at all but does anyone know if this may have
implications in the field of cryptography, because it involves prime numbers?
Also, the word "cryptic" in the title. ;)

------
williamjennings
I just learned about Math 55 and would trade plenty to teach that course.

The problem here is that Mochizuki uses too many German Nazi references.

Those people had no academic integrity whatsoever; and their works are
exclusively stolen, through _war crimes and the Holocaust._

After you do the right thing, and translate Mochizuki's nomenclature into
French and Slavic; the proof is no longer valid, and the problem is
negligible.

For the record, Oswald Teichmüller is a fraudulent Nazi war criminal who
robbed Felix Hausdorff during the Holocaust.

The undeniable fact that Oswald Teichmüller is solely published in a _journal
of racial propaganda_ makes Mochizuki's choice of terminology questionable at
the very least.

The fact that I am being suppressed for this demonstrates unambiguously that
Mochizuki's proof is nothing more than media hype and puffery.

~~~
late2part
I dont understand what you mean. Can you clarify?

------
myth_buster
On a related note, Shinichi Mochizuki has been speculated to be Satoshi
Nakamoto.

[http://www.forbes.com/sites/timworstall/2013/05/19/ted-
nelso...](http://www.forbes.com/sites/timworstall/2013/05/19/ted-nelson-says-
that-bitcons-satoshi-nakamoto-is-shinichi-mochizuki/)

