Suppose someone offered you a bet on P = NP. How should you go about deciding whether or not to take it?
Does it make sense to assign probabilities to unproved mathematical conjectures? If so, what is the probability of P = NP? How should we compute this probability, given what we know so far about the problem? If the answer is no, what is a rational way to deal with mathematical uncertainty?
I actually thought of this in the sense of statements being partially true: We know godels incompleteness theorem (most likely you know it better than I do). I'm pretty sure it's provable that BB(10^10) does not have a lower bound. However, if you simulate minds/civilizations/AI/something, and ask them to bet on mathematical theorems (at first with less resources so they don't just solve it), and then ask them whether they think a certain unprovable theorem is true and let them bet on it, you might somehow know how true an unprovable statement is? I realize this comment is poorly written but I hope you understand my intuition.
Open problems are clearly defined problems1 that have not been solved. In older fields, such as Mathematics, the list is rather intimidating. Rationality, on the other, seems to have no list.
While we have all of us here together to crunch on problems, let's shoot higher than trying to think of solutions and then finding problems that match the solution. What things are unsolved questions? Is it reasonable to assume those questions have concrete, absolute answers?
The catch is that these problems cannot be inherently fuzzy problems. "How do I become less wrong?" is not a problem that can be clearly defined. As such, it does not have a concrete, absolute answer. Does Rationality have a set of problems that can be clearly defined? If not, how do we work toward getting our problems clearly defined?
See also: Open problems at LW:Wiki
1: "Clearly defined" essentially means a formal, unambiguous definition. "Solving" such a problem would constitute a formal proof.