Trees, Dictionaries, Pickles and Loops: Part II


Image: “Pickle Pile,” by Brandon Dimcheff, on Flickr.

 

In the last post, we talked about the relationship between math and computer science: specifically, about how much computer science involves math and is itself similar to math.

This post is about some of the ways modern math relies on computer science.

I’ve observed three ways in which computer science is used in math: to generate conjectures, to help people visualize ideas, and to create proofs.

Conjectures

Often, we recognize patterns before we understand them. Sometimes these patterns have real, underlying causes. I always sneeze when I take out old clothes from my closet. I was sniffling for an hour after I visited a used-book sale on Sunday. From this, you might propose a conjecture: I sneeze when I’m exposed to dust.

Of course, not all apparent patterns are reliable. If I told you that I was late to school twice this month, and that I was wearing a blue T-shirt both times, it may not be wise to conjecture that I always wear blue T-shirts when I’m late. If, instead, it happened twenty times, the pattern might deserve more attention. But if I was then late when I wore a purple sweater, you’d conclude that the conjecture was false.

Conjectures are like theories. They’re educated guesses about what patterns are real. In math, a computer is often used to come up with or check the validity of a conjecture, and then logical reasoning is used to prove it. In some fields of math, it can be extremely helpful or even necessary to use a computer to generate conjectures. One reason is that computers can help find the “purple sweaters”– counterexamples that allow you to dismiss a conjecture immediately. Computers are good for this because they check a lot of cases far more quickly than humans can.

Visualization

Sometimes, the easiest way to understand a concept is with a picture. When the picture is a circle or a trapezoid, a pencil and paper may be all you need. But it can be more difficult if you’re presented with something like this:

From Wikimedia Commons

Even if you know about quantum wavefunctions (I don’t), this description doesn’t do a great job of getting an image in your head.

But using a software like Mathematica, you can get a picture:

2D Wavefunction (2,2) Surface Plot

Of course, the picture can’t teach you about these functions all by itself. But it can do a lot to help you start thinking.
Proofs

…to be continued in the next post!