If we define probability to be continuous on [0,1], the math works.
In practice, however, the probability of Professor McGonagall showing up with a Time-Turner tomorrow, given that I see and talk to her and try out the time-turner myself and it has the expected results, including being able to solve NP-complete problems in constant time, remains zero. The odds of spontaneous creation of Professor McGonagall and a device which causes me to perceive that I have traveled through time and which solves any NP-hard problems that I choose to give it in finite time is epsilon. The odds of a self-consistent hallucination that the above events have happened is an epsilon of a higher order.
Therefore, given impossible evidence one should conclude that one is insane. Once one has concluded that one is insane, one should reconsider all of one’s prior judgements in light of the fact that one cannot tell real evidence from hallucinatory evidence- which brings all of the evidence regarding the impossibility of any event into question.
In other words, there is epsilon chance that all of your experience is fake, and therefore at least epsilon uncertainty in any prediction you make, even predictions about pure hypothetical situations where mathematical proof exists.
P(A)=0; P(B)=0 P(A|B)=1
If we define probability to be continuous on [0,1], the math works. In practice, however, the probability of Professor McGonagall showing up with a Time-Turner tomorrow, given that I see and talk to her and try out the time-turner myself and it has the expected results, including being able to solve NP-complete problems in constant time, remains zero. The odds of spontaneous creation of Professor McGonagall and a device which causes me to perceive that I have traveled through time and which solves any NP-hard problems that I choose to give it in finite time is epsilon. The odds of a self-consistent hallucination that the above events have happened is an epsilon of a higher order.
Therefore, given impossible evidence one should conclude that one is insane. Once one has concluded that one is insane, one should reconsider all of one’s prior judgements in light of the fact that one cannot tell real evidence from hallucinatory evidence- which brings all of the evidence regarding the impossibility of any event into question.
In other words, there is epsilon chance that all of your experience is fake, and therefore at least epsilon uncertainty in any prediction you make, even predictions about pure hypothetical situations where mathematical proof exists.