What is math?

My capstone class started today, Math 495: the Nature of Mathematics. We take a historical look at math, and the first assignment is to blog shortly on what is math and 5 milestones that they know
already.

Since, for some reason, I am trying to blog everyday this month, I thought I'd join them.

Mathematics to me is noun and verb.

Noun: it's hard to do better than Eugenia Cheng's description, "the logical study of logical things." Quantities, their properties, operations on them, their properties, treat the operations as the new object and repeat. Pretty soon you have cohomological sheaves and categories dancing with your sugar plum fairies. "Mathematics is beautiful."

Verb: reasoning about, playing with, representing or describing mathematical objects or posing or solving the problems and questions that you have about them. "Hey, watch out, I'm doing mathematics over here."

Students first responses:

Milestones:

1. Invention of number systems to record quantities, especially place value systems (Babylonians, Mayans, Indians)
2. First application problems connected with measurement. (Egyptian and Babylonian stand out to me.)
3. Mathematization: the study of the mathematics itself. Greeks, of course. Brahmagupta, Liu Hui, also.
4. The Persian synthesis of algebra and geometry. Al Khwarizmi in the House! (House of Wisdom that is.)
5. Descartes and the invention of analytic geometry. 

Very hard to pick five. Not having Gauss or Euler is painful. But I'm big on synthesis. My favorite modern mathematics is the Atiyah-Singer Index Theorem.

p.s. If you're interested in the course, here's the Google doc of what we did the last time around.
p.p.s. Found this movie poster on this page. Haven't ever heard of this movie, but I'll be looking for it!
p.p.p.s Erin found the movie as released, which lost the cool x+y title.