I would define "math" as the subset of logic dealing with topics that can be treated in a rigorous way and put into symbols. Usually but not always the questions addressed by math can be quantified, so most familiar mathematics deals with numbers; however, other relationships like sets and the division of physical space (axiomatic geometry) can also be treated mathematically. What distinguishes math from other fields like science or history is that it requires no observation but instead operates in an "ideal world" of abstractions.
Off the top of my head, some mathematical milestones:
1. Euclid codifies all the geometric knowledge accumulated to that point and sets a standard for mathematical rigor in his
Elements.
2. Descartes creates the coordinate plane and enables analytic geometry as well as our modern understanding of functions
3. Newton and Leibniz discover calculus at roughly the same time in response to practical problems involving rates of change
4. Gödel's Incompleteness Theorem proves that no axiomatic system will span the field of mathematics
5. Computers and calculators are put to use to massively simplify computation. Supercomputers perform tasks like cranking out trillions of digits of
pi and proving various number theory conjectures by brute force.
