You can’t have a consistent probability assignment that says axioms are 100% true but the millionth digit of pi has a 50% chance of being odd.
Why not? If you haven’t actually worked out the millionth digit of pi, then this probability assignment is consistent given your current state of knowledge. It’s inconsistent given logical omniscience, but then if you were logically omniscient you wouldn’t assign a 50% chance in the first place. The act of observing a logical fact doesn’t seem different from the act of making any other observation to me.
If you knew that the universe doesn’t obey Newtonian mechanics precisely, it would be inconsistent to assign them a high probability, but that doesn’t mean that an early physicist who doesn’t have that knowledge is violating the rules of probability by thinking that the universe does follow Newtonian mechanics. It’s only after you make that observation that such an assignment becomes inconsistent.
Why not? If you haven’t actually worked out the millionth digit of pi, then this probability assignment is consistent given your current state of knowledge. It’s inconsistent given logical omniscience, but then if you were logically omniscient you wouldn’t assign a 50% chance in the first place. The act of observing a logical fact doesn’t seem different from the act of making any other observation to me.
If you knew that the universe doesn’t obey Newtonian mechanics precisely, it would be inconsistent to assign them a high probability, but that doesn’t mean that an early physicist who doesn’t have that knowledge is violating the rules of probability by thinking that the universe does follow Newtonian mechanics. It’s only after you make that observation that such an assignment becomes inconsistent.