| Yes, it's much easier for humans to prove complex theorems than computers. Just as it is much easier for humans to build complex programs than computers. Often times, a proof is (much) more complicated than just mixing theorems A, B, and C together to get result D. Many times proofs will require developing entirely new mathematical structures, or can be done only in steps. As an example, a friend of mine just proved a great theorem about whether or not Hessenberg varieties can be paved by affines. Ignoring what this actually means, the proof involved firstly recognizing that a Hessenberg variety actually fits within a spectrum - it is somehow "between" other types of well- (or better-) understood varieties. These are paved by affines in (somewhat) obvious ways, and so it is natural to try to use the "between-ness" of the Hessenberg variety to construct a "between-ish" paving. This isn't a proof, but it is a direction (one that turned out to be fruitful). It would be very hard for a computer to have made this intuitive leap. And it almost works. Except now you can't reduce things like you would like, and you need to construct multi-level fiber bundles in order to get the triviality that you want. And wait - this only works if the Hessenberg variety that we started with has such-and-such property. So she's reduced the problem to a simpler problem. How does she know it is simpler? Intuition. A computer would have a very hard time recognizing whether or not this is actually simpler (even if it could get there). It turns out that this property holds when our Hessenberg variety is constructed from an element whose semisimple part under the Jordan decomposition is regular. So she has proved it for a huge class of varieties, but not all. Such a partial result would also be very difficult for a computer, since it isn't exactly what she set out to do. She's only partly there. But even this part is huge. And there isn't anything hand-wavy about her argument, either. It's perfectly grounded in well-established theorems, all of which she understands the proof for. It's just that the original concept was shakey - the "between-ness". She had to work to ground it. It would have been exceptionally hard for a computer to make several of the intuitive leaps necessary to come up with the ideas for the proof in the first place. |