Also I do wonder sometimes whether mathematicians don't actually understand some of the maths they work on
We don’t. One of the first steps to mathematical maturity is learning to let go of the need to understand, the need to visualize. Much of mathematics is a formal affair of making arguments to satisfy necessary and sufficient conditions. Trying to understand infinite-dimensional spaces or highly abstract sets and objects is too much, and unnecessary.
”Young man, in mathematics you don't understand things. You just get used to them.”
We don’t. One of the first steps to mathematical maturity is learning to let go of the need to understand, the need to visualize. Much of mathematics is a formal affair of making arguments to satisfy necessary and sufficient conditions. Trying to understand infinite-dimensional spaces or highly abstract sets and objects is too much, and unnecessary.
”Young man, in mathematics you don't understand things. You just get used to them.”
— John von Neumann