The Pythagoean tradition is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and Platonism. Later resurgence of ideas similar to those held by the early Pythagoreans are collected under the term Neopythagoreanism. The Pythagoreans were called mathematikoi, which means "those that study all.
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. The Pythagorean conception of substance, on the other hand, is of unknown origin, partly because various accounts of his teachings are conflicting. The Pythagorean account actually begins with Anaximander's teaching that the ultimate substance of things is "the boundless," or what Anaximander called the "apeiron." The Pythagorean account holds that it is only through the notion of the "limit" that the "boundless" takes form.
The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed and developed various rules of living which they believed would enable their soul to achieve a higher rank among the gods. Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics included various purifactory rites and practices as well as incubatory rites of descent into the underworld. Apart from being linked with this, Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls. Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the soul in terms of a pentemychos ("five-nooks," or "five hidden cavities") — the most likely origin of the Pythagorean use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (eugieia).
A Pythagorean initiatic tradition included a magical and mystical system built around the use of the letters of the Greek Alphabet. This is the source of the famour I-A-O formula of the Golden Dawn’s Second Order. Indeed, this is only the tip of the iceburg, and secret Pythagorean esoteric transmissions play an important role in the secret curriculum of S.L. MacGregor Mathers Alpha et Omega, which goes far beyond the simple Golden Dawn material published by Crowley and Regardie.