Shinichi Mochizuki and the impenetrable proof | Hacker News Reader

Shinichi Mochizuki and the impenetrable proof

nature.com

79 pointsrobinhouston69 comments

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.

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

> 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").

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.

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

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!

I'm not qualified to answer this, but I remember this rather interesting talk where Richard Taylor talks about this somewhat, and there is some discussion of the ABC Conjecture even if somewhat tangentially: https://www.youtube.com/watch?v=eNgUQlpc1m0&t=25m40s

There is a quote from Mochizuki himself in the article: [my proof] "constitutes a sort of faithful miniature model of the status of pure mathematics in human society"

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%20... http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20... http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20...

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

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.

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.

> 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.

A nitpick: More mathematics has been created/discovered/whatever since 1950 than before. It's not true that most of math predates the American academic system. The modern academic system, for all its flaws, has fostered an explosion of new ideas.

> 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.

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,...

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.

Sure, but most things we take for granted today started as abstract math. We wouldn't be communicating with each other right now if it wasn't for the abstract math developed by many others.

I guess now that you were being sarcastic in your first post. There may be an argument that the reaction may be sensible in relation to the institutions that exist, but those are not the only possible institutions.

Fwiw, he has been posted before:

https://hn.algolia.com/?query=Mochizuki&sort=byPopularity&pr...

I doubt any pre-tenure mathematician would spend the necessary time to check the proof, as it would be such an immense career risk. Post-tenure mathematicians are not really worried about being hired anywhere.

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...

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.

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.

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.

I agree. I am just reading through Gowers paper on automatic theorem proving. I think once the machines are capable of automatic definitions (according to some definition of beautiful which limits to what humans consider beautiful, for example one based on entropy), all the bets are off. Humans will then struggle to understand the concepts involved with their limited brains.

AIs are dynamic systems though. You can prod pieces and see what happens.

There's no interactive debugger for proofs written in prose.

> In the face of metahuman science, humans have become metascientists.

http://www.nature.com/nature/journal/v405/n6786/full/405517a...

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).

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.

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.

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

Out of curiosity, how old was your advisor at the time (ie is he in his mid career at his peak, toward the end or is he on the younger side etc.)? And is the gap in capability just because of his experience, or is he just a much better scientist than the average researchers?

My advisor at the time was 60-ish, and had tenure for 30 years or so. The gap in capability was (to my eye) mostly due to his experience.

I'm guessing that he means that his work is closely related to arithmetic geometry (which I was once told was one of the hardest fields in pure math), and so someone with no background would have a lot of trouble getting it. Presumably it takes even longer to become an expert in arithmetic geometry (15 years?).

I also happen to believe in 10x programmers, in fact I think I am one. HN places a big emphasis on denying the importance of virtuosity or intelligence in programming, and characterizing it as a craft. It's very hard to argue against since nerds have had excessive modesty (often literally) beaten into them. At least you get the occasional boastful athlete who says "I'm just really talented".

It's plausible if by "maths graduate student" we understand "average maths graduate student". I.e. a different kind of student from that maths graduate student which the "expert in arithmetic geometry" once was.

We cannot interpret that as containing the claim that when any old randomly chosen maths graduate student becomes an expert in arithmetic geometry, they suddenly gain a 40X efficiency in understanding.

Think of it like this: Someone in Sacramento can get to Milwaukee in 3 days (driving). Someone in Chicago can get to Milwaukee in 3 hours (or less). That's a 24x factor.

When any old randomly chosen person Sacramento gets to Chicago, they do not gain a 24x efficiency in traveling. Rather, they've covered most of the distance to Milwaukee already.

Don't think of it as as 40x gap, think of it as a 10 year - 500 hours gap.

The expert has the 100 years - 500 hours study that are needed.

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.

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?

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.

"gâchis" doesn't mean mess but waste.

Definition 3:

https://fr.wiktionary.org/wiki/g%C3%A2chis#Nom_commun

it seems that Ivan Fesenko can be the Hardy to Shinichi Mochizuki, or that's how I read it anyway...

It's only an obligation to the extent that he wants anyone else to appreciate the work. Don't read it as "he should do X" but as "he should do X if he wants others to read his proof".

Whether he does or not is a different question. I think it's likely that he does, but perhaps he's content knowing that he devised a proof himself.

And, for what it's worth, communicating results and impact are both usually part of a researchers job description. That might not matter so much thanks to tenure, but it's still there in spirit.

Any obligation, no.

But the consensus among the mathematical community is that Mochizuki probably doesn't have a proof, that his work likely has some fundamental error, that it is probably (but not certainly) not worth taking all that seriously.

If Mochizuki doesn't mind this, then that's fine. But if he wants the respect that he claims to deserve, then the 'roadshow' can't be omitted.

> the consensus among the mathematical community is that Mochizuki probably doesn't have a proof

If that's the consensus (and I don't think it is, it's probably what mathematicians under influence of cognitive dissonance think the consensus is), then it is sadly not based on empirical evidence, which is that 4 other people studied the proof and certified it.

Explaining your work is part of the job (as it is for any academic discipline, or indeed most professional occupations).

That is akin to asking, did he owe to it to us to write up his proof rather than lock up reams of papers in his cabinet? As others have elaborated, academics owe it to the community to not just spend their time 'working', but communicating. In fact, where does 'working' end and communicating begin? At writing it up on reams of paper? Typing it up and posting it to Arxiv? Going through peer review? Lecturing about your work and convincing people? It is arbitrary to draw a line at 'posting it to Arxiv'. 'working' only ends when your results are effectively communicated to the community. It is an incomplete project till then. The whole objective of research is to advance the field for the community, not just in one's own head!

To use a software project analogy, it's the difference between working in isolation and dumping a source tarball on a personal website, and creating an nice introductory website, putting the source on github and accepting comments/pull-requests. One has more chances to get used and studied than the other.

At least that's how I understood it.

He's not obligated to do anything, but it's not surprising that other mathematicians are not too eager to retrace his steps to figure out if the proof is correct, given its length and unique notation.

Op clearly said "Researchers should not", so i am curious to hear on what base they make that claim.

it is not a proof until you have convinced at least one other [qualified] person. related rant [1]

[1] http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-be...

A proof only requires a valid transformation from axioms to conclusion. No person required.

until an automated theorem prover can be made to certify these transformations as valid, a person has to do this work manually. this is particularly true in this specific instance since an enormous, alien framework has been erected to set up what defines a valid transformation within it.

on point [1], though i suppose mathematical proofs don't quite align with this characterization.

[1] https://www.quora.com/Is-it-possible-to-have-a-world-without...

This is a simplified picture of mathematical practice. Not everything is formal in the sense of an logical derivation, and there are phrases like "trivially..." that show that the proof relies on being read by a competent mathematician.

The standards of what makes a good proof are also malleable. Here is a paper that discusses some of these issues in passing: https://dl.dropboxusercontent.com/u/10561191/Published/ProbP...

"Trivially" should be reserved for conclusions that follow directly from a simple transformation of an axiom or proved theorem.

Depends on your notion of proof. That is one notion.

There's the human version, that the other commentors talk about.

There's also https://en.wikipedia.org/wiki/Interactive_proof_system for an interesting formal version that is not `transforming axioms'.

In this case, the proof should be formal since it is a statement about the natural numbers. (as far as I can tell)

Doesn't really matter too much what the proof is about, does it? You can still choose different notions of proof.

The thing is that his work is basically worthless unless he communicates it to others.

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.

> 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!

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.

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".)

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.

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

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

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.)

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.

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.

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