## Freedom in Mathematics

When I was in primary school, I hated math. It bored me to no end. To me, all it was was some heavyset woman making us do endless addition or subtraction or multiplication drills, for no reason other than the fact that we were ordered to. What was worse, all I ever saw of mathematics was a single, linear path. You started with a problem, and then you proceeded by the same old steps through the same old terrain until you found the solution. Even if the numbers involved changed, the methods didn’t, and after a few years, I started to realize that it was really all just repeating the same few problems over and over again, in different guises.

It wasn’t until I entered college that I started to re-discover the beauty of numbers and patterns (which the public school system had blinded me to, but that’s another post entirely). And today, I came to an amazing realization.

You see, I’ve been taking Differential Equations for the past few months, and it’s always struck me as the way math is supposed to be. Differential Equations is a class where it must be acknowledged that there are some problems we don’t know how to solve in the traditional way, or that we can’t solve at all. It’s the logical opposite of algebra.

You see, algebra was the first subject that really made me despise math. There was no freedom in the solutions: all you did was shuffle some numbers around, and eventually figured out the value of the variable in question. There was no room for beauty or creativity; it was just a very roundabout way of doing regular arithmetic.

Not so with calculus. Not so with differential equations. You see, in calculus, there is more freedom. You’re usually not searching for a number, but for a function. And whether or not you can find the solution depends on how creatively you can construct and reconstruct the problem. For example, in order to find the area under the curve defined by y = exp(2x), when x varies between -1 and 1, you have to use an integral. But you can’t just integrate right away. First, since there’s only a fairly small types of integrals that can actually be evaluated, you have to rebuild the problem from the ground up. Usually, this particular problem would be solved by saying that u = 2x, which turns the problem into the one of integrating y = exp(u), which is solvable.

That’s why I’ve always loved calculus: there’s always room for creativity and exploration. And the more complex the mathematics, the more ways there are to solve any given problem.

Thinking about all of this led me to an epiphany while I was walking home from my philosophy class: once you get past arithmetic, mathematics is not about doing the same thing over and over to slightly different problems in order to get slightly different answers. In fact, most problems don’t have a single answer. For example, take Schrödinger’s Equation, which is the basis of most of quantum mechanics:

What the variables represent isn’t terribly important. What’s much more important is what the equation represents. If this were an algebraic expression, it would simply represent an answer that you haven’t found yet. But Schrödinger’s equation, and other equations like it (such as the Einstein Field Equations of general relativity), they simply represent constraints that select certain solutions and correct, and others as incorrect. They’re not number just waiting to be found, they’re more like intelligent little computers that decide what is real and what is not, mathematically.

To understand what I mean, consider a much simpler differential equation (and please forgive the ugliness of the notation, but I’m working from some pretty severe graphical limitations):

y’ = k * y

This is not just a number in disguise. Instead, it’s a pattern that any function f(x) must fit in order to be called a solution of the equation. Contained in that tiny equation is the “truth filter” that passes through any exponentially growing or decaying solution. Likewise, Schrödinger’s equation is merely a filter that determines whether or not a given function can occur in real life or not. According to Schrödinger’s equation, electrons can change energy levels and emit photons, and particles can occasionally jump through barriers.

That is the true beauty of mathematics: every equation, every principle, every theorem is nothing more and nothing less than a little filter that produces reality from all the infinite possibilities inherent in the universe. Mathematics is incredibly intelligent in that regard, a perfect structure that selects real reality from the infinitude of possible realities.

And that is why I’m a math major.