Love by the numbers.

Love by the numbers.

A mathematician's guide to the news.
April 16 2003 2:28 PM

Love by the Numbers

Can a few differential equations describe the course of a marriage?

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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.

The example of the marble brings home the fact that not all equilibria are alike. Put a marble at the bottom of a bowl, and it's likely to stay there. But balance it atop a hill, and any perturbation, however slight, will send it rolling down. The first kind of equilibrium is called stable, the second unstable. Stable equilibria are important because they're the places where the system "likes" to end up. Drop a bucket of marbles in a landscape; after a while, you'll find a lot at the bottoms of valleys, few if any balanced on peaks.

Gottman and his collaborators believe that the development of a marriage is governed—or at least can be described—by a differential equation. On the face of it, this makes sense: The amount of negative emotional expression a husband directs at his wife undoubtedly contributes to changes in her reciprocal feelings. But is this change entirely determined by the current state of the marriage? That seems hard to swallow; but The Mathematics of Marriage argues that just such an assumption yields results that match well with Gottman's decades of clinical data on couples.

The idea that marriages obey differential equations might not be so scary; after all, this only seems to say that the course of a marriage is as regular, in the long term, as weather.

But Gottman is quite specific about the differential equations he has in mind. And according to these equations, marriage isn't like the weather. It's like the marble.

Marriages, Gottman's group says, have equilibria. Where they are and what they're like depend on the characteristics of the individuals and of the marriage. Some couples may have all their stable equilibria in states where the marriage is desperately unhappy; luckier couples may have all their equilibria in agreeable states. And some couples—the most interesting ones, from the point of view of marital therapy—have both happy and unhappy equilibria.

This model has one flaw that should be obvious. The marble, once it's at a stable equilibrium, doesn't move. But a marriage, as we know, can shift in an instant from blissful to miserable and vice versa. If the cold equations are the whole story, how can this be?

The answer to this question is the most interesting idea in The Mathematics of Marriage. The key idea is that the differential equations describing the marriage can change with time. The results are easiest to envision in marble world. Imagine a landscape with two equilibria—one ("happy marriage") in the bowl of a volcano, and the other ("screaming-and-throwing-plates marriage") in a deep valley. Both are stable equilibria for the marble; where the marble ends up depends on its starting position ("wedding day"). Suppose the marble ends up in the volcano. And suppose that, with time, the bowl begins to fill in with dirt, growing shallower and shallower. Now the center of the bowl remains an equilibrium, and the marble stays put, until a certain critical moment is reached—the dirt piles higher than the bowl's rim, and the peak of the volcano switches from concave to convex. At that moment, the equilibrium ceases to be stable, and the marble rolls down to the unhappy valley. Likewise, a marriage, under the pressure of quite gradual changes in circumstance, can suddenly collapse. What's more, returning the circumstances to their prior state should not be expected to repair the marriage; digging a new hole in the volcano won't make the marble jump out of the valley!

The mathematical formalism addressing such phenomena, catastrophe theory, was extremely faddish not so long ago; as a result, attempts to apply catastrophe theory are often met with automatic skepticism. But the Gottman group's appeal to catastrophe in the marriage model seems to me quite reasonable, at least in principle. And, as honest theoreticians must, they offer empirical predictions, which can be confirmed or rejected by experiment—for instance, that a marriage is less likely to succeed when the spouses are individually more prone toward negative emotional expression.

The theory's attractiveness is hard to deny. It neatly presents marriage as a process both mathematical and unpredictable, both stable and prone to catastrophe. Even the John Nash character in A Beautiful Mind would have to agree—love is like that.