Right; sorry for not phrasing that in a way that sounded like agreement with you. We should be less that totally certain about mathematical statements in real life, but when setting up the formalism for probability, we’re “inside” math rather than outside of it; there isn’t going to be a good argument for assigning less than probability 1 to logical truths. Only bad things happen when you try.
This does change a bit when we take logical uncertainty into account, but although we understand logical uncertainty better these days, there’s not a super strong argument one way or the other in that setting—you can formulate versions of logical induction which send probabilities to zero immediately when things get ruled out, and you can also formulate versions in which probabilities rapidly approach zero once something has been logically ruled out. The version which jumps to zero is a bit better, but no big theoretical advantage comes out of it afaik. And, in some abstract sense, the version which merely rapidly approaches zero is more prepared for “mistakes” from the deductive system—it could handle a deductive system which occasionally withdrew faulty proofs.
Right; sorry for not phrasing that in a way that sounded like agreement with you. We should be less that totally certain about mathematical statements in real life, but when setting up the formalism for probability, we’re “inside” math rather than outside of it; there isn’t going to be a good argument for assigning less than probability 1 to logical truths. Only bad things happen when you try.
This does change a bit when we take logical uncertainty into account, but although we understand logical uncertainty better these days, there’s not a super strong argument one way or the other in that setting—you can formulate versions of logical induction which send probabilities to zero immediately when things get ruled out, and you can also formulate versions in which probabilities rapidly approach zero once something has been logically ruled out. The version which jumps to zero is a bit better, but no big theoretical advantage comes out of it afaik. And, in some abstract sense, the version which merely rapidly approaches zero is more prepared for “mistakes” from the deductive system—it could handle a deductive system which occasionally withdrew faulty proofs.