> 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.
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).
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).