Elizabeth Bear, a SF writer whose book Carnival I’ve just started reading, has a blog. I was very pleasantly surprised to see a series of entries tagged Math Is For Girls. As it turns out, Bear has decided to teach herself some basic mathematics on her own, from a textbook. She chronicles her progress in her journal. And, after fielding too many calculus questions of the “is it going to be on the exam?” variety, it’s such a joy to read a post like this one:
People keep asking me why I’m teaching myself math at the ripe old age of 35.
And you know, it strikes me as odd that so many people think you need a reason to learn something. Because it’s kind of its own goal, isn’t it? But then I realized that there’s a better answer.
And the answer is, of course, because I am a writer.
And then they look at me funny.
But it’s true. If you’re interested in writing books (well, in writing anything, I would guess, but books are what I know from), you might as well get invested in the whole lifetime learner thing now, because you are gonna need it. See, the problem with writing is that you need to know everything. […]
Now imagine how much research goes into a novel.
I’ve walked through or at least driven past every real place I describe in Blood & Iron. (I have not however ridden a horse the entire length of Broadway.) I have however ridden a horse, and read accounts of it by people who actually know what they are doing. I’ve swung a sword. I’ve fired a gun.
I am not really a novelty seeker, but I try to make a point of, whenever an opportunity to do something new comes up, giving it a whirl. (I’ll never water ski again, though. Ow.)
And now, proof that writer’s research is not always that different from research in mathematics.
And the thing is, underneath that research there needs to be a layer of general knowledge that tells you when to stop and look something up. Because it’s not the things you don’t know that trip you up. It’s the things you *think* you know. That you don’t stop to think about.
Because often, you are wrong.
The “things you think you know” is a trap that math researchers know all too well.
Let me explain. When we write a math paper, we usually begin with a rough outline: the general scheme of the proof, the key ingredients, the main building blocks. We then have to fill in the details. Sometimes it works out more or less as expected. Other times, the building blocks don’t fit together and have to be adjusted or replaced. Then there are times when the whole scheme has to be overhauled and we start over again.
Not all of these steps are equally difficult. Some are very hard. They require substantial new ideas and a lot of hard work, and they will be thought of as the authors’ main contribution. But then there’s also the narrative in between, where we might use a reasonably standard argument to get from Hard Step A to Hard Step B, or adapt something from existing literature, or just need a certain amount of preparation before the real work begins.
Needless to say, we focus on the hardest parts. That’s where we double-check our arguments, and triple-check them, and then check them some more. Once we believe we’ve solved them, the temptation is to consider our work basically done.
And then we find out later that actually it was the “standard argument” in the “easy part” where we should have been paying more attention.
It doesn’t help that when we submit papers for publication, the referees tend to focus on the same key parts and might not check the “easy parts” at all. (That is, if they’re knowledgeable about the subject. If they’re not, they might only read the introduction and do a perfunctory check.)
I have one published paper (about 3-body quantum scattering, joint with Christian Gerard) where the proof of the main result is incorrect. We did work out the “hard part” – something called a Mourre estimate, in case you are interested. What we really needed, though, were certain standard consequences of the Mourre estimate, and for that we referred to a theorem in a well-known reference book. We later found out that the theorem was incorrect as stated.
This happens more often than you might think. We rely on other people’s results without checking their correctness. We trust that the peer review system works, and frankly, we would never have enough time to verify everything that we use. We sometimes skip the details of a lengthy calculation if it looks similar to another lengthy calculation that we have already performed elsewhere. We assume that something is obvious because it looks obvious, and never bother to stop and think about it. And sometimes we’re wrong.
Mistakes like these are not always fatal. In the case of the Mourre estimate paper, we gave a corrected and simplified proof in a monograph we wrote later on. “Standard” arguments are often robust enough so that even if a gap is found, there will be some way around it. But once in a while, a paper has to be retracted. Plus, it’s embarrassing to have to explain to your graduate students that, well, the argument in your paper is incorrect and they should not use it in their thesis work.
But back to Elizabeth Bear:
And that’s why I’m learning math. And guitar. And why I practice archery. And why I juggle. Even though I do all of those things incredibly badly.
And it’s also why I go for long walks. And read constantly, about as many topics as I can sustain an interest in. And why I love reading blogs and books written by people who live in other countries.
Doesn’t that make you want to check her website? She has posted some short stories, too. What are you waiting for?