These books have been mentioned on the list before, but I'm recommending them again because a lot of new members have joined since we last talked about them. To motivate you to read these books, I've given some questions that each book helps answer or provide the necessary background knowledge to discuss. I'd say that the first book in this list should be required reading for all members of this list, while the other two are more optional. If anyone else has more book recommendations, please feel free to add to this list.
_An Introduction to Kolmogorov Complexity and Its Applications_, Ming Li and Paul Vitanyi (What is "measure", "prior", "universal distribution"? How can we explain why laws of physics exist and why we don't see random deviations from the laws of physics?) _Theory of Recursive Functions and Effective Computability_, Hartley Rogers (What does "computable" mean? What do we know about uncomputable mathematical structures? Should we assume that we're in a computable universe?) _The Foundations of Causal Decision Theory_, James Joyce (What justifies using numbers between 0 and 1 to represent degrees of belief and using probability theory to constrain/manipulate those numbers (i.e. probabilities)? Is the use of probabilities still appropriate if all possible universes exist, and if so how should it work? How should we make decisions if all possible universes exist?)