Thinking in a brand new way about Math, a way no one else has, to come up with novel new connections?

Arithmetic is to mathematics as typing is to being a writer. If the subject of an article is a brilliant author, then a cool new keyboard that makes it easier to type really isn't related, not even tangentially.

However, tangents themselves are part of geometry, which is tangentially related to Langlands' work.

Damnit, you win this round. :-)

Arithmetics is part of math. (And typing is not part of being a writer.)

Perhaps a more apt analogy is that mental arithmetic tricks are to math what stenography is to writing.

The Trachtenberg system is number theory.

Basically, every good scientific math publication does just that (on the level of the Trachtenberg system). What makes the Langlands program stand out is that it connects entire branches of mathematics previously thought to be unconnected; that means decades long legacies of journal publications of people working within their branch (each time through a novel discovery) and not noticing the connection. The two are not really comparable.

Perhaps there exists a framework isomorphic to Langlands' theories that unite the three disparate topics of this subthread:

1. The perfunctory machinery of arithmetic 2. The theoretical connective tissue between entire branches of mathematics 3. The "egotistical creatures", as he put it, who downvote and give someone a hard time for posting a comment.