Why I Love Math, Part 1: Because I Love Clever Movies
October 26, 2013 Leave a comment
It’s no secret, I’m a mathy person. In fact, I just spent several hours of my free time this weekend trying to learn some new math, even though I didn’t need it for work.
Understandably, this seems ridiculous to a lot of folks. I probably get asked, “how can you enjoy this stuff?” more often than I get asked any other question about math or physics.
Well, one of the reasons I love math is that I love surprises, and math is full of them. Two mathematical concepts that initially look very different might turn out to be exactly the same. A problem that looks literally impossible might turn out to have a really easy solution, if you’re clever about it. Or a very simple idea might turn out to have much more important consequences than you’d ever have thought.
Whenever I find a situation like this in mathematics, I get a real thrill. But it’s not some esoteric, high-level pleasure reserved for academics. On the contrary, it’s the same sense of “Oh wow! Oh awesome!” that anyone can get from watching a good, clever movie.
In fact, the three types of math surprises that I described* are all very similar to common movie structures. Consider the case where two things appear completely different, but in a shocking twist they turn out to be exactly the same. Well, we’ve all seen that one before, right? Spoiler alert: at the end of “The Usual Suspects,” the seemingly pitiful character Verbal turns out to have been the criminal mastermind Keyser Soze all along. The Planet of the Apes seems to be set somewhere really exotic… but no, it’s been Earth the whole time! And that guy Vader? Well, he actually is Luke’s father!
These moments are often iconic, and with good reason. They catch us completely off guard, but, when done correctly, they make some sort of satisfying sense, and all the pieces fall into place, just like in math. I can still remember feeling the same way as a kid when I first realized that “subtracting a negative number” was the same as adding a positive one. It was totally unexpected, but totally obvious and satisfying once I saw it.
“Clever solutions to impossible problems” is another common movie theme. In fact, it’s so common it’s basically its own genre: heist films. “Goodness!” we exclaim. “How could George Clooney ever possibly break into a casino vault that’s so heavily guarded? I bet it would be such a clever plan that I’d watch the movie just to find out what it was.” And math is the same way. Sometimes it’s just fun to look at a problem and say “this looks entirely unsolvable,” only to discover that some brilliant mathematician (lets be honest, it’s usually Euler or Gauss) has not only solved it, but done it in five lines.
And finally, there are the situations where a seemingly simple idea turns out to give birth to a huge variety of new and interesting things. In a movie, this means taking a very basic premise (“What if we could enter each other’s dreams?” “What would happen if you couldn’t form new memories?”) and pushing the ideas to their extreme. No one would watch “Inception” if it was just a series of people sharing dreams for fun. “Memento” would be dull if it was just a guy sitting around forgetting stuff. The magic isn’t in the basic idea. The magic is in what happens when you apply the idea to the situation where it is causes the most trouble. The same is true for mathematics: you can’t judge it because the basic ideas are boring. When you first met “negative numbers” in elementary school (probably in a game involving “magic peanuts“) you may have thought it was simple and stupid. But if you take them to a setting where they cause problems—a discussion about square roots, for example—then you realize that if you have negative numbers, you also have to have imaginary numbers. From there, next thing you know you’re constructing Euler’s Identity— widely regarded as one of the most beautiful and profound things in math.
As a matter of fact, I’d argue there is a sense in which the “plot twists” of math are better than anything you can find in a Hollywood thriller. Movies with a good twist ending almost always leave a few loose ends or contradictions. They almost have to, in order to fool you in the first place. The heist scheme in “Oceans Eleven,” for example, for all its brilliance, is literally impossible. How did they get the fake money into the real vault? They couldn’t have, and Soderbergh even concedes it in his DVD commentary. Of course, I still love the movie to death. It’s easy enough to ignore this problem when you watch it, because it’s intentionally well hidden. Nevertheless, a little part of my brain knows that the heist only looks beautiful because it’s too good to be true.
In mathematics, this never happens. Ever. If someone has a clever “heist” for solving an impossible problem, it’s guaranteed to be free from plot holes, or else it wouldn’t qualify as a solution (and hence if you DO find a “plot hole” in a famous proof, it might well make you famous too!) The same is true for the other kinds of twists: because math is founded on logic, it has to be internally consistent.
To me, this heightens the fun. I never walk away with that feeling that I was a little bit “cheated,” like when I watched “The Forgotten.” I tried so hard during the first half hour to figure out who had kidnapped Julianne Moore’s kid, and failed. But that’s probably because I had no way to know that “It was secretly aliens!” was an option.
In math, there are no secret aliens. No plot holes. The cleverness is always flawless, and that’s the fun.
*There are probably far more types of “mathematical plot twists,” but these strike me as the three biggest or most common categories. Of course I’d love to hear any others that folks can think of.