It usually seems to be a bad idea to try to solve problems intuitively or use our intuition as evidence to judge issues that our evolutionary ancestors never encountered and therefore were never optimized to judge by natural selection.
In mathematics, intuition is generally not used as evidence to support a conclusion, but instead as a tool with which to search for a rigorous way to solve a problem. First of all, this makes intuition a lot less dangerous. If a voter’s intuition tells him that some particular economic policy will be beneficial, then he is likely to rely on his intuition being right, and can harm public policy if he is wrong. If a mathematician’s intuition tells him that a certain way of attacking a problem is likely to be fruitful, he will fail to solve the problem if he is wrong. But if the mathematician intuitively feels that premise P is true, and he can use it to prove theorem T, he will not state T as fact. Instead, he will state that P implies T, and mention that he finds this especially interesting because he believes P to be true.
Secondly, this makes mathematical intuition trainable. Although our brains are not optimized for math, they are extremely adaptable. When a mathematician tries a fruitless path towards solving a problem as a result of bad intuition, he will notice that he has failed to solve it, update his intuitions accordingly, and try a different way. Similarly, he will notice when his intuition helps him solve a problem, and he’ll figure out what his intuition did right.
Would this be a valid rephrasing of your statement? “When you have done a certain number of problems and understood complex connected conceptions, your intuition becomes molded so that it becomes useful to trust them, but verify them as well.”
Pretty close, but my intuition can still be useful even in instances where it can be less reliable than “trust but verify” would suggest, because in a sufficiently difficult problem, the first possible solution that my intuition hits on is more likely than not to be wrong, but it’s still a lot better than chance. I trust that my intuitions are likely to help me find the right answer or a correct proof eventually if I work at it long enough. In these cases, I don’t assume that a possible solution suggested by my intuition is probably right, and that I just have to verify it. Instead, I assume that it is worth exploring since it has a reasonable probability of being either right or close to right.
In mathematics, intuition is generally not used as evidence to support a conclusion, but instead as a tool with which to search for a rigorous way to solve a problem. First of all, this makes intuition a lot less dangerous. If a voter’s intuition tells him that some particular economic policy will be beneficial, then he is likely to rely on his intuition being right, and can harm public policy if he is wrong. If a mathematician’s intuition tells him that a certain way of attacking a problem is likely to be fruitful, he will fail to solve the problem if he is wrong. But if the mathematician intuitively feels that premise P is true, and he can use it to prove theorem T, he will not state T as fact. Instead, he will state that P implies T, and mention that he finds this especially interesting because he believes P to be true. Secondly, this makes mathematical intuition trainable. Although our brains are not optimized for math, they are extremely adaptable. When a mathematician tries a fruitless path towards solving a problem as a result of bad intuition, he will notice that he has failed to solve it, update his intuitions accordingly, and try a different way. Similarly, he will notice when his intuition helps him solve a problem, and he’ll figure out what his intuition did right.
Would this be a valid rephrasing of your statement? “When you have done a certain number of problems and understood complex connected conceptions, your intuition becomes molded so that it becomes useful to trust them, but verify them as well.”
Pretty close, but my intuition can still be useful even in instances where it can be less reliable than “trust but verify” would suggest, because in a sufficiently difficult problem, the first possible solution that my intuition hits on is more likely than not to be wrong, but it’s still a lot better than chance. I trust that my intuitions are likely to help me find the right answer or a correct proof eventually if I work at it long enough. In these cases, I don’t assume that a possible solution suggested by my intuition is probably right, and that I just have to verify it. Instead, I assume that it is worth exploring since it has a reasonable probability of being either right or close to right.