I have always believed in numbers, in the equations and logics that lead to reason. But after a lifetime of such pursuits I ask, "What truly is logic?" … It is only in the mysterious equations of love that any logical reasons can be found.
—Russell Crowe playing mathematician John Nash in A Beautiful Mind
It's a nice thought—that some things, like love, are beyond the reach of mathematical techniques, or even, as this speech suggests, that mathematical truth is in the end subservient to the truths that love tells.
On the other hand, there's John Gottman. Gottman, a professor of psychology at the University of Washington, is probably best-known for the so-called "Love Lab." Here, couples get strapped into a bundle of physiological sensors, settle down in front of a one-way mirror, and have themselves a fight, while Gottman's researchers record their every criticism, apology, eye twitch, and pulse spike. By watching a couple for 15 minutes, Gottman says, he can predict success or failure of a marriage with 85 percent accuracy. And in a new book, The Mathematics of Marriage, his research group—which includes mathematician James Murray—argues that a marriage can, in fact, be modeled by a surprisingly simple ensemble of equations.
Can the speech from A Beautiful Mind be so wrong? Is the difference between endless love and a quick divorce no more than a numbers game?
Not quite. Just because marriage is amenable to mathematical analysis doesn't make it completely predictable, let alone logical. After all, the weather is subject to mathematical rules, too. And on the April day I'm writing this, there are 3 inches of snow on the ground. What kind of math allows such wild results?
To get some ideas, switch to a simpler problem. Suppose you put a marble down on a lumpy surface. What happens? It depends where you put the marble. If the marble is on a slope, it'll roll in whatever direction offers the steepest descent. But if you balance the marble at the top of a hill or place it at the bottom of a valley, it stays put. The progress of the marble is governed by a differential equation, which means, more or less, that the change in the marble's position is predictably determined by the marble's position at the moment.
[And now, the two-paragraph summary of a semester's course in differential equations. I ask my technically minded readers to forgive the many oversimplifications and omissions here, which are meant to get us quickly to the point of the exercise.]
Differential equations describe all kinds of natural phenomena, from epidemics (the change in the number of infected people depends on the number of currently infected people) to the orbits of the planets (the motion of a planet depends on the gravitational force on it, which in turn depends on the location of other planets, which in turn depends on the original planet's position, and so on). Some differential equations, like the ones governing the marble, yield predictable and regular results. Others, like those for the weather, are more chaotic. As another popular movie once put it, paraphrasing Edward Lorenz, a butterfly flapping its wings in Taiwan can set off a hurricane in Houston. More prosaically: Small changes in the state of the system can make for drastic, unforeseeable changes in the system's long-term behavior. Now things are starting to sound like marriage!
The first question we ask about any differential equation we meet is: What are the states of the system in which the system does not change? Such a state is called an equilibrium. For an epidemic, an equilibrium might be a low but stable incidence of the disease. The equilibria of the marble are the bottoms of valleys and the tops of hills.