We absolutely can do math in the face of some probability of being wrong about our proofs (and the proofs of our proof systems, and our proofs of the consistenc

We absolutely can do math in the face of some probability of being wrong about our proofs (and the proofs of our proof systems, and our proofs of the consistency of our logics, and so on). I know this because we do, in fact, do math and some of our proofs are almost certainly wrong. Obviously I couldn't tell you which generally accepted proofs are wrong, only that we've drawn a few black balls from the urn of double-checking-established-proofs already, and so we should not assume that we won't find more. Imre Lakatos' essay Proofs and Refutations is a fun read on the subject. A simple deductive proof of 2+2=4 (say in Peano Arithmetic) is clear enough that I'd give it a very small chance of being wrong, easily less than 10^-9 but certainly not less than 10^-100. To push something below a probability like 10^-100 one would have to be unreasonably confident in things like "I haven't developed a brain injury causing me to be very confident in 2+2=4 and confabulating all other evidence as needed". Such a thing is absurdly unlikely of course, but a 10^-100 occurrence is far, far less likely still. > Sure, you can be fooled into thinking a black ball was pulled out of an urn of white balls, but whether the urn has all white balls is just a fact, not a probability. And under proper conditions, we should be able to verify that fact. To doubt that is to entertain wild skeptical scenarios where we can't really do science. You can verify it enough put the odds of it low enough to arguendo it as true for most purposes, but that's because for most purposes a certain probability of being wrong is acceptable, and once you've pushed something far below that threshold then you can fairly safely ignore it. As an intuition pump, imagine that you have personally drawn the balls from the urn. You drew a black ball and carefully observed it. Then, someone approaches you and offers to make you a bet: if the urn had ever contained a black ball then he would pay you one dollar, otherwise you would owe him a hundred million dollars. He hands you a clear, unambiguous, and legally enforceable contract to that effect. Do you sign? If you believe that there is a 100% chance that you drew a black ball, then there's a zero percent chance that you'll lose this bet. It's a free dollar, why wouldn't you sign it? > To doubt that is to entertain wild skeptical scenarios where we can't really do science. > At any rate, I don't see that sort of reasoning as Bayesian. It's just radical skepticism. We can do science in the face of the probability that we might be wrong about anything and everything, the key thing is to keep in mind the bounds – at least roughly – of the different ways that we can be wrong. It does involve a bit of humility, but not a step change compared to the ordinary scientific humility of "this is the best answer we have so far, but it could be wrong".