If thinking about thinking is among your interests, a phenomenally rich (and LONG), widely-romping post from Scott Aaronson on "common knowledge," something called "Aumann's agreement theorem," Bayesian thinking, and much more here:
(it's actually from an earlier talk Scott gave at SPARC)
While this won't be everyone's cup-of-tea, and it is more epistemology-logic-cognition than it is mathematics, it is (to me) one of the most fascinating, remarkable posts I've ever read on a math-related blog! Indeed, several readings likely required to take in all the ideas Scott puts on display here.
Aumann’s Theorem predicts that all "rational disagreements" should "terminate in common knowledge of complete agreement." But of course that doesn't happen so much in real life, and in one passage (that reminds me of so much stuff on the internet ;-)) Aaronson writes,
"You could say that the 'failed prediction' of Aumann’s Theorem is no surprise, since virtually all human beings are irrational cretins, or liars. Except for you, of course: you’re perfectly rational and honest. And if you ever met anyone else as rational and honest as you, maybe you and they could have an Aumannian conversation. But since such a person probably doesn’t exist, you’re totally justified to stand your ground, discount all opinions that differ from yours, etc."Anyway, give it a gander; you'll probably know before you're half-way through it if it's the sort of mind-stretching thought-exercise that strikes your fancy or not. (I suspect I may still be re-reading it a week from now, trying to better grasp parts!) The piece also contains some key links to related material.