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How do these kinds of advancements in math happen? Is it a momentary spark of insight after thinking deeply about the problem for 20 years? Or is it more like b

How do these kinds of advancements in math happen? Is it a momentary spark of insight after thinking deeply about the problem for 20 years? Or is it more like brute forcing your way to a solution by trying everything?

getnormality 2y ago
In this case, a ton of progress had already been made. The conjecture had been proved in some cases, reduced to a simpler problem in others. This couple went the last mile of solving the simpler problem in some particularly thorny cases. You're really standing on the shoulders of giants when you rely on the classification of finite simple groups.
- btilly 2y ago
  You're really standing on the shoulders of giants when you rely on the classification of finite simple groups. Giants whose work has (dirty little secret) never truly been verified. The proof totals about 10,000 pages. At the end of the effort to prove it there were lots of very long papers, with a shrinking pool of experts reviewing them. There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere. Hopefully, the growing ease of formalization will lead to a verification some day. But even optimistically that is still a few years out.
  - isotypic 2y ago
    > There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere. My understanding was that the so called "second generation proof" of the classification of finite simple groups led by Gorenstein, Lyons, Solomon has been progressing slowly but steadily, and only the quasithin case had a significant (but now fixed) hole. Are there other significant gaps that aren't as well known?
    - btilly 2y ago
      Huh. Looking at it, they have made a lot more progress than I was aware of. I will correct my opinion. https://mathoverflow.net/questions/114943/where-are-the-second-and-third-generation-proofs-of-the-classification-of-fin https://mathoverflow.net/questions/114943/where-are-the-seco... was very informative on this. Still they have a couple more books of proof left, and I have to wonder how carefully it will be reviewed. This will still be a massive improvement, but I'd be a lot happier if the entire proof could be formalized. Plus there is still a possibility that there proves to be another significant hole. If any theorem needs to be formalized, this is the one. No other theorem is this big, this hard to prove, and this important to get right.
      - getnormality 2y ago
        At the moment we seem to have exactly one guy equipped to do this, and he'll be busy with something else for the next few years: https://www.reddit.com/r/math/comments/176vtju/kevin_buzzard_got_a_research_grant_to_begin_the/ https://www.reddit.com/r/math/comments/176vtju/kevin_buzzard...
jebarker 2y ago
My experience from proving a moderately complicated result in my PhD was that it's neither. There wasn't enough time to brute force by trying many complete solutions, but it also wasn't a single flash of insight. It was more a case of following a path towards the solution based on intuition and then trying a few different approaches when getting stuck to keep making progress. Sometimes that involves backtracking when you realize you took a wrong path.
- sweezyjeezy 2y ago
  Yeah agreed - there are actually many, smaller flashes of insight, but most of them don't lead to anything. I once joked that you could probably compress all the time I was actually going in the right direction in my PhD down to about a month or two. That's a bit glib, often seeing why an approach fails gives you a much better idea of what a proof 'has to look like' or 'has to be able to overcome'. But many months of my PhD were working on complete dead-ends, and I certainly had a few very dark days because of that. Research math takes a lot of perseverance.
  - jebarker 2y ago
    > Research math takes a lot of perseverance. Yup, I think stubbornness/perseverance is the most useful transferrable skill I got from doing research math. It's a double-edged sword though as I often just can't give up working on something when I really should in my tech job.
bloomingkales 2y ago
The structure of DNA was seen in a vision in James Watson's dream. Some say it's subconscious problem solving and I think most down to earth people agree with that, but some less down to earth people will absolutely attribute it to god (I'm in the latter). If we were to entertain a silly proposition, something in the universe could just move our story along, all of a sudden. These paradigm shifts just seem to appear.
- FartyMcFarter 2y ago
  Subconscious problem solving is definitely a thing as far as I'm concerned. It's happened several times that I struggled with a bug for hours, then suddenly came up with a key insight during the commute home (or while taking a shower or whatnot), while not actively thinking about the problem. I can't explain this any other way than some kind of subconscious "brainstorming" taking place. As a side note, this doesn't mean the hours of conciously struggling with the problem were a waste. I bet that this period of focus on the problem is what allows for the later insights to happen. Whether it's data gathering that alows the insights to happen, or even just giving importance to the problem by focusing on it. Most likely it's both.
  - fsckboy 2y ago
    an alternate hypothesis is that all thinking is unconscious with consciousness being the sum of many unconscious processes. So, while it's true as you say "that some kind of subconscious "brainstorming" taking place", that would just true all the time, and you only notice it when your conscious thoughts are of something else. (n.b. historically speaking in the literature, "subconscious" was the word used to describe Jung's "woo-woo" ideas about a collective subconscious shared across populations, and unconscious was the word used for ideas about a single brain a la Freud which is what we are talking about here)
    - deleted 2y ago
      [deleted]
    - lanstin 2y ago
deleted 2y ago
[deleted]

getnormality 2y ago

In this case, a ton of progress had already been made. The conjecture had been proved in some cases, reduced to a simpler problem in others. This couple went the last mile of solving the simpler problem in some particularly thorny cases. You're really standing on the shoulders of giants when you rely on the classification of finite simple groups.

btilly 2y ago
You're really standing on the shoulders of giants when you rely on the classification of finite simple groups. Giants whose work has (dirty little secret) never truly been verified. The proof totals about 10,000 pages. At the end of the effort to prove it there were lots of very long papers, with a shrinking pool of experts reviewing them. There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere. Hopefully, the growing ease of formalization will lead to a verification some day. But even optimistically that is still a few years out.
- isotypic 2y ago
  > There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere. My understanding was that the so called "second generation proof" of the classification of finite simple groups led by Gorenstein, Lyons, Solomon has been progressing slowly but steadily, and only the quasithin case had a significant (but now fixed) hole. Are there other significant gaps that aren't as well known?
  - btilly 2y ago
    Huh. Looking at it, they have made a lot more progress than I was aware of. I will correct my opinion. https://mathoverflow.net/questions/114943/where-are-the-second-and-third-generation-proofs-of-the-classification-of-fin https://mathoverflow.net/questions/114943/where-are-the-seco... was very informative on this. Still they have a couple more books of proof left, and I have to wonder how carefully it will be reviewed. This will still be a massive improvement, but I'd be a lot happier if the entire proof could be formalized. Plus there is still a possibility that there proves to be another significant hole. If any theorem needs to be formalized, this is the one. No other theorem is this big, this hard to prove, and this important to get right.
    - getnormality 2y ago
      At the moment we seem to have exactly one guy equipped to do this, and he'll be busy with something else for the next few years: https://www.reddit.com/r/math/comments/176vtju/kevin_buzzard_got_a_research_grant_to_begin_the/ https://www.reddit.com/r/math/comments/176vtju/kevin_buzzard...

jebarker 2y ago

My experience from proving a moderately complicated result in my PhD was that it's neither. There wasn't enough time to brute force by trying many complete solutions, but it also wasn't a single flash of insight. It was more a case of following a path towards the solution based on intuition and then trying a few different approaches when getting stuck to keep making progress. Sometimes that involves backtracking when you realize you took a wrong path.

sweezyjeezy 2y ago
Yeah agreed - there are actually many, smaller flashes of insight, but most of them don't lead to anything. I once joked that you could probably compress all the time I was actually going in the right direction in my PhD down to about a month or two. That's a bit glib, often seeing why an approach fails gives you a much better idea of what a proof 'has to look like' or 'has to be able to overcome'. But many months of my PhD were working on complete dead-ends, and I certainly had a few very dark days because of that. Research math takes a lot of perseverance.
- jebarker 2y ago
  > Research math takes a lot of perseverance. Yup, I think stubbornness/perseverance is the most useful transferrable skill I got from doing research math. It's a double-edged sword though as I often just can't give up working on something when I really should in my tech job.

bloomingkales 2y ago

The structure of DNA was seen in a vision in James Watson's dream. Some say it's subconscious problem solving and I think most down to earth people agree with that, but some less down to earth people will absolutely attribute it to god (I'm in the latter). If we were to entertain a silly proposition, something in the universe could just move our story along, all of a sudden. These paradigm shifts just seem to appear.

FartyMcFarter 2y ago
Subconscious problem solving is definitely a thing as far as I'm concerned. It's happened several times that I struggled with a bug for hours, then suddenly came up with a key insight during the commute home (or while taking a shower or whatnot), while not actively thinking about the problem. I can't explain this any other way than some kind of subconscious "brainstorming" taking place. As a side note, this doesn't mean the hours of conciously struggling with the problem were a waste. I bet that this period of focus on the problem is what allows for the later insights to happen. Whether it's data gathering that alows the insights to happen, or even just giving importance to the problem by focusing on it. Most likely it's both.
- fsckboy 2y ago
  an alternate hypothesis is that all thinking is unconscious with consciousness being the sum of many unconscious processes. So, while it's true as you say "that some kind of subconscious "brainstorming" taking place", that would just true all the time, and you only notice it when your conscious thoughts are of something else. (n.b. historically speaking in the literature, "subconscious" was the word used to describe Jung's "woo-woo" ideas about a collective subconscious shared across populations, and unconscious was the word used for ideas about a single brain a la Freud which is what we are talking about here)
  - deleted 2y ago
    [deleted]
  - lanstin 2y ago

deleted 2y ago

[deleted]