I think mathematics is a very creative science. There’s a story that students see, especially in K-12 education, that mathematics is and has always been there: they are led to believe that not everything in mathematics is a human creation. There’s also this debate of whether mathematics is created or discovered. I say it’s created, but the truth of the matter is, there’s no consequence of ending that sentence one way or the other. It doesn’t matter to the working mathematician. It’s more of a fun mind game, and then you set it aside and get back to work. But I tend to lean on the side that it is a created thing, that the ideas are all creations.

Some ideas in mathematics are pretty much inevitable. The counting numbers — 1, 2, 3 — I think that’s sort of an inevitable creation, but nonetheless, it is a creation. Certainly, negative numbers — that’s something that people made up. And then all the imaginary numbers, and then vector spaces in 5, 10, 12, 15 dimensions, or infinite dimensions — these are all things that people made up to model the universe, or just because they’re interested in these things themselves.

I don’t know to what extent math is taught as a creative subject. I was very fortunate that I had, in high school, a teacher that pushed the idea of creativity and wanted us to think of new ideas, new things — just make something up and see how it works. And that was, like, what? You can do that? Thinking about infinity as a number was a creation.

And then you have to start deciding, okay, what properties do we want it to have? How does it behave with the other numbers? Those are all decisions, and the guiding factor is to what extent it comports with what properties we already expect numbers to have, like the commutative property of addition. Things like that we want to preserve as much as possible.

There’s also the activity of creating mathematics, which is even harder to describe. It’s definitely like puzzle solving. You spend a lot of time having no idea what you’re thinking about. For me, there are two things that get my juices going: can I draw a picture of it? Whatever it is, can I make myself diagrams? That really seems to help. And the other thing I like to do is to write a computer program to model it. That activity helps a lot.

In graduate school I did spend a fair amount of time lying on a couch, staring at the ceiling with a cat on my stomach. It was just taking some time to think. It’s important in the creative process to set things aside and come back to them later. I don’t know the neurobiology of this — I’m not a biologist, or a psychologist — but I think your brain is still working on problems even when you’re not paying attention. 

That happens to me with the word puzzles I do every day. I can’t get all the words, so I set it aside, and I come back an hour later, and in an instant I see the answer. It’s there. How could I not have seen this before? Something’s happening in the background somehow. I don’t understand it, but it does seem to be there. The whole creative process is pretty mysterious.

It also helps sometimes to work problems out in groups. Being forced to explain something to somebody else helps to clarify what I’m thinking about. I may have this vague idea, but when you force me to say it to somebody else, I realize I don’t know what I’m talking about. They ask a very basic question, and I think that helps the creative juices flow.