I agree that you can never be „infinitly certain“ about the way the physical world is (because there‘s always a very tiny possibility that things might suddenly change, or everything is just a simulation, or a dream, or […] ), but you should assign probability 1 to mathematical statements for which there isn‘t just evidence, but actual, solid proof.
Suppose you have the choice beetween the following options:
A You get a lottery with a 1-Epsilon chance of winning.
B You win if 2+2=4 and 53 is a prime number and Pi is an irrational number.
Is there any Epsilon>0 for which you would chose option A? What if something really bad happens if you lose (like, all of humanity being tortured for [insert large number] years)?
I would chose option B for any Epsilon>0, which means assigning Bayes-probability 1 to option B.
Even if you believe that mathematical truths are necessarily true, you can still ask why you believe that they are necessarily true. What caused you to believe it? Likely whatever process it is is fallible.
Let’s suppose you believe that 2+2=4 follows axiomatically from Peano axioms or something. The question is what kind of evidence should convince you that 2+2=4 doesn’t follow from those axioms? According the post, it’d be exactly the same kind of evidence that convinced you 2+2=4 does follow from the axioms. Perhaps you wake up one day and find that when you sit down to apply the axioms, working through them step by step, you get 2+2=3. And when you open up a textbook it shows the same thing, and when you ask your math professor friend, and when you just think about it in your head.
I suppose the point is that how you interact with mathematical proofs isn’t much different from how you interact with the rest of the world. Mathematical results follow in some logically necessary ways, but there’s a process of evidence that causes you to have contingent beliefs even about things that themselves are seemingly necessarily could only be one way.
Cf. logical omniscience and related lines of inquiry.
I realize I haven’t engage with your Epsilon scenario. It does seem pretty hard to imagine and assign probabilities to, but actually assigning I seems like a mistake.
Assigning Bayes-probabilities <1 to mathematical statements (that have been definitly proven) seems absurd and logically contradictory, because you need mathematics to even asign probabilities.
If you assign any Bayes probability to the statement that Bayes probabilities even work, you already assume that they do work.
And, arguably, 2+2=4 is much simpler than the concept of Bayes-probability
(To be fair, the same might not be true for my most complex statement that Pi is irrational)
I agree that you can never be „infinitly certain“ about the way the physical world is (because there‘s always a very tiny possibility that things might suddenly change, or everything is just a simulation, or a dream, or […] ), but you should assign probability 1 to mathematical statements for which there isn‘t just evidence, but actual, solid proof.
Suppose you have the choice beetween the following options: A You get a lottery with a 1-Epsilon chance of winning. B You win if 2+2=4 and 53 is a prime number and Pi is an irrational number.
Is there any Epsilon>0 for which you would chose option A? What if something really bad happens if you lose (like, all of humanity being tortured for [insert large number] years)?
I would chose option B for any Epsilon>0, which means assigning Bayes-probability 1 to option B.
You might want to see How to Convince Me That 2 + 2 = 3
Even if you believe that mathematical truths are necessarily true, you can still ask why you believe that they are necessarily true. What caused you to believe it? Likely whatever process it is is fallible.
I’ll quote you what I commented elsewhere on this topic:
I realize I haven’t engage with your Epsilon scenario. It does seem pretty hard to imagine and assign probabilities to, but actually assigning I seems like a mistake.
Assigning Bayes-probabilities <1 to mathematical statements (that have been definitly proven) seems absurd and logically contradictory, because you need mathematics to even asign probabilities.
If you assign any Bayes probability to the statement that Bayes probabilities even work, you already assume that they do work.
And, arguably, 2+2=4 is much simpler than the concept of Bayes-probability (To be fair, the same might not be true for my most complex statement that Pi is irrational)