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The idea that there is a difference between "science" and "philosophy" is a very recent one, in many ways dating to the eighteenth century CE (i.e. only about 300 years ago). The word "philosophy" literally means "love of knowledge," and in the ancient world the people we might identify as Greek "scientists" were simply regarded as philosophers by their fellow Greeks, ones who happened to be especially interested in how the world worked and what things were made of. Unlike earlier thinkers, the Greek scientists sought to understand the operation of the universe on its own terms, without simply writing off the details to the will of the gods.
The importance of Greek scientific work is not primarily in the conclusions that Greek scientists reached, which ended up being factually wrong most of the time. Instead, its importance is in its spirit of rational inquiry, in the idea that the human mind can discover new things about the world through examination and consideration. The world, thought the Greek scientists, was not some sacred or impenetrable thing that could never be understood; they sought to explain it without recourse to supernatural forces. To that end, Greek scientists claimed that things like wind, fire, lightning, and other natural forces, were not necessarily spirits or gods (or at least tools of spirits and gods), but might just be natural forces that did not have personalities of their own.
The first known Greek scientist was Thales of Miletus (i.e. Thales, and the students of his who went on to be important scientific thinkers as well, were from the polis of Miletus in Ionia), who during the Archaic Age set out to understand natural forces without recourse to references to the gods. Thales explained earthquakes not as punishments inflicted by the gods arbitrarily on humanity, but as the result of the earth floating in a gigantic ocean that occasionally sloshed it around. He traveled to Egypt and was able to measure the height of the pyramids (already thousands of years old) by the length of their shadows. He became so skilled at astronomy that he (reputedly) successfully predicted a solar eclipse in 585 BCE.
Thales had a student, Anaximander, who posited that rather than floating on water as his teacher had suggested, the earth was held suspended in space by a perfectly symmetrical balance of forces. He created the first known map of the world that attempted to accurately depict distances and relationships between places. Following Anaximander, a third scientist, Anaximenes, created the theory of the four elements that, he argued, comprise all things – earth, air, fire, and water. Many centuries later, Galen of Pergamon, a Greek physician living under Roman rule, would explain human health in terms of the balance of those four forces (the four “humors” of the body), ultimately crafting a medical theory that would persist until the modern era.
In all three cases, the significance of the Greek scientists is that they tried to create theories to explain natural phenomena based on what they observed in nature itself. They were employing a form of what is referred to as inductive reason, of starting with observation and moving toward explanation. Even though it was (at it turns out) inaccurate, the idea of the four elements as the essential building-blocks of nature and health remained the leading explanation for many centuries. Other Greek scientists came along to refine these ideas, most importantly when two of them (Leucippus and Democritus) came up with the idea that tiny particles they called atoms formed the elements that, in turn, formed everything else. It would take until the development of modern chemistry for that theory to be proved correct through empirical research, however.