From an interview with Gödel's biographer, Rebecca Goldstein. (via MeFi)

He had meant his incompleteness theorems to prove the philosophical position to which he was, heart and soul, committed: mathematical Platonism, which is, in short, the belief that there is a human-independent mathematical reality that grounds our mathematical truths; mathematicians are in the business of discovering, rather than inventing, mathematics. His incompleteness theorems concerned the incompleteness of our man-made formal systems, not of mathematical truth, or our knowledge of it. He believed that mathematical reality and our knowledge of mathematical reality exceed the formal rules of formal systems. So unlike the view that says there is no truth apart from the truths we create for ourselves, so that the entire concept of truth disintegrates into a plurality of points of view, Gödel believed that truth - most paradigmatically, mathematical truth - subsists independently of any human point of view. If ever there was a man committed to the objectivity of truth, and to objective standards of rationality, it was Gödel. And so the usurpation of his theorems by postmodernists is ironic. Jean Cocteau wrote in 1926 that "The worst tragedy for a poet is to be admired through being misunderstood." For a logician, especially one with Gödel's delicate psychology, the tragedy is perhaps even greater.