As you say, the artistic use of cross-object synthesis and analogy definitely distinguishes the advanced students from the mechanical. Advanced students develop

As you say, the artistic use of cross-object synthesis and analogy definitely distinguishes the advanced students from the mechanical. Advanced students develop a sort of dialect that harmonizes the mathematical objects they encounter in a way that illuminates them all. More than any of that, though, I think what you can see (even here in this thread!) is that, while we pretend that mathematics is a single cultural practice of rational humans communicating with other rational humans, it's really many smaller communities of mathematicians, all of varying skills trying to communicate with each other. My mathematical language as a teacher of 12-18 year olds is very different from my mathematical language when I did computational geometry for a living. Because you have many communities of mathematicians producing new notation, you end up with dialects that all sort of meld together in the same way that reading Shakespeare is very different from reading Hemingway or Eco (in translation). The closest programming analogue would be C++, where you have several mutually unintelligible dialects spoken by different communities of programmers with different concerns.