Pythagoreanism

The claim

All things are number. Reality has a mathematical structure, discoverable in the ratios that govern musical harmony, and understanding it purifies the soul that is trapped in a cycle of rebirths.

What it held

The discovery behind Pythagoreanism is a physical one: that the concordant musical intervals correspond to simple whole-number ratios of string length. Halve the string and you get the octave, two to three gives the fifth, three to four the fourth. Something audible and beautiful turns out to be arithmetic.

From that the school drew a conclusion of enormous ambition — that number is not a description of things but their principle. The heavens are ordered by the same ratios, which is why the phrase harmony of the spheres exists. Ten is perfect because it is the sum of the first four numbers, drawn as the tetractys, and the Pythagoreans swore oaths by it.

But it was never only a theory. Membership meant a rule of life: silence for years before speaking, communal property, dietary restrictions including the famous prohibition on beans, and a doctrine of transmigration that made the treatment of animals a live question. Philosophy here is a purification, and mathematics is part of the practice rather than an application of it.

How it argued

Mathematical demonstration combined with initiation. Doctrines were attributed to the founder rather than the discoverer, which is why individual contributions are almost impossible to separate.

What it was arguing against

Nothing systematically — the school predates the habit of refutation. Later Pythagoreans defined themselves against materialist accounts of the soul.

Where it disagreed with itself

The split between the akousmatikoi, who preserved the sayings and the way of life, and the mathematikoi, who pursued the mathematics. Each claimed to be the true school and denied the other's credentials.

Common misconceptions

That the theorem is his. It was known in Babylon a millennium earlier, and no early source credits him with a proof. And the number mysticism was not opposed to real mathematics — the same people did both, which is precisely what makes them interesting.

How it began

Pythagoras left Samos for Croton around 530 BCE and founded a community that acquired real political power in the Greek cities of Italy — enough that it was violently suppressed, its meeting houses burned.

What became of it

The communities were destroyed in the fifth century and the doctrines dispersed. Neopythagoreanism revived them philosophically in the Roman period, by which time the ratios had become theology.

Lasting influence

Plato, decisively — the mathematical cosmology of the Timaeus is Pythagorean. Through him, the entire Western assumption that nature is written in mathematics, which Galileo and Kepler both state in almost these terms.

Why it still matters

The claim that physical reality has mathematical structure is now so completely assumed that its strangeness is invisible. This is where it was first asserted.

Put another way

A string that sounds beautiful only at exact fractions of its length, and a conclusion drawn from that about everything.

Where to start

Aristotle first, for what can be established, then Iamblichus for what the tradition became.

The surprising bit

The discovery that the square root of two cannot be expressed as a ratio broke the central doctrine, and the story goes that the man who revealed it, Hippasus, was drowned at sea for it.

In their own words

Connections

Influenced

Old Academy, Neoplatonism

The verdict

Half a mathematical research programme and half a religious order, and the two halves could not be separated even by its own members.

Sources

• Stanford Encyclopedia of Philosophy