[Question] Finding a quote: “proof by contradiction is the closest math comes to irony”

[Hazard]

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 $\sqrt{2}$ , 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!

[gwern]

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 $\sqrt{2}$ 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.

[Hazard]

Thank you Gwern! This was it.