Mathematical jargon

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The field 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 more rigorous arguments or more precise ideas. Much of this is common English, used in a mathematical or quasi-mathematical sense.

Note that some phrases, like " in general", will appear in more than one section.

Philosophy of mathematics

These terms discuss mathematics as mathematicians think of it; they connote common intellectual strategies or notions the investigation of which somehow underlies much of mathematics.

abstract nonsense
Also generalized abstract nonsense, a tongue-in-cheek reference to the prevalence of category theory in mathematics, which leads to arguments that establish a result without reference to any specifics of the present problem.
canonical
A reference to a standard or choice-free presentation of some mathematical object. The term canonical is also used more informally, meaning roughly "standard": for example, one might say that Euclid's proof is the 'canonical proof' of the infinitude of primes (and indeed this is possibly the canonical example of a canonical proof).
elegant
Also beautiful; an aesthetic term referring to the ability of an idea to provide insight into mathematics, whether by unifying disparate fields, introducing a new perspective on a single field, or providing a technique of proof which is either particularly simple, or captures the intuition as to why the result it proves is true. Gian-Carlo Rota distinguished between elegant and beautiful, saying that for example, some topics could be written about elegantly although the mathematical content is not beautiful, and some theorems or proofs are beautiful but may be written about inelegantly.
natural
Similar to "canonical" but more specific, this term makes reference to a description (almost exclusively in the context of transformations) which holds independently of any choices. Though long used informally, this term has found a formal definition in category theory.
pathological
An object behaves pathologically if it fails to conform to the generic behavior of such objects, fails to satisfy certain regularity properties (depending on context), or simply disobeys mathematical intution. These can be and often are contradictory requirements. Sometimes the term is more pointed, referring to an object which is specifically and artificially exhibited as a counterexample to these properties.
rigor (rigour)
Mathematics strives to establish its results using indisputable logic rather than informal descriptive argument. Rigor is the diametric opposite of the terms on this page.
well-behaved
An object is well-behaved if it does satisfy the prevailing regularity properties, or sometimes if it conforms to intuition (but intuition often suggests the opposite behavior as well).

Informalities

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.

almost all
In contexts which admit a notion of generality or genericity of objects, a property holds for almost all objects if it holds generically. For example, if it holds of all but finitely many integers, or for every real number except for a set of measure zero.
arbitrarily large, arbitrarily small, arbitrarily close
Notions which arise mostly in the context of limits, referring to phenomena which recur as the limit is approached.
arbitrary
A shorthand for the universal existential quantifier. An arbitrary choice is one which is made unrestrictedly, or alternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set.
for all sufficiently nice X
For all X which satisfy a set of conditions to be specified later. When working out a theorem, the conditions involved may be not yet known to the speaker; the intent is to restrict the set of X to which the theorem applies when the proof runs into difficulties.
in general
In a descriptive context, this phrase introduces a simple characterization of a broad class of objects, with an eye towards identifying a unifying principle. Concisely, this term introduces an "elegant" description which holds for "arbitrary" objects "modulo" "pathology".
sharp
Often, a mathematical theorem will establish constraints on the behavior of some object; for example, a function will be shown to have an upper or lower bound. The constraint is sharp if it cannot be made more restrictive without failing in some cases.
smooth
Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability to analyticity, and still others which are more complicated. Each such usage attempts to invoke the physically intuitive notion of smoothness.
sufficiently large, suitably small, sufficiently close
In the context of limits, these terms refer to phenomena which prevail as the limit is approached.
upstairs, downstairs
In a fiber bundle, the total space is often said to be upstairs, with the base space downstairs.
up to, modulo, mod out by
An extension to mathematical discourse of the notions of modular arithmetic. A statement is true up to a condition if the establishment of that condition is the only impediment to the truth of the statement.

Proofs and proof techniques

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

aliter
An obsolescent term which refers to an alternative method of proof.
if and only if (iff)
An abbreviation for logical equivalence of statements.
in general
In the context of proofs, this phrase is often seen in induction arguments when passing from the base case to the "induction step", and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence.
necessary and sufficient
A minor variant on "if and only if"; necessary means "only if" and sufficient means '"if".
one and only one
An especially precise existence statement; the object exists, and furthermore, no other such object exists.
by way of contradiction (BWOC), or "for, if not, ..." 
The rhetorical prelude to a proof by contradiction, preceding the negation of the statement to be proved.
Q.E.D.
A Latin abbreviation historically placed at the end of proofs, but less common currently.
required to prove (RTP)
Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desired theorem; thus, it is required to prove just these statements.
the following are equivalent (TFAE)
A particular definition is not always the most convenient for certain applications; often one proves theorems stating equivalent rephrasings of the definition.
wish to show, want to show (WTS)
If a proof proceeds along several steps, the goal of each stage of the argument is prefaced with this expression.
without (any) loss of generality (WLOG, WOLOG, WALOG), we may assume (WMA)
Sometimes a proposition can be more easily proved with additional assumptions on the objects it concerns. If the proposition as stated follows from this modified one with a simple and minimal explanation (for example, if the remaining special cases are identical but for notation), then the modified assumptions are introduced with this phrase and the altered proposition is proved.

Proof non-techniques

A few terms, such as those below, refer to the absence of correct proof technique.

back-of-the-envelope
An informal computation omitting much rigor without sacrificing correctness. Often this computation is "proof of concept" and treats only an accessible special case.
by inspection
A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctness of a proposed expression or deduction.
clearly, can be easily shown
A term which shortcuts around calculation the mathematician perceives to be tedious or routine, accessible to any member of the audience with the necessary expertise in the field; Laplace used obvious.
handwaving
A non-technique of proof mostly employed in lectures, where formal argument is not strictly necessary. It proceeds by omission of details or even significant ingredients, and is merely a plausibility argument.
in general
In a context not requiring rigor, this phrase often appears as a labor-saving device when the technical details of a complete argument would outweigh the conceptual benefits. The author gives a proof in a simple enough case that the computations are reasonable, and then indicates that "in general" the proof is similar.
morally true
Used to indicate that the speaker believes a statement should be true, given their mathematical experience, even though a proof has not yet been put forward. As a variation, the statement may in fact be false, but instead provide a slogan for or illustration of a correct principle.
trivial
Similar to clearly. A concept is trivial if it holds by definition, is immediately corollary to a known statement, or is a simple special case of a more general concept.

Miscellaneous

left-hand side, right-hand side (LHS, RHS)
transport of structure
wrt, with respect to