Love or Math


In the 1993 movie “Groundhog Day,” Bill Murray plays Phil Connors, a reporter who, confronted with living the same day over and over again, matures from an arrogant, self-serving professional climber to someone capable of loving and appreciating others and his world. Murray convincingly portrays the transformation from someone whose self-importance is difficult to abide into a person imbued with kindness.  It seems that the Nietzschean test of eternal return, insofar as it is played out in Punxsutawney, yields not an overman but a man of decency.

But there is another story line at work in the film, one we can see if we examine Murray’s character not in the early arrogant stage, nor in the post-epiphany stage, where the calendar is once again set in motion, but in the film’s middle, where he is knowingly stuck in the repetition of days. In this part of the narrative, Murray’s character has come to terms with his situation. He alone knows what is going to happen, over and over again.  He has no expectations for anything different.  In this period, his period of reconciliation, he becomes a model citizen of Punxsutawney. He radiates warmth and kindness, but also a certain distance.

The early and final moments of “Groundhog Day” offer something that is missing during this period of peace:  passion. Granted, Phil Connors’s early ambitious passion for advancement is a far less attractive thing than the later passion of his love for Rita (played by Andie MacDowell).  But there is passion in both cases.  It seems that the eternal return of the same may bring peace and reconciliation, but at least in this case not intensity.

And here is where a lesson about love may lie.  One would not want to deny that Connors comes to love Rita during the period of the eternal Groundhog Day.  But his love lacks the passion, the abandon, of the love he feels when he is released into a real future with her. There is something different in those final moments of the film.  A future has opened for their relationship, and with it new avenues for the intensity of his feelings for her. Without a future for growth and development, romantic love can extend only so far.  Its distinction from, say, a friendship with benefits begins to become effaced.

There is, of course, in all romantic love the initial infatuation, which rarely lasts.  But if the love is to remain romantic, that infatuation must evolve into a longer-term intensity, even if a quiet one, that nourishes and is nourished by the common engagements and projects undertaken over time.

This might be taken to mean that a limitless future would allow for even more intensity to love than a limited one.  Romantic love among immortals would open itself to an intensity that eludes our mortal race.  After all, immortality opens an infinite future.  And this would seem to be to the benefit of love’s passion.  I think, however, that matters are quite the opposite, and that “Groundhog Day” gives us the clue as to why this is.  What the film displays, if we follow this interpretive thread past the film’s plot, is not merely the necessity of time itself for love’s intensity but the necessity of a specific kind of time:  time for development.  The eternal return of “Groundhog Day” offered plenty of time.  It promised an eternity of it.  But it was the wrong kind of time.  There was no time to develop a coexistence.  There was instead just more of the same.

The intensity we associate with romantic love requires a future that can allow its elaboration.  That intensity is of the moment, to be sure, but is also bound to the unfolding of a trajectory that it sees as its fate.  If we were stuck in the same moment, the same day, day after day, the love might still remain, but its animating passion would begin to diminish.

This is why romantic love requires death.

If our time were endless, then sooner or later the future would resemble an endless Groundhog Day in Punxsutawney.  It is not simply the fact of a future that ensures the intensity of romantic love; it is the future of meaningful coexistence.  It is the future of common projects and the passion that unfolds within them.  One might indeed remain in love with another for all eternity.  But that love would not burn as brightly if the years were to stammer on without number.

Why not, one might ask?  The future is open.  Unlike the future in “Groundhog Day,” it is not already decided.  We do not have our next days framed for us by the day just passed.  We can make something different of our relationships.  There is always more to do and more to create of ourselves with the ones with whom we are in love.

This is not true, however, and romantic love itself shows us why.  Love is between two particular people in their particularity.  We cannot love just anyone, even others with much the same qualities.  If we did, then when we met someone like the beloved but who possessed a little more of a quality to which we were drawn, we would, in the phrase philosophers of love use, “trade up.”  But we don’t trade up, or at least most of us don’t.  This is because we love that particular person in his or her specificity.  And what we create together, our common projects and shared emotions, are grounded in those specificities.  Romantic love is not capable of everything. It is capable only of what the unfolding of a future between two specific people can meaningfully allow.

Sooner or later the paths that can be opened by the specificities of a relationship come to an end.  Not every couple can, with a sense of common meaningfulness, take up skiing or karaoke, political discussion or gardening.  Eventually we must tread the same roads again, wearing them with our days.  This need not kill love, although it might.  But it cannot, over the course of eternity, sustain the intensity that makes romantic love, well, romantic.

One might object here that the intensity of love is a filling of the present, not a projection into the future.  It is now, in a moment that needs no other moments, that I feel the vitality of romantic love.  Why could this not continue, moment after moment?

To this, I can answer only that the human experience does not point this way.  This is why so many sages have asked us to distance ourselves from the world in order to be able to cherish it properly.  Phil Connors, in his reconciled moments, is something like a Buddhist.  But he is not a romantic.

Many readers will probably already have recognized that this lesson about love concerns not only its relationship with death, but also its relationship with life.  It doesn’t take eternity for many of our romantic love’s embers to begin to dim.  We lose the freshness of our shared projects and our passions, and something of our relationships gets lost along with them.  We still love our partner, but we think more about the old days, when love was new and the horizons of the future beckoned us.  In those cases, we needn’t look for Groundhog Day, for it will already have found us.

And how do we live with this?  How do we assimilate the contingency of romance, the waning of the intensity of our loves?  We can reconcile ourselves to our loves as they are, or we can aim to sacrifice our placid comfort for an uncertain future, with or without the one we love.  Just as there is no guarantee that love’s intensity must continue, there is no guarantee that it must diminish.  An old teacher of mine once said that “one has to risk somewhat for his soul.” Perhaps this is true of romantic love as well. The gift of our deaths saves us from the ineluctability of the dimming of our love; perhaps the gift of our lives might, here or there, save us from the dimming itself.

 

* Text By TODD MAY, NYT, FEBRUARY 26, 2012

 Todd May is Class of 1941 Memorial Professor of the Humanities at Clemson University.  His forthcoming book, “Friendship in an Age of Economics,” is based on an earlier column for The Stone.

“In the spring,” wrote Tennyson, “a young man’s fancy lightly turns to thoughts of love.” And so in keeping with the spirit of the season, this week’s column looks at love affairs — mathematically. The analysis is offered tongue in cheek, but it does touch on a serious point: that the laws of nature are written as differential equations. It also helps explain why, in the words of another poet, “the course of true love never did run smooth.”

blancanieves

To illustrate the approach, suppose Romeo is in love with Juliet, but in our version of the story, Juliet is a fickle lover. The more Romeo loves her, the more she wants to run away and hide. But when he takes the hint and backs off, she begins to find him strangely attractive. He, on the other hand, tends to echo her: he warms up when she loves him and cools down when she hates him.

What happens to our star-crossed lovers? How does their love ebb and flow over time? That’s where the math comes in. By writing equations that summarize how Romeo and Juliet respond to each other’s affections and then solving those equations with calculus, we can predict the course of their affair. The resulting forecast for this couple is, tragically, a never-ending cycle of love and hate. At least they manage to achieve simultaneous love a quarter of the time.

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The model can be made more realistic in various ways. For instance, Romeo might react to his own feelings as well as to Juliet’s. He might be the type of guy who is so worried about throwing himself at her that he slows himself down as his love for her grows. Or he might be the other type, one who loves feeling in love so much that he loves her all the more for it.

Add to those possibilities the two ways Romeo could react to Juliet’s affections — either increasing or decreasing his own — and you see that there are four personality types, each corresponding to a different romantic style.

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My students and those in Peter Christopher’s class at Worcester Polytechnic Institute have suggested such descriptive names as Hermit and Malevolent Misanthrope for the particular kind of Romeo who damps out his own love and also recoils from Juliet’s. Whereas the sort of Romeo who gets pumped by his own ardor but turned off by Juliet’s has been called a Narcissistic Nerd, Better Latent Than Never, and a Flirting Fink. (Feel free to post your own suggested names for these two types and the other two possibilities.)

Although these examples are whimsical, the equations that arise in them are of the far-reaching kind known as differential equations. They represent the most powerful tool humanity has ever created for making sense of the material world. Sir Isaac Newton used them to solve the ancient mystery of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion applied to both.

In the 300 years since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. This is true for the equations governing the flow of heat, air and water; for the laws of electricity and magnetism; even for the unfamiliar and often counterintuitive atomic realm where quantum mechanics reigns.

1 Princesa-Tigre 1

In all cases, the business of theoretical physics boils down to finding the right differential equations and solving them. When Newton discovered this key to the secrets of the universe, he felt it was so precious that he published it only as an anagram in Latin. Loosely translated, it reads: “It is useful to solve differential equations.”

The silly idea that love affairs might progress in a similar way occurred to me when I was in love for the first time, trying to understand my girlfriend’s baffling behavior. It was a summer romance at the end of my sophomore year in college. I was a lot like the first Romeo above, and she was even more like the first Juliet.

The cycling of our relationship was driving me crazy until I realized that we were both acting mechanically, following simple rules of push and pull. But by the end of the summer my equations started to break down, and I was even more mystified than ever. As it turned out, the explanation was simple. There was an important variable that I’d left out of the equations — her old boyfriend wanted her back.

In mathematics we call this a three-body problem. It’s notoriously intractable, especially in the astronomical context where it first arose. After Newton solved the differential equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun), he turned his attention to the three-body problem for the sun, earth and moon. He couldn’t solve it, and neither could anyone else. It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.

Newton knew nothing about chaotic dynamics, but he did tell his friend Edmund Halley that the three-body problem had “made his head ache, and kept him awake so often, that he would think of it no more.”
I’m with you there, Sir Isaac.
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NOTES:
For models of love affairs based on differential equations, see Section 5.3 in Strogatz, S. H. (1994) “Nonlinear Dynamics and Chaos.” Perseus, Cambridge, MA.
For Newton’s anagram, see page vii in Arnol’d, V. I. (1988) “Geometrical Methods in the Theory of Ordinary Differential Equations.” Springer, New York.
Chaos in the three-body problem is discussed in Peterson, I. (1993) “Newton’s Clock: Chaos in the Solar System.” W.H. Freeman, San Francisco.
For the quote about how the three-body problem made Newton’s head ache, see page 158 in Volume II of Brewster, D. (1855) “Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton.” Thomas Constable and Company, Edinburgh.
For readers who are curious about the math used here:
In the first story above, Romeo’s behavior was modeled by the differential equation dR/dt = aJ. This equation describes how Romeo’s love (represented by R) changes in the next instant (represented by dt). The amount of change (dR) is just a multiple (a) of Juliet’s current love (J) for him. This equation idealizes what we already know – that Romeo’s love goes up when Juliet loves him – by assuming something much stronger. It says that Romeo’s love increases in direct linear proportion to how much Juliet loves him. This assumption of linearity is not emotionally realistic, but it makes the subsequent analysis much easier. Juliet’s behavior, on the other hand, was modeled by the equation dJ/dt = -bR. The negative sign in front of the constant b reflects her tendency to cool off when Romeo is hot for her. Given these equations and an assumption about how the lovers felt about each other initially (R and J at time t = 0), one can use calculus to inch R and J forward, instant by instant. In this way, we can figure out how much Romeo and Juliet love (or hate) each other at any future time. For this elementary model, the equations should be familiar to students of math and physics: Romeo and Juliet behave like simple harmonic oscillators.

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* Text by Steven Strogatz, May 26, 2009

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chica milkshake


Some Comments:

* I never could understand differential equations and I never could understand women. Oh well.
It’s easy to see the ebb & flow in practice but the trick is to see where you both are on the curve and adjust for it at the right time. Each woman is different so there’s probably some other underlying wave form that dampens or reinforces the peaks and valleys.
— Johnny E

* Rather wonderful isn’t it that God has given us the ability to see the very cosmic principles that influence our lives? Being aware of them can only lead to more freedom for the human mind and spirit.
— Steve

* OK, math unpredictable is fun. Still, there are only a small set of actions that are possible given the real world constraints of actions. The same stories occur again and again with only small varions from story to story where we fill in the names and the details.
I can appreciate math and life but I sense there is still more to grasp even as I try to run variables in my mind.
Keep us thinking. I am just frustrated at the prospect there must be many variables that we can not phantom … . Somehow, if we ever where to meet other intelligent life we would better appreciate the role of math in the differences and similarities reflected in the possible.
Hard to think it all through, but I am tempted to try… .
— Mark

* There are four factors necessary for having the best chance at a love that will last:
1. Mutual attraction.
2. Mutual good-heartedness.
3. Compatibility.
4. Maturity.
Maybe that’s why lasting love is so hard to find.
— Ronald E. Maxson

* Here is a mathematician who twigged in time: “There was an important variable that I’d left out of the equations.” Mathematicians often analyze models of reality, which first require that they determine what variables to consider and what to leave out. History, especially in economics and the biosciences, shows they can sometimes get this profoundly wrong. In economics there is, from time to time, a “meltdown,” that brings economists (or at least those who trust economists) to there senses. In the biosciences the meltdowns are less obvious. Nevertheless – Bravo differential equations!
Donald R. Forsdyke, Kingston, Canada
— Donald R. Forsdyke

* Although the concept might be beyond comprehension of the ‘present day scientists/ mathematicians’, in the ‘recent times’ it is accepted that the mysterious moon – that in the three-body problem made Newton’s head ache – evolved from earth-moon itself. And, reading between lines the ‘Hindu mythology’, one would find the same phenomenon is indicated with the help of cryptic clues in the story related with Androgynous Shiva losing His ‘better half’, or consort, in a ‘sacred fire’ that was arranged by her father and, therefore, at a later stage Shiva remarrying her as the daughter of Himalaya – as she was yet another form of His original wife itself!
Thus moon virtually is indicated as a reflection/ projection in space of the energy at the centre of earth that apparently acts as a protective layer, like a bullet-proof jacket as its model, to help sustain eternally not only earth but also the entire solar system…The ancient astronomers resorted to astrology and Palmistry etc. to deal with expected human behaviour, and not simple mathematics…
— JC Joshi

* Nice introduction to differential equations, though a small error might confuse people – it should be ” The amount of change (dR) that occurs every moment (dt) is just a multiple (a) of Juliet’s current love (J) for him . “
— LJ Graham

* Attending la creme de la creme instiution of higher education Shimer College, my astute classmates came to the conclusion that, “Most people say two plus two are four. By the time we leave here, we’ll say, ‘Oh yeah, want to make a bet?’”
And so an interesting love square/pentacle of the century was predicted.
Call them Romeo and Juliet, or Hermit and Malevolent Misanthrope.
Or you could call them MICKEY and MICKEY or DONALD and DONALD. Let me regress.
I had an interesting, influential experience as a child. Likening myself as a princess, I played in tower of blue with the tough kids, who kept shouting, “Donald Duck.” We were the elite, with wings we could access the tower complete with slide. At the bottom were the angry peasants, working class, shouting MICKEY mouse. These kids were blocked off from the ladder to the slide and lacked the wings and the latitude to enforce order. They were rioting, crying with disorder, “MICKEY mouse.” Their Furies had been awoken. They felt their day of reckoning had come. I stood atop the tower, smug at my position, watching the contorted angry expressions of the children without access, I wanted to help them. I wanted to say that I like the elitist Donalds, but I like you MICKEYS too. So mass chaos ensued. Tower children shouted, “Donald Duck,” while the children below angrily insisted “MICKEY Mouse.” I was captivated by this scene. I didn’t know what I should be shouting. Even as a child, I had a sense of “Don’t be a racist, be an elitist mentality” while similtaneously wanting to help the rest rise. I suppose that is why I am still single, I am searching for that right MICKEY, Donald combo.
— Eve of Blue Tower

* I have always advised my engineering students not to draw on the equations that we use in classes in order to understand social interactions, if for no other reason than that even though there are some amusing comparisons they will probably not get a useful answer and will waste time more usefully spent on social interactions trying to do math problems.
That said, one can’t help but wonder about the three (or more) body problem that is usual at that age, and their LaGrange points (points of stability between mutually attractive large objects). Perhaps the constant tweeting and texting that occurs these days is the maintenance of some kind of metastable equilibrium in an ages-old problem.
— Scott

* “It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.”
hence: literature, one solution to describing these iterations of love, or love stories –
thanks for a fab article!
— 411bee

* If you enjoyed this piece you will also enjoy Steven Strogatz in programs 12 and 13 of Mathematics Illuminated, a new 13-part multimedia resource that explores major themes in mathematics, from mankind’s earliest study of prime numbers to the cutting edge mathematics used to consider the shape of the universe.
The entire resource is free to all at http://www.learner.org/courses/mathilluminated/.
— Scott Roberts

* I think this theory is the perfect explanation why engineers and mathematicians should feel less confident in explaining love and human relationships:). Applying a clinical equation to human interaction probably works when you calculate how many people might show to a party, but not to predict relationships.
First, you cannot assume a relationship between J&R can exist in isolation from everything else and furthermore you cannot assume you know the number of variables: what do they both want in life, how important are to them the wishes of their family, what are those wishes, new potential love interests, incompatibility and so on. Romeo and Juliet might have experienced absolute love, but they didn’t make it past their teens so there is an inherent limitation of choosing them as examples. And why would you
assume Juliet to be the fickle one?
— Ashleen Hayes

* The problem with modeling real social dynamics is (1) it’s not deterministic, (2) it’s not autonomous, and (3) it’s not as simple as a 2-variable system. But that doesn’t keep us modelers from trying!
One of my favorite social models is by Sergio Rinaldi on cycles in the love poetry of Petrarch. He manages to get rather good fits to the data – and with an autonomous, deterministic differential equation model. But it has three variables ;-)
This article is available free on JStor.
— Sharon Lubkin

* The Romeo/Juliet ambivalent tryst represents about 10% of types of romantic relationships. Since 10% is relatively infrequent, it is probably easier to predict than say romantic relationships that are based on normal attachment processes (70-80%).
There are too many variables in the love equation to try to simplify it with a math model. (Although it is a way to amuse oneself.) People marry for many reasons besides love, for example, they might be pregnant, they wanted to married by a certain age, they receive pressure from family and/or peers, sex guilt, religious influences, fear of not obtaining a better catch, etc. You think the three-body problem is bad, try adding all the other influences!
It might help to read the psychology literature on romantic relationships. Even though I minored in math at Cal, and have tender feelings for the subject, I doubt romance will be explained in math departments.
— Joe

* Interesting article: the example seems to be (inadequate) mathematics to describe social interactions. The gender demographic of readers commenting is also interesting: 7 males (8 including this response), 2 indeterminate and 1 female. Hmmm…
– Mike
— mike mittleman

* This illustrates one problem with blind faith in the power of mathematics–you don’t always know if you have identified all the relevant variables for an equation. Solving problems completely can require an exceedingly large number of variables and exceedingly complex equations. At the end of the day, even supercomputers could turn out to be fallible for lack of the correct variables. As you point out, Isaac Newton recognized this when he tried to understand the movements of only three bodies. Imagine then, trying to understand what influences the behavior of “star-crossed lovers.”
— ttj

*Per Maxson:, there are four independent variables for lasting love:
1. Mutual attraction.
2. Mutual good-heartedness.
3. Compatibility.
4. Maturity.
Maybe that’s why lasting love is so hard to find.
If each variable can only be either there or not there, then the probability of getting all four as positive is 1/16 or about 6%. So lasting love on average only occurs 6% of all couples/marriages. Is this true? Can data support this?
— Manny

* Great correlation evaluated between the hearts and maths. Better if a survey is conducted or the real time analysis is done and results are published. That adds more weight to the theory.
— Harish

* So then now knowing this and having the ability to Meta-think about it, do I then doom myself to a near impossible 2nd order differential equation?
Let me explain. Since I can now know that I might be dooming myself to a SHO off set by pi/2 or something, then I will modify my behavior. This modification, this meta thinking, then would be a 2nd order term. How about solving that one?
Oh sure, if it gets bad enough, I can use a Bessel function, but then again approximating a woman as sphereical might be counter reproductive.
— Rob

* In the arithmetic of love, one plus one equals everything,
and two minus one equals nothing.
— Dru

* Professor Strogatz,
I find it curious that you deemed it necessary to point out the “tongue in cheek” nature of your delightful excursion into the mathematics of love. But I am grateful for your perspective which moves the discussion beyond the plane geometry of “three’s company” to the far more stimulating venue of differential calculus.
I would only add that your observation regarding the collective behaviour of Romeo and Juliet as harmonic oscillators confirms my own observation over the years that families that engage in music strengthen their emotional bonds (i.e., are more harmonious).
Leo Toribio
Pittsburgh, PA