Edit, 10-15-2016: I learned some more math since I wrote this. I still think functions are really important, and I’m leaving the post as it was. But there is math that doesn’t revolve around functions – they are not as completely ubiquitous as I once thought.

We spent the last two posts talking about functions. I used a lot of examples. If you’re sick of examples, bear with me – they’re building up to something cool.

Functions – maps – quietly hold math together, like nails in the wall of a building. The best way to see that is to see all the different places that functions show up. I’m hoping to give you a glimpse of what makes functions so transcendent.

Let’s look at some more weird functions in the world of numbers. That will help us understand the functions you’re used to (or hate). We’ll see how these functions fit in to the general idea of what a function is.

Then we’ll leave the world of numbers entirely.

Integers

Here’s a function from {0, 1, 2, 3, 4, 5} to the integers.

The integers are whole numbers, like 0, 1, 2, and 3, as well as negative numbers, like -7. There are infinitely many integers, but that doesn’t matter. We just want to associate every object in {0, 1, 2, 3, 4, 5} with one integer each.

This shouldn’t be too difficult. Just like in last post, we can associate 0 –> 0, 1 –> 4, 2 –> 2, 3 –> 2, 4 –> 2, 5 –> 1. We can even use the same picture to illustrate it:

We didn’t use any negative numbers here. But we could have, if we wanted to. Another perfectly good function would map 5 –> -20 instead of 5 –> 1.

There are, in fact, infinitely many choices for what to map 5 to. The point is, having infinitely many options is completely okay: everything still works smoothly.

If we mapped from the integers to the integers, instead of {0, 1, 2, 3, 4, 5} to the integers, that would be fine too. We’d have to cut off our picture somewhere, or else it would be infinitely wide. (That’s what we do for the next picture.) Conceptually, though, it’s just an extension of the same type of mapping we did here.

Real Numbers

So what if we map from the real numbers to the real numbers?

Don’t freak out. Please. I promise, this is the SAME stuff we’ve been doing this whole time.

For the last function we made a picture. The way we did that was by going along the horizontal line of orange numbers, and marking the point that corresponded to the blue number we wanted.

Now, we’re doing the same thing, but with real numbers instead of integers. To every real number on the horizontal line, my function associates a real number on the vertical line. We mark of that point with a blue dot. But since the real numbers don’t have holes, we get a smooth curve instead of separated X’s.

Keep in mind that this picture is not the function. The function is a rule that associates real numbers to real numbers. This picture is just an illustration of that function.

Stardust

While we’re still talking about numbers, there is one more thing to consider: stardust.

(This is not a real math term, as far as I know. My professor coined it, and I like it, so I’m keeping it.)

The function I drew earlier had a pattern. Specifically, each real number got mapped to the square of that number. The nice-ness of that pattern was something we could see, through the smoothness of the curve in our picture.

Almost any function you learn about in high school is “nice” in this kind of way. You can draw pictures, describe them, and write down formulas for them. As a student, that’s all you’re exposed to. So common sense would tell you to assume that all functions follow some sort of pattern.

But functions don’t have to be that way. They’re just maps: rules that associate objects to other objects. Those rules don’t have to make sense.

A “stardust” function is what you’d get if you mapped each real number to some other real number in some crazy way. That is, you’d make a map without an obvious pattern or neat formula.

When mapping the reals to the reals, I can’t write down* a stardust function or draw a picture of one. I can’t use it to predict a dodgeball’s trajectory or model a trend in the stock market. They’re infinite and complicated, without a nice summary like “f(x) = x^2.”

But what’s weird is that, in the universe of math, they exist. Even if I don’t know how to specify any particular one, the definition of functions tells me that infinitely many functions that are just stardust.

Maps and Mathematics

In the next Making Maps post, we’ll try to see why mathematicians seem to like functions so much. I think the ubiquity of maps can tell us a lot about what math is and how people think about it.

 

* I’m not sure if this claim gets me in trouble without the axiom of choice. If it does, assume I just mean functions that don’t form a curve in R^2 and don’t have any particularly simple or useful pattern.