Mathematical jargon

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", 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" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes; indeed, this is a 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 or imagination 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 intuition. 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 use of such logic in a proof.

strong

A theorem is said to be strong if it deduces restrictive results from general hypotheses. One celebrated example is Donaldson's theorem, which puts tight restraints on what would otherwise appear to be a large class of manifolds. This (informal) usage reflects the opinion of the mathematical community: not only should such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area.

well-behaved

An object is well-behaved (in contrast with being pathological) if it does satisfy the prevailing regularity properties, or sometimes if it conforms to intuition (but intuition often suggests the opposite behavior as well).

Descriptive 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. Note that many of the terms are completely rigorous in context.

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, one might say "almost all integers are not zero." This phrase is not informal in the context of real numbers or other measure spaces, where it means "except for a set of measure zero".

arbitrarily large, arbitrarily small, arbitrarily close

Notions which arise mostly in the context of limits, referring to the recurrence of a phenomenon as the limit is approached. A statement such as that predicate P is satisfied by arbitrarily large values, can be expressed in more formal notation by ∀x : ∃y ≥ x : P(y).

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.

factor through

A term in category theory referring to composition of functions. If we have three objects A, B, C, a map f:A→C which is written as a composition with g:A→B, h:B→C is said to factor through any (and all) of B, g, and h.

generic

This is similar to almost all, with usage mostly in non-measure theoretic contexts. For example, a property which holds on a dense G_δ (intersection of countably many open sets) is said to hold generically.

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".

left-hand side, right-hand side (LHS, RHS)

Most often, these refer simply to the left-hand or the right-hand side of an equation; for example, x = y + 1 has x on the LHS and y +1 on the RHS. Occasionally, these are used in the sense of lvalue and rvalue: an RHS is primitive, and an LHS is derivative.

resp.

(Respectively) A convention to shorten parallel expositions. "A (resp. B) [has some relationship to] X (resp. Y)" means that A [has some relationship to] X and also that B [has (the same) relationship to] Y.

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.

strong

A theorem, result, or condition is called stronger than another one if a proof of the second can be easily obtained from the first. An example is the sequence of theorems: Fermat's little theorem, Euler's theorem, Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (see above) is a stronger result than a non-sharp one. Finally, the adjective strong or the adverb strongly may be added to a mathematical notion to indicate a related stronger notion; for example, a strong antichain is an antichain satisfying certain additional conditions, and likewise a strongly regular graph is a regular graph meeting stronger conditions. When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as "antichain").

sufficiently large, suitably small, sufficiently close

In the context of limits, these terms refer to some (unspecified, even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as that predicate P holds for sufficiently large values, can be expressed in more formal notation by ∃x : ∀y ≥ x : P(y).

upstairs, downstairs

A descriptive term referring to notation in which two objects are written one above the other; the upper one is upstairs and the lower, downstairs. For example, in a fiber bundle, the total space is often said to be upstairs, with the base space downstairs. In a fraction, the numerator is occasionally referred to as upstairs and the denominator downstairs, as in "bringing a term upstairs".

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.

weak

The converse of "strong" above.

Proofs and rigorous 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.

diagram chase

Given a commutative diagram of objects and morphisms between them, if one wishes to prove some property of the morphisms (such as injectivity) which can be stated in terms of elements, then the proof can proceed by tracing the path of elements of various objects around the diagram as successive morphisms are applied to it. That is, one chases elements around the diagram, or does a diagram chase.

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.

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". For example, "For a field K to be algebraically closed it is necessary and sufficient that it have no finite field extensions" means "K is algebraically closed if and only if it has no finite extensions". Often used in lists, as in "The following conditions are necessary and sufficient for a field to be algebraically closed...".

need to show (NTS), required to prove (RTP), wish to show, want to show (WTS)

Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desired theorem; thus, one needs to show just these statements.

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.

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.

transport of structure

It is often the case that two objects are shown to be equivalent in some way, and that one of them is endowed with additional structure. Using the equivalence, we may define such a structure on the second object as well, via transport of structure. For example, any two vector spaces of the same dimension are isomorphic; if one of them is given an inner product and if we fix a particular isomorphism, then we may define an inner product on the other space by factoring through the isomorphism.

without (any) loss of generality (WLOG, WOLOG, WALOG), we may assume (WMA), it may be assumed that (WOLOGIMBAT)

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.

Informal proof techniques

Some terms are techniques for the avoidance of rigorous proof, though are not logical fallacies. They suggest the content of a correct proof without supplying it.

back-of-the-envelope calculation

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. Hasse's local-global principle is a particularly influential example of this.

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.

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