"So two and two now make five?"

... If you speak of a revolution in mathematics, you are likely to be treated to the ironic response, "Oh, so two and two now make five?" We can answer: "Why not? Suppose we ask for the delivery of two articles each weighing two pounds; they are delivered in a box weighing one pound; then in this package two pounds and two pounds will make five pounds!"

"But you get five pounds by adding three weights, 2 and 2 and 1.""

"True, our operation '2 and 2 make 5' is not an addition in the usual sense. But we can define the operation to make the result hold true. We can imagine a packaging such that the operation might be, for instance:

• 2 and 2 make 5; 2 and 3 make 6 (if the box weighs one pound in each case),
• 3 and 3 make 8; 3 and 4 make 9 (if the box weighs two pounds), etc.

"The symbol 'etc.' here stands for a rule of the game, which would define how much a and b would 'make' for all pairs of integers a and b. We would thus define an operation and we could study its properties."

This answer would undoubtably seem frivolous to our our questioner. It would seem to him to be a game; the study we could make of this game would not seem to be a part of mathematics; besides, we would not have invented anything, for we would merely be thinking of the sum of three terms (of which one was understood), while he was thinking of a real addition of two terms.

Lucienne Felix (1960) The Modern Aspect of Mathematics, Basic Books: 3-4.

I am not sure what I think of Nicolas Bourbaki, a man who never existed. But this is the sort of mathematics some need to understand if they are going to help keep the Internet running or explore the solar system.