An Abundance of Katherines (24 page)

This graph is quite familiar to Colin—it’s a graph of a short relationship in which he’s dumped by the girl (we know that the girl dumps Colin because the graph is above the
x
-axis between the first kiss and the dumping). It’s the graph that tells an outline of the story of Colin’s life. Now all we need to do is figure out how to modify it so as to flesh out some details.

 
One of the great themes of twentieth-century mathematics has been the drive to study everything in “families.” (When mathematicians use the word “family,” they mean “any collection of like or related objects.” E.g., a chair and a desk are both members of the “furniture family.”)

Here’s the idea: a line is nothing more than a collection (a “family”) of points; a plane is simply a family of lines, and so forth. This is supposed to convince you that if one object (like a point) is interesting, then it will be even more interesting to study a whole family of similar objects (like a line). This point of view has come to completely dominate mathematical research over the last sixty years.

This brings us to the third piece of Colin’s Eureka puzzle. Every Katherine is different, so each dumping that Colin receives at the hands of a new Katherine is different from all the previous ones. This means that no matter how carefully Colin crafts a
single
function, a
single
graph, he’ll only ever be learning about a
single
Katherine. What Colin really needs is to study all possible Katherines and their functions, all at once. What he needs, in other words, is to study the
family
of all Katherine functions.

And this, at last, was Colin’s complete insight: that relationships can be graphed, that graphs come from functions, and that it might be possible to study all such functions at once, with a single (very complicated) formula, in such a way that would enable him to predict when (and, more importantly, whether), any prospective Katherine would dump him.
⁸¹

Let’s give an example of what this might mean; in fact, we’ll talk about the first example that Colin tried. The formula looks like this:

In explaining this expression, I certainly have a lot of questions to answer: first off, what on earth is
D
? It’s the Dumper/Dumpee differential: you can give anybody a score between 0 and 5 depending on where they fall on the spectrum of heartbreak. Now, if you’re trying to predict how a relationship between a boy and girl will work out, you begin by taking the boy’s D/D differential score and subtracting from it the girl’s D/D differential score and calling the answer
A
. (So if the boy is a 2 and the girl is a 4, you get D =-2.)

Now, let’s see what effect this has on the graph. In the example I just gave where the boy gets a 2 and the girl gets a 4, so that
D
= -2, we have whose graph looks like this:

As you can see, the relationship doesn’t last too long, and the girl ends up dumping the boy (a situation Colin is quite familiar with).

If, instead, the boy was a 5 and the girl was a 1, we’d have D = 4, so that

which has the following graph:

This relationship is even shorter, but it seems even more intense (the peak is remarkably steep), and this time the boy dumps the girl.

Unfortunately, this formula has problems. For one thing, if D = 0, that is, if they’re equal Dumpers or Dumpees, then we get

whose graph is just a horizontal line, so you can’t tell where the relationship begins or ends. The more basic problem is that it’s patently absurd to suggest that relationships are so simple, that their graphs are so uniform, which is what Lindsey Lee Wells eventually helps Colin to figure out. And so Colin’s final formula ends up being far more subtle.

But the main point is already visible in this case: because
D
can vary, this
single
formula is capable of specifying a whole
family
of functions, each of which can be used to describe a different Colin-Katherine affair. So all Colin needs to do now is add more and more variables (more ingredients along the lines of
D
) to this formula so that the family of functions it encompasses is bigger and more complicated, and there fore has a hope of encapsulating the complex and challenging world of Katherine-dumpings, which is what Colin eventually realizes thanks to Lindsey’s insight.

 
So that’s the story of Colin Singleton and his Eureka moment and the Theorem of Underlying Katherine Predictability. I should briefly point out that although no reasonable adult mathematician (at least not one with a soul) would seriously suggest that you can predict romance with a single formula, there actually has been some recent work that points in this direction. To be specific, psychologist John Gottman (and longtime head of the University of Washington’s “Love Lab”) and a group of coauthors, including the mathematician James Murray, have published a book entitled
The Mathematics of Marriage
that purports to use math to predict whether marriages will break up. The basic philosophy is, in its outline, not unlike Colin’s The-o rem, but the math that goes into it is far more sophisticated, and the claimed outcome is far more modest (these people aren’t pretending that they can predict
every
divorce, just that they can make some educated guesses
⁸²
.
⁸⁷

There’s one last thing I’d like to mention: notwithstanding John’s notorious tendency to cannibalize his friends’ lives for literary material, and notwithstanding the fact that I was somewhat accelerated in school as a kid, Colin’s character was in no way inspired by me. For one thing, I’ve only ever kissed two girls named Katherine. Interestingly, though, throughout my whole career as a pathological Dumper, the Katherines were the only two women who ever dumped me. Strange. It almost makes me wonder if there’s a formula out there somewhere . . .

—Daniel Biss
Assistant Professor, University of Chicago,
and Research Fellow at the Clay Mathematics Institute

acknowledgments

1. My incomparable editor and friend, Julie Strauss-Gabel, who worked on this book when she was, literally,
in labor.
I rely so much on Julie’s editing that—this is a true story—I once made her edit an e-mail I wrote the woman with whom I was then “just friends” and with whom I am now “living in holy matrimony.” Which reminds me . . .

2. Sarah. (See dedication.)

3. My mentor and collaborator and alter ego and BFF Ilene Cooper, who is responsible for most of the good things that have ever happened to me. And also, come to think of it, helped me woo Acknowledgee #2.

4. My dear friend Daniel Biss, who fortunately for me is one of the best mathematicians in America—and also one of the best teachers on the subject. I could never have imagined this book without Daniel, let alone written it.

5. My family—Mike, Sydney, and Hank Green.

6. Sarah Shumway, my very talented
in loco editoris
at Dutton. Also everyone else at Dutton, particularly Margaret “Double Letters” Woollatt.

7. My man in the United Arab Emirates, Hassan al-Rawas, who has been providing me with Arabic translations and his wonderful friendship for many years now.

8. Adrian Loudermilk.

9. Bill Ott, 10. Lindsay Robertson, 11. Shannon James and Sam Hallgren, 12. David Levithan and Holly Black, 13. Jessica Tuchinsky, 14. Bryan Doerries, 15. Levin O’Connor and Randy Riggs, 16. Rosemary Sandberg, 17.
Book-list
, 18. All librarians everywhere, and of course . . .

19. The Katherines. I wish I could name them all, but (a) I lack the space, and (b) I fear the libel suits.