This morning (I wrote) that:

[F]or anyone active in mathematical research today, most mathematical learning has happened outside the classroom.

This is not all that unusual in the sciences. In fact, in most experimental sciences, enormous amounts of training happens "on the job". In other words, one is learning by observing others "do" science and mimicking them. One gets lifted onto the "shoulders of giants" without having to climb all the way up there!

Now, this similarity may leave the wrong impression that mathematics is only learnt through "apprenticeship"; in other words that Ekalavyas are not possible in mathematics. By Ekalavya, I do not refer to the gruesome and disgusting part of the story that popular mythology is so fond of. Rather, I am struck by the idea of a student who is inspired to self-study in a remote land which has little or no local expertise. This too is non-classroom learning!

Ramanujan is one of the well-known Ekalavyas but there are many less famous examples. These are mathematicians who took up a subject that was relatively unknown in their geographic vicinity (in the era before modern communications) and read (and worked hard to understand!) whatever was available, to teach themselves, and ended up with a novel viewpoint not known to the remote specialists!

Such people are not *common* in mathematics; the
point is that they exist! I think this is because
mathematicians insist on precise communication. One goal of
excellently written mathematics has always been that it should
be *possible* for someone with access to the relevant
literature (and adequate patience and intelligence) to verify
its correctness without asking experts for clarifications.

In the sciences, it is almost impossible to make a dent without an early apprentice-ship. In fact, it always surprises me how different string-theorists are from mathematicians in this respect, considering that most physicist see them as "too mathematical". In fact, the sociology of doing string-theory is a lot like the experimental sciences even though the theory is still far removed from experiments!