A significant reason I’ve had problems with advanced Calculus is that my brain starts inventing too many justifications for things, and then I become unable to distinguish between remembered rules which are valid and ones which my mind invented without sufficient justification.
That may reflect more of a lack of sufficient practice on your part than anything else. It takes a long time to become familiar enough with a topic that your brain can start intuitively and spontaneously generating good ideas on that topic. As an example, despite having spent several years playing chess, I still have to consider every position carefully and with deliberation; although there have been cases in which the move which immediately springs to mind is correct, I’ve found that in general the opposite is true. However, there is evidence that top grandmasters do not view chess positions this way; their play is based a lot more on “feeling” than “thinking”. (I don’t have the source for it, but I definitely remember reading something about it in both GEB and Thinking, Fast and Slow.) Clearly, this means that despite having played chess for so long, I have still not yet reached the level at which intuition can play a significant role in my calculations. Based on what you’ve written here, I would judge it likely that you are in a similar situation with respect to calculus.
(Also see this. I think that the “post-rigorous” stage described in this post matches nicely with what Jonah said above.)
I feel a link to an old comment of mine belongs somewhere under this top-level post, and this subthread might be the best place for it, so I’m putting it here...
If you’re right, in chess it requires years and years of domain specific practice to get pattern recognition skills adequately prepared so that scrupulous thought is not required when evaluating moves. That doesn’t seem like an argument against the importance of scrupulous thought to me, it seems like the opposite. Scrupulous thought is very hard to avoid relying on.
I think you’re wrong however. I think once you reach a certain level of familiarity with a subject, the distinction between pattern recognition and scrupulous reasoning itself breaks down. I don’t think chess experts only use the raw processing power of their subconscious minds when evaluating the board, I think they alternate between making bottom-up assessments and top-down judgements. The accounts given in the neurology books are reactions to the popular perception that reasoning abilities are all that matters in chess, but if they’ve given you the impression that reasoning isn’t important in chess then I feel like they may have gone too far in emphasizing their point. Expert chess players certainly feel like they’re doing something important with their conscious minds. They give narrative descriptions of their rounds regularly. I acknowledge that explicit thought is not all there is to playing chess, but I’m not prepared to say experts’ accounts of their thoughts are just egoist delusions, or anything like that.
I suppose one point I’m trying to make here is that biased stupid thought and genius insightful thought feel the same from the inside. And I think even geniuses have biased stupid thoughts often, even within their fields of expertise, and so the importance of rigor should not be downplayed even for them. Genius isn’t a quality for avoiding bad thoughts, it’s quality that makes someone capable of having a few good thoughts in addition to all their other bad ones. When genius is paired with good filters, then it produces excellence regularly. Without good filters, it’s much less reliable.
Finally, when you’re dealing with theories about the universe the situation is different than when dealing with strategy games. You can’t make a dumb subargument and then a smart subargument and have the two statements combine to produce a moderately valuable hypothesis. If you start driving down the wrong street, correctly following the rest of a list of directions will not be helpful to you. Rigor is important throughout all steps of the entire process. No mistakes can lead to success without first being undone (or at least almost none will—there are always exceptions).
I think even geniuses have biased stupid thoughts often, even within their fields of expertise, and so the importance of rigor should not be downplayed even for them.
To use the chess analogy once more: this seems to conflict with the fact that in chess, top grandmasters’ intuitions are almost always correct (and the rare exceptions almost always involve some absurd-looking move that only gets found after the fact through post-game computer analysis). Quite often, you’ll see a chess author touting the importance of “quiet judgment” instead of “brute calculation”; that suggests extremely strongly to me that most grandmasters don’t calculate out every move—and for good reason: it would be exhausting!
Likewise, I’m given to understand many mathematicians also have this sort of intuitive judgment; of course, it takes a long time to build up the necessary background knowledge and brain connections for such judgment, but then, Jonah never claimed otherwise. From the post itself:
It took me 10,000+ hours to learn how to “see” patterns in evidence in the way that I can now. Right now, I don’t know how to communicate how to do it succinctly. It’s too much for me to do as an individual: as far as I know, nobody has ever been able to convey the relevant information to a sizable audience!
If we could find a way to quickly build up the type of judgment described above, it could very well change the way people do things forever, but alas, we’re not quite there. That’s the whole point of Jonah’s request for collaboration. (In an ideal world, I’d participate, but as a 17-year-old I doubt I’d have much to contribute and a lot of my time is used up preparing for college at this stage anyway, so… yeah. Unfortunate.)
I was not aware most grandmasters’ first instincts ended up being correct usually, interesting.
Likewise, I’m given to understand many mathematicians also have this sort of intuitive judgment; of course, it takes a long time to build up the necessary background knowledge and brain connections for such judgment, but then, Jonah never claimed otherwise. From the post itself:
I’ve been changing my position somewhat thoughout this conversation, just so it’s clear. At this point, I guess what I think is that a hard distinction between “reasoning” and “pattern recognition” doesn’t make much sense. It seems like successful pattern recognition is to a significant extent comprised of scrupulously reasoned ideas that have been internalized. If someone hypothetically refused to use explicit reasoning while being taught to recognize certain patterns, I’d expect that person to have a more difficult time learning. Reasoning about ideas in the way that is slow and deliberative eventually makes patterns easier to recognize in the way that is fast and intuitive. If someone doesn’t incorporate slow thought originated restrictions into their fast pattern matching capabilities, then they will start believing in faces that appear in the clouds, assuming that they ever learn to pattern match at all.
That may reflect more of a lack of sufficient practice on your part than anything else. It takes a long time to become familiar enough with a topic that your brain can start intuitively and spontaneously generating good ideas on that topic. As an example, despite having spent several years playing chess, I still have to consider every position carefully and with deliberation; although there have been cases in which the move which immediately springs to mind is correct, I’ve found that in general the opposite is true. However, there is evidence that top grandmasters do not view chess positions this way; their play is based a lot more on “feeling” than “thinking”. (I don’t have the source for it, but I definitely remember reading something about it in both GEB and Thinking, Fast and Slow.) Clearly, this means that despite having played chess for so long, I have still not yet reached the level at which intuition can play a significant role in my calculations. Based on what you’ve written here, I would judge it likely that you are in a similar situation with respect to calculus.
(Also see this. I think that the “post-rigorous” stage described in this post matches nicely with what Jonah said above.)
Thanks :-). I was going to respond along these lines before seeing that you had spoken my mind.
I feel a link to an old comment of mine belongs somewhere under this top-level post, and this subthread might be the best place for it, so I’m putting it here...
If you’re right, in chess it requires years and years of domain specific practice to get pattern recognition skills adequately prepared so that scrupulous thought is not required when evaluating moves. That doesn’t seem like an argument against the importance of scrupulous thought to me, it seems like the opposite. Scrupulous thought is very hard to avoid relying on.
I think you’re wrong however. I think once you reach a certain level of familiarity with a subject, the distinction between pattern recognition and scrupulous reasoning itself breaks down. I don’t think chess experts only use the raw processing power of their subconscious minds when evaluating the board, I think they alternate between making bottom-up assessments and top-down judgements. The accounts given in the neurology books are reactions to the popular perception that reasoning abilities are all that matters in chess, but if they’ve given you the impression that reasoning isn’t important in chess then I feel like they may have gone too far in emphasizing their point. Expert chess players certainly feel like they’re doing something important with their conscious minds. They give narrative descriptions of their rounds regularly. I acknowledge that explicit thought is not all there is to playing chess, but I’m not prepared to say experts’ accounts of their thoughts are just egoist delusions, or anything like that.
I suppose one point I’m trying to make here is that biased stupid thought and genius insightful thought feel the same from the inside. And I think even geniuses have biased stupid thoughts often, even within their fields of expertise, and so the importance of rigor should not be downplayed even for them. Genius isn’t a quality for avoiding bad thoughts, it’s quality that makes someone capable of having a few good thoughts in addition to all their other bad ones. When genius is paired with good filters, then it produces excellence regularly. Without good filters, it’s much less reliable.
Finally, when you’re dealing with theories about the universe the situation is different than when dealing with strategy games. You can’t make a dumb subargument and then a smart subargument and have the two statements combine to produce a moderately valuable hypothesis. If you start driving down the wrong street, correctly following the rest of a list of directions will not be helpful to you. Rigor is important throughout all steps of the entire process. No mistakes can lead to success without first being undone (or at least almost none will—there are always exceptions).
To use the chess analogy once more: this seems to conflict with the fact that in chess, top grandmasters’ intuitions are almost always correct (and the rare exceptions almost always involve some absurd-looking move that only gets found after the fact through post-game computer analysis). Quite often, you’ll see a chess author touting the importance of “quiet judgment” instead of “brute calculation”; that suggests extremely strongly to me that most grandmasters don’t calculate out every move—and for good reason: it would be exhausting!
Likewise, I’m given to understand many mathematicians also have this sort of intuitive judgment; of course, it takes a long time to build up the necessary background knowledge and brain connections for such judgment, but then, Jonah never claimed otherwise. From the post itself:
If we could find a way to quickly build up the type of judgment described above, it could very well change the way people do things forever, but alas, we’re not quite there. That’s the whole point of Jonah’s request for collaboration. (In an ideal world, I’d participate, but as a 17-year-old I doubt I’d have much to contribute and a lot of my time is used up preparing for college at this stage anyway, so… yeah. Unfortunate.)
I was not aware most grandmasters’ first instincts ended up being correct usually, interesting.
I’ve been changing my position somewhat thoughout this conversation, just so it’s clear. At this point, I guess what I think is that a hard distinction between “reasoning” and “pattern recognition” doesn’t make much sense. It seems like successful pattern recognition is to a significant extent comprised of scrupulously reasoned ideas that have been internalized. If someone hypothetically refused to use explicit reasoning while being taught to recognize certain patterns, I’d expect that person to have a more difficult time learning. Reasoning about ideas in the way that is slow and deliberative eventually makes patterns easier to recognize in the way that is fast and intuitive. If someone doesn’t incorporate slow thought originated restrictions into their fast pattern matching capabilities, then they will start believing in faces that appear in the clouds, assuming that they ever learn to pattern match at all.