It is not perfectly clear whether this post is describing what the actual structure of your beliefs should be, or how you should present them. I think I disagree strongly in both cases.
Suppose I believe that my friend Albert has very recently converted to Christianity, because (1) Albert has told me so and (2) our mutual friend Beth tells me he told her so. These are both good evidence. Neither is conclusive; sometimes people make jokes, for instance. Neither is a crux; if it turned out, say, that I had merely had an unusually vivid dream in which Albert told me of his conversion, I would become less sure that it had actually happened but Beth’s testimony would still make me think it probably had.
In this situation I could, I guess, say that I believe in Albert’s conversion “because Albert and Beth both told me it was so”. But that’s purely artificial; one could do that with any set of justifications for a belief. And merely contradicting this belief would not suffice to change my opinion; removing either Albert’s or Beth’s testimony, but not the other, would falsify “both told me” but I would still believe it.
This is unusual mostly in being an artificially clean and simple case. I think most beliefs, or at any rate a large fraction, are like this. A thing affects the world in several ways, many of which may provide separate channels by which evidence reaches you.
This is true even in what’s maybe the cruxiest of all disciplines, pure mathematics. I believe that there are infinitely many prime numbers because of Euclid’s neat induction proof. But mathematics is subtle and sometimes people make mistakes, and maybe you could convince me that there’s a long-standing mistake in that proof that sometimes [EDIT: of course this should have said “somehow”] I and every other mathematician had missed. But then I would probably (it might depend on the nature of the mistake) still believe that there are infinitely many prime numbers because there are other quite different proofs, like the one about the divergence of ∑1n=∏(1−1p)−1 or the one using (2nn) that proves an actual lower bound on how many primes there are, or the various proofs of the Prime Number Theorem. To some extent I would believe it merely because of the empirical evidence of the density of prime numbers, which (unlike say the distribution of zeros of the zeta function, the empirical evidence of which is also evidence that there are infinitely many primes) seems to be of a very robust kind. To make me change my mind about there being infinitely many prime numbers the proposition you would have to refute is something like “mathematics is not all bullshit”.
(Sometimes a thing in pure mathematics has only a single known proof, or all the known proofs work in basically the same way. In that case, there may be an actual crux. But for theorems people actually care about this state of affairs often doesn’t last; other independent proofs may be found.)
Outside mathematics things are less often cruxy, and I think usually sincerely so.
Finding cruxes is a useful technique, but there is not the slightest guarantee that there will be one to find.
Perhaps one should present one’s beliefs cruxily even when they aren’t actually cruxy, either in order to give others the best chance of presenting mind-changing evidence or to look open-minded? I don’t think so; if your beliefs are not actually cruxy then lying about them will make it less likely that your mind gets changed when the evidence doesn’t really support your current opinion, and if you get caught it will be bad for your reputation.
It is not perfectly clear whether this post is describing what the actual structure of your beliefs should be, or how you should present them. I think I disagree strongly in both cases.
Suppose I believe that my friend Albert has very recently converted to Christianity, because (1) Albert has told me so and (2) our mutual friend Beth tells me he told her so. These are both good evidence. Neither is conclusive; sometimes people make jokes, for instance. Neither is a crux; if it turned out, say, that I had merely had an unusually vivid dream in which Albert told me of his conversion, I would become less sure that it had actually happened but Beth’s testimony would still make me think it probably had.
In this situation I could, I guess, say that I believe in Albert’s conversion “because Albert and Beth both told me it was so”. But that’s purely artificial; one could do that with any set of justifications for a belief. And merely contradicting this belief would not suffice to change my opinion; removing either Albert’s or Beth’s testimony, but not the other, would falsify “both told me” but I would still believe it.
This is unusual mostly in being an artificially clean and simple case. I think most beliefs, or at any rate a large fraction, are like this. A thing affects the world in several ways, many of which may provide separate channels by which evidence reaches you.
This is true even in what’s maybe the cruxiest of all disciplines, pure mathematics. I believe that there are infinitely many prime numbers because of Euclid’s neat induction proof. But mathematics is subtle and sometimes people make mistakes, and maybe you could convince me that there’s a long-standing mistake in that proof that sometimes [EDIT: of course this should have said “somehow”] I and every other mathematician had missed. But then I would probably (it might depend on the nature of the mistake) still believe that there are infinitely many prime numbers because there are other quite different proofs, like the one about the divergence of ∑1n=∏(1−1p)−1 or the one using (2nn) that proves an actual lower bound on how many primes there are, or the various proofs of the Prime Number Theorem. To some extent I would believe it merely because of the empirical evidence of the density of prime numbers, which (unlike say the distribution of zeros of the zeta function, the empirical evidence of which is also evidence that there are infinitely many primes) seems to be of a very robust kind. To make me change my mind about there being infinitely many prime numbers the proposition you would have to refute is something like “mathematics is not all bullshit”.
(Sometimes a thing in pure mathematics has only a single known proof, or all the known proofs work in basically the same way. In that case, there may be an actual crux. But for theorems people actually care about this state of affairs often doesn’t last; other independent proofs may be found.)
Outside mathematics things are less often cruxy, and I think usually sincerely so.
Finding cruxes is a useful technique, but there is not the slightest guarantee that there will be one to find.
Perhaps one should present one’s beliefs cruxily even when they aren’t actually cruxy, either in order to give others the best chance of presenting mind-changing evidence or to look open-minded? I don’t think so; if your beliefs are not actually cruxy then lying about them will make it less likely that your mind gets changed when the evidence doesn’t really support your current opinion, and if you get caught it will be bad for your reputation.