Agree with Nisan’s intuition, though I also agree with Wei Dai’s position that we shouldn’t feel sure that Bayesian probability is the right way to handle logical uncertainty. To more directly answer the question what it means to assign a probability to the twin prime conjecture: If Omega reveals to you that you live in a simulation, and it offers you a choice between (a) Omega throws a bent coin which has probability p of landing heads, and shuts down the simulation if it lands tails, otherwise keeps running it forever; and (b) Omega changes the code of the simulation to search for twin primes and run for one more step whenever it finds one; then you should be indifferent between (a) and (b) iff you assign probability p to the twin prime conjecture. [ETA: Argh, ok, sorry, not quite, because in (b) you may get to run for a long time still before getting shut down—but you get the idea of what a probability over logical statements should mean.]
but you get the idea of what a probability over logical statements should mean
Not from your example, I do not. I suspect that if you remove this local Omega meme, you are saying that there are many different possible worlds in your inner simulator and in p*100% of them the conjecture ends up being proven… some day before that world ends. Unless you are a Platonist and assign mathematical “truths” independent immaterial existence.
Agree with Nisan’s intuition, though I also agree with Wei Dai’s position that we shouldn’t feel sure that Bayesian probability is the right way to handle logical uncertainty. To more directly answer the question what it means to assign a probability to the twin prime conjecture: If Omega reveals to you that you live in a simulation, and it offers you a choice between (a) Omega throws a bent coin which has probability p of landing heads, and shuts down the simulation if it lands tails, otherwise keeps running it forever; and (b) Omega changes the code of the simulation to search for twin primes and run for one more step whenever it finds one; then you should be indifferent between (a) and (b) iff you assign probability p to the twin prime conjecture. [ETA: Argh, ok, sorry, not quite, because in (b) you may get to run for a long time still before getting shut down—but you get the idea of what a probability over logical statements should mean.]
Not from your example, I do not. I suspect that if you remove this local Omega meme, you are saying that there are many different possible worlds in your inner simulator and in p*100% of them the conjecture ends up being proven… some day before that world ends. Unless you are a Platonist and assign mathematical “truths” independent immaterial existence.
Retracted my comment for being unhelpful (I don’t recognize what I said in what you heard, so I’m clearly not managing to explain myself here).
Thanks for trying, anyway :)