The important thing about Pythagoras’ emigration from Ionian Samos to Croton in Italy (on the outposts of Greek civilisation) is that he was joining his kind, that is to say religious enthusiasts keen on mystery rites, Hades, the Underworld, and the cult of the dead – the perfect place for a Mystery Man Religious Guru to find tenure and start his own mystery religion, which he did. We don’t know much about it. (On Pythagoreanism in general, I could hardly hope to compete with this, so I’ll just link to it instead.) Reincarnation and the immortality of the soul were central dogmas. Herodotus thinks the idea comes from Egypt but, with respect to Pythagoras, doesn’t want to name names. Hmmm….
The Egyptians are the first to have enunciated this theory that the human soul is immortal, and that as and when the body fails it repeatedly clothes itself within another creature that is being born, and when it has done the round of all the terrestrial, marine and flying animals, it once again puts on a human body that is being born; and the circuit takes it 3000 years. There are some among the Greeks who adopted this theory as their own, some earlier, some later. Their names I know but I don’t write down. (Herodotus c. 490-410 BC, Histories, II. 123 – quoted in Catherine Osborne, The Presocratics, p. 99.)
Relatedly, he appears to have espoused vegetarianism and was something of an activist for the prevention of cruelty to animals, though other evidence indicates he was especially devoted to ritual sacrifice and purification. Contradictory? Probably not, actually. Anyhow, for my money these influences on Pythagoras were Orphic.
Enough prolegomena. On the mystical mathematics of Pythagoras, the proof of the
Pythagorean theorem, the discovery of harmonic ratios and the like….
It probably deserves the sacrifice of a hundred oxen. It opens up the possibility that the entire world is based on hidden mathematical ratios that are not immediately obvious. (Osborne, p. 105.)
Not just in the realm of geometry however, although there is a visual aesthetic appeal to geometric form, especially in the knowledge of the mathematical ratios underlying musical harmony, beauty, number and physical reality converge. And so
it transpires that the beauty of harmony is an objective fact about nature, and it is a matter for calculation. It turns out to be a quantity as well as a quality. Perhaps, then, everything about the world, and every kind of good quality, is actually a matter of patterns of number? Science has followed this path with great success in the two and a half millennia since Pythagoras discovered it. Could moral, aesthetic, religious, and political values also be like that? (Osborne, p. 106.)