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Pythagoreanism is an early Pre-Socratic Greek school of philosophy based around the metaphysical beliefs of Pythagoras and his followers. Their views and methods were influential on many later movements including Platonism, Neo-Platonism and Cynicism.

The early Pythagoreans (the first society was established in about 530 B.C.) met in the Greek Achaean colony at Croton in Southern Italy, but after becoming caught up in some fierce local fighting, the movement dispersed and those that survived fled back to the Greek mainland and settled around Thebes and Phlius.

Pythagoras himself wrote nothing down, and we must rely on the second-hand accounts of his followers and commentators, Parmenides, Empedocles, Philolaus (c. 480 - 385 B.C.) and Plato, but accounts are often sketchy and sometimes contradictory.

Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. Pythagoras (along with his teacher Pherecydes of Syros), was one of the first Western philosophers to believe in metempsychosis (the transmigration of the soul and its reincarnation after death). He also subscribed to the views of another of his teachers, Anaximander, that the ultimate substance of things is what he described as "apeiron" (variously described as "the boundless" or "the undefined infinite"). Pythagoras believed that the apeiron had inhaled the void from outside, filling the cosmos with vacuous bubbles that split the universe into many inter-connected parts separated by "void", and that this play of apeiron and peiron takes place according to a natural harmony. Always somehow underlying all these theories is the assumption that numbers and mathematics constitute the true nature of things.

The Pythagoreans were well-known in antiquity for their vegetarianism, which they practiced for religious, ethical and ascetic reasons. Women, who were held to be different from men, but not necessarily inferior, were given equal opportunity to study as Pythagoreans, although they had to also learn practical domestic skills.

Pythagoreanism developed at some point into two separate schools of thought:

  • the "akousmatikoi" (or "listeners"), who focused on the more religious and ritualistic aspects of Pythagoras' teachings;
  • the "mathematikoi" (or "learners"), who extended and developed the more mathematical and scientific work he began.

The akousmatikoi claimed that the mathematikoi were not genuinely Pythagorean, but followers of the "renegade" Pythagorean Hippasus (c. 500 B.C.) The mathematikoi, on the other hand, allowed that the akousmatikoi were indeed Pythagorean, but felt that they were more representative of Pythagoras' real views. The mathematikoi group eventually became closely associated with Plato and Platonism, and much of Pythagoreanism seems to overlap Platonism. The akousmatikoi became wandering ascetics, finally joining the Cynicism movement of the 4th Century B.C.

Neo-Pythagoreanism was a revival, in the 2nd Century B.C. - 2nd Century A.D. period, of various ideas traditionally associated with the followers of Pythagoras. Notable Neo-Pythagoreans include 1st Century Apollonius of Tyana (c. 40 - 120 A.D.), and their meetings were mainly held in Rome.

Ultimately, Pythagoreanism has been a dynamic force on Western culture. It has creatively influenced philosophers, theologians, mathematicians and astronomers, as well as musicians, composers, poets and architects of the Middle Ages.

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