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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... A vocabulary is a set of words known to a person or other entity, or that are part of a specific language. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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.

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).

Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Canonical is an adjective derived from canon. ... Euclid, is also referred to as Euclid of Alexandria, (Greek: , 330 BC – 275 BC), a Greek mathematician, who lived in the city of Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC), is often considered to be the father of geometry. His most popular work, Elements... Most mathematicians derive aesthetic pleasure from their work, and from mathematics in general. ... Gian-Carlo Rota (April 27, 1932 – April 18, 1999, known as Juan Carlos Rota to Spanish speakers) was an Italian-born American mathematician and philosopher. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ... In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ... Look up Rigour in Wiktionary, the free dictionary. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ...

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 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.

factor through

A term in category theory referring to composition of functions. If we have three objects A, B, C, and two maps, and , then we say that f factors through g if there exists some map such that . Likewise, if h rather than g is given first, we say that f factors through h.

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.

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.

sufficiently large, suitably small, sufficiently close

In the context of limits, these terms refer to phenomena which prevail as the limit is approached.

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.

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.

In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the phrase arbitrarily large is used in contexts such as: is true for arbitrarily large which is actually shorthand for: for every , is true for at least one . ... Choices and actions are considered to be arbitrary when they are done not by means of any underlying principle or logic, but by whim or some decidedly illogical formula. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... For heuristics in computer science, see heuristic (computer science) Heuristic is the art and science of discovery and invention. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... In mathematics, LHS is informal shorthand for the left-hand side of an equation. ... In computer science, a value may be a number, literal string, array and anything that can be treated as if it were a number. ... In computer science, a value may be a number, literal string, array and anything that can be treated as if it were a number. ... In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In mathematics, the phrase sufficiently large is used in contexts such as: is true for sufficiently large which is actually shorthand for: there exists an such that is true for all . ... In mathematics, the term transport of structure is a descriptive shorthand for the process of defining additional structure on an object by comparing it to another object on which such a structure already exists. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... A cake divided into four equal quarters. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

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. In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ...

aliter

An obsolescent term which refers to an alternative method of proof.

diagram chasing

An argument to show that a diagram is commutative by tracking the value of an element through compositions in the diagram.

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...".

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), 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.

Wikipedia does not yet have an article with this exact name. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... This is a page about mathematics. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ... A principle, especially in software development, that values avoiding the hazards of redundancy. ... Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument... Q.E.D. is an abbreviation of the Latin phrase (literally, which was to be demonstrated). In simple terms, the use of this Latin phrase is to indicate that something has been definitively proven. ... Without loss of generality or simply WLOG is a frequently used expression in mathematics. ...

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. A logical fallacy is an error in logical argument which is independent of the truth of the premises. ...

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.