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
The Tetractys
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.)
Zero Theology 
IIRC, didn’t the Pythagorean project to reduce the whole of reality to beautiful geometric ratios hit the buffers when some Pythagorean clever clogs proved that the square root of 2 is irrational?
Incidentally, I know a *beautiful* geometric proof that the golden ratio is irrational. Much nice than the root 2 proof, which has always struck me as a bit of a hack. May blog it at some point.
Ah right, the “Pythagorean catastrophe” is what Penrose called it in his massive tome. And of course I would love to see the *beautiful* proof. And might I point out that if a mathematical proof can be deemed “beautiful” then of course the Pythagorean *instinct* still stands, once one is prepared to accept the reality of numbers beyond integers and rationals.
Finally got round to this (after an abortive attempt last month involving saving the scanned diagrams to /tmp and then rebooting… *facepalm*)
In fact, part of the beauty of mathematics is that everything— even chaos— is so wonderfully systematic, describable, behaves in patterns. That the whole ebb and flow of the universe as we know it seems to be orderly in some sort of strange way. I think I may have had a conversation like this with some hermeneut at seminary in which we agreed that mathematics, taken to its logical end, can make you a theist, but no further… 😉
Pingback: A golden proof | Curlew River