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. 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 but 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". 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 ∀x : ∃y ≥ x : P(y). See also frequently. The statement that quantity f(x) depending on x "can be made" arbitrarily large, corresponds to ∀y : ∃x : f(x) ≥ y.
- 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.
- eventually, definitely
- In the context of limits, this is shorthand for sufficiently large arguments; the relevant argument(s) are implicit in the context. As an example, one could say that "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 we have three objects A, B, and C and a map which is written as a composition with and, then f is said to factor through any (and all) of, and .
- finite
- Next to the usual meaning of "not infinite", in another more restrictive meaning that one may encounter, a value being said to be "finite" also excludes infinitesimal values and the value 0. For example, if the variance of a random variable is said to be finite, this implies it is a positive real number.
- frequently
- In the context of limits, this is shorthand for for arbitrarily large arguments and its relatives; as with eventually, the intended variant is implicit. As an example, one could say that "The function sin(x) is frequently zero", where "frequently" means "for arbitrarily large x".
- 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 objects, 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 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 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."
- 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.
- 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, 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) X (resp. Y)" means that A X and also that B Y. For example, squares (resp. triangles) have 4 sides (resp. 3 sides); or compact (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 nonnegative real numbers x, the exponential function ex, where e = 2.7182818..., gives an upper bound on the values of the quadratic function x2. 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.
- 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. An example is the sequence of theorems: Fermat's little theorem, Euler's theorem, Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (see above) is a stronger result than a non-sharp one. Finally, the adjective strong or the adverb strongly may be added to a mathematical notion to indicate a related stronger notion; for example, a strong antichain is an antichain satisfying certain additional conditions, and likewise a strongly regular graph is a regular graph meeting stronger conditions. When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as "antichain").
- sufficiently large, suitably small, sufficiently close
- In the context of limits, these terms refer to some (unspecified, even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as that predicate P holds for sufficiently large values, can be expressed in more formal notation by ∃x : ∀y ≥ x : P(y). 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, 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.
- 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.
Read more about this topic: List Of Mathematical Jargon