The most recent brough-ha-ha from one of Mark's posts seems to center on rationality and philosophy, but underlying all the stuff in the "new rationality" is understanding the process of updating our current knowledge with new information through Bayes Theorem (LW calls the process belief updating or bayesian updating). Bayes Theorem is both very useful and very interesting historically and philosophically, and not without its philosophical issues. (Uninformative priors anyone?) I want to point everyone to two books by the philosopher of science Ian Hacking that open up the landscape of probability and induction with a good, strong dose of philosophical reflection along the way. Hacking writes clearly and, for my money, does a great job of keeping the open questions out in the open.
The first is Hacking's introductory text on probability and inductive logic. Induction is about reasoning in the absence of complete information. It's about risk, gambling, and figuring-out. The book focuses on the problem of induction, providing an education in quantitative induction while not covering-up the various disagreements about forumlating and solving the problem. In the introduction he says this about the philosophical content of the book:
There is a famous problem in philosophy called the problem of induction. It comes at the end of the book. There are ethical questions about risk. Some philosophers say we should always act so as to maximize the common good. Others say that duty and right and wrong come before cost-benefit thinking. These questions arise in Chapter 9.
There are even some probability arguments for, and against, religious belief. One comes up in Chapter 10.
There are philosophical arguments about probability itself. Right now there are big disagreements over the basic ideas of inductive inference. Different schools of thought approach practical issues in different ways. Most beginning statistics courses pretend that there is no disagreement. This is a philosophy book, so it put the competing ideas up front. It tries to be fair to all parties.
The second, The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, is an earlier work that explores the birth of the modern understanding of probability and induction in Europe in the mid-seventeenth century. The new edition includes material on current philosophical trends. If you want to understand the development (and continuing arguments) in probability and induction, it's worth the read.