Love by the Numbers
Can a few differential equations describe the course of a marriage?
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.
Jordan Ellenberg is a professor of mathematics at the University of Wisconsin. His book How Not To Be Wrong is forthcoming. He blogs at Quomodocumque.