Once in a while, I feel a bit tired and discouraged, especially when I find myself stuck while trying to solve a particular problem. I attack it from different angles, try special cases, write down thousands of lines of code, and the little brat doesn’t give in. Sometimes, the best conclusion I am able to obtain is “none of these methods work”. Frustration sets in and, many years ago, I used to think to myself: “I wish I were as smart as person X; they never run into trouble like this.” But guess what? They do! An extremely successful researcher, whom I greatly respect and admire, once told me this:
At least half of my research has been unsuccessful.
…This is the flavor of mathematical discovery. It is an uneven process that often becomes hectic, with periods of sleepless of half-sleepless nights. It requires the kind of passionate concentration in the grip of which you forget about everything else for a while. To be successful at it, you must have “fire in your belly.” And it certainly helps if your basic inclination is to persist and not give up in the face of difficulties, not to become dejected in case of setbacks, but to try again and again until you manage to find the right way.
In the same book, Balas refers to a former colleague and collaborator of his, Grigore Moisil (a famous Romanian algebraist), who had some interesting views on how to do mathematics:
Mathematics is not necessarily done at your desk. Mathematics is done when you wake up in the morning and do not immediately get out of bed; it’s done in the bathtub; it’s done while sitting on the toilet; it’s done while you are dressing; and it’s done while you are taking a walk.
I agree with Moisil. That’s part of the fun with math; you can work on it just about anywhere. (I’m particularly fond of all kinds of waiting rooms: doctor’s/dentist’s office, airport gates, nail salon while waiting for wifey, etc.) In fact, changing your work environment may actually help. I’ve had many good ideas away from my desk.
Do you have a motivational quote or text to which you refer when feeling a bit discouraged? I’d love to read about them in the comments.