Real, and pivotal. The irrationality of √2 is a genuine theorem and one of the earliest proofs by contradiction; the whole-number ratios of musical consonance are real; and the cult of proof shaped Greek mathematics permanently.
Almost nothing is firsthand. Pythagoras left no writings; "Pythagoras" means the brotherhood as much as the man. The theorem named for him was known to Babylonians a millennium earlier; the irrationality proof is credited to Hippasus.
A religion with theorems. Number-worship, transmigration of souls, dietary taboos, secrecy. The brotherhood reportedly suppressed the irrationality result because it shattered their doctrine — proof used, then feared.
Their creed: every length is a ratio of whole numbers. The diagonal of a square of side 1 has length √2. Suppose it were a fraction p/q in lowest terms — and follow the logic. Step through the argument; each line forces the next, until the assumption collapses into a flat contradiction. This is reductio ad absurdum: assume, derive nonsense, conclude the assumption was impossible.
"There is geometry in the humming of the strings; there is music in the spacing of the spheres." — attributed to the Pythagoreans
The evidence for the creed is real and beautiful. Stop a vibrating string at a simple fraction of its length and the two notes are consonant — and the simpler the ratio, the sweeter the harmony. Pick a ratio; see the lengths and the interval it names.
The conviction that reality has a rational, mathematical structure accessible to proof. Wrong in its literal cult form, right as the founding faith of mathematical science.
√2's irrationality: assume it's a ratio, derive that both terms are even though the fraction was in lowest terms — impossible. Reductio, the spine of deduction.
Gate kept on. Here the two layers must be held apart cleanly. The mathematics is real: the irrationality of √2 is a correct, profound theorem and among the first proofs by contradiction; the whole-number ratios of consonance are genuine; and the Pythagorean insistence that nature is mathematically intelligible became the working faith of all of physics. But the history is fog and legend: Pythagoras wrote nothing, so "he" is inseparable from a secretive brotherhood that attributed all its findings back to the founder; the "Pythagorean theorem" (a²+b²=c²) was used by Babylonian scribes over a thousand years before him; and the irrationality discovery is traditionally credited not to Pythagoras but to Hippasus of Metapontum — whom legend says the brotherhood drowned at sea for revealing that the cosmos contained a number their doctrine forbade. Strip the myth and a hard truth remains: a community devoted to proof followed its own method to a conclusion it could not bear, and the tool was stronger than the creed. ⚑ That is exactly why proof belongs at the front of this lineage. [[thales-of-miletus]] ← · next → Parmenides.