Lesson from my father

 I was perhaps 9 or 10 years old when I told my father that I was due for a raise in my weekly allowance, from 15 cents to 25 cents. (Yes, this either dates me, or says something about our socioeconomic status, or both.)

I explained to him that 25 cents was the going rate.

"How do you know that?" he asked.

"All my friends are getting 25 cents," I replied.

"Who, for example?"

"Suzy next door."

"How do you know that Suzy next door is getting 25 cents?" he asked.

"She told me."

"So all you know is that Suzy told you she's getting 25 cents. You don't know that she's getting 25 cents. For all you know, she's telling her parents that she wants a raise since you're getting 25 cents."

"Are you saying my friends are lying?" A weak and angry retort, but it was the best I could do. I had already lost. I think we compromised at 20 cents.

It's true that my training as a mathematician, especially at math summer camp when I was in high school, helps me distinguish true from false, and true from "someone told me it was true". So I'm continually surprised when I see mathematicians fall for false "proofs" or fake news. But then I remember that I had the additional benefit of learning from a father who was a newspaper reporter back in the day when journalism was supposed to be about facts, not opinion.

My Superpower, Part 4

I was sitting in the Common Room of the Princeton math department, reading a journal article and minding my own business. A few other grad students were hanging around. A fellow student, whom I'll call C, bounded in and announced to his friends that he had found a major flaw in an important and influential work by a famous Princeton professor, in a paper that had been published many years earier in the Annals of Mathematics.

They heartily congratulated him, gleefully discussed how famous this would make C, and suggested ways to celebrate.

I looked up. From my prior experience with C, I thought it extremely unlikely that he had found a major mistake in a high-profile paper that had been around for awhile and had been subject to extensive scrutiny.

My father was a journalist, and he had trained me from a young age in critical thinking skills. He taught me that I shouldn't believe that something is true just because someone said it was true. Be skeptical. Though it took me awhile to learn to be skeptical of famous professors, I was quite ready to be skeptical of C.

I asked to see the paper, and I asked C to point to the mistake. It was near the beginning. I read the line in question, and realized that C had thought that (a+b)² was a² + ab + b², rather than the correct a² + 2ab + b².

I took C aside and gently discussed it with him. He agreed that it was his mistake, not the famous professor's. I was pleased that I had succeeded in telling him in a way that allowed him to save face with his friends. I don't always manage to do that.

So what is my superpower? Bullshit detection.

My Superpower, Part 3

Back at math camp as a counselor, some of the other counselors were standing around our kitchen after dinner, trying to prove a certain result about finite groups. Buoyed by my success in the group theory course a few years earlier, I tried to join them. After they went down a dead end, I suggested that we try to prove or disprove it for some concrete examples. My fellow counselors disdained that approach, which would have gotten their hands dirty with computation. They preferred to go straight for the loftier goal of proving it in general.

I loved group theory, I enjoyed getting to know individual finite groups of small order, and I thought of them as my friends. Seeing where the difficulties were likely to be, I played around with groups of order 24 and 36.

Eventually I showed the other counselors a counterexample to what they were trying to prove. I was pleased that I had learned from Loomis how illuminating a counterexample can be. But for my friends, a counterexample made the problem uninteresting, and they turned up their noses and went on to something else. (Would they have been more grateful for my counterexample had I been someone they respected more? I don't know.)

My Superpower, Part 2

After class, a fellow student whom I'll call M brought Professor Loomis a proof that M knew was incorrect. A group of us students gathered round to help Loomis find the flaw in the proof. Every step was intuitively correct, but it led to an incorrect conclusion. How intriguing! Did this reveal a fundamental flaw in the fabric of the universe? Of course not. But what was wrong with M's proof?

There were a dozen or so simple steps, and for each one, I knew an airtight justification for why it was true. Each one, except for one that clearly had the ring of truth.

I pointed to that step and said "I don't understand why this is true." The other students chimed in to explain it to me. (Some people now call this "mansplaining".)

Loomis exclaimed, "That's it! She's got it!" I felt like Eliza Doolittle with Henry Higgins, and wondered if we should burst into song. I fell head over heels in love with him.

Loomis gave us a counterexample for that step. I wish I had thought to do that! I hadn't thought any further ahead than "I don't understand why this is true."

My Superpower

This begins a series of vignettes on the same theme.

Part 1:

I stared at the test question. The test was in a group theory course I was taking at a math summer camp for talented high school students. We were asked to prove that a certain statement was true. I struggled with the problem for awhile, and finally wrote in the answer space, "I must have a fundamental misunderstanding of group theory, because it seems to me that ..." and I went on to explain how I concluded something that was in direct contradiction to what we had been asked to prove.

After the test, the students gathered around to discuss it. My friends had all come up with proofs for that question. No one else seemed to have a fundamental misunderstanding of group theory, the way I had. I sighed.

That test question shook my self-confidence. I thought I knew right from wrong, at least in mathematics. That was the beauty of math; you know whether you're right. I had thought I was good at it. Clearly, I was wrong. How depressing.

I was awed by the brilliance of the top students. Many of the best ones had impressive backgrounds---they had highly educated parents, went to unusually good high schools, and had already taken courses at universities. I wasn't in their league. I managed to do well on some of the tests by doing well on the problems they looked down on and didn't bother with, namely the routine ones that required very fast and accurate computation by hand. I was great at those (but that wasn't my superpower).

One of the counselors told me, "Congratulations! You're the only one who aced the test." I didn't know what "aced" meant, and I was too embarrassed to ask. When the professor handed back the graded tests, I had a score of 20 points. That seemed odd. Twenty points was the maximum. Why hadn't he subtracted points for my fundamental error in understanding?

The professor pointed out to the class that he had made a mistake, and accidentally asked us to prove something that wasn't true. I was the only one who realized it. Those who gave an incorrect proof had one point taken off.

I felt bad about getting credit for realizing the problem was wrong, since I had realized no such thing. It hadn't occurred to me that such an intelligent professor could make a mistake. I only thought that I must be stupid.

While I wondered if it was fair for my fellow students to lose a point due to the professor's mistake, I was happy to be the only one who aced the test. And I was amazed that such smart students could write incorrect proofs and not realize they were flawed.