I think it's pretty obvious that philosophy has made profound contributions to human knowledge and culture. John Locke's Two Treatises on Civil Government , for example, lay the foundation for the political system in the United States, and that, despite its flaws, seems like a good thing. But maybe you weren't thinking about ethics or politicial theory. Well, Rene Descartes's Meditations on First Philosophy helped to establish a conception of the world and what is required for knowledge of it that made it possible for empircal science to grow and flourish, and that was a pretty good thing, too. But maybe that isn't what you had in mind, either. To speak for myself, I tend to think of philosophy (outside ethics) as what something is before it's science. Indeed, in Descartes's time, there wasn't a division between philosophy and science. There was just "natural philosophy", and both the Meditations and his work on optics were part of natural philosophy as he understood it. But as a result...

In college, I had a professor who was both a devout Jew and a Kant scholar. Kant, you may know, was pretty anti-Semitic, which wasn't uncommon then, of course. I asked him how he handled that. He said to me, "One wouldn't expect a geometer to be a triangle" by which, I take it, he meant that someone who can think profoundly about moral questions need not be very good at putting theory into practice.

Philosophers do spend a good deal of time worrying about this matter. Indeed, it is characteristic of many areas of philosophy to be particularly interested in the "unprovable assumptions" with which arguments begin. Two examples: Perceptually-based beliefs---such as that there is a window in front of me---form the starting point for many of our beliefs. (Empiricists hold that all beliefs must be grounded there, but let's set that aside.) But it seems clear, at least to some of us, that these beliefs are not reached by argument from other beliefs. In that sense, they cannot be "proved" on the basis of anything else. How then should we understand how we arrive at such judgements? What is it for one of them to count as known? These are basic questions in the philosophy of perception. In mathematics, theorems are proven from axioms. Axioms, on the other hand, are accepted as true without proof. On what ground do we accept such axioms as, say, that, if there are two sets A and B, then there is a...