Isn't that a bit of a strawman? Bayesian epistemology as stated would require someone to believe something like "drawing without replacement from this urn, I got 100 white balls and 1 black ball, therefore there is a nonzero probability that this urn contains only white balls", which is not a belief I can imagine anybody to seriously hold.
There are three concepts involved: an evidence "e", for example "I extracted 1 black ball"; a hypothesis "h", for example "the urn does not contain only white balls"; a theory "h <- e" that, by means of logic or otherwise, deduces the hypothesis from the evidence. Your (qsort) theory is that if you see a black ball then the hypothesis is correct.
Everybody, including you, me, and Deutsch, agree that if the probability of the evidence goes up, then the probability of the hypothesis goes up as well.
What Deutsch and Popper/Miller are also saying, however, is that if the probability of the evidence goes up then the probability of the theory goes down (proof in the paper).
I need to study the paper more carefully, because I am not 100% sure that it is strictly correct (there are many factors that would transform a < into <= if they were 1, and I suspect that some are), but I believe the weaker statement that if the evidence goes up the probability of the theory does not go up at all.
In any case, this conclusion is consistent with what all scientists have believed forever: the only way to increase confidence in a theory is to try to break it. Or at least they believed this until fact checkers and censorship came along and threw the baby away with the bathwater.
I have no idea whether it's possible to create a machine that can generate conscious simulations without itself being conscious. But there's no Bayesian reasoning to apply to such scenarios, since we have no idea as of yet. They're philosophical arguments. And applying to that to current technology is misplaced.
[1] https://www.lesswrong.com/posts/6FmqiAgS8h4EJm86s/how-to-con...
It is extremely close though.
[1] - https://www.kevindorst.com/stranger_apologies/the-gamblers-f...
Before you might have given a nonzero probability because you didn't know whether there was a black ball. But once you know there is, you can't have some nonzero belief in there only being white balls.
Still, I've been to enough magic shows, been fooled by enough subtly-wrong mathematical proofs, been utterly convinced that my program's behavior was impossible, etc that I side with the bayesian perspective here. I don't think there's a single thing that I'm justified in saying is true with 100% probability.
Or, in essay form: https://www.lesswrong.com/posts/QGkYCwyC7wTDyt3yT/0-and-1-ar...
Not even simple deductive proofs? 2+2=4 by definition. There's no probability for it being wrong. We couldn't do math if that wasn't the case. It's not even correct to assign those kinds of statements a probability. Now there is a possibility for humans to make mistakes with complicated proofs, as you pointed out. But the correct proof doesn't have a probability, only the belief in it being correct.
For empirical matters, I'm not sure what sort of probability you could assign to some skeptical scenario like it's all a dream, or nobody exists outside your mind, or we live inside a simulation. We take it that there's a world with other people for granted.
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.
At any rate, I don't see that sort of reasoning as Bayesian. It's just radical skepticism.
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".
In the case of having a brain injury that makes you incorrectly think a simple deductive proof is true when it's not, then bayesian reasoning isn't going to fare any better. We're in hinge proposition territory where we can't reasonably doubt what gives the basis for reasoning. 2+2=4 by definition of basic arithmetic. The cost of doubting that is is to doubt any consistent reasoning, including probabilities.
> 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?
The only reason not to sign is because there's strong reason to believe they have a trick up their sleeve they can fool you with. But if we take something verifiable by everyone, like the roughly spherical shape of the Earth, or the speed of light in a vacuum, there is no reason not to take the bet. It would be a free dollar.
> 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.
We can't be wrong about everything, because that would mean science was impossible. That would be the radical skepticism I was mentioning, and you can't use Bayesian reasoning with radical skepticism.
Which is why Kant had to come up with categories of thought to rescue rationality in the face of Humean skepticism. You can't give a probability to the liklihood of the sun continuing to undergo fusion tomorrow if the laws of nature could change on us at any time for no reason (tomorrow is Thanksgiving for the turkey scientist who is confident about continued survival Hume would say).