The idea that "when you extend that" is something human beings can do is an absolute revelation to young mathematicians. The idea that our notations are, at lea

The idea that "when you extend that" is something human beings can do is an absolute revelation to young mathematicians. The idea that our notations are, at least in some sense, choices that we make that come with tradeoffs is a huge point of mathematical maturity for them, and usually causes my students to look back with either deep awe or deep suspicion when they realize that even our choice of base 10 is a choice among many and we can make other choices. As von Neumann said, very much something to get used to.