October 18th, 2007: A bunch of people sent me examples of situations in which addition is non-commutative: nonabelian groups, rotating objects in 3D space (try it! Pick up a book and rotate it 90 degrees left and then 90 degrees away from you, and then try the rotations in the opposite order: the book ends up in different positions, so the order in which you "add" the rotations together matters), non-commutative geometry and so on. That last one is interesting because it has a real-world application too: walking on a curved surface (like on a hill, or on our planet). If you start at the equator and go 1000k north and then 100k east, you'll end up in a different position than if you go east first and then north. So there's some math that doesn't commute and has the added benefit of having real-world applications! So that's something, huh?
Thanks to everyone who emailed me explanations! I haven't written you all back but I've read them all. I'm getting to the point where I can't respond to all the email I receive, and I'm already about a month behind, but I'm hoping to catch up.