Brilliant Math Genius

The other day, I showed up at Ed's house, found the front door unlocked, raced up the stairs to his room, met him in the hall, and said, "Your girlfriend is a fucking brilliant math genius!"

Let's back it up.

I'm not a math genius. I am pretty good at math, and getting better all the time. I also think that almost anyone can learn and enjoy math, and that most people who think they are "bad at math" just haven't had good experiences with it and thus haven't stuck to it. (This applies double to my mom, who loves formal logic and puzzles and therefore is obviously math-inclined despite apparently never having enjoyed math itself.)

But math can really kick you around. Unlike history or literature, where you have progressively increased knowledge leading to mastery and insights, math can make you feel really stupid when you don't get it. (I might not understand the causes of the Great Depression, but this makes me feel, at worst, ignorant. When I do know some history, I am unaware of my lack of insight about it. You don't have to have the proper insights to understand what others have written.) The flip side of feeling like an idiot when you don't understand math, though, is feeling like a genius when you do.

Last week, I had a section on transition matrices and change of basis. (I promise I won't talk about this in detail.) I did all of the homework pretty quickly by stumbling (by some insight I couldn't recreate until recently) onto a method that worked, but when I triumphantly showed it to Ed, he didn't like my method (he's not a fan of the augmented matrix; he prefers an algebraic approach).

When we sat down to figure out the actual algebra, we found the section of the book rather inscrutable. (I have since seen the completely clear presentation in his linear algebra book, so I have a basis for comparison. No pun intended.) What we found by deduction from what I'd done in my homework didn't seem to make any sense, and Ed didn't feel it was consistent with the book's text. He hypothesised (hypothesied? hypothesized?) that someone different had written the questions from whoever wrote the text, and that the two didn't match. He also found that the questions and answers were not themselves consistent.

"But I did them all the same way," I protested, "based on what I read. And my answers match the ones in the back."

I said this over and over, but I couldn't show how it was true, because I had lost the understanding of what I'd done, somehow. I'd lost the knowledge of how it made sense.

I don't want to make it sound like I just rolled over, either. We worked on this hard for a long time, on two different days. We came to a conclusion about how most of the problems worked, but the conclusion was just stupid. It would sometimes require things to be multiplied backwards, in the direction that doesn't work for matrices. And it was just plain not sensible, even though it was workable in a lot of instances.

I was troubled. The night before my breakthrough, I felt that I thought about these problems all night while I slept. By morning, I had a new tack to try, and I hastened to the library to work it out.

By 90 or so minutes later, I was ready. I had figured out how to show that the book was consistent, correct, and sensible, and I was eager to demonstrate this to Ed. We had been stupid and missed a kind of obvious point. (For my fellow linear algebra students, I will state the central insight thusly: a transition matrix does not change a basis to another basis; it changes a vector stated in terms of one basis to the same vector in terms of a different basis.)

Hence my rushing up the stairs, declaration of myself as a genius, etc.

Ed was skeptical. We'd worked pretty hard and pretty clearly on this, after all.

"We're going to run this like a game," I said. "And games have rules, right?"

"Does this mean our relationship is a game?" he asked.

"Yes, but you knew that already."

"I guess you're right."

"OK," I said. "These are the rules. You can't go ahead of me, and you can't go sideways from me. You have to let me show you this. You can only interrupt me if you think I've done something wrong, or if you don't understand what I'm doing."

"No going forward, no going sideways. But I can go backwards? Got it."

We chatted a little more while he made some tea. He asked if I was going to be crushed if he didn't agree with me, once I had showed him my argument. I told him I'd only be crushed if I still thought I was right, but that I was confident we could come to an agreement either way. He was pretty sure I was going to turn out to be wrong.

I wasn't. In fact, once I gave him the central insight, the rest pretty much fell into place, though I went through my demo anyway. It was terrifically fun. Some of my algebra was wrong (for instance, in one place I divided each side of an equation by a vector, which is an undefined operation), but it led to the place we'd ended up the other day anyway, except that now that place didn't seem nonsensical. (The algebra worked in the other direction, just not in the way I'd derived it. What I provided was not in any way a proof, but I did show some relationship between our work the other day and the sensical way that the book was really working.)

Like I said. Brilliant math genius.