Pythagoras c. 570 bc onwards

It's hard to separate Pythagoras from the Pythagorean School which followed him as there are no direct writings of Pythagoras.

He was a mixture of mystic and rationalist. Even in his work on mathematics the motivation for his rational exploration of numbers was seemingly connected with a mystical numerology.

In terms of the Pythagorean 'religion', he believed in the transmigration of souls. The religion founded in his name by the Pythagorean order included such rules as abstaining from beans, not picking up what was fallen and rolling together your bedclothes when you rise so as not to leave the impress of the body.

Socially, the society he founded admitted men and women on equal terms, held property in common and deemed any discoveries they made as collective (and ultimately attributable to Pythagoras).

Russell argues that he was one of the inspirations for the concept of the gentleman, encompassing the valuing of contemplation over action, the idea that to observe was higher than to do and to contemplate higher than to observe. Russell also argues that it was this attitude that led to the development of pure mathematics, such thought being prestigious precisely because it was disconnected from the world of the senses. This led to an emphasis on the ideal rather than the real in much of Western philosophy. There are obviously connections with Plato here.

Of course it is mathematics for which Pythagoras is most well known. He believed that numbers were at the heart of everything and discovered the importance of numbers in music and shapes, which still influence our thought and language today. Of course he (or his followers) also discovered the proposition about right-angled triangles which bears his name. Geometry and mathematics remained an important part of Greek philosophy.