Why is probability an under-emphasized subject in most math programs? Why does it seem that the hot topics these days are category theory and algebra? What do you think about the following: A student learns probability (both on the applied side and the pure side) and picks up the real analysis on the way. That way there is a purpose for learning the real analysis.
Can we do this with many mathematical subjects (e.g. first teach from an applied perspective, and learn the pure stuff because it is needed)?