# 9

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I've got a paraphrased quote floating around in my mind, and I'm trying to track down the source. I think it was an article online but I have no idea where.

There was a sentence like "proof by contradiction is the closest math comes to irony." They then laid out a demonstration of a polynomial root that was imaginary and said "We've found a number that, when squared, is negative! These numbers are quite peculiar and further study is required of them."

It was then paralleled with a standard proof by contradiction of the irrationality of , except the proof was ended with "We've found a number that is both odd and even! These numbers are quite peculiar and further study is required."

Does anyone know the source I'm referring to? Thanks!

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gwern

### Dec 26, 2019

14

You probably read my "One Man's Modus Ponens" page, where I quote a Timothy Gowers essay on proof by contradiction and he says (and then goes on to discuss two ways to regard the irrationality of as compared with complex numbers):

...a suggestion was made that proofs by contradiction are the mathematician’s version of irony. I’m not sure I agree with that: when we give a proof by contradiction, we make it very clear that we are discussing a counterfactual, so our words are intended to be taken at face value. But perhaps this is not necessary. ...

...Integers with this remarkable property are quite unlike the integers we are familiar with: as such, they are surely worthy of further study.

...Numbers with this remarkable property are quite unlike the numbers we are familiar with: as such, they are surely worthy of further study.