For Godel-Bayes issues, you can start with the responses to my post on the subject. (I’ve since learned and remembered more about Godel.)
We should have the ability to talk about subjective uncertainty in, at the very least, particular proofs and probabilities. I don’t know that we can. But I like the following argument, which I recall seeing here somewhere:
If there exists a perfect probability calculation based on a set of background information, it must take this uncertainty into account. Therefore, applying this uncertainty again to the answer would mean double-counting the evidence, which is strictly verboten. We therefore cannot use this line of reasoning to produce a contradiction. Barring other arguments, we can assume the uncertainty equals a really small fraction.
E.g., suppose a guy comes out tomorrow with a proof of the Riemann Hypothesis. What are the chances he is wrong? Surely not zero.
But the chance that the Riemann Hypothesis itself is wrong, if it has a proof? Well, that kinda seems like zero. (But then, how would we know that? It does seem like we have to filter through our unreliable senses.)
Hrmm… I’m still taking high school geometry, so “infinite set of axioms” doesn’t really make sense yet. I’ll try to re-read that thread once I’ve started college-level math.
For Godel-Bayes issues, you can start with the responses to my post on the subject. (I’ve since learned and remembered more about Godel.)
We should have the ability to talk about subjective uncertainty in, at the very least, particular proofs and probabilities. I don’t know that we can. But I like the following argument, which I recall seeing here somewhere:
If there exists a perfect probability calculation based on a set of background information, it must take this uncertainty into account. Therefore, applying this uncertainty again to the answer would mean double-counting the evidence, which is strictly verboten. We therefore cannot use this line of reasoning to produce a contradiction. Barring other arguments, we can assume the uncertainty equals a really small fraction.
E.g., suppose a guy comes out tomorrow with a proof of the Riemann Hypothesis. What are the chances he is wrong? Surely not zero.
But the chance that the Riemann Hypothesis itself is wrong, if it has a proof? Well, that kinda seems like zero. (But then, how would we know that? It does seem like we have to filter through our unreliable senses.)
Hrmm… I’m still taking high school geometry, so “infinite set of axioms” doesn’t really make sense yet. I’ll try to re-read that thread once I’ve started college-level math.