The doctrine that things are numbers is the most influential idea associated with Pythagoreanism, and the earliest good witness to it is Aristotle, writing in the fourth century BCE about people he calls the so-called Pythagoreans rather than about Pythagoras himself. On his account they held that the principles of number are the principles of all things, that the elements of number are the odd and the even, and that the whole heaven is a harmony and a number.
Behind this lies a genuine discovery about musical intervals: that the concordant intervals correspond to simple ratios of string length, the octave as two to one, the fifth as three to two, the fourth as four to three. Whether Pythagoras made the discovery cannot be established, and the anecdote about him hearing the ratios in the sound of hammers in a forge is physically wrong. But the observation itself is correct and consequential. It showed that something as immediate as consonance is governed by exact proportion, and it made plausible the far larger claim that proportion governs everything.
From that came the harmony of the spheres: the heavenly bodies move at speeds standing in ratios to one another and therefore produce sound, unheard because it has been constant since birth. Philolaus of Croton, in the later fifth century, built the first cosmology of this kind that can be attributed to a named person, and it is genuinely radical, the earth is not at the centre but moves, along with the sun, moon, planets and a counter-earth, around a central fire. This is not heliocentrism, but it removed the earth from the middle of things nearly two thousand years before Copernicus, who cited Philolaus by name.
The tetractys, the triangular arrangement of ten points in rows of one, two, three and four, functioned as a symbol of the whole scheme, containing the ratios of the musical concords in its rows. Oaths were reportedly sworn by it.
A separate tradition holds that a Pythagorean, usually named as Hippasus, discovered incommensurability, that the diagonal of a square has no common measure with its side, and was drowned for revealing it. The mathematical result is real and belongs to this milieu. The drowning is a good story with no early evidence behind it, and it is told in several incompatible versions.