The Greeks and Proof Writing.
Throughout my mathematical career and my experiences with proof writing, I'm convinced the Greek mathematician like Thales, Euclid, and Pythagorean made the one of the biggest contributions to communicating in mathematics. Before the Greeks, civilizations like the Egyptians and Babylonians relied on repeated observations which is what we now call inductive reasoning. The Greeks however, not only gave us geometrical theorems, but also dabbed in number theory as well as formalizing the proof process.
First, a conjecture is proposed and any mathematician will admit to trying lots and lots of examples. Sometimes, these examples will lay the groundwork for how the proof will flow. Other times, the actual proof is so abstract that even the simplest of examples can be troublesome to construct. Nevertheless, if the conjecture or proposition is proved using arbitrary elements of the given set following the axioms, then it becomes a theorem. Theorems laid by the Greeks are still in practice today. In my mind, theorems are like shortcuts; once someone has proved that the theorem holds, we can skips all steps in between and cut to the chase if we are given a problem with the same hypothesis. For example, a theorem may start out with "suppose we have a right triangle, then..."; already this grants us many avenues in which to work with the triangle since there are many theorems pertaining to right triangles.
Over the years, proof writing has become an art form. This may be because of Thales and his proof by picture. With explanation, a picture is a powerful tool. However, this limits one to thinking abstractly. Maybe that was his way of convincing his counterparts that he was right. Euclid was famous for Euclid's lemma and other number theory problems but he also made of compilation of the work of famous mathematicians that came before him. This collection was probably the most influential piece of literature for other mathematicians that came after Euclid.
All in all, the Greeks contribution to proof writing has evolved mathematics into a more diverse field.