1. What’s going on mathematically?
After working towards proving something for a while, I find a counterexample. This involves an insight, followed by a verification.
2. What is the emotional and logistical context?
These counterexamples usually show up suddenly. The most dramatic and surprising cases are after working towards a particular result for weeks, because my expectation is that I’ve been getting closer and closer to a complete proof. So counterexamples hit when I’m hopeful, maybe even overly idealistic.
3. What thoughts are there?
The initial insight is a surprising “Aha” moment, accentuated by the fact that most counterexamples have a simplicity and necessity that seems to stab directly into the essence of the problem. This is immediately followed by some concerned analysis of the situation – does this mean I just wasted three weeks? is there a way to fix it? But before a complete reassessment, there’s a careful verification, to prove that the counterexample is a counterexample, i.e. to prove myself wrong.
4. What quality of awareness?
It’s like the rug has been pulled out from underneath me. There’s a shock and surprise, grounded in certainty, that then trickles outward along logic pathways and finds a deserted city. Or worse, the city I thought I knew is now filled with people that speak a language I don’t understand. On a deep level, it’s an unsettled, shifting, almost paranoid wandering in this strange new city, searching for any familiar faces. But on the shallow level, there’s a sharp certainty and cleanness, as my proved counterexample resonates within itself.
These are the times when I’m most aware of the non-logical, heuristic, mysterious “intuition” I have built up about the math I do. I had a mathematical worldview in which Proposition X was true – this sense of the way things work guided me, helped me make sense of it all. But now that I have found a counterexample, it’s not just the statement of Proposition X, but the whole worldview, that needs to be adjusted.
5. What emotions?
Of course, I usually feel disappointed and frustrated, depending on the severity of the situation. At worst, it can devolve into fatigue and meaninglessness. (I’m fortunate that the most time I’ve thrown away on a false proposition is 2.5 weeks; I’m sure it gets much worse than that.) There’s also an undeniable sense of finality, that comes with proving any result – “at least now I know for sure.” It is a very strange feeling, to prove yourself wrong. This certainty is a feeling I almost only get from math, and for some reason I feel it more strongly when I’ve been proven wrong than when I’ve been proven right.
I’ll usually take a break from the problem for a bit, and then I feel some revulsion towards it. Maybe it’s a feeling of being betrayed, but I don’t want anything to do with the question. This goes away soon, though.
6. What does it resolve to, after how much time?
A good mathematician would say that in every counterexample there’s new ideas to follow up. Maybe I just need to tweak my hypotheses; maybe the counterexample is pointing towards the essence of what’s going on; maybe the fact that Proposition X fails is a “good” thing, that e.g. allows for more interesting behavior. I can usually start to pick up the pieces after a few hours.
7. How frequent is this flavor?
Oh, I’m such a bad research mathematician, this happens way too much.
8. What are good/bad ways to change or follow it up?
Bad: take it personally and get discouraged. Good: take a deep breath and get to work picking up the pieces. Mathematical intuition isn’t built overnight, and without surprises math would be boring.