What does 'complex' mean?

I enjoy occasionally perusing the math and science questions posted on Quora. Every once in a while, I even take the time to answer some.

A side benefit of the questions is that they remind me what 'lay' people think about mathematics. For example, a recent question asked, "What are some real-life applications of integration and differentiation?" A lot of people answered, as I would, everything1. Which is presumably not the answer that the person posting, or the precocious child whining, "When will I ever use this?!" had in mind. I have to admit, my perception isn't quite like this cartoon, but I do tend to think of a lot of things in life mathematically. Even the loosest mathematical analogy tends to aid thought.

Another recent post asked: "What are the most complex equations behind popular games?" Again, the answers here struck me. Especially these two. The two formulas are far from 'complex,' in the sense I would use. They're applications of the four basic arithmetic operations, and the second one has a max thrown in.

But what do I mean when I call an equation 'complex?' Is Navier-Stokes 'complex?' It's simple enough to write down. But doing anything with it requires a great deal of work, either numerical or theoretical. Similarly for Maxwell's equations, Schrödinger's equation, etc. (Apparently when I think of 'complex' equations, I think of partial differential equations.) But agent-based models are also very complex, and their 'equations' usually amount to simple local rules that have to play out, much like a cellular automata, one step at a time. One of the answers on Quora gets at this.

Interestingly enough, the math I do on a regular basis2, which usually involves some flavor of discrete probability theory, doesn't lend itself to complicated equations. Sure, you have to be comfortable with factoring distributions, and keeping track of conditioning properties. But once you speak the language, those really aren't that bad. There's no one complex equation in sight, just the massive edifice laid down by many brilliant minds that a lesser mind like mine can take advantage of.

That's complex in it's own way.

  1. Okay, technically we'd have to allow sums and differences into our toolbox to cover everything (practical). But sums are just integrals and differences are just derivatives.

  2. I don't always do math, but when I do, I do probability (and statistics).