Thelanguage of mathematics has a widevocabulary of specialist and technical terms. It also has a certain amount ofjargon: 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 forrigorous arguments or precise ideas. Much of this uses common English words, 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.
Atongue-in-cheek reference tocategory theory, 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 is also known asgeneral abstract nonsense orgeneralized abstract nonsense.
[The paper ofEilenberg andMac Lane (1942)] introduced the very abstract idea of a 'category' — a subject then called 'general abstract nonsense'!
[Grothendieck] raisedalgebraic 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.
A reference to a standard or choice-free presentation of somemathematical 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 thatEuclid's proof is the "canonical proof" ofthe infinitude of primes.
There are two canonical proofs that are always used to show non-mathematicians what a mathematical proof is like:
—The proof that there are infinitely manyprime numbers.
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, theprime number theorem — originally proved using techniques ofcomplex analysis — was once thought to be a deep result untilelementary proofs were found.[1] On the other hand, the fact thatπ is irrational is usually known to be a deep result, because it requires a considerable development ofreal analysis before the proof can be established — even though the claim itself can be stated in terms of simplenumber theory andgeometry.
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, thoughGian-Carlo Rota distinguished betweenelegance of presentation andbeauty of concept, saying that for example, some topics could be written about elegantly although the mathematical content is not beautiful, and sometheorems or proofs are beautiful but may be written about inelegantly.
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 ofprojective 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....
A proof or a result is called "elementary" if it only involves basic concepts and methods in the field, and is to be contrasted withdeep results which require more development within or outside the field. The concept of "elementary proof" is used specifically innumber theory, where it usually refers to a proof that does not resort to methods fromcomplex analysis.
A result is called "folklore" if it is non-obvious and non-published, yet 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.
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.
Similar to "canonical" but more specific, and which makes reference to a description (almost exclusively in the context oftransformations) which holds independently of any choices. Though long used informally, this term has found a formal definition in category theory.
An object behaves pathologically (or, somewhat more broadly used, in adegenerated way) if it either fails to conform to the generic behavior of such objects, fails to satisfy certain context-dependent regularity properties, or simply disobeysmathematical intuition. 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 atriangle havingangles which sum to π radians, a single straight line conforms to this definition pathologically.
Since half a century we have seen arise a crowd of bizarrefunctions 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....
[TheDirichlet 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...isthe one provided by Weierstrass....This function iscontinuous but notdifferentiable.
Note for that latter quote that as the differentiable functions aremeagre in the space of continuous functions, asBanach 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.
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.
An object is well-behaved (in contrast with beingPathological) 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), the term "smooth" can also be used to the same effect.
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.
A shorthand term for "all except for aset ofmeasure zero", when there is ameasure to speak of. For example, "almost allreal numbers aretranscendental" because thealgebraic real numbers form acountablesubset 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 areodd". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously withgeneric, below.
Notions which arise mostly in the context oflimits, referring to the recurrence of a phenomenon as the limit is approached. A statement such as that predicateP is satisfied by arbitrarily large values, can be expressed in more formal notation by∀x : ∃y ≥x :P(y). See alsofrequently. The statement that quantityf(x) depending onx "can be made" arbitrarily large, corresponds to∀y : ∃x :f(x) ≥y.
A shorthand for theuniversal 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. Also much in general-language use among mathematicians: "Of course, this problem can be arbitrarily complicated".
In the context of limits, this is shorthand meaningfor 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 "forsufficiently largex."
A term incategory theory referring to composition ofmorphisms. If for threeobjectsA,B, andC a map can be written as a composition with and, thenf is said tofactor through any (and all) of,, and.
finite
When said of the value of a variable assuming values from the non-negativeextended reals the meaning is usually "not infinite". For example, if thevariance of arandom variable is said to be finite, this implies it is a non-negative real number, possibly zero. In some contexts though, for example in "a small but finite amplitude", zero and infinitesimals are meant to be excluded. When said of the value of a variable assuming values from the extended natural numbers the meaning is simply "not infinite". When said of a set or a mathematicalobject whose main component is a set, it means that thecardinality of the set is less than.
frequently
In the context of limits, this is shorthand forarbitrarily large arguments and its relatives; as witheventually, the intended variant is implicit. As an example, thesequence is frequently in theinterval (1/2, 3/2), because there are arbitrarily largen for which the value of the sequence is in the interval.
formal, formally
Qualifies anything that is sufficiently precise to be translated straightforwardly in aformal system. For example. aformal proof, aformal definition.
This term has similar connotations asalmost all but is used particularly for concepts outside the purview ofmeasure theory. A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if itscomplement satisfies some (context-dependent) notion of smallness. For example, a property which holds on adenseGδ (intersection of countably manyopen sets) is said to hold generically. Inalgebraic geometry, one says that a property of points on analgebraic variety that holds on a denseZariski 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 ofobjects, with an eye towards identifying a unifying principle. This term introduces an "elegant" description which holds for "arbitrary" objects. Exceptions to this description may be mentioned explicitly, as "pathological" cases.
Norbert A'Campo of the University of Basel once asked Grothendieck about something related to thePlatonic 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.
Most often, these refer simply to the left-hand or the right-hand side of anequation; for example, has on the LHS and on the RHS. Occasionally, these are used in the sense oflvalue and rvalue: an RHS is primitive, and an LHS is derivative.
nice
A mathematicalobject is colloquially callednice orsufficiently 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 forpathological. For example, one might conjecture that adifferential operator ought to satisfy a certain boundedness condition "for nice test functions," or one might state that some interestingtopological invariant should be computable "for nicespacesX."
Anything that can be assigned to avariable and for whichequality with another object can be considered. The term was coined when variables began to be used forsets andmathematical structures.
onto
A function (which in mathematics is generally defined as mapping the elements of one setA to elements of anotherB) is called "A ontoB" (instead of "A toB" or "A intoB") only if it issurjective; 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 isreflexive), then the qualificationproper requires the objects to be different. For example, aproper subset of a setS is a subset ofS that is different fromS, and aproperdivisor of a numbern is a divisor ofn that is different fromn. Thisoverloaded word is also non-jargon for aproper morphism.
regular
A function is calledregular if it satisfies satisfactory continuity and differentiability properties, which are often context-dependent. These properties might include possessing a specified number ofderivatives, with the function and its derivatives exhibiting somenice property (seenice above), such asHölder continuity. Informally, this term is sometimes used synonymously withsmooth, below. These imprecise uses of the wordregular are not to be confused with the notion of aregular topological space, which is rigorously defined.
resp.
(Respectively) A convention to shorten parallel expositions. "A (resp.B) [has some relationship to]X (resp.Y)" means thatA [has some relationship to]X and also thatB [has (the same) relationship to]Y. For example,squares (resp. triangles) have 4 sides (resp. 3 sides); orcompact (resp.Lindelöf) spaces are ones where every opencover has a finite (resp. countable) open subcover.
sharp
Often, a mathematical theorem will establish constraints on the behavior of someobject; for example, a function will be shown to have anupper or lower bound. The constraint issharp (sometimesoptimal) if it cannot be made more restrictive without failing in some cases. For example, forarbitrary non-negative real numbersx, theexponential functionex, wheree = 2.7182818..., gives an upper bound on the values of thequadratic functionx2. This is not sharp; the gap between the functions is everywhere at least 1. Among the exponential functions of the form αx, setting α = e2/e = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α3 = 8 < 32. In applied fields the word "tight" is often used with the same meaning.[2]
Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability toanalyticity, 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 bestrong if it deduces restrictive results from general hypotheses. One celebrated example isDonaldson'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 calledstronger 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,Euler's theorem,Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (seesharp above) is a stronger result than a non-sharp one. Finally, the adjectivestrong or the adverbstrongly may be added to a mathematical notion to indicate a related stronger notion; for example, astrong antichain is anantichain satisfying certain additional conditions, and likewise astrongly regular graph is aregular 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").
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 predicateP holds for sufficiently large values, can be expressed in more formal notation by ∃x : ∀y ≥x :P(y). See alsoeventually.
upstairs, downstairs
A descriptive term referring to notation in which twoobjects are written one above the other; the upper one isupstairs and the lower,downstairs. For example, in afiber bundle, the total space is often said to beupstairs, with the base spacedownstairs. In afraction, thenumerator is occasionally referred to asupstairs and thedenominatordownstairs, as in "bringing a term upstairs".
An extension to mathematical discourse of the notions ofmodular arithmetic. A statement is trueup 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 ofequivalence classes, especially incategory theory, where theequivalence relation is (categorical) isomorphism; for example, "The tensor product in a weakmonoidal category is associative and unital up to anatural isomorphism."
vanish
To assume the value 0. For example, "The function sin(x) vanishes for those values ofx that are integer multiples of π." This can also apply to limits: seeVanish at infinity.
Accurately and precisely described or specified. For example, sometimes a definition relies on a choice of someobject; the result of the definition must then be independent of this choice.
The formal language ofproof 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.
In the context of proofs, this phrase is often seen ininduction 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.
A minor variant on "if and only if"; "A isnecessary (and sufficient) forB" means "A if (only if)B". For example, "For afieldK to bealgebraically closed it is necessary and sufficient that it have no finitefield 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, oneneeds to show just these statements.
(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 theHalmos end-of-proof mark, a square sign ∎.
sufficiently nice
A condition onobjects in the scope of the discussion, to be specified later, that will guarantee that some stated property holds for them. Whenworking out 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 asnormal subgroup) are equally useful in practice; one introduces a theorem stating an equivalence of more than two statements with TFAE.
It is often the case that twoobjects 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, viatransport of structure. For example, any twovector spaces of the samedimension areisomorphic; if one of them is given aninner product and if we fix a particular isomorphism, then we may define an inner product on the other space byfactoring through the isomorphism.
LetV be a finite-dimensional vector space overk....Let (ei)1≤i ≤n be abasis forV....There is an isomorphism of thepolynomial algebrak[Tij]1≤i,j ≤n onto thealgebra Symk(V ⊗ V*)....It extends to an isomorphism ofk[GLn] to the localized algebra Symk(V ⊗ V*)D, whereD = det(ei ⊗ ej*)....We writek[GL(V)] for this last algebra. By transport of structure, we obtain alinear algebraic groupGL(V) isomorphic toGLn.
Sometimes aproposition 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.
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 asproof by exhaustion).
by example
Aproof 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 evaluatedby inspection. It is also applied to solving equations; for example to find roots of aquadratic equation by inspection is to 'notice' them, or mentally check them. 'By inspection' can play a kind ofgestalt role: the answer or solution simply clicks into place.
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 usedobvious (French:évident).
[4] Given acommutative diagram of objects and morphisms between them, if one wishes to prove some property of the morphisms (such asinjectivity) which can be stated in terms ofelements, 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, onechases elements around the diagram, or does adiagram chase.
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.
morally true
Used to indicate that the speaker believes a statementshould 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'slocal-global principle is a particularly influential example of this.[citation needed]
Usually applied to a claim within a larger proof when the proof of that claim can be produced routinely by any member of the audience with the necessary expertise, but is not so simple as to beobvious.
Similar toclearly. A concept is trivial if it holds by definition, is an immediatecorollary to a known statement, or is a simple special case of a more general concept.
This section features terms used across different areas inmathematics, or terms that do not typically appear in more specialized glossaries. For the terms used only in some specific areas of mathematics, see glossaries inCategory:Glossaries of mathematics.
1. Acanonical map is a map or morphism between objects that arises naturally from the definition or the construction of the objects being mapped against each other.
2. Acanonical form of an object is some standard or universal way to express the object.
correspondence
Acorrespondence from a set to a set is a subset of a Cartesian product; in other words, it is a binary relation but with the specification of the ambient sets used in the definition.
Afunction is an ordered triple consisting of sets and a subset of the Cartesian product subject to the condition implies. In other words, it is a special kind ofcorrespondence where given an element of, there is a unique element of that corresponds to it.
The word fundamental is used to describe a theorem with a given area of mathematics considered to be the most central theorem of that particular area (e.g.Fundamental Theorem of Arithmetic forArithmetic).
A synonym for afunction between sets or amorphism in a category. Depending on authors, the term "maps" or the term "functions" may be reserved for specific kinds of functions or morphisms (e.g., function as an analytic term and map as a general term).
A "multivalued function” from a setA to a setB is a function fromA to the subsets ofB. It has typically the property that, for almost all pointsx ofB, there is a neighbourhood ofx such that the restriction of the function to the neighbourhood can be considered as a set of functions from the neighbourhood toB.
Aprojection is, roughly, a map from some space or object to another that omits some information on the object or space. For example, is a projection and its restriction to a graph of a function, say, is also a projection. The terms “idempotent operator” and “forgetful map” are also synonyms for a projection.
Jackson, Allyn (2004), "Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck",AMS Notices,51 (9, 10) (PartsI andII).
