The language of mathematics has a wide 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 rigorous 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.
Philosophy of mathematics abstract nonsenseA tongue-in-cheek reference to category 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 as general abstract nonsense or generalized abstract nonsense.
canonicalA 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.
deepA 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 – originally proved using techniques of complex analysis – 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. elegantAn 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.
elementaryA 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. folklore 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.
naturalSimilar 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. pathologicalAn 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. These can be contradictory requirements, and sometimes 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 angles 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-behavedAn object is well-behaved (in contrast with being Pathological) 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.
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 A shorthand term for "all except for a set of measure zero", when there is a measure to speak of, with the phrases almost surely and almost everywhere having related meanings. For example, "almost all real numbers are transcendental" because the algebraic real numbers form a countable 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, such as "almost all prime numbers are odd". 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 A shorthand for the universal 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". eventuallyIn 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 x." factor through A term in category theory referring to composition of morphisms. If for three objects A, B, and C a map can be written as a composition with and , then f is said to factor through any (and all) of , , and . finite When said of the value of a variable assuming values from the non-negative extended reals the meaning is usually "not infinite". For example, if the variance of a random 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 mathematical whose main component is a set, it means that the cardinality of the set is less than . 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 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, a formal definition. generic This term has similar connotations as almost all but is used particularly for concepts outside the purview of measure theory. A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if its complement satisfies some (context-dependent) notion of smallness. For example, a property which holds on a dense Gδ (intersection of countably many open sets) is said to hold generically. In algebraic geometry, one says that a property of points on an algebraic variety 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 , 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.
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, has on the LHS and 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 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 pathological. For example, one might conjecture that a differential operator 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." Anything that can be assigned to a variable and for which equality with another object can be considered. The term was coined when variables began to be used for sets and mathematical structures. 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, 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. A characteristic that a mathematical object may have or not; for example "being positive". Properties are often expressed with formulas and are used for specifying sets and subsets, typically with set-builder notation.