Pythagoreans and the problem of irrational numbers

Nino asked:

Why did irrational numbers bother the Pythagoreans mathematical worldview?

Answer by Craig Skinner

The Pythagoreans’ worldview combined mysticism, numerology, mathematics and philosophy.

They felt that somehow numbers were the essence of all things. As Aristotle later said:

‘The Pythagoreans… devoted themselves to mathematics; they were the first to advance this study… they thought its principles were the principles of all things.’

Each number had personality and symbolic meaning eg 1(energy), 2 (fertility), 3 (time), 4 (space), 5 (the elements), 6 (resurrection) etc. Odd and even numbers were of different gender.

Each number was exact and well-defined. So, by numbers they meant the whole numbers (1, 2, 3…one million…etc) or ratios of whole numbers (hence rational numbers) such as 1/2, 4/5, 137/233 etc. All reality could be represented, and all magnitudes expressed, as whole numbers or rational numbers. Irrational numbers (not expressible as such a ratio) were an impossibility, an absurdity.

Bizarrely, Pythagoras’s own theorem forced an irrational number on them. Consider the right angled triangle with short sides each one unit. Clearly the length of the hypotenuse is square root of two (hereafter “root 2”). Hence, for the Pythagoreans, root 2 must be expressible as a ratio between two whole numbers. They couldn’t come up with which two, but in due course this might be known. Sadly, it can be easily proven that root two is not rational (proof follows in a moment). When the sect grasped this, all members were sworn to secrecy about these numbers which challenged their doctrines. Hippasus blabbed about it, and was drowned.

Geometry was clearly superior to arithmetic – root 2 could be represented exactly by a line but not by numbers. Plato’s model for his views on forms and universals was ideal geometrical objects, and, famously, the sign over his Academy said ‘Let no one ignorant of geometry enter this house.’

Of course things have moved on since the Greeks, with successive acceptance (sometimes resisted by many mathematicians) of irrational numbers, negative numbers, zero, transcendental numbers, imaginary numbers and nimbers.

Proof that root 2 is irrational:

Assume it is expressible as a ratio of two whole numbers. Let these, reduced to their lowest common form by division, be m and n.

It follows that m and n are not both even (if they were we could divide both by 2 till one or both is odd). So one or both of m and n must be odd.

So, root 2 = m/n

(squaring both sides), 2 = m²/n²

Hence m² = 2n²

Hence m² is even

Hence m is even

Hence m = 2p

Hence m² = (2p)² = 4p²

Hence 2n² (= m²) = 4p²

Hence n² = 2p²

Hence n² is even

Hence n is even

Contradiction (m and n can’t both be even)

Hence assumption that root 2 = m/n is incorrect (reductio ad absurdum).