Students seek professors. The spend a few years doing whatever it is professors and students do. At the end, the professor pronounced the student "graduated". Society at large recognizes the value of the student who has had this experience. After several gradations of graduation, the student might decide to pursue the path pf becoming a professor too. Or not.

So underneath all the Latin and posturing, there really is a continuous arc from from Pythagoras and his order or Plato and his academy, to massively online courses, for profit colleges, and Ivy Leagues. Everything that has happened in between can be described as addition of third parties to the interaction of professor and student. This is necessary when it is a value multiplier. Three professors and a secretary can serve more students than four professors.  But how does the process end?