Mathematical jargon

The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.

Some phrases, like "in general", appear below in more than one section.

[The paper of Eilenberg and Mac Lane (1942)] introduced the very abstract idea of a 'category' — a subject then called 'general abstract nonsense'!

[Grothendieck] raised algebraic geometry to a new level of abstraction...if certain mathematicians could console themselves for a time with the hope that all these complicated structures were 'abstract nonsense'...the later papers of Grothendieck and others showed that classical problems...which had resisted efforts of several generations of talented mathematicians, could be solved in terms of...complicated concepts.

There are two canonical proofs that are always used to show non-mathematicians what a mathematical proof is like:

The beauty of a mathematical theory is independent of the aesthetic qualities...of the theory's rigorous expositions. Some beautiful theories may never be given a presentation which matches their beauty....Instances can also be found of mediocre theories of questionable beauty which are given brilliant, exciting expositions....[Category theory] is rich in beautiful and insightful definitions and poor in elegant proofs....[The theorems] remain clumsy and dull....[Expositions of projective geometry] vied for one another in elegance of presentation and in cleverness of proof....In retrospect, one wonders what all the fuss was about.

Mathematicians may say that a theorem is beautiful when they really mean to say that it is enlightening. We acknowledge a theorem's beauty when we see how the theorem 'fits' in its place....We say that a proof is beautiful when such a proof finally gives away the secret of the theorem....

Many of the results mentioned in this paper should be considered "folklore" in that they merely formally state ideas that are well-known to researchers in the area, but may not be obvious to beginners and to the best of my knowledge do not appear elsewhere in print.

Since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose....Nay more, from the logical point of view, it is these strange functions which are the most general....to-day they are invented expressly to put at fault the reasonings of our fathers....

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Wikipedia