Tangential, but may generalize: I strongly disagree that, in math, looking for a proof is a better use of time than trying to articulate the intuitions behind the conjecture. Those intuitions are the reason the conjecture exists in the first place! They are the only conjecture-specific clue you have about how to go about a proof. Ignore the intuitions, and you’re stuck searching through proof-space with generic heuristics.
This is a trap I’ve fallen into many times in my own research: coming up with an intuitive conjecture, and then looking for a proof without paying attention to the intuitions which led to the conjecture in the first place. Every time it’s happened, I spent anywhere from days to months crunching symbols before stepping back and saying “why do I believe this in the first place?” And once I ask that question, I often go “oh, this suggests a whole different approach”, and things get much easier.
Or as Wheeler put it, “never make a calculation until you know the answer”. Same with proofs. It’s a lesson which has been burned into me by repeated failure.
Tangential, but may generalize: I strongly disagree that, in math, looking for a proof is a better use of time than trying to articulate the intuitions behind the conjecture. Those intuitions are the reason the conjecture exists in the first place! They are the only conjecture-specific clue you have about how to go about a proof. Ignore the intuitions, and you’re stuck searching through proof-space with generic heuristics.
This is a trap I’ve fallen into many times in my own research: coming up with an intuitive conjecture, and then looking for a proof without paying attention to the intuitions which led to the conjecture in the first place. Every time it’s happened, I spent anywhere from days to months crunching symbols before stepping back and saying “why do I believe this in the first place?” And once I ask that question, I often go “oh, this suggests a whole different approach”, and things get much easier.
Or as Wheeler put it, “never make a calculation until you know the answer”. Same with proofs. It’s a lesson which has been burned into me by repeated failure.