Yesterday on a hike someone named Sara, who is not a mathematician nor a scientist, asked me what I do. I said I worked on theoretical computer science, that is, the mathematics of computer science. She asked: "What is it that you people do, exactly? I can understand what a musician, say, does in his work, but I have no idea what mathematicians do."
I started trying to answer her: "You like games. You enjoy puzzles. Consider the Rubik's cube, and think about what your brain does as it is trying to figure it out. That is very similar to what a mathematician does while he or she is working on solving a problem. In addition, instead of working with a physical object, it's as though they had a picture of the Rubik's cube in their head, and they work with their representation of the mathematical world inside their head, so it's a bit more abstract." Omer Tamuz intervened: "yes, abstraction is very important. Here is an example. For every group of six people, if you look at which pairs are friends and which are not, there always exist either three who are all friends or three, none of which are mutual friends." He tried to explain it more clearly, I criticized that choice of an example, he asked me what would be a better example of abstraction, I suggested commutativity of addition and of multiplication, he suggested that removing three apples from eight apples leaves five apples, just in the same way that removing three oranges from eight oranges leaves five oranges, I criticized those examples for not being impressive enough, suggested the probabilistic method, it reminded him of Noga Alon's simple and beautiful first lecture in a course, which, upon my request, he proceeded to tell me about in full detail, and suddenly we came upon the others, including Sara, who had stopped for a break.
Sara had long since left Omer and me to join the others, but we had not even noticed her absence. She commented how amazing it was that after she had asked just one simple question, we took off with it, started discussing math, and an hour later we were still going at it. I was a little embarrassed. That doesn't seem like quite the right way to talk to a non-scientist about what we do, does it?
I am afraid that it reminds me of the 19th century cartoon L'idée fixe du savant Cosinus -- a must-read for all French-speaking mathematicians.