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Nisan comments on Explanations for Less Wrong articles that you didn't understand - Less Wrong Discussion
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Can someone explain to me the significance of problems like Sleeping Beauty? I see a lot of digital ink being spilled over them and I can kind of see how they call into question what we mean by "probability" and "expected utility", but I can't quite pin down the thread that connects all of them. Someone will pose a solution to a paradox X, and then another reply with a modified version X' that the previous solution fails on, and I tend to have trouble seeing what the exact thing is people are trying to solve.
I don't know about academic philosophy, but on Less Wrong there is the hope of one day coming up with an algorithm that calculates the "best", "most rational" way to act.
That's a bit of a simplification, though. It is hoped that we can separate the question of how to learn (epistemology) and what is right (moral philosophy) from the question of given one's knowledge and values, what is the "best", "most rational" way to behave? (decision theory).
The von Neumann–Morgenstern theorem is the paradigmatic result here. It suggests (but does not prove) that given one's beliefs and values, one "should" act so as to maximize a certain weighted sum. But as the various paradoxes show, this is far from the last word on the matter.