The phenomenon definitely happens. There’s no question that insights pop into mind out of context for at least some rather large chunk of the population. Ask just about any math professor or graduate student: I’m willing to give 85% odds that they’ll indicate that key insights to problems they had been working on have occurred to them more than once during times when they weren’t thinking about the problems in question.
The question was whether the insight is more likely to pop into mind when stepping away from the problem.
I seem to recall an experiment in which subjects told to think about a hard problem for two hours had a significantly lower success rate than subjects told to think about it for half an hour, go play sudoku for an hour, then think about it for another half an hour. But grep and google are coming up empty for me at the moment, so take this with a grain of salt.
Do you think that the point about mathematicians coming consistently to this strategy does not constitute evidence? It certainly isn’t overwhelming evidence, but it seems suspicious that virtually all mathematicians who talk about mathematical process talk about the importance of walking away from problems. I’m personally not aware of a single mathematician who thinks that such a practice is unnecessary.
(My dissertation was on mathematicians’ methods for navigating struggle in their research, and as part of that I did a fair amount of looking both at mathematicians’ accounts and at summaries of such accounts. The closest thing to denying this phenomenon I’ve encountered is the strong insistance of a very tiny minority of mathematicians that “intuition” has nothing to do with mathematics—but those same people still reported needing multiple problems to work on in parallel so that they could turn their attention away from a given problem they were stuck on.)
Mathematicians’ claims may too be explained by selective memory effects mentioned by fubarobfusco in the first comment in this thread. The question is how to discriminate between the case when the mathematicians’ testimonies are reflecting an existing phenomenon and the case when they result from a bias. Even if the insights were less likely to materialise after stepping away, there would be plenty of cases of this happening, so the fact that virtually every mathematician can remember few of them wouldn’t be surprising.
Even if the insights were less likely to materialise after stepping away, there would be plenty of cases of this happening, so the fact that virtually every mathematician can remember few of them wouldn’t be surprising.
Point taken. I guess the likelihood ratio for this strategy being actively helpful is closer to 1 than I had previously thought.
However, it’s not just a few incidences. It’s remarkably frequent. And it’s also still valuable to note that problem-solving can occur in the background without the need for conscious attention. Even if the background process turns out not to be as efficient as conscious reflection, freeing up attention while still working on the problem looks like an obvious win to me.
The question was whether the insight is more likely to pop into mind when stepping away from the problem.
I seem to recall an experiment in which subjects told to think about a hard problem for two hours had a significantly lower success rate than subjects told to think about it for half an hour, go play sudoku for an hour, then think about it for another half an hour. But grep and google are coming up empty for me at the moment, so take this with a grain of salt.
Do you think that the point about mathematicians coming consistently to this strategy does not constitute evidence? It certainly isn’t overwhelming evidence, but it seems suspicious that virtually all mathematicians who talk about mathematical process talk about the importance of walking away from problems. I’m personally not aware of a single mathematician who thinks that such a practice is unnecessary.
(My dissertation was on mathematicians’ methods for navigating struggle in their research, and as part of that I did a fair amount of looking both at mathematicians’ accounts and at summaries of such accounts. The closest thing to denying this phenomenon I’ve encountered is the strong insistance of a very tiny minority of mathematicians that “intuition” has nothing to do with mathematics—but those same people still reported needing multiple problems to work on in parallel so that they could turn their attention away from a given problem they were stuck on.)
Mathematicians’ claims may too be explained by selective memory effects mentioned by fubarobfusco in the first comment in this thread. The question is how to discriminate between the case when the mathematicians’ testimonies are reflecting an existing phenomenon and the case when they result from a bias. Even if the insights were less likely to materialise after stepping away, there would be plenty of cases of this happening, so the fact that virtually every mathematician can remember few of them wouldn’t be surprising.
Point taken. I guess the likelihood ratio for this strategy being actively helpful is closer to 1 than I had previously thought.
However, it’s not just a few incidences. It’s remarkably frequent. And it’s also still valuable to note that problem-solving can occur in the background without the need for conscious attention. Even if the background process turns out not to be as efficient as conscious reflection, freeing up attention while still working on the problem looks like an obvious win to me.