Motivated by shame, I’ve taken up Dan Aaron’s challenge to read more original work by giants who laid the foundations of my field. Being a hack mathematician with physicist tendencies, I’ve turned to René Descartes—the man who married geometry and algebra to give us Cartesian coordinates, among other very important things—and his Discourse on Method. In the first part of the Discourse, he situates the reader a bit, explains his personal history, and how it led him to his approach to problem-solving. He tries very hard to be humble and sometimes he almost succeeds, but we know Descartes was a genius. He knew it, too. And even he can’t hide his high opinion of himself.
Descartes claims that the work of several authors is usually inferior in quality to that wrought by a single hand: it’s the old “too many cooks spoil the soup” theory. That’s how he justifies denouncing centuries of philosophers and their work: he had to do it start on his own from scratch; it’s impossible to sort through the ideas of others with any systematic clearity. But he stays out of the kitchen in his metaphor, instead traveling to the city:
Thus we notice that buildings conceived and completed by a single architect are usually more beautiful and better planned than those remodeled by several persons using ancient walls that had originally been built for quite other purposes. Similarly, those ancient towns which were originally nothing but hamlets, and in the course of time have become great cities [think London], are ordinarily very badly arranged compared to one of the symmetrical metropolitan districts which a city planner has laid out on an open plain according to his own designs [Brasilia, say: it looks like a bird or airplane]. It is true that when we consider their buildings one by one, there is often as much beauty in the first city as in the second, or even more; nevertheless, when we observe how they are arranged, here a large unit, there a small; and how the streets are crooked and uneven, one would rather supppose that chance and not the decisions of rational men had arranged them.
Now, Descartes’ philosophy aside, there’s a pretty interesting observation in there. Even though a very complicated system, such as a city, maybe be very orderly on small-scales, the over-all effect is painfully complex. I’ve declared more than a few times, proudly to visitors, that the streets of Boston were once the random paths of wandering cattle. So this Fathers’ Day, when my dad and sister came to visit my new digs, I had to remind them to look in all—not both—directions when crossing the street. “The streets around here are so wacky, that the cars going one way can be stopped at a red and still you can be hit in three other directions.” We all made it across, a little bit hurried but safe. Meanwhile, Manhattans live on a grid and seem to like it. Some have even claimed to prefer it.
A similar phenomenon exists in economics. A while back someone was awarded a Nobel Prize for showing that even though each person may act rationally, the market as a whole still might act irrationally. I don’t know much about how this works, but I can tell you about an analogue in math.
There are several competing definitions of chaos in mathematics, depending on just which subfield you subscribe to. One of the most tractable definitions comes from one-dimensional, real, discrete dynamics; that is, the study of iterated functions that live on the real number line. Simply, take a function, which is only a fancy word for a rule, which takes in a number (like 3) and spits out another (say 1.5) and repeat it over and over again. We say that a function is chaotic if it exhibits the following three behaviors:
- The function has a dense set of periodic points. Periodic points are orderly; their trajectories are predictable. They start at one location. The function moves a periodic point to another point; another application of the function moves it again to another, and another, and another. Eventually, though, the point needs to wind up where it started. Think of periodic points as travelers on a multi-city, round-trip itinerary. A salesman might start in Boston, go to Denver, follow-up in Phoenix, stop short to family visit in D.C., but at the end of it all, he’s got to go home to Boston. In the language of discrete dynamics, our salesman is a periodic point. Dense is just a mathematician’s way of saying most. So, if you blindly pick out a number, it’ll most likely be a periodic point. And if it’s not, there are plenty of periodic points nearby. On a plane, not every passenger is needs to be a travelling salesman like the one in our example. But near each passenger there should be a few of them close by.
- The function displays sensitivity with respect to initial conditions. This requirement ensures that points which start out close don’t stay that way forever. You can think of functions which are sensitive to initial conditions as those maps which mix points up. Sensitive functions are not very tolerant of approximation. They hate playing horse-shoes, for example. And they’re very hard to plot on computers due to rounding errors. Even though we might be very accurate, a chaotic function will churn the points about so wildly that we cannot guarantee that anything we learn about one point will shed any insight on the whereabouts its neighbor.
- The function is topologically transitive. Topological transitivity is a mathematician’s way of saying that the function meanders. Pick any two points A and B. If the function is topologically transitive, then I can find another point as close to A as you want that eventually makes its way as close to B as you want.
Notice that the definition of a chaotic map (or system or function—these terms all refer to the same thing) demands lots of order. Periodic points are simple. We know exactly where they go and exactly where they’ll end up: they travel in loops forever. However, chaos requires a little bit more.
So far, we require chaotic maps to be, on the one hand, very orderly—by way of a multitude of periodic points—, and simulatenously jumbled, on the other hand—in an indirect way, through its sensitivity. In a modern treatment, we could stop here. But to really drive things home, let’s add in a third requirement.
One of the simplest examples of a chaotic maps is the so-called logistic map. It’s a quadratic (there are squares in there):
Its continuous conterpart crops up in population dynamics as one of the simplest, foundational models. Some examples within the logistic equation’s domain include colonies of bacteria, blades of grass on a lawn, or frogs in a pond with no predators to eat them.
So, as long ago as 1637, Descartes noticed, very carelessly, that order can breed chaos. Let this be a warning to those of you neat-freaks who work tirelessly to assure everything is in its place. Also, Descartes might argue that cities provide evidence against the precepts of Intelligent Design. I know that’s how I read it.