I watched the movie Proof last night. I highly recommend it. It's not often that mathematicians get so much screen time.
The nature of mathematical creativity plays a large role in the movie. One comment was especially intriguing. (Minor spoiler ahead.) In describing the process of solving a particularly difficult problem, one character says, “It was like...connecting dots. Some nights I...I could connect three or four of them. And some nights they'd be really far apart. I'd have no idea how to get to the next one, if there was the next one.”
This description rings both true and false for me. The true part is I'd have no idea how to get to the next one, if there was the next one. This nicely summarizes what it's like to work on a hard problem. You don't know what to do next, and you're not sure that a solution even exists.
But the connecting dots part doesn't feel right. It makes the process sound linear, when the reality is anything but. In my experience, the really hard problems—the ones that require the most creativity—are most often solved by epiphany, by having the whole solution burst into your head, seemingly in an instant. That's when the connecting dots part happens, but it all happens at once, not a few a time.
Of course, the catch is that epiphany can't be forced. I'm reminded of the old joke about the lottery. A man prays fervently every night, “Please, Lord, let me win the lottery tomorrow.” After twenty years, a heavenly voice responds, “Would you buy a ticket already!”
No, you can't just sit around waiting for an epiphany to happen. Epiphanies are hard work! TANSTAAFL. So, how do you go about searching for epiphany? Every epiphany I've ever had has been the result of a two-pronged effort.
The first prong is building up sweat equity. I thoroughly explore the space around the problem, playing with the problem from many different angles. Of course, I'm actively trying to solve the problem during this time, because I might get lucky and stumble across a solution. But I don't worry too much if I don't seem to be getting anywhere. This part of the process is not so much about connecting the dots as about building lots and lots (and lots) of potential connections.
The second prong is staying alert. That's easy to say, but really hard to do over long periods of time. Eventually, I hope, some stray thought or some environmental trigger will start a cascade, where some subset of those potential connections will become actual connections, creating a path to the solution. Sometimes this happens when you're relaxed and your mind is drifting, such as in the shower or while driving. I remember once in grad school trying all afternoon to track down a bug. That night my wife and I went over to a friend's house for dinner. Everybody was shocked when, in the middle of beef stroganoff, I suddenly shouted out, “That's it!”
Other times the epiphany can happen while you're actively working on something else. For example, the key insight behind TradeMaximizer came while I was trying to put together a homework for my Algorithms class. I had been working off and on for weeks trying to find cycles in the trade graph. But I had set that aside and was coming up with homework problems, when I stumbled across a stray comment that talked about converting a directed graph into a bipartite graph. That sentence was enough to set off the chain reaction of connections and, within seconds, TradeMaximizer was born.
Probably the most famous story about scientific epiphany is that of Archimedes' bath. Funny story: Shortly after announcing TradeMaximizer to the math trade community on BoardGameGeek, I received a peevish email from another BoardGameGeek user who thought he also had a solution. He described how he had come to BoardGameGeek, eager to share his discovery, only to find my announcement first. He complained “Argh! Way to shout ‘Eureka!’ while Archimedes is drying himself off!” (The other solution turned out to be bogus.)