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Benja comments on Results from MIRI's December workshop - Less Wrong
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Comments (43)
I do not understand what you mean by "probability" here. Suppose I use one criterion to estimate that the twin-prime conjecture is true with probability 0.99, but a different criterion gives me 0.9999. In what situation would my choice of the criterion matter?
Are we talking about some measure over many (equally?) possible worlds in some of which the TPC is true and in others false (or maybe unprovable)? What would I do differently if I am convinced that one criterion is "right" and the other is "wrong" vs the other way around? Would I spend more time trying to prove the conjecture if I thought it is more likely true, or something?
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 :)