5 ms·

I don't understand how something, that anyone understands, can be this cryptic. Someone some time ago mentioned that all of Einstein's major papers on special

I don't understand how something, that anyone understands, can be this cryptic. Someone some time ago mentioned that all of Einstein's major papers on special and general relativitiy can be covered in a semester or two. Isn't the hard part discovering this stuff? How can anyone discover mathematical proofs but it is so impossible to communicate them that nobody in the world can follow your train of thought? I mean people are really, really good at communicating. It's kind of what we do. I don't understand how anyone can publish proofs that don't succumb easily to understanding by experts who concentrate on them for a long time, perhaps with tiny hints by the author. It's not as though we're looking at fragments written in a long-lost language we have no access to, from a culture nobody has access to, on unknown subjects, and with the authors dead hundreds of years ago! The author is right there. The subject isn't even art (like poetry criticism or something), it's fully rigorous. Understanding someone's proof (for devoted experts) should be easy, right? What gives?

mmarx 11y ago
The papers on Inter-universal Teichmueller theory span over 500 pages alone, and they build on material developed earlier that needs to be understood too. Those papers were published a little over 3 years ago, so claiming that they can't be understood “by experts who concentrate on them for a long time” might be a bit premature.
- logicallee 11y ago
  but the article doesn't make it sound like it's not "fully understood." the article makes it sound like not even the summary is known - for example, what parts of the other papers are used, what isn't. As though the experts didn't even known what parts of theories were being referred to. very bizarre.
  - jsprogrammer 11y ago
    The summary is known. Have you read the abstracts?
    - logicallee 11y ago
      Did you even read our article, jsprogrammer? That's what I'm reacting to. It says: >A consensus emerged that the highlight of the workshop was a lecture on 9 December by Kiran Kedlaya, an arithmetic geometer from the University of California, San Diego. He zeroed in on a result from a 2008 paper by Mochizuki that linked the statement of the abc conjecture to another branch of maths called topology. The link was immediately recognised as a crucial step in Mochizuki’s grand strategy. This is why I'm making the reference to ancient lost texts. I mean, if the result of a 2008 paper is crucial, you'd think the guy would have just mentioned this fact. It's as though they're investigating some long dead guy's hidden thoughts. This is a direct quote: "There is still no clear answer to lingering questions about how things are ultimately going to fit together." What? Wouldn't the guy just tell you how they're ultimately going to fit together? I'm utterly perplexed at the process the article describes, when the author is available and can give hints.
      - shmageggy 11y ago
        What I understood from http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-works... (posted elsewhere here), is that the problem stems from differing motivations. On one hand, Mochizuki wants to make his proof fullyt complete and correct (being both fully general and covering all special cases), but the community wants broad strokes intuitions. From Mochizuki's perspective, he has told everyone how things fit together, but fit together means something different for others. The workshop attendees want to know what things they can gloss over to get the big picture, but as of yet, nobody who understands the proof has succeeded in communicating that information, probably because they don't share the same motivation.
      - jsprogrammer 11y ago
        I did read the article. It wasn't clear that you were referring to that specific paragraph. > I mean, if the result of a 2008 paper is crucial, you'd think the guy would have just mentioned this fact. Maybe he did? Or maybe it is just not crucial.∎ The article just states "The link was immediately recognised as a crucial step in Mochizuki's grand strategy."...that is just the writer's opinion. Linking these papers to topology is trivial. The word "topological" alone occurs 111 times in the first paper. A lot of what I'm reading in this article, and others, is people complaining about not understanding something that they (probably/apparently) have not even read.
- jug 11y ago
  Personally I find this more mind boggling than that it can't be communicated easily. Clearly, a proof of 500 pages will be extremely difficult to verify or even understand. However... That a concept taking just a paragraph to explain* for people who know math requires a proof of over 500 pages. THAT is what I find mind boggling and I find hard to understand. That it can just balloon in size like that, especially given that mathematics is a language of logic. * https://en.wikipedia.org/wiki/Abc_conjecture https://en.wikipedia.org/wiki/Abc_conjecture
  - im2w1l 11y ago
    The halting problem proves that mathematics is hard. There is no algorithm that can prove/disprove any decidable statement. Not even a a superslow algorithm. Maybe ballooning is related to that...
bad_alloc 11y ago
> I mean people are really, really good at communicating. On my experience, people are usually really bad at communicating. It requires additional effort to imagine what information might be required for somebody else to follow your train of thought. Think of code documentation: What the original author percieves as obvious is cryptic to somebody else because context has been omitted for the sake of brevity. The author needs to invest a lot of time to actually dump his entire mindstate into documentation. Even if he does make an effort, not everybody is a good writer. If he doesn't care about others understanding his stuff (which also seems to be the case with Mochizuki) it gets even worse. Also remember that Mochizuki basically worked alone on his stuff for 20 odd years. He built his own world to work in and it's just as hard for collegues to catch up to his level in that world as it is for e.g. non-programmers to understand programming (which takes years).
- klibertp 11y ago
  > context has been omitted for the sake of brevity Such omissions should be forbidden. Writing down the context in comments and in any other ways available should be a minimum requirement for any new code written. I do enjoy a good puzzle once in a while, but I don't want to reverse engineer someone's brain every time I read code. We should really focus more on helping other people understand the code we write. Some tools can help, but if the original author keeps most of the information to himself it's going to be really hard to follow the code.
raverbashing 11y ago
I agree It's a bit obscurantist, to be honest. While people make source code to be as clear as possible, mathematicians go the other way. One idea would be to invert the proof, going top-down, instead of bottom up (in the explanation), then clearly dividing the subproofs instead of one big "wall of text" (yes, I know, there are numbered proofs/lemmas, but still it is not very well organized) Edit: looking at the paper above it seems the author really tries to be helpful, the paper seems very organized (but it is long)

mmarx 11y ago

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

logicallee 11y ago
but the article doesn't make it sound like it's not "fully understood." the article makes it sound like not even the summary is known - for example, what parts of the other papers are used, what isn't. As though the experts didn't even known what parts of theories were being referred to. very bizarre.
- jsprogrammer 11y ago
  The summary is known. Have you read the abstracts?
  - logicallee 11y ago
    Did you even read our article, jsprogrammer? That's what I'm reacting to. It says: >A consensus emerged that the highlight of the workshop was a lecture on 9 December by Kiran Kedlaya, an arithmetic geometer from the University of California, San Diego. He zeroed in on a result from a 2008 paper by Mochizuki that linked the statement of the abc conjecture to another branch of maths called topology. The link was immediately recognised as a crucial step in Mochizuki’s grand strategy. This is why I'm making the reference to ancient lost texts. I mean, if the result of a 2008 paper is crucial, you'd think the guy would have just mentioned this fact. It's as though they're investigating some long dead guy's hidden thoughts. This is a direct quote: "There is still no clear answer to lingering questions about how things are ultimately going to fit together." What? Wouldn't the guy just tell you how they're ultimately going to fit together? I'm utterly perplexed at the process the article describes, when the author is available and can give hints.
    - shmageggy 11y ago
      What I understood from http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-works... (posted elsewhere here), is that the problem stems from differing motivations. On one hand, Mochizuki wants to make his proof fullyt complete and correct (being both fully general and covering all special cases), but the community wants broad strokes intuitions. From Mochizuki's perspective, he has told everyone how things fit together, but fit together means something different for others. The workshop attendees want to know what things they can gloss over to get the big picture, but as of yet, nobody who understands the proof has succeeded in communicating that information, probably because they don't share the same motivation.
    - jsprogrammer 11y ago
      I did read the article. It wasn't clear that you were referring to that specific paragraph. > I mean, if the result of a 2008 paper is crucial, you'd think the guy would have just mentioned this fact. Maybe he did? Or maybe it is just not crucial.∎ The article just states "The link was immediately recognised as a crucial step in Mochizuki's grand strategy."...that is just the writer's opinion. Linking these papers to topology is trivial. The word "topological" alone occurs 111 times in the first paper. A lot of what I'm reading in this article, and others, is people complaining about not understanding something that they (probably/apparently) have not even read.
jug 11y ago
Personally I find this more mind boggling than that it can't be communicated easily. Clearly, a proof of 500 pages will be extremely difficult to verify or even understand. However... That a concept taking just a paragraph to explain* for people who know math requires a proof of over 500 pages. THAT is what I find mind boggling and I find hard to understand. That it can just balloon in size like that, especially given that mathematics is a language of logic. * https://en.wikipedia.org/wiki/Abc_conjecture https://en.wikipedia.org/wiki/Abc_conjecture
- im2w1l 11y ago
  The halting problem proves that mathematics is hard. There is no algorithm that can prove/disprove any decidable statement. Not even a a superslow algorithm. Maybe ballooning is related to that...

bad_alloc 11y ago

> I mean people are really, really good at communicating. On my experience, people are usually really bad at communicating. It requires additional effort to imagine what information might be required for somebody else to follow your train of thought. Think of code documentation: What the original author percieves as obvious is cryptic to somebody else because context has been omitted for the sake of brevity. The author needs to invest a lot of time to actually dump his entire mindstate into documentation. Even if he does make an effort, not everybody is a good writer. If he doesn't care about others understanding his stuff (which also seems to be the case with Mochizuki) it gets even worse. Also remember that Mochizuki basically worked alone on his stuff for 20 odd years. He built his own world to work in and it's just as hard for collegues to catch up to his level in that world as it is for e.g. non-programmers to understand programming (which takes years).

klibertp 11y ago
> context has been omitted for the sake of brevity Such omissions should be forbidden. Writing down the context in comments and in any other ways available should be a minimum requirement for any new code written. I do enjoy a good puzzle once in a while, but I don't want to reverse engineer someone's brain every time I read code. We should really focus more on helping other people understand the code we write. Some tools can help, but if the original author keeps most of the information to himself it's going to be really hard to follow the code.

raverbashing 11y ago

I agree It's a bit obscurantist, to be honest. While people make source code to be as clear as possible, mathematicians go the other way. One idea would be to invert the proof, going top-down, instead of bottom up (in the explanation), then clearly dividing the subproofs instead of one big "wall of text" (yes, I know, there are numbered proofs/lemmas, but still it is not very well organized) Edit: looking at the paper above it seems the author really tries to be helpful, the paper seems very organized (but it is long)