
Shinichi Mochizuki and the impenetrable proof - robinhouston
http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509
======
powera
OK, I have no idea how the proof works, but I think I read the abstracts well
enough to do something that might qualify as pretending to pretend to know how
the proof works: (please note: I'm not qualified to pretend to know how this
works, I have to pretend twice to get anything that sounds like both math and
English)

* Part 1: All chaotic systems are isomorphic to an elliptic curve [traditionally y2 = x3 + ax + b] for some extended definition of elliptic curves

* Part 2: A general method of constructing isomorphisms of chaotic systems to extended elliptic curves

* Part 3: Using the method from Part 2, construct a more understandable model of the chaotic structure of the natural numbers

* Part 4: Using the model constructed in part 3, construct a proof for abc

Hopefully if you understand any of this you can point out why I'm obviously
wrong.

~~~
unexistance
A for effort, since apparently mathematicians themselves are quite reluctant
to do this...

------
ArtB
> And most mathematicians have been reluctant to invest the time necessary to
> understand the work because they see no clear reward: it is not obvious how
> the theoretical machinery that Mochizuki has invented could be used to do
> calculations.

I find this _deliciously_ ironic since that is the position most students have
towards mathematics in general (replace "calculations" with "anything relevant
in their lives").

~~~
embiaa
I am a research mathematician. I don't think this quote is an accurate
reflection of how 99% of experts feel about Mochizuki's work. Most experts
expect that if Mochizuki's work is correct (and even if not) it contains a lot
of valuable ideas. Proofs of this kind are almost never mathematical dead ends
- they are difficult because they require fresh insights and these insights
can always be applied to other areas.

Mathematics is not solely about the proof. Good mathematics is about the
communication of the proof and the ideas in it. I think it is fair to say
Mochuzuki's work is not being communicated effectively. Though I am not saying
the problem lies with Mochizuki alone.

~~~
iso-8859-1
After Perelman's proof, there have been some "filling in of details". [1]
Would you say that Perelman's proof is comparable, and that he could also have
been more pedagogical? Do you see any parallel at all?

It sounds to me as if you are implying, that it is Mochizuki's responsebility
to be pedagogical. If being pedagogical is good (because it is more social?),
how is mathematics different from any other discipline? Surely one ought to be
social in every regard.

If the proof turn out to be correct, would you still say Mochizuki
communicated it wrong?

[1]:
[https://en.wikipedia.org/wiki/Grigori_Perelman#Verification](https://en.wikipedia.org/wiki/Grigori_Perelman#Verification)

~~~
embiaa
From what I understand (not exactly my area) Perelman's proof was quite
intelligible. Yes there were lots of details to fill in, but for the expert it
was clear how this should be done. Perelman's proof also had a long background
(it implemented ideas outlined by Hamilton earlier) so for experts it made a
lot of sense. I don't think there is a lot of similarity between the two
situations.

Research is a social activity. Being a successful researcher means being
social. What social means depends on the norms of the relevant field. Yes, we
should reflect on those norms and allow innovators to push boundaries but for
the science to evolve it has to take everyone with it.

Mochizuki and those around him have a responsibility only if they want take
part in the mathematical community - which I think and hope they do.

I am taking the word of experts in the area of arithmetic geometry who say
there isn't being enough done to communicate his ideas. Regardless of whether
he is right or wrong (really it isn't about this - it is about whether his new
ideas have merit - the proof of the abc conjecture would be strong evidence
for this) I think the current situation speaks for itself.

Edit: Also, you are entirely correct, we are humans, we should be social in
every regard!

------
Mithaldu
Links to the papers themselves:

[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)
[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)

Looks like understanding these doesn't only require reading 500 pages, but
also 4 other preceding papers of this author.

------
yarvin9
It somehow fits that this is halfway up the front page, and there are no
comments on it.

A good metric for the vitality of an academic field is the level of genuine
interest researchers take in the substance of their peers' work. A good proxy
for this metric is the use of preprint servers. Math and physics dominate,
obviously. Interestingly, these are also the fields which the Soviet
scientific system screwed up the least.

Even in math, as the case of Mochizuki (and de Branges) shows, there are
limits.

What's the actual tangible reward for investing a year or two of your career
in learning Mochizuki's world? Suppose you showed that inter-universal
geometry was really, truly, a new major subfield of mathematics? Would math
departments all over the world hire the faculty needed to make this subfield a
reality? Making you, the reviewer, who didn't do original work but only
checked someone else's work, a big shot? Not #1 in IUG, but maybe #2 or #3?

I'm not a math guy, but it's hard to see. And yet, 40 years ago, this might
well have been the outcome. Conclusion: maybe we really do live on Trantor.

~~~
jsprogrammer
There is a lot to read and it has only been posted here for around three
hours. What kind of substantial comments are you expecting in such a short
amount of time? The linked article itself is a decent read and the papers
(conveniently not directly linked to in the article) themselves are quite
long.

>What's the actual tangible reward for investing a year or two of your career
in learning Mochizuki's world?

I'd guess that would greatly depend on what Mochizuki's world allows one to
do.

~~~
yarvin9
> I'd guess that would greatly depend on what Mochizuki's world allows one to
> do.

Um, it's abstract math, it's not going to help you build a flying car. Isn't
proving a major conjecture enough?

Most of mathematics predates the modern American academic system. This immense
body of work wasn't developed by people with a careful eye on the best way to
get a good tenure-track job. It was developed by people who did math for one
main reason: they were fascinated by math.

I am not a mathematician, but I haven't seen anyone suggest that Mochizuki's
world is _boring_. It's difficult for me to imagine 19th-century
mathematicians resisting the opportunity to become students again in a new,
promising, and unfamiliar world.

The 21st-century reaction seems to be: I already got that degree, you want me
to start over? WTF? Why? What's in it for me? How do I make a name for myself
by studying someone else's theory, which might not even be true? And it's a
pretty sensible reaction, given the institutions we have.

~~~
hyperpape
> Isn't proving a major conjecture enough?

Not necessarily. And not because of a lack of tangible rewards, but because of
a lack of mathematical ones.

Mathematicians are not just after true statements, or proof, but
understanding, and new theories that open up new areas of study. For that
reason, a new proof of an old theorem is often quite interesting.

I seem to recall reading that Wiles' proof of Fermat's last theorem was
sensational, but ultimately less exciting than it could be--the proof did not
create a new way of viewing or systematizing things so much as apply very
specialized machinery to a specific problem. Take that with a grain of salt--I
don't have a hundredth of the mathematical background to judge it myself.

~~~
ninguem2
Your general statements are correct but your example of Wiles is quite wrong.
Wiles ideas and their developments in the general area of "modularity of
Galois representations" (which this comment box is too small to explain) led
to a huge explosion of results. Taylor's work on the Artin conjecture and the
Sato-Tate conjecture, Khare-Winterberger proof of the Serre conjecture,...

~~~
hyperpape
Thanks for the correction. My memory is hazy, and I must have read that about
some other result, or merged different comments into something no one said.

------
patcon
The situation described strikes me as a contemporary analogue of the sorts of
dilemmas we might expect in a future of increasingly complex AIs -- more and
more solutions to problems that even the best among us can't possibly
understand fully...

~~~
336f5
It also sounds like a motivating example for machine-checked proofs: if one
could feed Mochizuki's proof into Coq or something and be assured that it was
correct, even if only in a purely formal sense, I suspect there would be much
more interest in grappling with the concepts to understand what the proof is
doing, whether it's acceptable, and _why_ the proof is correct. As it stands,
there's the risk of a wild goose chase.

~~~
jordigh
If Mochizuki were able to write a machine-checkable proof, he would also be
able to write a human-checkable proof, which is far easier to write.

~~~
kinghajj
Well, supposedly he _did_ write a human-checkable proof--it's just that the
reviewer must be familiar with several of his novel ideas. Like any human-
intended proof, there's an assumption of foreknowledge.

------
NhanH
Tangentially to the topic at hand, but coming from the article:

> Mochizuki has estimated that it would take an expert in arithmetic geometry
> some 500 hours to understand his work, and a maths graduate student about
> ten years.

That's a 40x gap (500 hours assuming 40-hour work week is about 3 months-ish).
Assuming that math grad student is active albeit junior researchers, that's a
huge gap. I thought our notion of 10x programmer is already something
considered extremely wildly stupid and doesn't exist? (I know the last
sentence sounds a bit snarky, but it's too amusing to not point out).

~~~
yaks_hairbrush
40x sounds pretty conservative to me. I did graduate school in math (did not
get a Ph.D.) and my advisor indicated that he assigns problems assuming a 200x
multiplier between his capability and the students'. That is, he'd assume that
something he could see how to do and could execute inside of a week would be a
good thesis problem to keep a Ph.D. student busy for two years.

As far as the 10x programmer stuff... Obviously there are 10x and probably
even 100x programmers. They're the experts, and there are not many of them.

~~~
kaitai
Yep -- my advisor figured he could do my thesis problem in a week. It did take
me only 1.5 years to find a counterexample and kill my own thesis, but these
are estimates after all.

~~~
backlava
If he could have proven the theorem in a week despite its having a
counterexample, that is impressive.

------
shas3
I think a big problem is that he is not doing enough legwork to get out and
talk about his ideas. Researchers should not publish major papers without
doing a 'roadshow', especially when it is so dense with new ideas as
Mochizuki's abc-conjecture papers. Accounts of proofs and discoveries always
have a climax where the author delivers a lecture announcing the discovery.
Like Wiles announcing a proof (later shown to be flawed) of Fermat's last
theorem at a lecture in Cambridge's Isaac Newton Institute for Mathematical
Sciences in 1993. It is immensely helpful to talk through the proof in person
facing an audience of experts.

~~~
Mithaldu
I don't understand. He has already spent 10+ years of the time of his life
putting it together, has given it to the world freely, on his own and made no
demands whatsoever. Are you possibly arguing that he has any obligation at all
to do anything more? If so, on what basis?

~~~
jordigh
The sad reality is that without a good presentation, genius will take a long
time to be discovered, if indeed it is genius.

There are many examples, but doubtlessly the most dramatic ones are Ramanujan
and Galois. Ramanujan had no training and at first glance his work looked
indistinguishable from the flood of crank mathematics that most professional
mathematicians are familiar with. Were it not for Hardy's work in giving form
to Ramanujan's ideas, they may have been lost forever.

In Galois's case, he was _terrible_ at explaining his ideas, brilliant as they
were. The anecdote of him throwing the eraser at his examiners is an example
of his frustrations in trying to communicate to others. He himself was aware
of his bad presentation, as he even called his work "gâchis" (mess).

Even now, with the hindsight of knowing what he's talking about, it's
extremely hard to read his original papers. For example, he doesn't write down
formulas and he doesn't fix notation. He just _describes_ them in very
ambiguous terms, talking about "this" or "that" where it's difficult to always
determine what "this" or "that" refers to. This is why it took over 30 years
after his death for Liouville to notice that Galois was a genius.

~~~
LunaSea
"gâchis" doesn't mean mess but waste.

~~~
jordigh
Definition 3:

[https://fr.wiktionary.org/wiki/g%C3%A2chis#Nom_commun](https://fr.wiktionary.org/wiki/g%C3%A2chis#Nom_commun)

------
glxc
When someone works really hard on a problem and then puts it out there openly,
you have to respect their work. And from their perspective, they've put
weeks/months/10 years into a piece of work, and it can come off really bad if
you approach them and ask them to explain it to you so that you can understand
it in a fraction of the time. For complex ideas, you can talk at somebody and
they can get the general gist of it, but for many technical things you don't
understand it until it "clicks" from gears moving in your own head. I
understand this case is pretty drastic, but his claim that one needs to break
down the barriers in their mind shows how much he expects from an individual
to understand his work.

~~~
danidiaz
> it can come off really bad if you approach them and ask them to explain it
> to you so that you can understand it in a fraction of the time

But it works like that with software libraries. Even if they are the product
of arduous work, they should come with documentation, examples, introductory
material. Why should proofs be any different?

The Curry-Howard isomorphism should apply to documentation, too!

------
NickHaflinger
Wasn't this a plot device in 'The Hitchhiker's Guide to the Galaxy', where
they gave some scientist an award on the off-chance what he said even made
sense.

------
andyjohnson0
I had a look at one of the papers and, although I have absolutely no ability
to understand any of it, I get a kind of quiet joy from the knowledge that
reality has such fractal-like _depth_ that things like this can be
created/discovered. And that some of us can dive down there and bring them
back, even if I can't.

(Also, I like the term "Frobenioid". As in "precise specification of the
relevant monoids/Frobenioids within each Θ±ell NF-Hodge theater".)

------
ilitirit
I first read about his proof a year or two ago. I'm surprised so little
progress has been made. Is there no formal collaborative effort in trying to
verify his proof?

This seems like an opportunity to me. Host the proof (and ones like it) and
allow commentators to annotate and explain parts, and then replace all the
ambiguous parts with formal explanations.

------
davisr
Ted Nelson believes that Mochizuki is the real Satoshi behind Bitcoin, because
of the way both projects were dumped into the world's lap. I just saw his
video last night:
[https://www.youtube.com/watch?v=emDJTGTrEm0](https://www.youtube.com/watch?v=emDJTGTrEm0)

~~~
psychometry
Ridiculous. The mathematics of one has nothing to do with the other.

~~~
kazinator
All the same, couldn't Mochizuki could chew up the Bitcoin mathematics for
breakfast and spit it out? Being Satoshi could just be a kind of footnote in
his life. (I don't suspect this myself, but it's can't be dismissed _that_
easily; certainly not on grounds of different mathematics.)

~~~
AngrySkillzz
No. Cryptography is complicated, it's not just something you dabble in and
then put together a complex protocol like Bitcoin with few mistakes. Not even
mentioning the large programming task to implement that effectively.
Cryptography is it's own sizable field of study that takes a lot of work to
master.

~~~
jordigh
The bitcoin protocol is nowhere nearly as complicated as Mochizuki's proof.
The Bitcoin whitepaper can be understood without difficulty by most "laymen",
i.e. common programmers of modest ability such as myself. And Satoshi's
original code was decidedly amateurish. Satoshi was clever in solving the
consensus problem, but he's not some master cryptographer.

