The Number That Broke a Religion

The Pythagoreans believed that number was the substance of reality, and by number they meant whole numbers and their ratios. A musical interval was a ratio. A planetary orbit was a ratio. Any two lengths, they held, had a common measure: some unit, however small, that fitted a whole number of times into each.

Their own theorem killed it.

Take a square with sides of one. The diagonal, by the theorem named after them, has length √2. And √2 has no fractional name.

The proof is short enough to fit on a napkin. Suppose it did: suppose √2 = a/b, where a and b are whole numbers with no common factor, which we can always arrange by cancelling. Square both sides: 2 = a²/b², so a² = 2b². That makes a² even, and a square is even only if its root is, so a is even. Write a = 2k. Then 4k² = 2b², so b² = 2k², so b² is even, so b is even.

Both a and b are even. But we began by cancelling every common factor, and now they share one. The assumption has broken itself.

Notice what the argument depends on. The step that does the work is 'in lowest terms', which looked like housekeeping at the start and is in fact the precise thing the contradiction destroys. A proof by contradiction is only as good as the impossibility it reaches, and here the impossibility is exact: not implausible, not surprising, but flatly inconsistent with a stipulation made three lines earlier.

The Greek response is the impressive part. They did not abandon the geometry or bury the result. They built a theory of proportion that works for magnitudes with no common measure (Eudoxus's, preserved as Book V of Euclid) which is one of the most sophisticated pieces of reasoning in ancient mathematics and was not really improved on until Dedekind in 1872. The story about Hippasus being drowned at sea for revealing the secret is almost certainly later invention, and it survives because it dramatises something real: the result cost them a metaphysics.

The word 'irrational' has since drifted. It means having no ratio. It does not mean unreasonable, and irrational numbers are not in any way unruly: √2 is computable to as many digits as you want, constructible with a ruler and compass, and sitting there as the diagonal of any square you care to draw.

What is strange is how common they are. The rationals can be arranged in an endless list; the reals cannot. So the numbers with exact fractional names are a vanishingly thin scattering inside the ones without. The surprise of √2 is not that such a number exists but that the first one anyone met was the diagonal of a square, which is to say that the exception found the Greeks rather than the other way round.

Proof by contradiction is now standard equipment. There are infinitely many primes: suppose finitely many, multiply them together, add one, and the result is divisible by none of them. There is no largest even perfect number, there is no rational square root of any non-square, there is no program that decides halting: all the same shape.

It is worth being careful about how badly the shape transfers outside mathematics, because it is used constantly and almost always in a weakened form. Assume the defendant is innocent; derive that an extraordinary coincidence must have occurred; conclude guilt. That is not a reductio. It reaches an improbability, not an impossibility, and an improbability leaves the assumption standing.

The difference has sent people to prison. A one-in-seventy-three-million figure was quoted in the Sally Clark trial for two cot deaths in one family, and the jury heard it, in effect, as the probability that she was innocent. Those are different quantities, the figure itself was wrong, and the conviction was eventually overturned. The mathematical form of the argument requires every step to be forced. When the steps are merely likely, what you have is an argument about probabilities wearing the costume of a proof.