The first is that they’re getting stuck because the dead end thoughts are becoming too salient and crowding out other thoughts.
I agree this is a factor in problem-solving. I’ve found it to be important too. However, I suspect this can’t be the main reason behind the “magic genie” phenomenon because if it were, you’d expect that mind-quieting meditations between bursts of mathematical effort would be vastly more productive than spending hours on problems and taking breaks. E.g., spending fifteen minutes on a tough problem and then spending five minutes meditating, cycled three times, would produce vastly better results than thinking about the problem for an hour. I’m not aware that this is the case, nor that the 20% most prodigious mathematicians are above average in their interest in mindfulness.
I think that the problem really is on your mind, just not at the forefront.
I think you must be right! Something is processing the problem and producing a solution later down the road.
The classic example of this is the benzene ring and the daydream about snakes eating their own tails.
Yeah, I love that example! Oddly, it turns out that most mathematicians do not relate to the supposed experience of a solution presenting itself in a dream. Many, including Hadamard, have mentioned waking up to have the answer to a problem they’ve been working on trumpeted into mind, but not clearly as a result of dreamed experiences. Because of this (and also because of people like Stephen LaBerge talking about problem-solving in lucid dreams), I started digging into known examples of insights derived from dreams. This one was the only one I could find—and it turns out that it might not have ever happened!
I’ve gotten a whole lot more mileage out of looking at how thoughts and feelings relate to other thoughts and feelings, and not so much where they physically live. Have you actually found otherwise?
By and large, no. But in this domain, yes!
The unit that spawned this line of inquiry was originally based entirely on my experience in applying mindfulness to martial arts. In aikido, you can do some really remarkable things, some of which seem downright implausible, by slipping into an appropriately mindful state. This usually gets expressed very mystically (“You must have no sense of yourself, no desire to throw your opponent, and only then will they fall—but you must not strive for mindfulness with this as your goal”), so I figured I’d give a shot at operationalizing it when teaching aikido. This was really quite successful, allowing me to give people progress comparable (claims my brain from past dojo experience) to the first year of training in just a few months of weekly sessions.
This gave me some pretty good epistemic tools that have proven to be critical for me as I’ve grown as a rationalist. So, I thought to share them, and one of our wonderful volunteers (Dan Keys) pointed out to me that much of what I was talking about seemed to be the division between the sympathetic and parasympathetic nervous systems. As I dug into that, I found that the correspondence was surprisingly exact. Basically, it sounds like the Pareto efficiency of mindfulness consists of being in parasympathetic mode most of the time.
This allowed for some hacks that I found immediately useful. For instance, I knew that people expressed stress physically by doing things like rubbing their necks and breathing more shallowly, but now I know that the whole host of how the two branches of the autonomic nervous system oppose one another can offer signals for where one is on the PNS/SNS spectrum. This generated a wide range of hypotheses, some of which I’m still testing. For instance, it turns out that the parasympathetic side governs sexual arousal, but the sympathetic side governs orgasm. There should be something in this space that allows one to use sexuality to hack mindfulness. (This might explain what tantra was trying to accomplish.)
However, in general I agree with you. Internal motive structures are usually much more useful to understand in my experience than neural structures are. As you rightly point out, the layer of complexity at which we intervene seems to be at thoughts and feelings, not at neurology.
...but if we can pinpoint some neural structure that does this background thinking, and we know roughly how it interacts with the autonomic nervous system and possibly some other things, that could result in something insanely useful such as using body posture to massively accelerate the rate of problem incubation.
Speaking of mathematicians and dreams, I found a hilarious quote some time ago:
“Once in my life I had a mathematical dream which proved correct. I was twenty years old. I thought, my God, this is wonderful, I won’t have to work, it will all come in dreams! But it never happened again.”
--Stanislaw Ulam; January 14, 1974, in “Conversations with Gian-Carlo Rota”; as quoted on pg262 of Turing’s Cathedral (2012) by George Dyson
I agree this is a factor in problem-solving. I’ve found it to be important too. However, I suspect this can’t be the main reason behind the “magic genie” phenomenon because if it were, you’d expect that mind-quieting meditations between bursts of mathematical effort would be vastly more productive than spending hours on problems and taking breaks. E.g., spending fifteen minutes on a tough problem and then spending five minutes meditating, cycled three times, would produce vastly better results than thinking about the problem for an hour. I’m not aware that this is the case, nor that the 20% most prodigious mathematicians are above average in their interest in mindfulness.
I agree that it’s not the main thing, but not with your analysis. For one, this “mindfulness” thing is never really unpacked well. It could be that the habit of not focusing and bouncing between ideas (that’s considered “unmindful”, right?) is what it takes to not get stuck in ruts, and that the helpful mindfulness related bit is meta-awareness that “I’m noticing that I’m stuck”—and then fixing it instead of freaking out about it.
Instead of practicing mindfulness by itself, I’d hold mindfulness as an ideal and attack the specific blocks more directly.
Oddly, it turns out that most mathematicians do not relate to the supposed experience of a solution presenting itself in a dream. Many, including Hadamard, have mentioned waking up to have the answer to a problem they’ve been working on trumpeted into mind, but not clearly as a result of dreamed experiences
This actually doesn’t change my hypothesis much. I’m hypothesizing something that happens without requiring awareness. I occasionally notice myself making strange metaphorical connections that were there outside my awareness for some time and finally got bumped in. I very much expect this to happen completely outside awareness a lot. Heck, Milton Erickson was famous for doing this on purpose as a technique in therapy!
By and large, no. But in this domain, yes! [...]what I was talking about seemed to be the division between the sympathetic and parasympathetic nervous systems.
Which domain exactly?
Interesting. I’ll have to look into that distinction more, and generally spend more time in that perspective. I have gotten similar benefits, just on a fairly small scale—it’d just allow me to make sense of things there were a bit elusive and import it back to the individual thoughts/feelings level model.
Alas, no. But you are certainly not the first person to ask! I get this request on almost a weekly basis. I’m tempted to make a blog just for the social acclaim, but given the opportunity cost I’ll hold off until I can see a way to leverage something like that into something more clearly tied into world-saving.
I agree this is a factor in problem-solving. I’ve found it to be important too. However, I suspect this can’t be the main reason behind the “magic genie” phenomenon because if it were, you’d expect that mind-quieting meditations between bursts of mathematical effort would be vastly more productive than spending hours on problems and taking breaks. E.g., spending fifteen minutes on a tough problem and then spending five minutes meditating, cycled three times, would produce vastly better results than thinking about the problem for an hour. I’m not aware that this is the case, nor that the 20% most prodigious mathematicians are above average in their interest in mindfulness.
I think you must be right! Something is processing the problem and producing a solution later down the road.
Yeah, I love that example! Oddly, it turns out that most mathematicians do not relate to the supposed experience of a solution presenting itself in a dream. Many, including Hadamard, have mentioned waking up to have the answer to a problem they’ve been working on trumpeted into mind, but not clearly as a result of dreamed experiences. Because of this (and also because of people like Stephen LaBerge talking about problem-solving in lucid dreams), I started digging into known examples of insights derived from dreams. This one was the only one I could find—and it turns out that it might not have ever happened!
By and large, no. But in this domain, yes!
The unit that spawned this line of inquiry was originally based entirely on my experience in applying mindfulness to martial arts. In aikido, you can do some really remarkable things, some of which seem downright implausible, by slipping into an appropriately mindful state. This usually gets expressed very mystically (“You must have no sense of yourself, no desire to throw your opponent, and only then will they fall—but you must not strive for mindfulness with this as your goal”), so I figured I’d give a shot at operationalizing it when teaching aikido. This was really quite successful, allowing me to give people progress comparable (claims my brain from past dojo experience) to the first year of training in just a few months of weekly sessions.
This gave me some pretty good epistemic tools that have proven to be critical for me as I’ve grown as a rationalist. So, I thought to share them, and one of our wonderful volunteers (Dan Keys) pointed out to me that much of what I was talking about seemed to be the division between the sympathetic and parasympathetic nervous systems. As I dug into that, I found that the correspondence was surprisingly exact. Basically, it sounds like the Pareto efficiency of mindfulness consists of being in parasympathetic mode most of the time.
This allowed for some hacks that I found immediately useful. For instance, I knew that people expressed stress physically by doing things like rubbing their necks and breathing more shallowly, but now I know that the whole host of how the two branches of the autonomic nervous system oppose one another can offer signals for where one is on the PNS/SNS spectrum. This generated a wide range of hypotheses, some of which I’m still testing. For instance, it turns out that the parasympathetic side governs sexual arousal, but the sympathetic side governs orgasm. There should be something in this space that allows one to use sexuality to hack mindfulness. (This might explain what tantra was trying to accomplish.)
However, in general I agree with you. Internal motive structures are usually much more useful to understand in my experience than neural structures are. As you rightly point out, the layer of complexity at which we intervene seems to be at thoughts and feelings, not at neurology.
...but if we can pinpoint some neural structure that does this background thinking, and we know roughly how it interacts with the autonomic nervous system and possibly some other things, that could result in something insanely useful such as using body posture to massively accelerate the rate of problem incubation.
Speaking of mathematicians and dreams, I found a hilarious quote some time ago:
--Stanislaw Ulam; January 14, 1974, in “Conversations with Gian-Carlo Rota”; as quoted on pg262 of Turing’s Cathedral (2012) by George Dyson
I agree that it’s not the main thing, but not with your analysis. For one, this “mindfulness” thing is never really unpacked well. It could be that the habit of not focusing and bouncing between ideas (that’s considered “unmindful”, right?) is what it takes to not get stuck in ruts, and that the helpful mindfulness related bit is meta-awareness that “I’m noticing that I’m stuck”—and then fixing it instead of freaking out about it.
Instead of practicing mindfulness by itself, I’d hold mindfulness as an ideal and attack the specific blocks more directly.
This actually doesn’t change my hypothesis much. I’m hypothesizing something that happens without requiring awareness. I occasionally notice myself making strange metaphorical connections that were there outside my awareness for some time and finally got bumped in. I very much expect this to happen completely outside awareness a lot. Heck, Milton Erickson was famous for doing this on purpose as a technique in therapy!
Which domain exactly?
Interesting. I’ll have to look into that distinction more, and generally spend more time in that perspective. I have gotten similar benefits, just on a fairly small scale—it’d just allow me to make sense of things there were a bit elusive and import it back to the individual thoughts/feelings level model.
I wish to subscribe to your newsletter. Do you have a blog or summat?
Alas, no. But you are certainly not the first person to ask! I get this request on almost a weekly basis. I’m tempted to make a blog just for the social acclaim, but given the opportunity cost I’ll hold off until I can see a way to leverage something like that into something more clearly tied into world-saving.
Did Ramanujan receive solutions to math problems in dreams?