The

_{δ} (pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...

. For example, one might conjecture that a ^{x}'', where ''e'' = 2.7182818..., gives an upper bound on the values of the ^{2}. This is not sharp; the gap between the functions is everywhere at least 1. Among the exponential functions of the form α^{''x''}, setting α = ''e''^{2/''e''} = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α^{3} = 8 < 3^{2}. In applied fields the word "tight" is often used with the same meaning.
; 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, stronger: 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. A theorem, result, or condition is further called ''stronger'' than another one if a proof of the second can be easily obtained from the first but not conversely. An example is the sequence of theorems: Fermat's little theorem,

I

an

II

. *. *. *. *. *. * . *. *{{citation , title = The Seventeen Provers of the World , editor-last = Wiedijk , editor-first = Freek , year = 2006 , publisher = Birkhäuser , isbn = 978-3-540-30704-4 , url-access = registration , url = https://archive.org/details/seventeenprovers00free . Jargon

language of mathematics
The language of mathematics or mathematical language is an extension of the natural language (for example English) that is used in mathematics and in science for expressing results (scientific laws, theorems, proofs, logical deductions, etc) wit ...

has a vast vocabulary
A vocabulary is a set of familiar words within a person's language. A vocabulary, usually developed with age, serves as a useful and fundamental tool for communication and acquiring knowledge. Acquiring an extensive vocabulary is one of the ...

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
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as ma ...

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.
Philosophy of mathematics

;abstract nonsense In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, "abs ...

:A tongue-in-cheek reference to category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it's also known as ''general abstract nonsense'' or ''generalized abstract nonsense''.
; canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...

:A reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonical form, or canonical ordering). The same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes.
; deep:A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...

— originally proved using techniques of complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

— was once thought to be a deep result until elementary proofs were found. On the other hand, the fact that π is irrational is usually known to be a deep result, because it requires a considerable development of real analysis before the proof can be established — even though the claim itself can be stated in terms of simple number theory and geometry.
; elegant: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 by providing a technique of proof which is either particularly simple, or which captures the intuition or imagination as to why the result it proves is true. In some occasions, the term "beautiful" can also be used to the same effect, though Gian-Carlo Rota distinguished between ''elegance of presentation'' and ''beauty of concept'', 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.
; elementary
Elementary may refer to:
Arts, entertainment, and media Music
* ''Elementary'' (Cindy Morgan album), 2001
* ''Elementary'' (The End album), 2007
* ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977
Other uses in arts, entertainment, a ...

:A proof or a result is called "elementary" if it only involves basic concepts and methods in the field, and is to be contrasted with deep results which require more development within or outside the field. The concept of "elementary proof" is used specifically in number theory, where it usually refers to a proof that does not resort to methods from complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

.
; folklore :A result is called "folklore" if it is non-obvious, non-published, yet somehow generally known to the specialists within a field. In many scenarios, it is unclear as to who first obtained the result, though if the result is significant, it may eventually find its way into the textbooks, whereupon it ceases to be folklore.
; natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans a ...

:Similar to "canonical" but more specific, and which 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
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...

:An object behaves pathologically (or, somewhat more broadly used, in a ''degenerated'' way) if it either fails to conform to the generic behavior of such objects, fails to satisfy certain context-dependent regularity properties, or simply disobeys mathematical intuition
Logical Intuition, or mathematical intuition or rational intuition, is a series of instinctive foresight, know-how, and savviness often associated with the ability to perceive logical or mathematical truth—and the ability to solve mathematical c ...

. In many occasions, these can be and often are contradictory requirements, while in other occasions, the term is more deliberately used to refer to an object artificially constructed as a counterexample to these properties. A simple example is that from the definition of a triangle having angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...

s which sum to π radians, a single straight line conforms to this definition pathologically.
:Note for that latter quote that as the differentiable functions are meagre in the space of continuous functions, as Banach found out in 1931, differentiable functions are colloquially speaking a rare exception among the continuous ones. Thus it can hardly be defended any-more to call non-differentiable continuous functions pathological.
; rigor (rigour):The act of establishing a mathematical result using indisputable logic, rather than informal descriptive argument. Rigor is a cornerstone quality of mathematics, and can play an important role in preventing mathematics from degenerating into fallacies.
; well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...

:An object is well-behaved (in contrast with being ''pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...

'') if it satisfies certain prevailing regularity properties, or if it conforms to mathematical intuition (even though intuition can often suggest opposite behaviors as well). In some occasions (e.g., analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

), the term " smooth''"'' can also be used to the same effect.
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 mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...

: A shorthand term for "all except for a set of measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...

", when there is a measure to speak of. For example, "almost all real numbers are transcendental" because the algebraic real numbers form a countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...

subset of the real numbers with measure zero. One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing ...

". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with ''generic'', below.
; arbitrarily large: 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 . See also ''frequently''. The statement that quantity ''f''(''x'') depending on ''x'' "can be made" arbitrarily large, corresponds to .
; arbitrary
Arbitrariness is the quality of being "determined by chance, whim, or impulse, and not by necessity, reason, or principle". It is also used to refer to a choice made without any specific criterion or restraint.
Arbitrary decisions are not necess ...

: A shorthand for the universal quantifier
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...

. 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. Also much in general-language use among mathematicians: "Of course, this problem can be arbitrarily complicated".
; eventually:In the context of limits, this is shorthand meaning ''for sufficiently large arguments''; the relevant argument(s) are implicit in the context. As an example, the function log(log(''x'')) ''eventually'' becomes larger than 100"; in this context, "eventually" means "for sufficiently large
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have p ...

''x''."
; factor through: A term in category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

referring to composition of morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...

s. If for three objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...

''A'', ''B'', and ''C'' a map $f\; \backslash colon\; A\; \backslash to\; C$ can be written as a composition $f\; =\; h\; \backslash circ\; g$ with $g\; \backslash colon\; A\; \backslash to\; B$ and $h\; \backslash colon\; B\; \backslash to\; C$, then ''f'' is said to ''factor through'' any (and all) of $B$, $g$, and $h$.
; finite: "Not infinite". For example, if the variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

of a random variable is said to be finite, this implies it is a non-negative real number.
; frequently: In the context of limits, this is shorthand for '' arbitrarily large arguments'' and its relatives; as with ''eventually'', the intended variant is implicit. As an example, the sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

$(-1)^n$ is frequently in the interval (1/2, 3/2), because there are arbitrarily large ''n'' for which the value of the sequence is in the interval.
;
; formal, formally: Qualifies anything that is sufficiently precise to be translated straightforwardly in a formal system. For example. a formal proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the seq ...

, a formal definition.
;
; generic: This term has similar connotations as ''almost all'' but is used particularly for concepts outside the purview of measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many s ...

. A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if its complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...

satisfies some (context-dependent) notion of smallness. For example, a property which holds on a dense Gintersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...

of countably many open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...

s) is said to hold generically. In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, one says that a property of points on an algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...

that holds on a dense Zariski open set is true generically; however, it is usually not said that a property which holds merely on a dense set (which is not Zariski open) is generic in this situation.
; 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. This term introduces an "elegant" description which holds for "arbitrary
Arbitrariness is the quality of being "determined by chance, whim, or impulse, and not by necessity, reason, or principle". It is also used to refer to a choice made without any specific criterion or restraint.
Arbitrary decisions are not necess ...

" objects. Exceptions to this description may be mentioned explicitly, as "pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...

" cases.
; 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
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

; 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.
; nice: A mathematical object is colloquially called ''nice'' or ''sufficiently nice'' if it satisfies hypotheses or properties, sometimes unspecified or even unknown, that are especially desirable in a given context. It is an informal antonym for differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retu ...

ought to satisfy a certain boundedness condition "for nice test functions," or one might state that some interesting topological invariant should be computable "for nice spaces ''X''."
; onto: A function (which in mathematics is generally defined as mapping the elements of one set ''A'' to elements of another ''B'') is called "''A'' onto ''B''" (instead of "''A'' to ''B''" or "''A'' into ''B''") only if it is surjective; it may even be said that "''f'' is onto" (i. e. surjective). Not translatable (without circumlocutions) to some languages other than English.
; proper: If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is reflexive), then the qualification ''proper'' requires the objects to be different. For example, a ''proper'' subset of a set ''S'' is a subset of ''S'' that is different from ''S'', and a ''proper'' divisor of a number ''n'' is a divisor of ''n'' that is different from ''n''. This overloaded word is also non-jargon for a proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ...

.
; regular : A function is called ''regular'' if it satisfies satisfactory continuity and differentiability properties, which are often context-dependent. These properties might include possessing a specified number of derivatives, with the function and its derivatives exhibiting some ''nice'' property (see ''nice'' above), such as Hölder continuity. Informally, this term is sometimes used synonymously with ''smooth'', below. These imprecise uses of the word ''regular'' are not to be confused with the notion of a regular topological space, which is rigorously defined.
; resp.: (Respectively) A convention to shorten parallel expositions. "''A'' (resp. ''B'') as some relationship to''X'' (resp. ''Y'')" means that ''A'' as some relationship to''X'' and also that ''B'' as (the same) relationship to''Y''. For example, square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ...

s (resp. triangles) have 4 sides (resp. 3 sides); or compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...

(resp. Lindelöf) spaces are ones where every open cover has a finite (resp. countable) open subcover.
; 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'' (sometimes ''optimal'') if it cannot be made more restrictive without failing in some cases. For example, for arbitrary
Arbitrariness is the quality of being "determined by chance, whim, or impulse, and not by necessity, reason, or principle". It is also used to refer to a choice made without any specific criterion or restraint.
Arbitrary decisions are not necess ...

non-negative real numbers ''x'', the exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

''equadratic function
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomi ...

''x''Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congr ...

, Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (see ''sharp'' 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
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
The size of the largest antichain in a partially ordered set is known as its wid ...

satisfying certain additional conditions, and likewise a strongly regular graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that:
* Every two adjacent vertices have co ...

is a regular graph
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegre ...

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
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have p ...

, 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''). See also ''eventually''.
; 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
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and ...

, the total space is often said to be ''upstairs'', with the base space ''downstairs''. In a fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

, the numerator is occasionally referred to as ''upstairs'' and the denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

''downstairs'', as in "bringing a term upstairs".
; up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...

, modulo, mod out by: An extension to mathematical discourse of the notions of modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...

. A statement is true ''up to'' a condition if the establishment of that condition is the only impediment to the truth of the statement. Also used when working with members of equivalence classes
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

, especially in category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, where the equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...

is (categorical) isomorphism; for example, "The tensor product in a weak monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left a ...

is associative and unital up to a natural isomorphism
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...

."
; vanish: To assume the value 0. For example, "The function sin(''x'') vanishes for those values of ''x'' that are integer multiples of π." This can also apply to limits: see Vanish at infinity.
; weak, weaker: The converse of strong.
; well-defined: Accurately and precisely described or specified. For example, sometimes a definition relies on a choice of some object; the result of the definition must then be independent of this choice.
Proof terminology

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 is used to announce to the reader an alternative method, or proof of a result. In a proof, it therefore flags a piece of reasoning that is superfluous from a logical point of view, but has some other interest. ; by way of contradiction (BWOC), or "for, if not, ...": The rhetorical prelude to aproof by contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also know ...

, preceding the negation of the statement to be proved.
; if and only if (iff): An abbreviation for logical equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending ...

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"; "''A'' is ''necessary'' (''sufficient'') for ''B''" means "''A'' if (only if) ''B''". For example, "For a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a gras ...

''K'' to be algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, becau ...

it is necessary and sufficient that it have no finite field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield o ...

s" 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: A statement of the existence and uniqueness of an object; the object exists, and furthermore, no other such object exists.
; Q.E.D.
Q.E.D. or QED is an initialism of the Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in pri ...

: (''Quod erat demonstrandum''): A Latin abbreviation, meaning "which was to be demonstrated", historically placed at the end of proofs, but less common currently, having been supplanted by the Halmos end-of-proof mark, a square sign ∎.
; sufficiently nice: A condition on objects in the scope of the discussion, to be specified later, that will guarantee that some stated property holds for them. When working out
Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness.
It is performed for various reasons, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic ...

a theorem, the use of this expression in the statement of the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intent is to collect the conditions that will be found to be needed in order for the proof of the theorem to go through.
; the following are equivalent (TFAE): Often several equivalent conditions (especially for a definition, such as normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...

) are equally useful in practice; one introduces a theorem stating an equivalence of more than two statements with TFAE.
; 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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

are isomorphic; if one of them is given an inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

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): Sometimes a proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the ...

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 techniques

Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for filling in tedious details. ; angle chasing: Used to describe a geometrical proof that involves finding relationships between the various angles in a diagram. ;back-of-the-envelope calculation
A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic o ...

: An informal computation omitting much rigor without sacrificing correctness. Often this computation is "proof of concept" and treats only an accessible special case.
; brute force: Rather than finding underlying principles or patterns, this is a method where one would evaluate as many cases as needed to sufficiently prove or provide convincing evidence that the thing in question is true. Sometimes this involves evaluating every possible case (where it is also known as proof by exhaustion
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equiv ...

).
; by example: A ''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.
; 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. If an expression can be evaluated by straightforward application of simple techniques and without recourse to extended calculation or general theory, then it can be evaluated ''by inspection''. It is also applied to solving equations; for example to find roots of a quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quad ...

by inspection is to 'notice' them, or mentally check them. 'By inspection' can play a kind of '' gestalt'' role: the answer or solution simply clicks into place.
; by intimidation: Style of proof where claims believed by the author to be easily verifiable are labelled as 'obvious' or 'trivial', which often results in the reader being confused.
; 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
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summ ...

used ''obvious'' (French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...

: ''évident'').
; complete intuition : commonly reserved for jokes (puns on complete induction).
; diagram chasing
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...

:Numerous examples can be found in , for example on p. 100. Given a commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...

of objects and morphisms between them, if one wishes to prove some property of the morphisms (such as injectivity
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposit ...

) 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''.
; handwaving
Hand-waving (with various spellings) is a pejorative label for attempting to be seen as effective – in word, reasoning, or deed – while actually doing nothing effective or substantial. Cites the ''Random House Dictionary'' and ''The Dictionary ...

: 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.
; index battle: for proofs involving objects with multiple indices which can be solved by going to the bottom (if anyone wishes to take up the effort). Similar to diagram chasing.
; left as an exercise to the student: Usually reserved for shortcuts which can be ''clearly'' filled-in by any member of the audience with the necessary expertise, but are not so ''trivial'' as to be solvable ''by inspection''.
; trivial: Similar to ''clearly''. A concept is trivial if it holds by definition, is an immediate corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving anothe ...

to a known statement, or is a simple special case of a more general concept.
See also

* Glossary of mathematicsNotes

References

*. * . * (PartI

an

II

. *. *. *. *. *. * . *. *{{citation , title = The Seventeen Provers of the World , editor-last = Wiedijk , editor-first = Freek , year = 2006 , publisher = Birkhäuser , isbn = 978-3-540-30704-4 , url-access = registration , url = https://archive.org/details/seventeenprovers00free . Jargon