Regarding your last statement, what do you see as the distinction? Take for example eg continuity. Students are first taught it means you can graph a function

Regarding your last statement, what do you see as the distinction? Take for example eg continuity. Students are first taught it means you can graph a function in a single stroke of the pen. Later comes epsilon and delta, this is more abstract but at that stage the understanding is that "nearby values map to nearby values" (or some equivalent). If the student dives in from there, they're taught the "existence" of real numbers (or any mathematical term) is rather a consequence of a system of symbols and relations that increasingly look nothing like "numbers", instead describing more of a process. Later that "consequence" and "relation" themselves are formalities. "Pure" math occasionally delivers strange consequences in this sense. But it always boils down to a process that something or another must interpret and carry out. So I wonder whether the edifice is meaningfully a thing in and of itself. Methods developed in ancient China and India etc would have been useful to the Greeks and vice versa, however all of them though worked by means of the human brain. "Line" has a distinct meaning to us, the axioms of geometry don't create the line, they allow us to calculate some properties more efficiently. We always need to interpret the result in terms we understand, don't we?