The ancient Greeks created mathematics, which is to say, the science of mathematics. Babylonians and ancient Egyptians performed remarkable mathematical computations. But it was the Greeks who put forth conjectures about numbers and geometric objects which they then endeavored to prove or refute.
Once a good number of related theorems have been established, it’s natural to consolidate the principles and assumptions used in their proofs. The result is a system of axioms from which the theorems are derived.
The most well-known axiom system is of course Euclid’s system of plane geometry.
There is, however, a problem that pervades Euclid’s system. His proofs have gaps: they make assumptions that go beyond his axioms. For example, in proving Proposition 1, that an equilateral triangle can be constructed on a line segment, Euclid assumed that circles intersect at a point. Though obvious, it’s doesn’t follow from his axioms. In The Foundations of Geometry (1902), David Hilbert put forth a more rigorous set of axioms; but he needed 20 axioms compared to Euclid’s five.
The ultimate solution to the problem of gaps is the formal axiom system, a system in which:
Axioms are sentences and proofs are sequences of sentences.
Sentences are defined rigorously, so that there’s an “effective procedure” for determining whether a string of symbols is a sentence or not, an effective procedure being a set of instructions that can be carried mechanically and terminates in a yes-or-no answer.
Examples of axiom systems in mathematics:
Peano axioms for natural numbers
Zermelo-Fraenkel axioms for set theory
Axioms of modern algebra for fields, rings, and groups
A Philosopher's View 