Class of 2021: On behalf of the faculty, I welcome you to Franklin & Marshall College and to the many intellectual adventures that await you here.
I think it’s perfect that it’s a math professor addressing you today, because, after all, the vast majority of you --- over 400 of you --- have signed up to take calculus this semester. I know that many of you must therefore be incredibly eager to begin your mathematical studies, but I am guessing that even more of you are filled with dread at the thought of listening to me give a math talk now.
When I tell people I’m a mathematician, the response I most often get is, “oh, you must be so smart --- I never could do math.” My colleagues in other fields get their own knee-jerk responses, too. Psychologists hear, “Are you going to psychoanalyze me?” A professor of Judaic Studies tells me people assume that he must therefore be a rabbi. English professors inspire fears that they’ll correct everyone else’s grammar. Economists are asked for predictions about the stock market. The list goes on and on.
What these responses have in common, in addition to being wrong and misguided, is that the comments address only the most superficial aspects of our disciplines. They only go skin deep, so to speak. But you, coming to F&M, are here to learn muscle-deep, bone-deep. Because this is a liberal arts college, we will ask you to explore many subjects. And when you do, you won’t get some breezy whirlwind tour. We want you to understand the roles that evidence, observation, and analysis play in each subject. We want you to try to understand the core questions that drive our disciplines. You think you came to F&M because a college education opens doors, but we believe that your education here ought to open universes.
All of this explains why I am, right now, going to introduce you to a core question of my own research, in an area of math called “chaos theory”.
My research has looked at the question, “what makes a system chaotic?” I’m going try to unpack the definition with you by using the typical example of a weather system, but also by explaining why your lives here at F&M ought to be chaotic, and I will end with a poem that might be very familiar to you but that you probably never thought of as being a poem about math.
So, what makes a system chaotic? There are three conditions that the system needs to fulfill.
The first is called “sensitive dependence on initial conditions”. In weather, we sometimes call this the butterfly effect, the notion that a very small change, like a butterfly flapping its wings here in Lancaster, could eventually (if we look ahead as far as we could) have a huge effect, altering the course of a tropical cyclone in the Mediterranean. In your own lives here, you might find similarly that small changes have a way of cascading. Maybe you sit next to somebody new, and ages and ages from now this person becomes your best friend, your business partner, your spouse. Or maybe by accident you go to the Brooks Seminar Room instead of the Bonchek Seminar room, where a course on Islamic Law and Ethics winds up changing the way you look at the world. Or maybe you drop by office hours one day on a whim, and that short conversation with your Connections professor makes all the difference to that course, and also to your future major. Small changes -- even changes between things that are really about the same -- can lead to big changes ages and ages from now. That’s the first condition of chaos.
We say the second condition is that “periodic points are dense”. To mathematicians, “periodic points” are circumstances that repeat, that come back to themselves again and again. And we use “dense” in the sense of the woods; these repetitions are close by, everywhere you look. With weather, every summer gets hot; every winter gets cold. But you might also notice daily weather cycles, with temperatures rising through the day and falling at night, and maybe even seven-day weather cycles (where storms arrive every Thursday, say).
At F&M, we gather every August for convocation; each semester builds in intensity so that April is always a hurricane of activity, and May is always a calm after the storm. But if you look carefully, and you should carefully, you’ll see intellectual themes repeat themselves as well. The tetrahedron that plays a starring role inOrganic Chemistry will also shape the way you look at crystals in The Dynamic Earth and at human movement in Modern Dance. The conventions of social status that you pick apart in Social Anthropology will return for discussion in Caribbean Literature and in Comparative Politics. Even if you doubt the topics you’re learning will ever come back, you’ll see the same ideas arising again and again, in unexpected ways. That periodic points are dense is the second condition of chaos.
The third condition--- the one most relevant to my own research --- is called “transitivity”. It says that starting from just about anywhere, the system could wind up just about anywhere else. Today’s weather is cool and rainy, but if we wait long enough, as day leads on to day, we know we’ll see days that are hot and humid, we’ll see ice storms, we’ll see cloudless days of perfect temperature.
Your own lives here are full of transitivity now, just as your professor’s lives were when we were sitting in our freshman classes. When I arrived at college, I knew exactly what I wanted to do: I wanted to study foreign languages so that I could become a Washington DC tour guide. But my dad convinced me I ought to take “one last math class”, and I agreed, and one thing led to another, and now I’m addressing you as a math professor.
In the same way, you, sitting in these chairs in front of us, have unexpected delights waiting ahead for you. Even choosing your major won’t force your destiny into some major-shaped concrete box: some of my own favorite math majors have gone on to start their own companies, to become doctors and opera singers, and even work at the Center for Conflict Resolution. It’s the chaotic nature of transitivity: starting from just about anywhere, your path through life might take you just about anywhere else.
Which leads us to a poem written by a guy who dropped out of school twice. He worked, according to one source, in a “slew of unfulfilling jobs”, including becoming an unsuccessful poultry farmer. Eventually he started writing poetry that transformed his times and our world. To me, his best-loved poem perfectly illustrates that sometimes small changes have big effects, that our thoughts and ideas return periodically to the same themes, and that starting from anywhere we can go just about anywhere else. This is how Robert Frost described mathematical chaos. He wrote,
Two roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveler, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;
Then took the other, as just as fair,
And having perhaps the better claim,
Because it was grassy and wanted wear;
Though as for that the passing there
Had worn them really about the same,
And both that morning equally lay
In leaves no step had trodden black.
Oh, I kept the first for another day!
Yet knowing how way leads on to way,
I doubted if I should ever come back.
I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I—
I took the one less traveled by,
And that has made all the difference.
Class of 2021, your own adventures await you. We, your faculty, are eager to see where they take you. Welcome to Franklin & Marshall.