6 ms·

Huang's comment is also very interesting: > Regarding how long it took me to find this proof, here is the timeline for those who are interested. https://www.s

by zcbenz 7y ago

Huang's comment is also very interesting: > Regarding how long it took me to find this proof, here is the timeline for those who are interested. https://www.scottaaronson.com/blog/?p=4229#comment-1813116 https://www.scottaaronson.com/blog/?p=4229#comment-1813116

krastanov 7y ago
The gargantuan intellectual effort behind this is incredibly humbling. I can hardly imagine the breadth of knowledge one needs to be able to keep in their mind to tackle this problem.
- JJMcJ 7y ago
  Not sure on this, but it is said the Fermat's Last Theorme proof involved most every area of modern mathematics. Andrew Wiles was so productive he dribbled out older results to hide the fact he was 100% on the Fermat problem for years. Huang and Knuth are pretty smart as well it seems like.
  - hilbertseries 7y ago
    The proof of Fermats last theorem did not involve many areas of mathematics. It was primarily proved via algebraic geometry. That’s also not true about wiles productivity... he first discovered the proof in 93 but then when he talked to people about it an error was found. He eventually fixed the error in 94 and then in 95 it was published. No mathematician waits to publish the proof of one of the greatest unsolved problems in mathematics.
    - insulanus 7y ago
      The above comment is talking about Wiles slowly giving out unrelated results he had saved up before starting on Fermat's last Theorem in earnest.
- tempguy9999 7y ago
  Here's a quote from Richard Bellman, from his paper on the birth of dynamic programming <<<What is worth noting about the foregoing development is that I should have seen the application of dynamic pro- gramming to control theory several years before. I should have, but I didn’t. It is very well to start a lecture by saying, ‘Clearly, a control process can be regarded as a multistage decision process in which???,’ but it is a bit misleading. Scientific developments can always be made logical and rational with sufficient hindsight. It is amazing, however, how clouded the crystal ball looks beforehand. We all wear such intellectual blinders and make such inexplicable blun- ders that it is amazing that any progress is made at all>>> In simpler, personal terms, "oh you bloody idiot, why didn't you XYZ...?" I was considering a problem related to a new data structure. There was a part that was plain ugly, just stank. I thought I could do better, so expressed the problem as a programmer might, and looked for regularity. There was a lot of regularity, so there was an underlying structure which I could grasp at, but not reach. Grab, grab, grab. It was like mist. I realised it involved the edges of an n-dimensional cube. I could see the 2D case, it's trivial, so what did I do? Well I needed to generalise it so I went for the 4D cube. I spent literally weeks muttering to myself like a nutter, half the time wandering like a zombie. And eventually it clicked. I may, just may have found something new and genuinely useful in combinatorics but that's not the point. The point is, had I gone from the 2D case to the 3D case the structure I was looking for would have dropped out in an afternoon. Weeks wasted. It was so simple, you bloody idiot, why didn't you... I'm not a mathematician, but I guess a lot of maths is like that. (Edit: quoted Bellman extract to delineate it properly)
jxramos 7y ago
I find it interesting too in so far as it references Math Overflow... > Yet at that time, I was hoping for developing something more general using the eigenspace decomposition of the adjacency matrix, like in this unanswered MO question: > https://mathoverflow.net/questions/331825/generalization-of-cauchys-eigenvalue-interlacing-theorem https://mathoverflow.net/questions/331825/generalization-of-... There's some serious stuff on Math Overflow and the Mathematics stack exchange (https://math.stackexchange.com/ https://math.stackexchange.com/). I haven't figured out what the difference between the two sites is however in focus
- mauricioc 7y ago
  The line can be blurry sometimes, but it's roughly the difference between research and teaching. MathOverflow is meant to help mathematics researchers. At its best, it's like being able to walk over to a colleague's room in a prestigious math department to ask for insight. Math.SE, on the other hand, is for "people studying math at any level and professionals in related fields" and should be used for questions that didn't originate in research, such as hard textbook exercises. Both are super interesting, and even mathematicians will have plenty of questions that more suitable for Math.SE than MathOverflow.
  - jxramos 7y ago
    Thank you for clarifying. I've been incredibly impressed with the adoption by the mathematics community with these tools.
- _Nat_ 7y ago
  [SE.Math](https://math.stackexchange.com/ https://math.stackexchange.com/) is for pretty much any sort of math question you might have. At least for traditional mathematics -- statistics, numerical algorithms, data science, AI, etc., tend to better fit on other StackExchange sites. [MathOverflow](https://mathoverflow.net/ https://mathoverflow.net/) is the sorts of math questions that will probably sound like gibberish to anyone who doesn't have a graduate-level education in Math. And I mean specifically in Math, not in Engineering or Physics. Usually, if you want to ask a math question -- even a reasonably complex one that someone would need a lot of education to understand -- it'll go on SE.Math. MathOverflow's more of a specialty site for mathematicians. To note it, the distinction between SE.Math and MathOverflow has at least two parallels: 1. [SE.ComputerScience](https://cs.stackexchange.com/ https://cs.stackexchange.com/) vs. [SE.TheoreticalComputerScience](https://cstheory.stackexchange.com/ https://cstheory.stackexchange.com/) ; and 2. [SE.Physics](https://physics.stackexchange.com/ https://physics.stackexchange.com/) vs. [PhysicsOverflow](https://www.physicsoverflow.org/ https://www.physicsoverflow.org/) . In all three cases, one site's more appropriate for most users, ranging from beginners to experts, while the other site's more specifically for research-level topics in the field.
j7ake 7y ago
"2013-2018: I revisited this conjecture every time when I learn a new tool, without any success though. But at least thinking about it helps me quickly fall asleep many nights." I believe Feynmann said the best people always had the top unsolved problems in the top of their minds and whenever a new technique or result arrived, they would drop everything and try out the new attack on the old problem.
- vanderZwan 7y ago
  I'm kind of surprised it helped him fall asleep. Whenever my brain starts to think about technical problems in the middle of the night I know I have a few hours of wakeful puzzling ahead of me
  - j7ake 7y ago
    I find reading technical work makes my brain fall asleep very quickly. I am sure if I read Knuth's books at night I would be asleep within fifteen minutes. I am not sure thinking about technical work would help sleep, because thinking could be orthogonal to reading.
    - vanderZwan 7y ago
      Oh, I'm with you regarding technical reading being a good way to fall asleep! But reading someone else's train of thought is a different way to engage the brain than trying to figure out a mathematical proof on your own - that's a lot more like what pondering over programming problems would be like for us I think
  - H8crilA 7y ago
    I had a friend at university who studied mathematics with me (I studied both maths and computer science, but he was a genuine mathematician, and a pretty good one, much better than me), and he told me that he often takes problems "to bed". After that he would often wake up with at least a half solution in his head. Sometimes a full solution. Ramanujan was like that too. He believed that a goddess sends him solutions in his sleep. Not sure if the goddess is real, but the formulas certainly were real. http://www.oceanvivasilver.com/5-namakkals-infinity-dream-srinivasa-ramanuja http://www.oceanvivasilver.com/5-namakkals-infinity-dream-sr...

by zcbenz 7y ago

krastanov 7y ago

The gargantuan intellectual effort behind this is incredibly humbling. I can hardly imagine the breadth of knowledge one needs to be able to keep in their mind to tackle this problem.

JJMcJ 7y ago
Not sure on this, but it is said the Fermat's Last Theorme proof involved most every area of modern mathematics. Andrew Wiles was so productive he dribbled out older results to hide the fact he was 100% on the Fermat problem for years. Huang and Knuth are pretty smart as well it seems like.
- hilbertseries 7y ago
  The proof of Fermats last theorem did not involve many areas of mathematics. It was primarily proved via algebraic geometry. That’s also not true about wiles productivity... he first discovered the proof in 93 but then when he talked to people about it an error was found. He eventually fixed the error in 94 and then in 95 it was published. No mathematician waits to publish the proof of one of the greatest unsolved problems in mathematics.
  - insulanus 7y ago
    The above comment is talking about Wiles slowly giving out unrelated results he had saved up before starting on Fermat's last Theorem in earnest.
tempguy9999 7y ago
Here's a quote from Richard Bellman, from his paper on the birth of dynamic programming <<<What is worth noting about the foregoing development is that I should have seen the application of dynamic pro- gramming to control theory several years before. I should have, but I didn’t. It is very well to start a lecture by saying, ‘Clearly, a control process can be regarded as a multistage decision process in which???,’ but it is a bit misleading. Scientific developments can always be made logical and rational with sufficient hindsight. It is amazing, however, how clouded the crystal ball looks beforehand. We all wear such intellectual blinders and make such inexplicable blun- ders that it is amazing that any progress is made at all>>> In simpler, personal terms, "oh you bloody idiot, why didn't you XYZ...?" I was considering a problem related to a new data structure. There was a part that was plain ugly, just stank. I thought I could do better, so expressed the problem as a programmer might, and looked for regularity. There was a lot of regularity, so there was an underlying structure which I could grasp at, but not reach. Grab, grab, grab. It was like mist. I realised it involved the edges of an n-dimensional cube. I could see the 2D case, it's trivial, so what did I do? Well I needed to generalise it so I went for the 4D cube. I spent literally weeks muttering to myself like a nutter, half the time wandering like a zombie. And eventually it clicked. I may, just may have found something new and genuinely useful in combinatorics but that's not the point. The point is, had I gone from the 2D case to the 3D case the structure I was looking for would have dropped out in an afternoon. Weeks wasted. It was so simple, you bloody idiot, why didn't you... I'm not a mathematician, but I guess a lot of maths is like that. (Edit: quoted Bellman extract to delineate it properly)

jxramos 7y ago

I find it interesting too in so far as it references Math Overflow... > Yet at that time, I was hoping for developing something more general using the eigenspace decomposition of the adjacency matrix, like in this unanswered MO question: > https://mathoverflow.net/questions/331825/generalization-of-cauchys-eigenvalue-interlacing-theorem https://mathoverflow.net/questions/331825/generalization-of-... There's some serious stuff on Math Overflow and the Mathematics stack exchange (https://math.stackexchange.com/ https://math.stackexchange.com/). I haven't figured out what the difference between the two sites is however in focus

mauricioc 7y ago
The line can be blurry sometimes, but it's roughly the difference between research and teaching. MathOverflow is meant to help mathematics researchers. At its best, it's like being able to walk over to a colleague's room in a prestigious math department to ask for insight. Math.SE, on the other hand, is for "people studying math at any level and professionals in related fields" and should be used for questions that didn't originate in research, such as hard textbook exercises. Both are super interesting, and even mathematicians will have plenty of questions that more suitable for Math.SE than MathOverflow.
- jxramos 7y ago
  Thank you for clarifying. I've been incredibly impressed with the adoption by the mathematics community with these tools.
_Nat_ 7y ago
[SE.Math](https://math.stackexchange.com/ https://math.stackexchange.com/) is for pretty much any sort of math question you might have. At least for traditional mathematics -- statistics, numerical algorithms, data science, AI, etc., tend to better fit on other StackExchange sites. [MathOverflow](https://mathoverflow.net/ https://mathoverflow.net/) is the sorts of math questions that will probably sound like gibberish to anyone who doesn't have a graduate-level education in Math. And I mean specifically in Math, not in Engineering or Physics. Usually, if you want to ask a math question -- even a reasonably complex one that someone would need a lot of education to understand -- it'll go on SE.Math. MathOverflow's more of a specialty site for mathematicians. To note it, the distinction between SE.Math and MathOverflow has at least two parallels: 1. [SE.ComputerScience](https://cs.stackexchange.com/ https://cs.stackexchange.com/) vs. [SE.TheoreticalComputerScience](https://cstheory.stackexchange.com/ https://cstheory.stackexchange.com/) ; and 2. [SE.Physics](https://physics.stackexchange.com/ https://physics.stackexchange.com/) vs. [PhysicsOverflow](https://www.physicsoverflow.org/ https://www.physicsoverflow.org/) . In all three cases, one site's more appropriate for most users, ranging from beginners to experts, while the other site's more specifically for research-level topics in the field.

j7ake 7y ago

"2013-2018: I revisited this conjecture every time when I learn a new tool, without any success though. But at least thinking about it helps me quickly fall asleep many nights." I believe Feynmann said the best people always had the top unsolved problems in the top of their minds and whenever a new technique or result arrived, they would drop everything and try out the new attack on the old problem.

vanderZwan 7y ago
I'm kind of surprised it helped him fall asleep. Whenever my brain starts to think about technical problems in the middle of the night I know I have a few hours of wakeful puzzling ahead of me
- j7ake 7y ago
  I find reading technical work makes my brain fall asleep very quickly. I am sure if I read Knuth's books at night I would be asleep within fifteen minutes. I am not sure thinking about technical work would help sleep, because thinking could be orthogonal to reading.
  - vanderZwan 7y ago
    Oh, I'm with you regarding technical reading being a good way to fall asleep! But reading someone else's train of thought is a different way to engage the brain than trying to figure out a mathematical proof on your own - that's a lot more like what pondering over programming problems would be like for us I think
- H8crilA 7y ago
  I had a friend at university who studied mathematics with me (I studied both maths and computer science, but he was a genuine mathematician, and a pretty good one, much better than me), and he told me that he often takes problems "to bed". After that he would often wake up with at least a half solution in his head. Sometimes a full solution. Ramanujan was like that too. He believed that a goddess sends him solutions in his sleep. Not sure if the goddess is real, but the formulas certainly were real. http://www.oceanvivasilver.com/5-namakkals-infinity-dream-srinivasa-ramanuja http://www.oceanvivasilver.com/5-namakkals-infinity-dream-sr...