13 ms·

An ABC proof too tough even for mathematicians

282 points by ot 14y ago

dsrguru 14y ago
The more mathematically-inclined HNers might be interested in Brian Conrad and Terrence Tao's comments at the bottom of this previous HN article: http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-a... Edit: Minhyong Kim's initial thoughts seem very interesting as well! http://mathoverflow.net/questions/106560/what-is-the-underlying-vision-that-mochizuki-pursued-when-trying-to-prove-the-abc/106658#106658 http://mathoverflow.net/questions/106560/what-is-the-underly... And for the less mathematically-inclined: http://news.ycombinator.com/item?id=4477241 http://news.ycombinator.com/item?id=4477241
dbaupp 14y ago
Another article with slightly more background on the ABC problem itself (and possibly slightly less sensationalist). http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 http://www.nature.com/news/proof-claimed-for-deep-connection... And the MathOverflow discussion referenced: http://mathoverflow.net/questions/106560/what-is-the-underlying-vision-that-mochizuki-pursued-when-trying-to-prove-the-abc http://mathoverflow.net/questions/106560/what-is-the-underly...
- podperson 14y ago
  I think implying the article is sensationalist is a little harsh. It's definitely a click-grabbing headline, but overall a decent article on a tough subject.
  - abraxasz 14y ago
    Agreed, I was very please to see a mainstream journal take the time to describe a potentially groundbreaking mathematical discovery
  - revelation 14y ago
    Seemed extremely hand-wavey to me, and the obscure pictures of 3D wireframe 'A', 'B' and 'C' by the side certainly didn't help. It tried to explain both the dynamics in math research and the problem at hand, and in the end explained neither.
    - bstpierre 14y ago
      It's the Boston Globe, hand-wavey is all they can really be expected to publish regarding hard abstract math. I thought they did a decent enough job for a newspaper.
Xcelerate 14y ago
This article seems to suggets that mathematicians are all too eager to drop his work at the slightest whiff of any flaw. Could someone more knowledgable on the subject explain to me why this is? It is clear that he has already done some very great things in mathematics, so even if there was a flaw in his proof, I would think his papers would still have many deep insights that no else had thought of. I mean, it's not like mathematicians are pressed for time -- if I was one I would certainly dedicate a lot of time to studying something interesting like this.
- gwern 14y ago
  Consider the incentives involved in studying his work carefully for years, and those of being in academia.
  - _delirium 14y ago
    I don't think it's even just the incentives of academia, but of life/interest as well. Tenured professors could choose to devote their next few years to learning inter-universal geometry. But they have to guess: is this likely to be a more interesting/useful thing to learn than something else they could learn in those same years? It seems so far that most are heuristically guessing no.
- shardling 14y ago
  That wasn't the takeaway I got from the article at all -- and the original proof of Fermat's Theorem by Wiles had flaws that took some work to patch over, but in no way diminished his accomplishment. I think you've just misinterpreted some of the language used.
  - Xcelerate 14y ago
    "Mochizuki has almost certainly made significant discoveries—but without the allure of a proof, it’s possible no one will take the time to understand them." "if Mochizuki is able to build up sufficient trust with his peers" "if an error is found early in the vetting process, the math world will likely move on without bothering to explore the rest of the mathematical universe Mochizuki has created" I was basing my post on these quotes.
mememememememe 14y ago
Will the solution(s) to ABC proof be a nightmare to all security protocols relying on prime number factorization, such as RSA?
- vhf 14y ago
  No, it should not.
- sold 14y ago
  Whether ABC (or a similar conjecture) is proven has practically no cryptographic impact. This is because you can already write and execute algorithms (whose correctness depends on unsolved conjectures) without proving the conjectures. See http://math.stackexchange.com/questions/69540/would-a-proof-to-the-riemann-hypothesis-affect-security http://math.stackexchange.com/questions/69540/would-a-proof-...
- cperciva 14y ago
  No. A proof is a nice thing to have, but everybody assumes that ABC is true already.
- kainosnoema 14y ago
  Initially I had the same concern, but after digging a bit deeper, I don't believe this would be an immediate outcome. From the little I can gather, the ABC conjecture is really just a statement about the relationship between the operations of addition and multiplication on triples of coprime positive integers. While prime factors are involved in the conjecture, it doesn't seem that the proof of the conjecture would naturally lead to prime factorization in polynomial time. I'm not a mathematician though, and given how radical Mochizuki's proof sounds, I imagine it may (if it stands) eventually lead to more efficient factorization algorithms. Advances in computing power seem to be a more worrisome threat to RSA, especially for RSA-1024 whose factorization is probably already feasible (http://www.cs.tau.ac.il/~tromer/papers/cbtwirl.pdf http://www.cs.tau.ac.il/~tromer/papers/cbtwirl.pdf). And once advances in quantum computing become reality, quantum algorithms should be able to very quickly break even RSA-2048 (http://en.wikipedia.org/wiki/Shors_algorithm http://en.wikipedia.org/wiki/Shors_algorithm).
elliptic 14y ago
Is this situation similar to that of Louis de Branges & the Riemann Hypothesis a few years back? I.e, a well-respected mathematician (de Branges had settled the Bieberbach conjecture in the 80s) releases a proof of an important unsolved problem using his own poorly understood mathematical technology? Edit - lest this sound too negative, one should realize that the Bieberbach proof took a long time to be accepted.
bnegreve 14y ago
Would it be possible to use proof assistants like Coq [1] to verify this kind of proofs ? If not, does anyone know why ? [1] http://en.wikipedia.org/wiki/Coq http://en.wikipedia.org/wiki/Coq
- cperciva 14y ago
  Proofs at this level are nowhere near formal enough for existing mechanical tools to understand. I've seen researchers cite "translated proof X into an automatically-verifiable form" as a major achievement -- even for very simple proofs.
  - kvb 14y ago
    This seems overly pessimistic. I'm sure it'd take a lot of work, but consider results like Gonthier's proof of the Odd Order Theorem (see http://research.microsoft.com/en-us/news/features/gonthierproof-101112.aspx http://research.microsoft.com/en-us/news/features/gonthierpr...), which formalized a book-length proof.
  - makmanalp 14y ago
    Non-math person here. Wait, so you're saying it's easier for humans to prove complex theorems than computers? That's nuts. Doesn't that mean that verifying proofs is then based on a super handwavy process where a bunch of researchers come to a consensus and say "yes, this is correct"? Ugh! What if they all overlook some crucial aspect because of the way they were trained, or their entrenched / ingrained assumptions? When you describe a proof with words like you say, aren't there quibbles over semantic things? But I thought the whole point of a proof was that you leave nothing to semantics. Everything logically follows from each other. I thought the main issue with proofs systems was that encoding the concepts into code was tedious. Could we maybe have a common database of well-known proofs, axioms, concepts etc, so that you wouldn't have to rewrite common ideas from scratch? Like a package manager with proof-libraries? Or is this an issue with the expression power of proofs languages?
    - impendia 14y ago
codeulike 14y ago
If a programmer locked himself away for 14 years and then emerged and announced he'd written a completely bug free OS, there would be skepticism. Code needs to be battle tested by other people to find the bugs. Mathematics is the same, to an extent; one guy working alone for 14 years is likely to have missed ideas and perspectives that could illuminate flaws in his reasoning. Maths bugs. If he's produced hundreds of pages of complex reasoning, on his own, however smart he is I'd say there's a high chance he's missed something. Humans need to collaborate in areas of high complexity. With a single brain, there's too high a chance of bias hiding the problems.
- ekianjo 14y ago
  I do not think it is fair to compare Software and Abstract Mathematics. Software is mostly an Engineering problem, while Mathematics is mostly Science and Conjecture. And even if they locked themselves for 14 years, there is no "single mind at work" there. Mathematics are built "on the shoulder of giants" through generations of brain dedicated to it. There is no such thing as pure invention - new discoveries lead to new theorems, new leads that other people take on.
  - georgeorwell 14y ago
    Proofs are programs. https://www.google.com/search?q=proofs+are+programs https://www.google.com/search?q=proofs+are+programs
    - ekianjo 14y ago
      Let me say it in another way. Abstract Maths is like having a Organic Chemist working on understanding the transformation of food at the molecular level. A Software Programmer is like a cook who knows how to mix and prepare food in order to make a delicious dish out of it.
      - fusiongyro 14y ago
        "Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians."– E. W. Dijkstra
ph0rque 14y ago
...the proof itself is written in an entirely different branch of mathematics called “inter-universal geometry” that Mochizuki—who refers to himself as an “inter-universal Geometer”—invented and of which, at least so far, he is the sole practitioner. In this universe, at least...
ArtB 14y ago
Wouldn't the easiest way to check this proof be to enter it into something like Coq? That way you'd only have to understand how to translate each step rather than learn each field.
- deleted 14y ago
  [deleted]
- Evbn 14y ago
  You would still have to write 100 pages of Coq code, and write every definition properly, which may be just as hard as learning the field.
dbz 14y ago
Can anyone explain what "inter-universal geometry" is?
- adaml_623 14y ago
  Mochizuki
- hakaaak 14y ago
  Different universes in mathematics play by different rules and have different components. Inter-universal Geometry is how these sets of rules and components can relate and is the bridge to understanding more complex theory.
genuine 14y ago
Here is the link to the proof: 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... And the preceding three for context: 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...
sek 14y ago
Just read his Wikipedia entry: > Mochizuki attended Phillips Exeter Academy and graduated in 2 years. He entered Princeton University as an undergraduate at age 16 and received a Ph.D. under the supervision of Gerd Faltings at age 23. He is 43 Years old now, I assume he is 100% committed to Mathematics. These people fascinate me, having a feedback loop that is unbreakable. Especially for topics where you have a knowledge of something and almost nobody else is the world is capable of understanding you. It's like Star Trek for the mind.
sek 14y ago
A Youtube video with a pretty accessible explanation. http://www.youtube.com/watch?v=RkBl7WKzzRw http://www.youtube.com/watch?v=RkBl7WKzzRw
pfanner 14y ago
I'm a physics student. Sometimes I'm thinking if I should completely change my path to math. I always sucked at it but it seems to be so huge, exciting and powerful.
atas 14y ago
"Release early release often" applies to Math as well. Wouldn't it be better for everyone if he hadn't been so secluded and published some of his work in the meantime?
- taejo 14y ago
  My impression was that he had published some material on IUT before, but nobody had any reason to read it.

282 points by ot 14y ago

The more mathematically-inclined HNers might be interested in Brian Conrad and Terrence Tao's comments at the bottom of this previous HN article: http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-a... Edit: Minhyong Kim's initial thoughts seem very interesting as well! http://mathoverflow.net/questions/106560/what-is-the-underlying-vision-that-mochizuki-pursued-when-trying-to-prove-the-abc/106658#106658 http://mathoverflow.net/questions/106560/what-is-the-underly... And for the less mathematically-inclined: http://news.ycombinator.com/item?id=4477241 http://news.ycombinator.com/item?id=4477241

dbaupp 14y ago

Another article with slightly more background on the ABC problem itself (and possibly slightly less sensationalist). http://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 http://www.nature.com/news/proof-claimed-for-deep-connection... And the MathOverflow discussion referenced: http://mathoverflow.net/questions/106560/what-is-the-underlying-vision-that-mochizuki-pursued-when-trying-to-prove-the-abc http://mathoverflow.net/questions/106560/what-is-the-underly...

podperson 14y ago
I think implying the article is sensationalist is a little harsh. It's definitely a click-grabbing headline, but overall a decent article on a tough subject.
- abraxasz 14y ago
  Agreed, I was very please to see a mainstream journal take the time to describe a potentially groundbreaking mathematical discovery
- revelation 14y ago
  Seemed extremely hand-wavey to me, and the obscure pictures of 3D wireframe 'A', 'B' and 'C' by the side certainly didn't help. It tried to explain both the dynamics in math research and the problem at hand, and in the end explained neither.
  - bstpierre 14y ago
    It's the Boston Globe, hand-wavey is all they can really be expected to publish regarding hard abstract math. I thought they did a decent enough job for a newspaper.

Xcelerate 14y ago

This article seems to suggets that mathematicians are all too eager to drop his work at the slightest whiff of any flaw. Could someone more knowledgable on the subject explain to me why this is? It is clear that he has already done some very great things in mathematics, so even if there was a flaw in his proof, I would think his papers would still have many deep insights that no else had thought of. I mean, it's not like mathematicians are pressed for time -- if I was one I would certainly dedicate a lot of time to studying something interesting like this.

gwern 14y ago
Consider the incentives involved in studying his work carefully for years, and those of being in academia.
- _delirium 14y ago
  I don't think it's even just the incentives of academia, but of life/interest as well. Tenured professors could choose to devote their next few years to learning inter-universal geometry. But they have to guess: is this likely to be a more interesting/useful thing to learn than something else they could learn in those same years? It seems so far that most are heuristically guessing no.
shardling 14y ago
That wasn't the takeaway I got from the article at all -- and the original proof of Fermat's Theorem by Wiles had flaws that took some work to patch over, but in no way diminished his accomplishment. I think you've just misinterpreted some of the language used.
- Xcelerate 14y ago
  "Mochizuki has almost certainly made significant discoveries—but without the allure of a proof, it’s possible no one will take the time to understand them." "if Mochizuki is able to build up sufficient trust with his peers" "if an error is found early in the vetting process, the math world will likely move on without bothering to explore the rest of the mathematical universe Mochizuki has created" I was basing my post on these quotes.

mememememememe 14y ago

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

vhf 14y ago
No, it should not.
sold 14y ago
Whether ABC (or a similar conjecture) is proven has practically no cryptographic impact. This is because you can already write and execute algorithms (whose correctness depends on unsolved conjectures) without proving the conjectures. See http://math.stackexchange.com/questions/69540/would-a-proof-to-the-riemann-hypothesis-affect-security http://math.stackexchange.com/questions/69540/would-a-proof-...
cperciva 14y ago
No. A proof is a nice thing to have, but everybody assumes that ABC is true already.
kainosnoema 14y ago
Initially I had the same concern, but after digging a bit deeper, I don't believe this would be an immediate outcome. From the little I can gather, the ABC conjecture is really just a statement about the relationship between the operations of addition and multiplication on triples of coprime positive integers. While prime factors are involved in the conjecture, it doesn't seem that the proof of the conjecture would naturally lead to prime factorization in polynomial time. I'm not a mathematician though, and given how radical Mochizuki's proof sounds, I imagine it may (if it stands) eventually lead to more efficient factorization algorithms. Advances in computing power seem to be a more worrisome threat to RSA, especially for RSA-1024 whose factorization is probably already feasible (http://www.cs.tau.ac.il/~tromer/papers/cbtwirl.pdf http://www.cs.tau.ac.il/~tromer/papers/cbtwirl.pdf). And once advances in quantum computing become reality, quantum algorithms should be able to very quickly break even RSA-2048 (http://en.wikipedia.org/wiki/Shors_algorithm http://en.wikipedia.org/wiki/Shors_algorithm).

elliptic 14y ago

Is this situation similar to that of Louis de Branges & the Riemann Hypothesis a few years back? I.e, a well-respected mathematician (de Branges had settled the Bieberbach conjecture in the 80s) releases a proof of an important unsolved problem using his own poorly understood mathematical technology? Edit - lest this sound too negative, one should realize that the Bieberbach proof took a long time to be accepted.

bnegreve 14y ago

Would it be possible to use proof assistants like Coq [1] to verify this kind of proofs ? If not, does anyone know why ? [1] http://en.wikipedia.org/wiki/Coq http://en.wikipedia.org/wiki/Coq

cperciva 14y ago
Proofs at this level are nowhere near formal enough for existing mechanical tools to understand. I've seen researchers cite "translated proof X into an automatically-verifiable form" as a major achievement -- even for very simple proofs.
- kvb 14y ago
  This seems overly pessimistic. I'm sure it'd take a lot of work, but consider results like Gonthier's proof of the Odd Order Theorem (see http://research.microsoft.com/en-us/news/features/gonthierproof-101112.aspx http://research.microsoft.com/en-us/news/features/gonthierpr...), which formalized a book-length proof.
- makmanalp 14y ago
  Non-math person here. Wait, so you're saying it's easier for humans to prove complex theorems than computers? That's nuts. Doesn't that mean that verifying proofs is then based on a super handwavy process where a bunch of researchers come to a consensus and say "yes, this is correct"? Ugh! What if they all overlook some crucial aspect because of the way they were trained, or their entrenched / ingrained assumptions? When you describe a proof with words like you say, aren't there quibbles over semantic things? But I thought the whole point of a proof was that you leave nothing to semantics. Everything logically follows from each other. I thought the main issue with proofs systems was that encoding the concepts into code was tedious. Could we maybe have a common database of well-known proofs, axioms, concepts etc, so that you wouldn't have to rewrite common ideas from scratch? Like a package manager with proof-libraries? Or is this an issue with the expression power of proofs languages?
  - impendia 14y ago

codeulike 14y ago

If a programmer locked himself away for 14 years and then emerged and announced he'd written a completely bug free OS, there would be skepticism. Code needs to be battle tested by other people to find the bugs. Mathematics is the same, to an extent; one guy working alone for 14 years is likely to have missed ideas and perspectives that could illuminate flaws in his reasoning. Maths bugs. If he's produced hundreds of pages of complex reasoning, on his own, however smart he is I'd say there's a high chance he's missed something. Humans need to collaborate in areas of high complexity. With a single brain, there's too high a chance of bias hiding the problems.

ekianjo 14y ago
I do not think it is fair to compare Software and Abstract Mathematics. Software is mostly an Engineering problem, while Mathematics is mostly Science and Conjecture. And even if they locked themselves for 14 years, there is no "single mind at work" there. Mathematics are built "on the shoulder of giants" through generations of brain dedicated to it. There is no such thing as pure invention - new discoveries lead to new theorems, new leads that other people take on.
- georgeorwell 14y ago
  Proofs are programs. https://www.google.com/search?q=proofs+are+programs https://www.google.com/search?q=proofs+are+programs
  - ekianjo 14y ago
    Let me say it in another way. Abstract Maths is like having a Organic Chemist working on understanding the transformation of food at the molecular level. A Software Programmer is like a cook who knows how to mix and prepare food in order to make a delicious dish out of it.
    - fusiongyro 14y ago
      "Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians."– E. W. Dijkstra

ph0rque 14y ago

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

ArtB 14y ago

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

deleted 14y ago
[deleted]
Evbn 14y ago
You would still have to write 100 pages of Coq code, and write every definition properly, which may be just as hard as learning the field.

dbz 14y ago

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

adaml_623 14y ago
Mochizuki
hakaaak 14y ago
Different universes in mathematics play by different rules and have different components. Inter-universal Geometry is how these sets of rules and components can relate and is the bridge to understanding more complex theory.

genuine 14y ago

Here is the link to the proof: 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... And the preceding three for context: 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...

sek 14y ago

Just read his Wikipedia entry: > Mochizuki attended Phillips Exeter Academy and graduated in 2 years. He entered Princeton University as an undergraduate at age 16 and received a Ph.D. under the supervision of Gerd Faltings at age 23. He is 43 Years old now, I assume he is 100% committed to Mathematics. These people fascinate me, having a feedback loop that is unbreakable. Especially for topics where you have a knowledge of something and almost nobody else is the world is capable of understanding you. It's like Star Trek for the mind.

sek 14y ago

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

pfanner 14y ago

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

atas 14y ago

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

taejo 14y ago
My impression was that he had published some material on IUT before, but nobody had any reason to read it.