My first reaction to the prompt, "What is math?" is a line of numbers running across my mind in a horizontal style not unlike the one in the Matrix movies. Once the number's dance is done, images of shells with the Fibonacci sequence outlined, the unit circle, and a triangle with the Pythagorean theorem flash into view. For me, math has been a source of intrigue, education, and frustration. In the past I have focused on the puzzles that math offers rather than the history behind it, but there are a few names and times that have stuck with me in math's history.
The discoveries/milestones that strike me as "Top 5" worthy in the history of mathematics (that I can remember) are:
1. Use of numbers
2. "Discovery" of zero
3. The Fibonacci Sequence
5. The Fundamental Theorem of Calculus
I'll admit that I don't have the knowledge of when all of these discoveries/milestones hit. I recall learning that numbers first started being recorded for trade in the sense of "1 cow for 10 chickens." To count, add, and subtract numbers, and to figure out a way to write it down, may seem trite, but it was (in my opinion) the first great leap that started our current concept of mathematics.
Similarly, the "discovery" of zero by the Hindus and Mayans (independently) added another key component to today's mathematics. Recognizing that zero isn't just a numbers absence, but is a number in itself, influences much of how I look at math.
The Fibonacci Sequence has always fascinated me because it has an elegance to it and because it is found everywhere in nature--and that it has evolved in nature because of it's natural efficiency.
I think that pi deserves to be in the top 5 because it is the foundation of the circle, and the Fundamental Theorem of Calculus (which I remember was cemented in history by Issac Newton) because it has led to many breakthroughs in modern (and ancient) mathematics, and because calculus is in so many of our actions on a daily basis.