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smoofra comments on No Universal Probability Space - Less Wrong
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Yet it seems that math is what it should be. Bayesian probability, as it's used in probabilistic inference, is usually founded on the same Kolmogorov axioms, standard mathematical probability theory. I don't see any problems with the mathematical part, I dispute your characterization of Bayesian probability as an inherently informal theory (hence it was taken in quotation marks in my comment).
Any theory about the real world is inherently informal.
Do you disagree that Bayesian probability theory is about as informal as physics, or do you disagree with my characterization of physics as informal? If it's the latter, then we don't disagree on anything except the meaning of words.
A theory about the real world may be perfectly formal, it just won't have a perfectly formal applicability proof. On the other hand, if you can show that a theory is applicable with probability of 1-2^{-10000}, it's as good as formally proven to apply.
I disagree that it's correct terminology to call a theory informal, just because it's can't be formally proven to apply to the real world.
It's not the lack of a proof that makes it informal, it's that the elements themselves of the theory aren't precisely, formally, mathematically defined. A valid proposition in measure-theoretic probability is a subset of the measure space. nothing else will do. Propositions in Bayseian probability are written in natural language, about events in the real world.
I'm using the word "formal" in the sense that it is used in mathematics. If you're going to say that propositions written in natural language, about events in the real world are "formal" in that sense, then you're just refusing to communicate.