Mathematics Is Smarter Than I Am

A mathematics professor uttered this sentence the other day, and is absolutely correct. I had trouble explaining this to people, but thanks to him, I have the right words now.

Mathematics is not a collection of symbols so you can “show off” how smart you are to your friends.

Mathematics is a technology. It helps you do things you would never be able to otherwise.

How does it work? Well, the “internal smarts” of mathematics comes from its collection of theories, rules, etc. While we are developing a model, the moment we connect our thinking with one of these prepared rules, we are getting something for free. These premade rules might take us to different form (using other rules) let us simplify things, and finally we reach a clean computational stop. These rules, statements that form the internal smarts are easier to remember than going back, say to number theory, and / or computing the first, obvious thing you see in front of you (the worst you can do in any problem is trying all solutions).

One of my teachers used to say “when you are stuck, just write something you know is true”.

That is, take a mathematical statement out of your bag, write it down, and try to connect it with your problem. Try to find that entry point to the “internal smarts” of mathematics.