6 ms·

Shinichi Mochizuki and the impenetrable proof

79 points by robinhouston 11y ago

yarvin9 11y ago
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.
- pmiller2 11y ago
  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.
- jsprogrammer 11y ago
  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.
  - Mithaldu 11y ago
    Fwiw, he has been posted before: https://hn.algolia.com/?query=Mochizuki&sort=byPopularity&prefix&page=0&dateRange=all&type=story https://hn.algolia.com/?query=Mochizuki&sort=byPopularity&pr...
  - yarvin9 11y ago
    > 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.
    - jsprogrammer 11y ago
      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.
    - hyperpape 11y ago
      > 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.
Mithaldu 11y ago
Links to the papers themselves: 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 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.
NhanH 11y ago
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 11y ago
  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.
  - NhanH 11y ago
    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?
    - yaks_hairbrush 11y ago
      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.
  - kaitai 11y ago
    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.
patcon 11y ago
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 11y ago
  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 11y ago
    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 11y ago
      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.
- the8472 11y ago
  AIs are dynamic systems though. You can prod pieces and see what happens. There's no interactive debugger for proofs written in prose.
- asgard1024 11y ago
  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.
davisr 11y ago
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 11y ago
  Ridiculous. The mathematics of one has nothing to do with the other.
  - kazinator 11y ago
    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 11y ago
      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 11y ago
        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.
ArtB 11y ago
> 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").
- asgard1024 11y ago
  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"
- embiaa 11y ago
  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 11y ago
    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
shas3 11y ago
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 11y ago
  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?
  - Rangi42 11y ago
    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.
    - Mithaldu 11y ago
      Op clearly said "Researchers should not", so i am curious to hear on what base they make that claim.
  - jordigh 11y ago
    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.
glxc 11y ago
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 11y ago
  > 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!
powera 11y ago
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 11y ago
  A for effort, since apparently mathematicians themselves are quite reluctant to do this...
NickHaflinger 11y ago
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.
ilitirit 11y ago
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.
deleted 11y ago
[deleted]
andyjohnson0 11y ago
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".)

79 points by robinhouston 11y ago

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.

pmiller2 11y ago
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.
jsprogrammer 11y ago
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.
- Mithaldu 11y ago
  Fwiw, he has been posted before: https://hn.algolia.com/?query=Mochizuki&sort=byPopularity&prefix&page=0&dateRange=all&type=story https://hn.algolia.com/?query=Mochizuki&sort=byPopularity&pr...
- yarvin9 11y ago
  > 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.
  - jsprogrammer 11y ago
    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.
  - hyperpape 11y ago
    > 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.

Mithaldu 11y ago

Links to the papers themselves: 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 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.

NhanH 11y ago

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 11y ago
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.
- NhanH 11y ago
  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?
  - yaks_hairbrush 11y ago
    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.
- kaitai 11y ago
  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.

patcon 11y ago

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 11y ago
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 11y ago
  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 11y ago
    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.
the8472 11y ago
AIs are dynamic systems though. You can prod pieces and see what happens. There's no interactive debugger for proofs written in prose.
asgard1024 11y ago
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.

davisr 11y ago

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 11y ago
Ridiculous. The mathematics of one has nothing to do with the other.
- kazinator 11y ago
  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 11y ago
    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 11y ago
      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.

ArtB 11y ago

> 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").

asgard1024 11y ago
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"
embiaa 11y ago
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 11y ago
  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

shas3 11y ago

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 11y ago
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?
- Rangi42 11y ago
  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.
  - Mithaldu 11y ago
    Op clearly said "Researchers should not", so i am curious to hear on what base they make that claim.
- jordigh 11y ago
  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.

glxc 11y ago

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 11y ago
> 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!

powera 11y ago

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 11y ago
A for effort, since apparently mathematicians themselves are quite reluctant to do this...

NickHaflinger 11y ago

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.

ilitirit 11y ago

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.

deleted 11y ago

[deleted]

andyjohnson0 11y ago

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".)