The history of Pythagoreanism is perhaps the most controversial subject in all Greek philosophy, and much about it must remain obscure. For this there are several good reasons, which are themselves not without interest. The subject is not only obscure but highly complex, and its complexity demands above all a clear statement at the outset of what is to be attempted and the outline of a plan of campaign.

First, is it justifiable to put a general account of the Pythagoreans at this early point in the exposition? Pythagoras was a contemporary of Anaximenes, but his school existed, and its doctrines developed and diverged, for the next two hundred years. Little can be attributed with certainty to the founder himself, and much Pythagorean teaching is associated with the names of philosophers of the late fifth or early fourth century. There is, however, no doubt that Pythagoras inaugurated a new tradition in philosophy, sharply divided in purpose and doctrine, as in external organization, from anything that we have met hitherto, and that from his time onwards this new current is something to be reckoned with. The Italian outlook exists in contrast to the Ionian, and an individual philosopher is likely to be influenced by sympathy with, or reaction against, the one or the other. Pythagoras himself is mentioned by the contemporary writer Xenophanes and by Heraclitus not many years after his death, and for an understanding of the development of thought during the fifth century it is important to have some idea of the main features of Pythagorean teaching which were certainly known to the philosophers of the period.

The attempt might be made to treat at this point only the earliest phase of the school, leaving until their proper chronological place the developments and divergences that culminated in a Philolaus and an Archytas and the use which they made of the latest mathematical and astronomical discoveries. This, however, would immediately meet the difficulty that our sources are in many cases too vague to allow of certain decision concerning the chronological sequence of doctrines or their attribution to a particular thinker. Moreover although divergences occurred, and strongly individual philosophers arose within the school, it was characteristic of the Pythagoreans to combine progressive thought with an immense respect for tradition.