I think the situation is much more complicated than this, at least for experts.
I agree that the situation is more complicated, I disagree that it is “much more complicated”. Yes, mathematicians rely on intuition to feel in the gaps in proofs and to seek out the errors in proofs. And yet, it is uncontroversial that having a proof should make you much more confident in a mathematical statement than just having an intuition. In reality, there is a spectrum that goes roughly “intuition that T is correct” → “informal argument for T” → “idea for how to prove T” → “sketch of a proof of T” → “unvetted proof of T” → “vetted, peer-reviewed proof of T” → “machine verifiable formal proof of T”.
Why do you believe this? Have you actually tried?
Have I actually tried what? As to why I believe this, I think I already gave an “explicit reasoning” argument: and, yes, my intuition and life experience confirm this although this is not something that I can transmit to you directly.
This seems to imply a fairly different cognitive algorithm than “combine your intuition and your explicit reasoning” (which, to be clear, is a thing I actually do, but probably a different way than you), namely “let your explicit reasoning veto anything it doesn’t think is worth doing.”
This is a wrong way to look at this. Intuition and explicit reasoning are not two separate judges that give two separate verdicts. Combining intuition and explicit reasoning doesn’t mean averaging the results. The way it works is, when your intuition and reasoning disagree, you should try to understand why. You should pit them against each other and let them fight it out, and in the end you have something that resembles a system of formal arguments with intuition answering some sub-queries, and your reasoning and intuition both endorse the result. This is what I mean by “understanding on an intellectual level”.
Okay, but Kaj and Val have both been saying (and I agree) that doing this runs the risk of making it harder to actually communicate Looking itself.
I don’t insist that you only use explicit reasoning. Feel free use metaphors, koans, poetry and whatnot. But you should also explain it with explicit reasoning.
For now I am basically content to have people either decide that Looking is worth trying to understand and trying to understand it, or decide that it isn’t. But I get the sense that this would be unsatisfying to you in some way.
Well, if you are saying “I don’t want to convince everyone or even the most” that’s your prerogative of course. I just feel that the point of this forum is trying to have discussions whose insights will percolate across the entire community. Also I am personally interested in understanding what Looking is about and I feel that the explanations given so far leave me somewhat confused (although this last attempt by Kaj was significant progress).
I agree that the situation is more complicated, I disagree that it is “much more complicated”. Yes, mathematicians rely on intuition to feel in the gaps in proofs and to seek out the errors in proofs. And yet, it is uncontroversial that having a proof should make you much more confident in a mathematical statement than just having an intuition. In reality, there is a spectrum that goes roughly “intuition that T is correct” → “informal argument for T” → “idea for how to prove T” → “sketch of a proof of T” → “unvetted proof of T” → “vetted, peer-reviewed proof of T” → “machine verifiable formal proof of T”.
Have I actually tried what? As to why I believe this, I think I already gave an “explicit reasoning” argument: and, yes, my intuition and life experience confirm this although this is not something that I can transmit to you directly.
This is a wrong way to look at this. Intuition and explicit reasoning are not two separate judges that give two separate verdicts. Combining intuition and explicit reasoning doesn’t mean averaging the results. The way it works is, when your intuition and reasoning disagree, you should try to understand why. You should pit them against each other and let them fight it out, and in the end you have something that resembles a system of formal arguments with intuition answering some sub-queries, and your reasoning and intuition both endorse the result. This is what I mean by “understanding on an intellectual level”.
I don’t insist that you only use explicit reasoning. Feel free use metaphors, koans, poetry and whatnot. But you should also explain it with explicit reasoning.
Well, if you are saying “I don’t want to convince everyone or even the most” that’s your prerogative of course. I just feel that the point of this forum is trying to have discussions whose insights will percolate across the entire community. Also I am personally interested in understanding what Looking is about and I feel that the explanations given so far leave me somewhat confused (although this last attempt by Kaj was significant progress).