The Department at Victoria University at Wellington compiles a quite interesting set of reflections by contemporary and 20th-century figures in answer to this question. My favorite is the one from John Campbell, now at Berkeley:
Philosophy is thinking in slow motion. It breaks down, describes and assesses moves we ordinarily make at great speed - to do with our natural motivations and beliefs. It then becomes evident that alternatives are possible.
My least favorite, since it makes philosophy out to be hopelessly conservative in its ambitions, is from David Lewis:
One comes to philosophy already endowed with a stock of opinions. It is not the business of philosophy either to undermine or to justify these pre-existing opinions, to any great extent, but only to try to discover ways of expanding them into an orderly system. It succeeds to the extent that (1) it is systematic, and (2) it respects those of our pre-philosophical opinions to which we are firmly attached. In so far as it does both better than any alternative we have thought of, we give it credence.
Thomas Nagel's account is bound to be highly contentious in certain circles:
Philosophy is different from science and from mathematics. Unlike science it doesnÍt rely on experiments or observation, but only on thought. And unlike mathematics it has no formal methods of proof. It is done just by asking questions, arguing, trying out ideas and thinking of possible arguments against them, and wondering how our concepts really work.
Those unhappy with Nagel, will be happier, I suspect, with Quine's take:
I see philosophy not as f groundwork for science, but as continuous with science. I see philosophy and science as in the same boat - a boat which f we can rebuild only at sea while staying afloat in it. There is no external vantage point, no first philosophy. All scientific findings, all scientific conjectures that are at present plausible, are therefore in my view as welcome for use in philosophy as elsewhere.
Sources, and more quotes, are at the Victoria site.
