My job is to come up with ideas. Sometimes we mathematicians call the things we think about and work with “objects,” which doesn’t mean triangles, spheres, or other shapes. Mathematical objects are big ideas about algebra, geometry, and logic, about the properties and definitions of numbers.
It’s not at all obvious how to go about thinking up some new twist on these things—the transformation from test-taker to theorem poser and then theorem prover is difficult to articulate. My ideas have always felt contingent and magical to me. I don’t think I’m alone, at least as far as the magic goes. Henri Poincaré, the father of chaos theory and the co-discoverer of special relativity, is famous for a story that appears in his 1908 book “Science and Method,” about an insight being jarred loose while boarding a bus: “At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it.” The Irish mathematician Sir William Rowan Hamilton, who devoted many years to searching for a way to multiply numbers in higher dimensions, had a similar epiphany, in 1843, just as he was strolling by the Brougham Bridge, in Dublin, while on a walk with his wife. He was so delighted that he stopped and carved the defining algebraic equation into the bridge: i2=j2=k2=ijk=-1. One person’s graffiti is another person’s breakthrough.
These stories suggest that an initial period of concentration—conscious, directed attention—needs to be followed by some amount of unconscious processing. Mathematicians will often speak of the first phase of this process as “worrying” about a problem or idea. It’s a good word, because it evokes anxiety and upset while also conjuring an image of productivity: a dog worrying a bone, chewing at it to get to the marrow—the rich, meaty part of the problem that will lead to its solution. In this view of creative momentum, the key to solving a problem is to take a break from worrying, to move the problem to the back burner, to let the unwatched pot boil.
All problem solvers and problem inventors have had the experience of thinking, and then overthinking, themselves into a dead end. The question we’ve all encountered—and, inevitably, will encounter again—is how to get things moving and keep them moving. That is, how to get unstuck. [Continue reading…]