Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

On a linear system like a scale, the whole is equal to the sum of the parts. That’s the first key property of linearity. The second is that causes are proportional to effects. Imagine pulling on the string of an archer’s bow. If it takes a certain amount of force to pull the string back a certain distance, it takes twice as much force to pull it back by twice that distance. Cause and effect are proportional. These two properties—the proportionality between cause and effect, and the equality of the whole to the sum of the parts—are the essence of what it means to be linear. Yet many things in nature are more complicated than this. Whenever parts of a system interfere or cooperate or compete with each other, there are nonlinear interactions taking place. Most of everyday life is spectacularly nonlinear; if you listen to your two favorite songs at the same time, you won’t get double the pleasure. The same goes for consuming alcohol and drugs, where the interaction effects can be deadly. By contrast, peanut butter and jelly are better together. They don’t just add up—they synergize. Nonlinearity is responsible for the richness in the world, for its beauty and complexity and, often, its inscrutability. For example, all of biology is nonlinear; so is sociology. That’s why the soft sciences are hard—and the last to be mathematized. Because of nonlinearity, there’s nothing soft about them.