Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, in modern Calabria (Italy) circa 530 BC. Early Pythagorean communities spread throughout Magna Graecia.
Already during Pythagoras' life it is likely that the distinction between the akousmatikoi ("those who listen"), who is conventionally regarded as more concerned with religious, and ritual elements, and associated with the oral tradition, and the mathematikoi ("those who love science") existed. The ancient biographers of Pythagoras, Iamblichus (c.245-c.325 AD) and his master Porphyry (c.234–c.305 AD) seem to make the distinction of the two as that of 'beginner' and 'advanced'. As the Pythagorean cenobites practiced an esoteric path, like the mystery schools of antiquity, the adherents, akousmatikoi, following initation became mathematikoi. It is wrong to say that the Pythagoreans were superseded by the Cynics in the 4th century BC, but it seems to be a distinction mark of the Cynics to disregard the hierarchy and protocol, ways of initiatory proceedings significant for the Pythagorean community; subsequently did the Greek philosophical traditions become more diverse. The Platonic Academy was arguably a Pythagorean cenobitic institution, outside the city walls of Athens in the 4th century BC. As a sacred grove dedicated to Athena, and Hecademos (Academos). The Academy, the sacred grove of Academos, may have existed, as the contemporaries seem to have believed, since the bronze age, even pre-existing the Trojan War. Yet according to Plutarch it was the Athenian strategos (general) Kimon Milkiadou (c.510–450 BC) who converted this, "waterless and arid spot into a well watered grove, which he provided with clear running-tracks and shady walks". Plato (less known as Aristocles) lived almost a hundred years later, circa 427 to 348 BC. On the other hand it seems likely that this was a part of the re-building of Athens led by Kimon Milkiadou and Themosticles, following the Achaemenid destruction of Athens in 480-479 BC - The Second Persian War. Kimon is at least associated with the building of the southern Wall of Themosticles, the city walls of ancient Athens. It seems likely that the Athenians saw this as a rejuvenation of the sacred grove of Academos.
Following political instability in Magna Graecia, some Pythagorean philosophers moved to mainland Greece while others regrouped in Rhegium. By about 400 BC the majority of Pythagorean philosophers had left Italy. Pythagorean ideas exercised a marked influence on Plato and through him, on all of Western philosophy. Many of the surviving sources on Pythagoras originate with Aristotle and the philosophers of the Peripatetic school.
As a philosophic tradition, Pythagoreanism was revived in the 1st century BC, giving rise to Neopythagoreanism. The worship of Pythagoras continued in Italy and as a religious community Pythagoreans appear to have survived as part of, or deeply influenced, the Bacchic cults and Orphism. Even the early christian cenobitic traditions may be seen in light of the matematikoi. The biblical Greek name for 'disciple' is 'mathetes' [1].