Mark Dominus writes about a notional "Journal of the Empty Set", which would publish papers on results about mathematical objects that don't actually exist. He asks: "But on the other hand, suppose you had been granted a doctorate on the strength of your thesis on the properties of objects from some class which was subsequently shown to be empty. Wouldn't you feel at least a bit like a fraud?"
There's a (perhaps apocryphal) story that's made the rounds in my department about one possible way to fail your thesis defense. Namely, it's said that a student went to their defense and began by defining the sort of group they were studying. There was a rather long list of conditions, and one of the examiners quickly proved there was no such group. The student therefore failed.
The good thing about combinatorics? That sort of thing doesn't happen. Usually we can actually construct explicit examples of the things we're talking about.
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Good thing about my thesis, I can prove that my own computation method is enough for certain special cases recognized by highly technical conditions.
Bad thing about my thesis, one of the examples of the computation methods demonstrates that for one of the TWO classes of groups that have cohomology rings fulfilling even the first of those technical conditions, the rest of the conditions fail.
And the case left? It's already in the literature, though badly justified.
But on the other hand, suppose you had been granted a doctorate on the strength of your thesis on the properties of objects from some class which was subsequently shown to be empty. Wouldn't you feel at least a bit like a fraud?
Well, it would depend on the circumstances. If you wrote a paper about projective planes of order 12, with the goal of contributing to a non-existence proof, and your methods did contribute something, then you should feel proud. If you expected the objects to exist, and they did not, then it depends on how interesting or surprising their non-existence is, and whether your results played any role.
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