No philosopher worth her salt would try to answer this question as formed because of the large number of buried assumptions in it that presuppose the form of the correct answer, and that answers not meeting that form are, by definition, incorrect. Your question is a sophisticated version of "have you stopped beating your wife?"
What "truth" has mathematics itself produced from that search which is well-defined, non-trivial, and correct?
Is it a loaded question? I think it isn't, and it has a lot of valid answers. Here are some recent ones: http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_ma.... Why the corresponding question for philosophers should be loaded? Are they exempt from settling things?
First, Goedl's Incompleteness Theorem comes out of philosophy through the early 20th century's focus on logic and positivism. I'm sure you consider that well-defined, non-trivial, and correct.
Second, there is no requirement for philosophers to "settle things", and it's undemonstrated that, in order to settle things, the answer must be well-defined, non-trivial, and correct. One example of that is Wittgenstein's dismissal of a significant number of previous philosophical problems as empty language games. The field advances; it doesn't "settle things" in the sense that mathematics lays down proofs upon which the edifice later builds. Sometimes the advance is dismissing large portions of what went before as mistaken.
I don't consider Godel's theorem to be philosophy. Its status is similar to Cantor's theorem stating that reals are uncountable. Surely is inspired by philosophy and has philosophical consequences, but it is a part of mathematics. Trivia: Godel used the Chinese remainder theorem in his proof http://mathoverflow.net/questions/19857/has-decidability-got.... I acknowledge that philosophy can be inspiration for mathematics, but this is rather unsatiating, as very many things can be.
You might not, but philosophers certainly do--I learned it in my philosophy classes on logic. I also learned there about Cantor's diagonalization proof. More generally, in the early 20th century there was a huge overlap between mathematics and philosophy. You had Russell and Whitehead's Principia Mathematica, you had the Vienna Circle, you had Carnap and logical positivism... It's really not possible to cleanly categorize Goedl's proof into either math or philosophy--they weren't disjunct categories then, and they're not now, either. More importantly, if you approached any of those figures and asked whether they were separate things, they'd have thought the question nonsensical. Philosophy was, and still is, the home of logic in the academy, and the fact that logic and math frequently seem like different sides of the same coin doesn't settle the issue either way.
