In this post and the post that will follow next week, I want to make you say "whoa." Specifically, I want to make you say "whoa" by showing you how complicated things in real life can sometimes be described by amazingly simple mathematical equations. These equations are simple enough that even the most mathematically phobic and/or inexperienced among you will be able to handle them.

With the goal of making you say "whoa" in mind, let's talk about The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In 1960, Nobel prize winning physicist and mathematician Eugene Wigner wrote a paper with this title. His main point was to ask the philosophical question "Why on earth does mathematics do such a good job describing stuff in the real world?" In 1980, applied mathematician Richard Hamming followed up Wigner's paper with a similarly-but-slightly-more-succinctly-titled homage, The Unreasonable Effectiveness of Mathematics.

Wigner and Hamming were trying to understand how it can be that by pushing around a bunch of abstract mathematical symbols, we can discover how electromagnetic waves behave, or what trajectory a launched missile will follow, or countless other phenomena. Wigner and Hamming were thinking mostly of physics and engineering, and indeed, these areas were the focal application areas of mathematics for quite a long time. Starting in the mid-to-late 20th century, though, there was explosive growth in the application of mathematics to biology. So in addition to describing physical and engineering systems, math does an uncanny job describing the human immune system, the spread of disease within a population (which I will post about sometime soon), the march of evolution, and much more. Even more recently, math has crossed over to the realm of describing social systems. Indeed, much of my own research is about aggregations formed by social organisms: think fish schools, bird flocks, and swarms of locusts.

Speaking of social phenomena, let's talk about Facebook. Since this is a blog, and since you are reading it on a computer, I dare not even bother to explain to you what Facebook is. Chances are that you are a Facebook user, whether you are the type who logs in every day or the type who created an account but has abandoned it. Yesterday, I came across this data on the number of active Facebook users over time. Now, I am not sure how "active" is defined, but still, I was curious about the data, so I did the simplest thing possible, namely, make a graph. Here's what I got:

Hmmm, I thought, those points look like they are tracing out a pretty well-defined curve. That's amazing! But what curve is it? (The mathematicians among you can likely tell from one glance at the picture.)

In the next post, we'll answer this question very explicitly. First we will have to go on a digression that relates to money and differential equations. Please don't be put off. Even if you are a total non-mathematician, you can still learn about differential equations. Seriously. Get yourself ready, and we'll talk about it next week.