At the time, I was reading part of Janis Voigtländer’s habilitation thesis. Unsure where to even start, I decided to just answer straightforwardly: “I’m reading a very long story about free theorems.”
Never one to shrink from a pedagogical challenge, I thought for a moment, then began: “Do you know what a function is?” He didn’t. “A function is like a machine where you put something in one end and something comes out the other end. For example, maybe you put a number in, and the number that is one bigger comes out. So if you put in three, four comes out, or if you put in six, seven comes out.” This clearly made sense to him, so I continued, “The type of a function machine tells you what kinds of things you put in and what kinds of things come out. So maybe you put a number in and get a number out. Or maybe you put in a list of numbers and get a number out.” He interrupted excitedly, “Or maybe you could put words in??” “Yes, exactly! Maybe you can put words in and get words out. Or maybe there is a function machine where you put other function machines in and get function machines out!” He gasped in astonishment at the idea of putting function machines into function machines.
“So,” I concluded, “a free theorem is when you can say something that is always true about a function machine if you only know its type, but you don’t know anything about what it does on the inside.” This seemed a bit beyond him (and to be fair, free theorems are only interesting when polymorphism is involved which I definitely didn’t want to go into). But the whole conversation had given me a different idea.