[grungegun]

>It didn't have very many interesting properties I could find

Usually notation is defined for higher order functionals when the need arises - when a new result is found to have interesting properties. Topological proofs on function spaces can use third order functions.

> Mathematics only works as well as it does because mathematicians are all working on pretty much the same thing

I don't think any person alive can understand all the major branches of mathematics well. By that measure mathematics is very broad.