Roger and Rick have lured me into this discussion, but I'm afraid I only have time for a drive-by potshot. Roger cited Casti's intriguing paper "The Outer Limits," which argues that nature might still be infinitely knowable even if our mathematical models for it are not.
This optimism seems arrogant to me. Goedel and Heisenberg didn't set out to devise theories that explain most of the cosmos while leaving some secrets intact. They set out to know everything, and the theories themselves pushed back.
Furthermore, it's not that the incompleteness theorem and uncertainty principle rebuke scientists for trying to nail down the humanistic unknowns on the timeworn Two Cultures checklist (art, morality, the soul). They're way less generous. They say, "I'm not gonna tell you whether that theorem is true" and "I'm not gonna tell you the spin of that electron."
More to the point, real-world tests such as the Bell experiment prove indeterminate theories like quantum mechanics are a better fit for our world than their deterministic competition. Present-day physicists believe the universe is slippery not because they distrust current theories, but because they do trust them.
This is why I have no interest in the supernatural. The natural is quite mysterious enough for me.
jon
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