Reductio ad Absurdum

§4.7

“Reductio ad absurdum” (“reductio” for short) is Latin for “reduction to absurdity”. It refers to the strategy of proving a proposition false by deducing an absurd conclusion (often a contradiction) from it in conjunction with other known premises. Such arguments are common in math. They were used to prove that the square root of two is not a rational number (and thus that there are such things as irrational numbers) and that there is no largest prime number. Here’s a non-mathematical example of a reductio.

There cannot be a chess-playing computer program good enough to win every game against any opponent, regardless of whether it plays as white or black. If there were such a program, it could be pitted against itself, and both sides would have to win. But it’s impossible for both sides to win in a chess game, therefore there cannot be a chess-playing program such as the one described.13FTNOTE TBA

And here is an example from mathematics, first in paragraph form and then laid out:

Suppose that the set of prime numbers is finite. If so, it must contain a largest member—call it L. We can then multiply all the prime numbers together. Call the result P. P will be divisible by each of the prime numbers, so P+1, won’t be divisible by any prime number. Therefore, it won’t be divisible by any number, which means that it will be prime. But P+1 is larger than L. So, L isn’t the largest prime number. But this is a contradiction, so the assumption that lead to it must be false, and the set of prime numbers must be infinite.