Ah, the old irresistible force acting upon immovable object argument.
This seems (dis)solvable by representing changing beliefs as shifting probability mass around. You might argue that after you've worked your way through the proof of T step by step, you've moved the bulk of probability mass to T (with respect to priors that don't favor either T or ~T too much). But if it were enough, we would expect to see all of the following: 1) people are always certain of their conclusions after they've done the math once; 2) people don't find errors in proofs that have been published for a long time; 3) there's no perceived value in checking each other's proofs; 4) If there is a certain threshold of complexity or length after which people would stop becoming certain of their conclusions, nobody has reached it yet.
None of this is true in our word, which supports the hypothesis that a non-trivial amount of probability mass gets stuck along the way, subjectively this manifests in you acknowledging the (small, but non-negligible) possibility of having erred in each part of the proof.
Now, the proper response to TM would be to shift your probability according to the weight of Ms. Math's authority, which is not absolute. If you're uncomfortably uncertain afterwards, you just re-examine your evidence paying more attention the hardest parts, and squeeze some more probability juice either way until you either are certain enough or until you spot an error.
I present here two puzzles of rationality you LessWrongers may think is worth to deal with. Maybe the first one looks more amenable to a simple solution, while the second one has called attention of a number of contemporary epistemologists (Cargile, Feldman, Harman), and does not look that simple when it comes to a solution. So, let's go to the puzzles!
Puzzle 1
At t1 I justifiably believe theorem T is true, on the basis of a complex argument I just validly reasoned from the also justified premises P1, P2 and P3.
So, in t1 I reason from premises:
(R1) P1, P2 ,P3
To the known conclusion:
(T) T is true
At t2, Ms. Math, a well known authority on the subject matter of which my reasoning and my theorem are just a part, tells me I’m wrong. She tells me the theorem is just false, and convince me of that on the basis of a valid reasoning with at least one false premise, the falsity of that premise being unknown to us.
So, in t2 I reason from premises (Reliable Math and Testimony of Math):
(RM) Ms. Math is a reliable mathematician, and an authority on the subject matter surrounding (T),
(TM) Ms. Math tells me T is false, and show to me how is that so, on the basis of a valid reasoning from F, P1, P2 and P3,
(R2) F, P1, P2 and P3
To the justified conclusion:
(~T) T is not true
It could be said by some epistemologists that (~T) defeat my previous belief (T). Is it rational for me to do this way? Am I taking the correct direction of defeat? Wouldn’t it also be rational if (~T) were defeated by (T)? Why ~(T) defeats (T), and not vice-versa? It is just because ~(T)’s justification obtained in a later time?
Puzzle 2
At t1 I know theorem T is true, on the basis of a complex argument I just validly reasoned, with known premises P1, P2 and P3. So, in t1 I reason from known premises:
(R1) P1, P2 ,P3
To the known conclusion:
(T) T is true
Besides, I also reason from known premises:
(ME) If there is any evidence against something that is true, then it is misleading evidence (evidence for something that is false)
(T) T is true
To the conclusion (anti-misleading evidence):
(AME) If there is any evidence against (T), then it is misleading evidence
At t2 the same Ms. Math tells me the same thing. So in t2 I reason from premises (Reliable Math and Testimony of Math):
(RM) Ms. Math is a reliable mathematician, and an authority on the subject matter surrounding (T),
(TM) Ms. Math tells me T is false, and show to me how is that so, on the basis of a valid reasoning from F, P1, P2 and P3,
But then I reason from::
(F*) F, RM and TM are evidence against (T), and
(AME) If there is any evidence against (T), then it is misleading evidence
To the conclusion:
(MF) F, RM and TM is misleading evidence
And then I continue to know T and I lose no knowledge, because I know/justifiably believe that the counter-evidence I just met is misleading. Is it rational for me to act this way?
I know (T) and I know (AME) in t1 on the basis of valid reasoning. Then, I am exposed to misleading evidences (Reliable Math), (Testimony of Math) and (F). The evidentialist scheme (and maybe still other schemes) support the thesis that (RM), (TM) and (F) DEFEATS my justification for (T) instead. So that whatever I inferred from (T) is no longer known. However, given my previous knowledge of (T) and (AME), I could know that (MF): F is misleading evidence. It can still be said that (RM), (TM) and (F) DEFEAT my justification for (T), given that (MF) DEFEAT my justification for (RM), (TM) and (F)?