If so, I’m really interested with techniques you have in mind for starting from a complex mess/intuitions and getting to a formal problem/setting.
This deserves its own separate response.
At a high level, we can split this into two parts:
developing intuitions
translating intuitions into math
We’ve talked about the translation step a fair bit before (the conversation which led to this post). A core point of that post is that the translation from intuition to math should be faithful, and not inject any “extra assumptions” which weren’t part of the math. So, for instance, if I have an intuition that some function is monotonically increasing between 0 and 1, then my math should say “assume f(x) is monotonically increasing between 0 and 1″, not “let f(x) = x^2”; the latter would be making assumptions not justified by my intuition. (Some people also make the opposite mistake—failing to include assumptions which their intuition actually does believe. Usually this is because the intuition only feels like it’s mostly true or usually true, rather than reliable and certain; the main way to address this is to explicitly call the assumption an approximation.)
On the flip side, that implies that the intuition has to do quite a bit of work, and the linked post talked a lot less about that. How do we build these intuitions in the first place? The main way is to play around with the system/problem. Try examples, and see what they do. Try proofs, and see where they fail. Play with variations on the system/problem. Look for bottlenecks and barriers, look for approximations, look for parts of the problem-space/state-space with different behavior. Look for analogous systems in the real world, and carry over intuitions from them. Come up with hypotheses/conjectures, and test them.
This deserves its own separate response.
At a high level, we can split this into two parts:
developing intuitions
translating intuitions into math
We’ve talked about the translation step a fair bit before (the conversation which led to this post). A core point of that post is that the translation from intuition to math should be faithful, and not inject any “extra assumptions” which weren’t part of the math. So, for instance, if I have an intuition that some function is monotonically increasing between 0 and 1, then my math should say “assume f(x) is monotonically increasing between 0 and 1″, not “let f(x) = x^2”; the latter would be making assumptions not justified by my intuition. (Some people also make the opposite mistake—failing to include assumptions which their intuition actually does believe. Usually this is because the intuition only feels like it’s mostly true or usually true, rather than reliable and certain; the main way to address this is to explicitly call the assumption an approximation.)
On the flip side, that implies that the intuition has to do quite a bit of work, and the linked post talked a lot less about that. How do we build these intuitions in the first place? The main way is to play around with the system/problem. Try examples, and see what they do. Try proofs, and see where they fail. Play with variations on the system/problem. Look for bottlenecks and barriers, look for approximations, look for parts of the problem-space/state-space with different behavior. Look for analogous systems in the real world, and carry over intuitions from them. Come up with hypotheses/conjectures, and test them.