‘There was this implicit belief, for a long time, that math was a sort of physics of bottlecaps. That any mathematical operation you could do on paper, no matter how complicated, could be reduced—in theory, anyway—to messing about with actual physical counters, such as bottlecaps, in the real world … when mathematicians began fooling around with things like the square root of negative one, and quaternions, then they were no longer dealing with things that you could translate into sticks and bottlecaps. And yet they were still getting sound results … Or at least internally consistent results … Meaning that math was more than a physics of bottlecaps.’

If the modern requirement of mathematics being simply internally consistent was a reaction, as Alan Turing’s character says, against math seen as a “physics of bottlecaps”, then this latter notion would be a reaction against the Greek notion of math as philosophy (abstract and for its own sake). The Greek notion and the modern notion are more similar to each other in their irrelevanting of physics to math—very interesting!