Math is a cool way for us to understand the world we live in. And to that end Business Insider recently spoke with , the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University.
Strogatz specializes in areas of nonlinear dynamics and complex systems, and he is the author of the wonderful "."
He talked to us about game theory, "elegant" math, math education, and the effectiveness of
This interview has been edited for clarity and length. It was originally published on June 8, 2016.
Elena Holodny: What's interesting in chaos theory right now?
Steven Strogatz: I’m often very interested in whatever my students get interested in. I primarily think of myself as a teacher and a guide. I try to help them — especially my Ph.D. students — become the mathematicians they’re trying to become. The answer often depends on what they want to do.
In broad terms, the question of how order emerges out of chaos. Even though we talk about it as “chaos theory,” I’m really more interested in the orderly side of nature than the chaotic side. And I love the idea that things can organize themselves. Whether those things are our system of morality or our universe or our bodies as we grow from a single cell to the people we eventually become. All this kind of unfolding of structure and organization all around us and inside of us, to me, is inspiring and baffling. I live for that kind of thing, to try to understand where these patterns come from.
Holodny: How do things organize themselves in nature even when there's no "central command" — like when birds fly in formation or people organize themselves in a power structure?
Strogatz: We’re learning a lot about this all the time — bird flocks, fish schools, herds of animals. You can have human organization both within companies or even frivolous examples like people in a soccer match who want to start doing the wave or clapping in unison. So we do sometimes spontaneously organize.
There are also cases when it’s really serious, like when buildings are on fire and people need to escape. You can study the motion of people as they escape the building. And they actually will go out like water flowing through a pipe … There’s a kind of “fluid dynamics of people” as well as of cars. When traffic engineers are trying to figure out how people are driving ... sometimes you’ll be stuck in traffic and there’s a jam, and you’re thinking, “Oh there must be an accident somewhere down the road,” but then you never see an accident and you wonder why was there this traffic jam? There was no reason for this.
So that’s another case of bizarre collective behavior of people that we get in these density waves on the highway. Density of traffic in some places and density somewhere else. And in that case it has to do with driver behavior. That people don’t want to get too close to the car in front of them. There are sort of mathematical rules that govern how fast you’re comfortable driving, depending on how far ahead of you the next car is, and also how dense the traffic is generally. And so you can write all of these things in math and then start to predict what will happen with thousands of people on a big long stretch of highway.
Holodny: To what degree are these computer simulations accurate? We don’t even know how to solve turbulence yet!
I don’t want to give the impression that we totally understand flocks. You’re right to be skeptical. There’s a lot that we’re just learning about this, but the field seems to be moving pretty fast.
Now, to come to the punch line, when Axelrod analyzed what programs tended to do well in this prisoner’s dilemma tournament, the ones that did well had four properties: Be nice, be provocable, be forgiving, and be clear. [Editor's note: "Nice" means to cooperate; "provocable" means to immediately defect in retaliation when the other player defects on you; "forgiving" means not to hold a grudge, e.g., someone resumes cooperating with you, and then you resume cooperating and don't continue to punish them.]
But you could say morality came from evolution — it's natural selection, which is all we're talking about here — trying to win at the game of life. If natural selection leads to morality, I think that's pretty interesting. And that came from math!
It sort of seems like our math can’t possibly be compatible with reality. Except that it is! And not just compatible, but remarkably powerful.
It's this spooky thing where you reason about perfect objects, like real numbers or perfect circles or equilateral triangles — we know that they don’t really exist, and yet by pretending that they do exist to a good approximation in the real world, you get predictions that work.
Holodny: Angus Deaton, who won the 2015 Nobel Prize in economics, is more of a data collector by style, and that’s more similar to biologists who look at the nitty-gritty details of a given population to predict what might happen next. As opposed to just building models, and then wondering why things in the real world don't look like your model.
That teacher could be a parent or an actual teacher. I think we’ve all had good teachers who inspired us to want to learn more math and helped us, and then we’ve also had some not-so-good teachers who started to turn us off. That’s not really different from anything else because in any profession; there are good dentists and bad dentists.
Holodny: What does it mean to you when a proof is “elegant” or “beautiful”?
A big part of teaching successfully is to have empathy. That you need to be sensitive to the students who aren't getting it, who don’t see why it's beautiful.
When we're doing our professional work in research, we're constantly saying stupid things, making mistakes, correcting each other, being embarrassed. Your face gets red. That's normal. I mean, that's why they call it trial and error. You have to take risks, and try things, and make mistakes to make progress.
But so much of school is, "Don’t make any mistakes, get it right." This class, I feel, is a necessary corrective to that to show, that this is a safe place where you can make all the mistakes you want as long as you can learn from them. And it’s fine and it’s actually good because mistakes are very instructive.
There’s a problem with this class, which is that we don’t cover very much. And I’m not teaching any preestablished body of material that we have to get through. You can say, fine, you can get away with this because this is the last [math] class these students are ever going to take — and how much math did they really learn? OK, those are fair questions. But the things that they did learn they learned really well. And more than that, they learned what it actually means to do math, honestly, for the first time.
Strogatz: That’s why they hate it! No one wants to feel like a trained monkey.
To do something creative or very original, you will make a mess, you will break things, you'll be confused, you'll be sloppy, you'll try stuff that turns out to be a dead end ... You have to be strong enough and brave enough to make a mess and to get stuck and to not give up.