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I have found over the years that the most productive research outcomes occur when there is a balance between optimism that one's preferred techniques will work (or that one's conjectured assertions are true), and pessimism that a proposed technique will encounter insuperable obstacles, or that a hoped-for result is unlikely to be true. Too much pessimism and one becomes discouraged and quits; but too much optimism and one wastes time chasing arguments or results that will never pan out. 1/5
Therefore, an ideal collaboration should contain at least one "pessimist" and one "optimist". I've served as both during various collaborations (and sometimes switched sides halfway through!). As a general rule, the junior collaborators tend to play the role of energetic optimists, and the senior ones as wary pessimists, though there are notable exceptions. 2/5
There was one time when I (and my coauthor) spent several years optimistically trying to prove a certain inequality, buoyed by a partial result we had already obtained, and some appealing near-misses in the literature. There didn't seem to be any way to adapt the known methods to attack the inequality itself. It was only after presenting our partial results in a talk that a colleague in the audience astutely asked if anyone had attempted to build a counterexample to the inequality. 3/5
This possibility simply had not occurred to me, but that evening I realized that there were some relevant counterexamples to related results in the literature that might be adapted to this problem, and a few weeks afterwards I had managed to do so and settle the inequality in the negative. 4/5
I have since tried to adopt my colleague's habit of tempering pure optimism with a sincere effort to locate counterexamples. Even if the result is true and counterexamples do not exist, these efforts often "map out the negative space" and leave a lot of clues as to how the proof of the positive result has to proceed, for instance by directly identifying the most dangerous putative counterexample scenarios and suggesting what the right "weapons" are to defeat them. 5/5
@tao Shelah talks of consistency results this way. They may be interesting on their own, of course, but also "forcing is necessary to tell us when we cannot prove a theorem".
(S. Shelah, "The future of set theory", in Set Theory of the Reals. Haim Judah, ed., Israel Mathematical Conference Proceedings, vol. 6, https://arxiv.org/abs/math/0211397)
@tao Reminds me of a quote by a mathematician (no memory where) suggesting that when working on a conjecture, switch every few weeks between trying to prove it, and trying to find a counterexample.
@tao Ideally a computer search should be automatically triggered for explicit counterexamples (along with semantic search for relevant results)
@tao Didn't Walter Hayman's thesis advisor ask Hayman to furnish a counter-example to the Bieberbach conjecture, which turned out to be true?