1. What’s going on mathematically?
Someone points out that I’ve made a very big mathematical mistake. Perhaps in a paper that I’ve submitted, or during a research talk. Mathematicians are devout truth-seekers, and mathematical truth can be harsh.
2. What is the emotional and logistical context?
The context is social. I’m presenting what I believe to be correct, with confidence. Someone points out a serious flaw.
3. What thoughts are there?
The flaw is usually presented with a counterexample. So first there is cold certainty (of the mistake), and then along with an unpleasant emotional response there are non-mathematical thoughts about the non-mathematical implications. Does this mean the paper is junk, or the theorem fails? Maybe I won’t get a good letter of recommendation then? Does this mean I’m actually an idiot and should quit mathematics? How can I acknowledge the mistake and recover in a way that saves some face?
4. What quality of awareness?
There is a shock and a vivid immediacy. I feel acutely aware, but with an instinctual fight-or-flight quality. My thoughts actually move very slowly, and I find myself focusing acutely on a single symbol on the blackboard, the texture of my seat, or my breathing.
5. What emotions?
There is usually an unpleasant mix of embarrassment, terror, panic, and disgust. Once there was a feeling of being betrayed. There is also a strong feeling of surrender and letting go, which is maybe the silver lining of this mathematical experience. I’m forced to surrender my pride and a bit of ego, in the face of incontrovertible mathematical truth.
6. What does it resolve to, after how much time?
It’s necessary to immediately admit that a mistake was made. There may be an effort to fix or learn from the mistake, but it might be clear that there is no fix. Some face-saving gestures, some wound-licking.
7. How frequent is this flavor?
Maybe once or twice a year for me. Of course, there are smaller mistakes all the time. I wonder if there are mathematicians that have never made such a big mistake.
8. What are good/bad ways to change or follow it up?
There can be something very freeing, about surrendering pride and ego and accepting reality. Mathematical truth is uncompromising and absolute, and mathematicians are harsh truth-seekers. When I can step back from the frustration and embarrassment, I can watch my ego dissolve a little, and find new freedom in the experience. When my mathematical world has just broken so dramatically, it seems like the real world is going to break — and yet it doesn’t.
The worst thing to do is get caught up in the emotions, to the detriment of others. Sometimes I find myself getting angry and trying to place blame on someone else. In the end, however, it’s absolutely clear who is responsible for the mistake: me.
I’ve had mistakes pointed out graciously and ungraciously, and there’s something to be said for compassion and understanding in these moments. Doing mathematics means being stuck, confused, and wrong most of the time. It is our job to clarify and get to the bottom of things, even if that means pointing out a fatal mistake to a colleague. By going through the process of making a big mistake and having it pointed out, I’ve developed a little more empathy for others in the same situation.