Poke: Are you sure about mineralogy and physics as foundations of modern geology?
Patrick:
I agree that something roughly along the lines of what you are discussing can be done and is unavoidable. I am primarily attempting to refute the proposal that it is or can be corrected to become Bayesian, and hence the proposal that the process that we use to do things like this stands with the same sort of logical foundations as Bayesian reasoning does. It definitely seems to me that strictly speaking, once you remove logical omniscience, unless you replace it with some very specific abstraction (most of which have their own problems) you need to assign probabilities to “The RH can be proved”, “The RH can be disproved”, “The RH is undecidable from ZFC”, “The RH can be proved AND the RH can be disproved”, “The RH can be proved AND the RH can be disproved AND The RH is undecidable from ZFC” “The RH can be disproved AND The RH is undecidable from ZFC” etc. In practice we can apportion zero probability to the latter cases, but only conditional upon the quality of our reflection being perfect, which we know to be false, and only after SOME reflection. It seems to me that as we assign probabilities we have to do reflection that moves our estimates continually.
Poke: Are you sure about mineralogy and physics as foundations of modern geology?
Patrick:
I agree that something roughly along the lines of what you are discussing can be done and is unavoidable. I am primarily attempting to refute the proposal that it is or can be corrected to become Bayesian, and hence the proposal that the process that we use to do things like this stands with the same sort of logical foundations as Bayesian reasoning does. It definitely seems to me that strictly speaking, once you remove logical omniscience, unless you replace it with some very specific abstraction (most of which have their own problems) you need to assign probabilities to “The RH can be proved”, “The RH can be disproved”, “The RH is undecidable from ZFC”, “The RH can be proved AND the RH can be disproved”, “The RH can be proved AND the RH can be disproved AND The RH is undecidable from ZFC” “The RH can be disproved AND The RH is undecidable from ZFC” etc. In practice we can apportion zero probability to the latter cases, but only conditional upon the quality of our reflection being perfect, which we know to be false, and only after SOME reflection. It seems to me that as we assign probabilities we have to do reflection that moves our estimates continually.