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Will_Sawin comments on Dutch Books and Decision Theory: An Introduction to a Long Conversation - Less Wrong
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How, exactly, to deal with logical uncertainty is an unsolved problem, no?
It's not clear why it's more of a problem for Bayesian than anyone else.
The post you linked to is a response to Eliezer arguing with Hanson, with Eliezer taking a pro-his-own-contrarianism stance. Of course he's aware of that contrarianism.
Your choice is either accepting that you will be sometimes inconsistent, or accepting that you will sometimes answer "I don't know" without providing a specific number, or both.
There's nothing wrong with "I don't know".
For Perfect Bayesian or for Subjective Bayesian?
Subjective Bayesian does believe many statements of kind P(simple math step) = 1, P(X|conjunction of simple math steps) = 1, and yet P(X) < 1.
it does not believe math statements with probability 1 or 0 until it investigates them. As soon as it investigates whether (X|conjunction of simple math steps) is true and determines the answer, it sets P(X)=1.
The problem with "I don't know" is that sometimes you have to make decisions. How do you propose to make decisions if you don't know some relevant mathematical fact X?
For example, if you're considering some kind of computer security system that is intended to last a long time, you really need an estimate for how likely it is that P=NP.
Then you need to fully accept that you will be inconsistent sometimes. And compartmentalize your belief system accordingly, or otherwise find a way to deal with these inconsistencies.
Well, they can wriggle out of this by denying P(simple math step) = 1, which is why I introduced this variation.
Doesn't this imply you'd be willing to accept P(2+2=5) on good enough odds?
This might be pragmatically a reasonable thing to do, but if you accept that all math might be broken, you've already given up any hope of consistency.