A Most Ingenious Proof

Professor Q was teaching group theory to undergrads. One day, while handing back homework, as he was about to hand me mine, Professor Q excitedly announced to the class that Alice's homework contained "the most ingenious incorrect double induction proof I've ever seen."

I froze in my seat, and stopped listening. All I could hear was the word "incorrect", reverberating in my head. I might not be as inventive or creative as the next mathematician, but I prided myself on getting it right. For me, submitting an incorrect proof was one of the worst sins a mathematician could commit.

Professor Q briefly explained the proof to the class, but I was so stunned by the public humiliation that I could only half listen. I turned to a fellow student and asked if he understood what was wrong with my proof, but he hadn't followed the explanation.

I spent the rest of the class looking at my homework, attempting to understand where I'd gone wrong, while trying not to show how upset I was.

I looked at my solution again that night in my dorm room.

Still no success.

Every few years, I would take out that problem set, and try to find the flaw in my proof. I knew that to figure it out would mean that I had reached a higher level of mathematicial maturity. It was important to do that on my own. Each time, I eventually put away the page, disappointed in myself for not finding the mistake.

Grad school was a particularly depressing time. Grad students (especially female ones) were at the lowest rung of the hierarchy at Princeton, even below undergraduates and postdocs, and the early 1980s might have been a particular low point in morale for math grad students.

One night I was working very late on my PhD thesis. It was perhaps two in the morning, and I was getting more and more depressed. I didn't feel I was making fast enough progress on my thesis, I was unhappy with the way I was treated at Princeton, and I felt terribly sorry for myself.

Always the problem solver, I decided to cheer myself up by looking at the incorrect proof, and finding the mistake. I knew I had made a lot of progress and learned a lot since my sophomore year of college. I was sure this was the right time to revisit that problem set. I would feel better, and perhaps finally appreciate how ingenious my incorrect proof had been.

I dug up the dreaded problem set, and read the proof. It wasn't very well written. But putting that aside, I could still figure out the logic of the argument. And I still couldn't figure out what was wrong with it. My depression hit a new low.

There was nothing I could do in the middle of the night, so I went to bed and slept fitfully. The next day I mustered up the courage to show the proof to a quite brilliant postdoc. I told him that it was an incorrect double induction proof, and I asked him to find the mistake. He read through the proof, handed it back to me, and said, "It's not very well written, but the proof is correct."

I handed the proof back to him, and said, "There's a particularly ingenious and subtle mistake. Please read it again."

He read it again. "There's nothing wrong with this proof," he said.

I insisted. "Professor Q said it's wrong, so it's definitely wrong! Please read it again," I pleaded as I thrust the homework back at him.

He refused to look at it and handed it back. "I don't care who says it's wrong. It's a correct proof."

I was taken aback. Whom should I believe, Professor Q or a young postdoc? Q was a Harvard professor! It hadn't occurred to me that he could be wrong about something he had stated so remarkably confidently.

I confess that I'm still afraid to show that proof to anyone, lest they find the truly ingenious, embarrassing, humiliating mistake.

The potato test

When I interviewed for a job at a British university, to my surprise they brought all the candidates together at the same time, and put us up in a grand historic mansion with beautiful gardens and grounds. This kid from Queens had come a long way.

It felt like something out of an Agatha Christie or Dorothy Sayers mystery. Was I in "And Then There Were None", where whoever survived till the end was presumed to be the murderer? Or was the survivor the one who got the job? Or both?

The weather was dreadful, with unrelenting rain. One reason the job ad had intrigued me was that someone once told me there were palm trees in the area. My queries about palm trees were met with quizzical looks. Some palm trees did grow nearby, but they were all quite short.

I was older than the other candidates, and the only female one, and I felt a bit like a mother hen. When the powers-that-be needed the candidates to meet in the lobby to go somewhere as a group, they put me in charge of corraling the others. And when one of the candidates did mind games on one of our rivals to undermine his confidence, I pushed back. Since I had a secure job and other offers, I could afford to be supportive of the other candidates, though my sense that we were in a murder mystery did lead me to keep a watchful eye on the bully. If we were in "The Lady Vanishes", I might be the lady.

The Vice Chancellor (the British equivalent of a university president) hosted a formal dinner for the candidates, the external assessors, and assorted university dignitaries, administrators, and faculty, at a different old country estate. I half-expected Miss Marple or Lord Peter Wimsey to show up and reveal all. I looked at the many pieces of silverware at my place setting, and decided to follow the lead of the locals before choosing which utensil to use when, just in case the hiring committee was watching us and keeping score.

I was seated next to the Vice Chancellor, who was quite charming and interesting. In his position as head of the university, he had traveled the world and met many fascinating people. Someone who impressed him greatly was Libyan leader Muammar Gaddafi. He told me what a nice person Gaddafi was. How does one respond to that, during a job interview? As tactfully as I could, I reminded the Vice Chancellor about the plane that exploded over Lockerbie, Scotland, to give him a chance to put his views in context.

We were served three or four different styles of potato. Mashed, roasted, scalloped, and boiled? One kind was on the plate, another was in dishes we passed around the table ourselves, and yet another was served to us from serving trays by the people who waited on us. Was I really supposed to eat that many different kinds of potato? Or stick to one? Was this another test, to see if we fit in, and were discriminating enough to know which potatoes to accept or reject? I ate them all.

I hadn't realized that the job offer would be made before I left, and that I was expected to respond on the spot. They kindly granted me an extension so I could think about it. They treated me so well, that I was sad when I turned down the offer.

Kangaroo attack?

When I was a visitor at an Australian university in 1989, I saw strange flyers on walls here and there on campus, warning about attacks on women. Detailed physical and demographic descriptions were given of the women who were attacked. So little was said about the attacker and the modus operandi that I realized that the flyers were consistent with the women having been struck by lightning. Or punched by a kangaroo.

Now, I knew (or at least strongly suspected) that the women hadn't been hit by lightning or punched by kangaroos. It would be odd to put up flyers for lightning strikes. I guess if an unusually large number of people in a short period of time were hurt by lightning, it might make sense to post flyers.

But why give detailed descriptions of the victims?

The fact that some of the victims wore jeans, or had Asian ancestry, or were of medium height didn't help me much. Nothing in the descriptions told me anything useful about what I should do differently (other than perhaps not walk alone on or near campus). It was winter and got dark early. There wasn't much I could do to ensure that I wouldn't look, to an attacker on a dark evening, like the victims.

As I walked from the math department to my lodging at the end of the day, I wondered what I should and shouldn't do. Should I be afraid of the boy walking towards me? Or of the older man with a beard? Should I turn and run when the mustachioed middle-aged man appeared suddenly around a bend in a dark alley?

Descriptions of the attackers might have been helpful. Or whether the women didn't get a good view of the attacker because he/she/it always approached from behind. Even better would have been improved lighting. (I recalled being told at Princeton that better lighting wasn't an option since bright lights would be unsightly.) I wondered whether the point of the flyers was to get women to stay inside, especially young women of Asian ancestry.

Luckily for me, lightning didn't strike while I was there, and I wasn't ambushed by kangaroos.