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 Göd

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. :-)