After I spoke at the 2005 "Mathematics and Narrative" conference in Mykonos, 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. Consider the following passage.

There are those who would believe that every polynomial equation with integer coefficients has a rational solution, a view that leads to some intriguing new ideas. For example, take the equation x² - 2 = 0. Let p/q be a rational solution. Then (p/q)² - 2 = 0, from which it follows that p² = 2q². The highest power of 2 that divides p² is obviously an even power, since if 2^k is the highest power of 2 that divides p, then 2^2k is the highest power of 2 that divides p². Similarly, the highest power of 2 that divides 2q² is an odd power, since it is greater by 1 than the highest power that divides q². Since p² and 2q² are equal, there must exist a positive integer that is both even and odd. Integers with this remarkable property are quite unlike the integers we are familiar with: as such, they are surely worthy of further study.

I find that it conveys the irrationality of √2 rather forcefully. But could mathematicians afford to use this literary device? How would a reader be able to tell the difference in intent between what I have just written and the following superficially similar passage?

There are those who would believe that every polynomial equation has a solution, a view that leads to some intriguing new ideas. For example, take the equation x² + 1 = 0. Let i be a solution of this equation. Then i² + 1 = 0, from which it follows that i² = -1. We know that i cannot be positive, since then i² would be positive. Similarly, i cannot be negative, since i² would again be positive (because the product of two negative numbers is always positive). And i cannot be 0, since 0² = 0. It follows that we have found a number that is not positive, not negative, and not zero. Numbers with this remarkable property are quite unlike the numbers we are familiar with: as such, they are surely worthy of further study.

• Timothy Gowers, Vividness in Mathematics and Narrative, in Circles Disturbed: The Interplay of Mathematics and Narrative