# List of 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....

[The Dirichlet function] took on an enormous importance...as giving an incentive for the creation of new types of function whose properties departed completely from what intuitively seemed admissible. A celebrated example of such a so-called 'pathological' function...is the one provided by Weierstrass....This function is continuous but not differentiable.

Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context.

Norbert A'Campo of the University of Basel once asked Grothendieck about something related to the Platonic solids. Grothendieck advised caution. The Platonic solids are so beautiful and so exceptional, he said, that one cannot assume such exceptional beauty will hold in more general situations.

The formal language of proof draws repeatedly from a small pool of ideas, many of which are invoked through various lexical shorthands in practice.

*needs to show*just these statements.

Let *V* be a finite-dimensional vector space over *k*....Let (*e*_{i})_{1≤ i ≤ n} be a basis for *V*....There is an isomorphism of the polynomial algebra *k*[*T*_{ij}]_{1≤ i, j ≤ n} onto the algebra Sym_{k}(*V* ⊗ *V*^{*})....It extends to an isomorphism of *k*[**GL**_{n}] to the localized algebra Sym_{k}(*V* ⊗ *V*^{*})_{D}, where *D* = det(*e*_{i} ⊗ *e*_{j}^{*})....We write *k*[**GL**(*V*)] for this last algebra. By transport of structure, we obtain a linear algebraic group **GL**(*V*) isomorphic to **GL**_{n}.

Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for filling in tedious details.

*proof by example*is an argument whereby a statement is not proved but instead illustrated by an example. If done well, the specific example would easily generalize to a general proof.