Comment by charlangas
2 days ago
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?
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.
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.
> 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.
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.
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.
> 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?
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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.
> The structure of DNA was seen in a vision in James Watson's dream
I believe this is apocryphal. Watson likely said this because he stole Rosalind Franklin's research.
I disagree. What he stole was the pictures she took. He really did come up with the structure.
Mind you he was helped by a nice coincidence. Franklin knew all 230 space symmetry groups, and so had to sort through them. Watson only really knew one - his PhD thesis was on a protein with the same group as DNA.
I can only think of that example because it’s the only one I know of where the scientist admitted to something divine.
This is probably something I need to research more, because the questions scientists are dealing with are too deep to ignore the one big question, and I wonder who struggled with it.
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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.
Subconscious synthesis of seemingly unrelated strands of thought is the basis of assessing an event's meaning, for me. A day or two will pass after the event, and I notice the meaning evolves even without new information and without conscious thinking about the event.
For me, the subconscious is the wellspring of sudden insights.
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)
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