As a high-school student, I used to go to the mathematics library at the University of Pennsylvania to look up and try to read articles articles in number theory. Usually I couldn't understand them at a first reading, so I'd photocopy them and take them home to puzzle over. I remember being completely flummoxed by a paper on Bell numbers that used the "umbral calculus"; I just didn't understand that you were supposed to move the exponents down as indices. That is, in an equation like
B4 = (B + 1)3
you were supposed to expand the right hand side, getting
B3 + 3B2 + 3B1 + 1
and then magically change this to
B3 + 3B2 + 3B1 + 1 .
I had nobody to ask about stuff like that. Although my high-school teachers were great, they didn't know about the umbral calculus.
Things like this permeate the mathematical literature. Take compactness, for example. Compactness is a marvelous tool that lets you deduce -- usually in a non-constructive fashion -- the existence of objects (particularly infinite ones) from the existence of finite "approximations". Formally, compactness is the property that a collection of closed sets has a nonempty intersection if every finite subcollection has a nonempty intersection; alternatively, if every open cover has a finite subcover.
Now compactness is a topological property, so to use it, you really should say explicitly what the topological space is, and what the open and closed sets are. But mathematicians rarely, if ever, do that. In fact, they usually don't specify anything at all about the setting; they just say "by the usual compactness argument" and move on. That's great for experts, but not so great for beginners.
I really wonder who was the very first to take this particular lazy approach to mathematical exposition. So far, the earliest reference I found was in a 1953 article by John W. Green in the Pacific Journal of Mathematics 3 (2), 393-402. On page 400 he writes
By the usual compactness argument ([2, p.62]), there does exist a minimizing curve K.
Can anybody find an earlier occurrence of this exact phrase?