two times two is five??? by Lewis Carroll

I recently came across an example of Carroll's cross between wit and logic in a 2008 Gresham lecture by Robin Wilson:

Honoured Sir,



Understanding you to be a distinguished algebraist (that is, distinguished from other algebraists by different face, different height, etc.), I beg to submit to you a difficulty which distresses me much.

If x and y are each equal to 1, it is plain that


2 × (x2 - y2) = 0, and also that 5 × (x - y) = 0.


Hence 2 × (x2 - y2) = 5 × (x - y).


Now divide each side of this equation by (x - y).


Then 2 × (x + y) = 5.


But (x + y) = (1 + 1), i.e. = 2. So that 2 × 2 = 5.

Ever since this painful fact has been forced upon me, I have not slept more than 8 hours a night, and have not been able to eat more than 3 meals a day.

I trust you will pity me and will kindly explain the difficulty to Your obliged, Lewis Carroll.