Here's the first "philosophical open question" that's currently driving me mad. Any ideas anyone?
Why is it epistemically OK to assume some necessary truths (2+2=4) but not others (the four color theorem)? We say that people who assume that 2+2=4 just because it feels obvious to them count as knowing, but those people who assumed the 4 color theorem in the same way would not count as knowing. Note that in this case, neither belief is consciously produced by any kind of method, which you might say is reliable in one case but not others (can imagine creatures for whom the suberpersonal/unconscious faculty that produces the feeling that the 4 color theorem is obvious is reliable).
It's tempting to be a Cornell Realist about justification here. You might say: there are some true/valid principles/methods everyone accepts- you are justified in believing whatever you can prove from these true principles using these truth preserving methods of inference.
But what you say about justification is tied to what you say about everything else by the fact Moore pointed out - that you can't say: p, but I'm not justified in believing that p. If this Moorean fact is indeed the central part of our practice of talking about justifications that it seems to be, every situation where you believe that p has to be one where (if you have the concept of justification) you'd either assent to 'I am justified in believing that p', or be inclined to stop believing that p. Taking the Cornell Realist line - the 4 color theorem is true, but I don't have a proof of it from things that most ppl would accept, so I'm not justified in believing it - sounds totally bizare. I mean, it seems to be a core part of the way that we use the word justification that saying 'P, but I'm not justified in believing that P' is not an option.
So, maybe it's best to say that one is justified in assuming any and all mathematical truths, should these feel brutely obvious to one (though, of course, you would not be justified in believing mathematical truths which you derive from false premises). Note that you could still criticize someone for generating a random mathematical sentence and then taking meds that would make them find that sentence obvious. Even it they pick a true proposition, and hence count as having fully justified beliefs after they take the meds, their conduct before hand involves a huge risk of producing false beliefs. Still, this is quite a radical and somewhat counterintuitive conclusion though!