Contradiction by Proof

Mathematician G. H. Hardy called proof by contradiction, or reductio ad absurdum, one of a mathematician’s finest weapons.

G. H. Hardy, pictured yesteryear.

Without getting all technical, it’s the technique of assuming the opposite of what you’re attempting to prove and then using logical steps to show that assumption leads to an absurd conclusion.

The classic example of the technique is the proof that the square-root of 2 is irrational (that is, it cannot be exactly expressed as a fraction). The proof rests on a piece of maths that we all learned in primary school; simplifying fractions. If we have a fraction such as 4/8 we can see that both the numerator and the denominator (the fancy book-larnin’ words for the top and bottom of the fraction) can be divided by 2, to give us 2/4, we can then divide by 2 again, to get 1/2, and that is the fraction in its simplest form.

I won’t work through the proof that the square-root of 2 is irrational (there are many worked examples a Google away) but it starts by assuming that it can be written as a fraction, a/b. Then, through some sub-GCSE algebraic juggling, we can show that both a and b can be simplified, to give c/d. So far so good, but then we can run through the exact same process again, to give an even simpler representation of root 2, e/f. Then again, to give g/h, and so on, forever, never arriving at a simplest form.

A fraction that can be simplified forever is an absurdity, but every step in the logic is mathematically correct, so the only possible conclusion is that our initial premise was wrong, and that the square-root of 2 cannot be represented as a/b and is, therefore, irrational.

Pythagoras, legend has it, believed that every number could be represented by simple fraction, so when his student, Hippasus, proved that the square-root of 2 was irrational, Pythagoras ordered him drowned at sea.

Hippasus being drowned at sea

I guess that’s a timely warning about what can happen if you completely wrap yourself up in an unfounded belief, and then find the absurdity at the end of it. Especially if it’s so absurd that comedians can get a laugh just by repeating it out loud.

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