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The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. The group strived to keep the discovery of irrational numbers a secret; and legends tell of a member being drowned, for breaching this secrecy.

The advent of modern mathematical science is generally believed to have begun with the Pythagoreans; although, it would be a mistake to believe that modern science was a "Greek invention". Rather, Greek history has survived in far greater detail than the histories of other ancient civilizations.

The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans. It was called "health".

Pythagorean Cosmology

Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. In the area of cosmology there is less agreement about what Pythagoras himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soul is too central to have been added by a later follower of Pythagoras. On the other hand it is impossible to determine the origin of the Pythagorean account of substance. It seems that the Pythagorean account begins with Anaximander's account of the ultimate substance of things as "the boundless." Another of Anaximander's pupils, Anaximenes, who was a contemporary of Pythagoras, gave an account of how Anaximander's "boundless" took form, through condensation and refraction. On the other hand, the Pythagorean account says that it is through the notion of the "limit" that the "boundless" takes form.

Diogenes Laertius (about 200 BC) quotes Alexanders (about 100 BC) book Successions of Philosophers[?] (and according to Diogenes Alexander has access to a book called The Pythagorean Memoir) in his account of how the pythagorean cosmology was constructed (Diogenes Laertius, Vitae philosophorum VIII[?], 24):

The principle of all things is the monad or unit; arising from this monad the undefined dyad or two serves as material substratum to the monad, which is cause; from the monad and the undefined dyad spring numbers; from numbers, points; from points, lines; from lines, plane figures; from plane figures, solid figures; from solid figures, sensible bodies, the elements of which are four, fire, water, earth and air; these elements interchange and turn into one another completely, and combine to produce a universe animate, intelligent, spherical, with the earth at its centre, the earth itself too being spherical and inhabited round about. There are also antipodes, and our ‘down' is their ‘up'.

This cosmology also inspired the arabic gnostic Monoimus to combine this system with monism and other things to form his own cosmology.

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