Abstract

There are many familiar theorems whose proofs use methods which are in some appropriate sense substantially more "abstract" than its statement. Some particularly well known examples come from the use of complex variables in number theory. Sometimes such abstraction can be removed - for example by the "elementary proof of the prime number theorem" - and sometimes no appropriate removal is known. The interest in removing abstraction typically varies, with no agreed upon criteria for appropriateness. E.g., the removal might sacrifice naturalness or intelligibility, or the result of the removal criticized as being merely a thinly disguised form of the original.