What kind of high-level mathematics is provably impossible for computers, but merely difficult for suitably-trained humans?
If you're thinking of either of Gödel's incompleteness theorems, you may be slightly mistaken about what they say. In general, a human operating by rigorous standards of proof is no more able to prove the completeness and consistency of certain formal systems—using the tools provided by those formal systems—than a machine can.
If, as a human, you somehow prove the consistency of these particular formal systems, you run smack into Gödel's second theorem: For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. Thus, any proof of consistency is self-defeating, whether it's made by neurons or silicon.
Really, math doesn't care what parts of the periodic table you use to prove things. :-)
Correct. For those that don't agree see the Incompleteness Theorem. It was once thought (by Hilbert no less) that computers would one day be able to derive all mathematical truths for us. Alas, G\:{o}del came along and ruined all of those grand plans by proving such an endeavor was in possible. He also showed the limits of human reason. Computers may or may not be "dumber" than humans, but we know that they can't be "smarter."
