By the first centuries BCE, moreover, it became fashionable to present Pythagoras in a largely unhistorical fashion as a semi-divine figure, who originated all that was true in the Greek philosophical tradition, including many of Plato's and Aristotle's mature ideas. A number of treatises were forged in the name of Pythagoras and other Pythagoreans in order to support this view.
It is stated that he was a disciple of Anaximander, his astronomy was the natural development of Anaximander's. Also, the way in which the Pythagorean geometry developed also bears witness to its descent from that of Miletos. The great problem at this date was the duplication of the square, a problem which gave rise to the theorem of the square on the hypotenuse, commonly known still as the Pythagorean proposition (Euclid, I. 47). If we were right in assuming that Thales worked with the old 3:4:5 triangle, the connection is obvious.
Pythagoras argued that there are three kinds of men, just as there are three classes of strangers who come to the Olympic Games. The lowest consists of those who come to buy and sell, and next above them are those who come to compete. Best of all are those who simply come to look on. Men may be classified accordingly as lovers of wisdom, lovers of honor, and lovers of gain. That seems to imply the doctrine of the tripartite soul, which is also attributed to the early Pythagoreans on good authority, though it is common now to ascribe it to Plato. There are, however, clear references to it before his time, and it agrees much better with the general outlook of the Pythagoreans. The comparison of human life to a gathering like the Games was often repeated in later days. Pythagoras also taught the doctrine of Rebirth or transmigration, which we may have learned from the contemporary Orphics. Xenophanes made fun of him for pretending to recognize the voice of a departed friend in the howls of a beaten dog. Empedocles seems to be referring to him when he speaks of a man who could remember what happened ten or twenty generations before. It was on this that the doctrine of Recollection, which plays so great a part in Plato, was based. The things we perceive with the senses, Plato argues, remind us of things we knew when the soul was out of the body and could perceive reality directly.
There is more difficulty about the cosmology of Pythagoras. Hardly any school ever professed such reverence for its founder's authority as the Pythagoreans. 'The Master said so' was their watchword. On the other hand, few schools have shown so much capacity for progress and for adapting themselves to new conditions. Pythagoras started from the cosmical system of Anaximenes. Aristotle tells us that the Pythagoreans represented the world as inhaling 'air' form the boundless mass outside it, and this 'air' is identified with 'the unlimited'. When, however, we come to the process by which things are developed out of the 'unlimited', we observe a great change. We hear nothing more of 'separating out' or even of rarefaction and condensation. Instead of that we have the theory that what gives form to the Unlimited is the Limit. That is the great contribution of Pythagoras to philosophy, and we must try to understand it. Now the function of the Limit is usually illustrated from the arts of music and medicine, and we have seen how important these two arts were for Pythagoreans, so it is natural to infer that the key to its meaning is to be found in them.
Pythagoras was famous as an expert on the fate of the soul after death, who thought that the soul was immortal and went through a series of reincarnations; as an expert on religious ritual; as a wonder-worker who had a thigh of gold and who could be two places at the same time; as the founder of a strict way of life that emphasized dietary restrictions, religious ritual and rigorous self discipline. It remains controversial whether he also engaged in the rational cosmology that is typical of the Presocratic philosopher/scientists and whether he was in any sense a mathematician. The early evidence suggests, however, that Pythagoras presented a cosmos that was structured according to moral principles and significant numerical relationships and may have been akin to conceptions of the cosmos found in Platonic myths, such as those at the end of the Phaedo and Republic. In such a cosmos, the planets were seen as instruments of divine vengeance (“the hounds of Persephone”), the sun and moon are the isles of the blessed where we may go, if we live a good life, while thunder functioned to frighten the souls being punished in Tartarus. The heavenly bodies also appear to have moved in accordance with the mathematical ratios that govern the concordant musical intervals in order to produce a music of the heavens, which in the later tradition developed into “the harmony of the spheres.” It is doubtful that Pythagoras himself thought in terms of spheres, and the mathematics of the movements of the heavens was not worked out in detail. But there is evidence that he valued relationships between numbers such as those embodied in the so-called Pythagorean theorem, though it is not likely that he proved the theorem.
Pythagoras: His Life, Teaching, and Influence