Kant claims in the preface that Hume interrupted his "dogmatic slumber." What was Kant's "dogmatic slumber" and how did Hume's attack on causal reasoning prompt Kant's critical philosophy?
Kant's philosophical development took place in the German tradition of rationalist metaphysics. In the Prolegomena, as in other of his mature works, Kant refers to this form of metaphysics as "dogmatic" because there is very little effort made to question the ground on which these metaphysical claims are justified. Hume awoke Kant from this "dogmatic slumber" (which is what Kant calls his metaphysical period in the preface) by showing the importance and the difficulty of justifying knowledge claims. Hume argues that we have no rational justification for believing that every effect has a cause, but that we simply believe this out of habit. Kant observes that Hume's reasoning could be applied generally to a priori knowledge, thus casting doubt on the rational justification of all metaphysics. Hume's skepticism prompts Kant to seek a more solid basis on which to ground metaphysics.
What is the difference between the a priori/a posteriori distinction and the analytic/synthetic distinction? How do both of these distinctions differ from the distinction between necessary and contingent truths?
The a priori/a posteriori distinction has to do with cognitions, or things that we know. It distinguishes between knowledge I can have prior to any experience and knowledge I gain from experience. The analytic/synthetic distinction has to do with judgments. It distinguishes between judgments that are trivial and judgments that bring together two different concepts. The necessary/contingent distinction has to do with whether a certain fact could have been otherwise. A priori truths are generally considered necessary, since they do not seem to hinge on the particularities of experience. However, saying a truth is a priori is a matter of discussing how we know it, and saying it is necessary is a matter of discussing its relation to other truths and to the world.
What is a "thing in itself"? Why can't we perceive it directly? How can we perceive it? How can we even know that things in themselves exist if we cannot perceive them?
Kant argues that while experience is made up entirely of appearances, these appearances are in some way caused by things in themselves. We cannot perceive things in themselves directly; what we perceive must first be interpreted by our senses, and then by our faculties of sensibility and understanding. Our senses and faculties are what make it possible to connect with the world outside our mind, but they also determine the way this connection is made. Though we cannot perceive things in themselves directly, we know they must exist because there must be some cause behind the appearances we meet with in experience. The existence of things in themselves is crucial to Kant's philosophy, but he insists that we cannot know anything about them.