Pythagoras and Early Pythagoreanism

Number as the Nature of Things

The Pythagoreans made a claim quite unlike the Milesians': the archē is not a substance at all but number. Things are what they are because of the numerical ratios in them, and to understand something is to grasp its proportion.

The evidence was music, and it is genuinely striking. Divide a vibrating string in simple whole-number ratios and you get the consonances:

Nothing in the sound of a note suggests arithmetic, yet the intervals that sound harmonious correspond exactly to the simplest ratios. Here was an audible quality with a hidden mathematical structure, and if music worked this way, why not everything? The harmony of the spheres extended the thought to the cosmos: the heavenly bodies move at proportional distances and produce a music we cannot hear only because we have never known its absence.

The Pythagoreans were a school and a religious community at once. They held metempsychosis — the transmigration of souls between bodies, including animal bodies — practised dietary restrictions, held property in common and imposed secrecy. Mathematics was pursued as purification: understanding proportion was a way of ordering the soul, not a career. This fusion of mathematics with a doctrine of salvation is genuinely alien to modern habits of thought and shaped Plato deeply.

Then their own methods refuted them. The diagonal of a unit square has length $\sqrt{2}$, and it can be proved, the proof survives essentially unchanged, that no ratio of whole numbers equals it. Assume $\sqrt{2} = p/q$ in lowest terms; then $p^2 = 2q^2$, so $p$ is even, so $q$ must be even too, contradicting lowest terms.

Incommensurable magnitudes exist. Geometry contains lengths that no whole-number ratio can express, so "all things are number" fails in exactly the domain that had made it plausible. The legend that the discoverer was drowned at sea is almost certainly false, but it registers something real about how unwelcome the result was.

Common pitfall: treating the irrationality of $\sqrt{2}$ as a technical curiosity. For the Pythagoreans it was a refutation of their metaphysics by their own best method, the first time in recorded thought that a proof destroyed the position of the people who produced it.