While it would incorrect to assume the organized study of the rational world began solely with the Ancient Greeks, their long ties to the idea of 'Western Culture' have long captivated the European imagination. Furthermore, direct lines of idealogical evolution can be drawn from early ideas about natural philosophy (which broadly speaking, attempted to rational analyze nature, both human and earthly) and the eventual development of the scientific revolution.
Web to explain relationships between various Greek pre-socratic philosophers.
Perhaps the most important concept developed by Ancient Greek natural philosophers is in trying to describe the world through reasonable as opposed to mythical explanations. The earliest recorded greek philosophers were most interested in cosmology, ontology and mathematics. The earliest known natural philosophical school/movement was that of the Milesian school where philosophers attempted to define all things by their quintessential substance such as Thales who believed the nature of all matter descended from water or Anaximander who proposed an unobserved, undefined element, which he called apeiron(ἄπειρον "having no limit"). He reasoned that if each of the four classical empedoclean elements (water, air, fire, and earth) are opposed to the other three, and if they cancel each other out on contact, none of them could constitute a stable, truly elementary form of matter.
Another early important Ancient greek natural philosopher was that of Pythagorus who attempted to bridge the mystic with the mathematic in a way that would prove highly influential in the greek world. While no primary sources of his are still available he was widely cited by later greek philosophers.
The most famous of Greek philosopher were thos from The Athenian school which largely starts with socrates who believed that philosophy and questioning was an important persuit for all people and that encouraging intellectual persuits was paramount to both individual hoping to live a more fulfilled life and society at large as a wise citizenry would be best for everyone. A direct causal line can be drawn from Socrates to other famous and influential greek philosphers such as plato and Aristotle. Aristotle himself would be incredibly important to the eventual development of the Scientific Revolution. While most of Aristotle's early 'scientific theories' were pure conjecture and mostly incorrect, his attempt to categorize the natural world and his subsequent writing would become incredibly important to later intellectuals all over the world, encouraging developments in fields such as The Humanities (as Aristotle was on the first to try and break down and analyze literature), physical science and mathematics
Cahan, David, ed. (2003). From Natural Philosophy to the Sciences: Writing the History of Nineteenth-Century Science. Chicago: University of Chicago Press
G. S. Kirk and J. E. Raven and M. Schofield, The Presocratic Philosophers (Cambridge University Press, 1983, 108-109
Reidwig, Christoph; Rendall, Steven (2005). Pythagoras. Cornell University Press.
Griffin, Jasper; Boardman, John; Murray, Oswyn (2001). The Oxford history of Greece and the Hellenistic world. Oxford [Oxfordshire]: Oxford University Press. p. 140.