Post Made Community Wiki by S. Carnahan
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Hey Tim. I'm not so sure whether my model of how "proving a theorem" works is the same as yours. But let me give some kind of an example of something and you can take it or leave it. I'm writing a paper with Toby Gee at the minute, and we're both number theorists, and the arguments in the paper are "robust" but the details need checking. We're now at the point where we're writing up technical calculations and these technical calculations are mostly in the area of representation theory of reductive algebraic groups, an area which I think it's fair to say that neither Toby nor I would call ourselves experts in. So we have this overall "robust" argument, and a write-up that exists but occasionally says "lemma: (statement in representation theory); proof: TO BE ADDED". We 'know' these lemmas are true because they fit into our overall picture, but occasionally when I write one of these things up I can't go from my intuition to a rigorous proof. Here's an example of an occasion when I got stuck:

This question of mine.

Ben Webster made a crucial remark that enabled me to finish the argument, so that lemma went from "must be true but proof not yet written or even discovered by authors" to "lemma proved".

So if I were interpreting your question in a particularly anal way, one might argue that had this lemma been the last of the lemmas we need to write up the proof of, then Ben's contribution might be "just what I needed to complete my research project". Unfortuately there are several more to go :-).

Having said all that, it's not clear to me that MO was "crucial" to solving the problem. I could have worked more on the problem until I'd done it myself. I could have asked one of the representation theorists in my own department. I could have left it and hoped that my co-author sorted it out. All of these would have been viable approaches. Why did I ask at MO? Simply because I am sick of writing this paper and asking at MO was by no means the only way of solving the problem, but I had high hopes that it would be the quickest.

Kevin