Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time).[1] However, since the concept of formal/syntactical correctness depends on both time and context, certain notations in mathematics that are flagged as abuse in one context could be formally correct in one or more other contexts. Time-dependent abuses of notation may occur when novel notations are introduced to a theory some time before the theory is first formalized; these may be formally corrected by solidifying and/or otherwise improving the theory. Abuse of notation should be contrasted with misuse of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration[2]).

One may encounter, in many textbooks, sentences such as "Let f(x) be a function ...". This is an abuse of notation, as the name of the function is f, and f(x) usually denotes the value of the function f for the element x of its domain. The correct phrase would be "Let f be a function of the variable x ..." or "Let xf(x) be a function ..." This abuse of notation is widely used,[5] as it simplifies the formulation, and the systematic use of a correct notation quickly becomes pedantic.

A similar abuse of notation occurs in sentences such as "Let us consider the function x2 + x + 1...", when in fact x2 + x + 1 is not a function. The function is the operation that associates x2 + x + 1 to x, often denoted as xx2 + x + 1. Nevertheless, this abuse of notation is widely used, since it can help one avoid the pedantry while being generally not confusing.

Many mathematical structures are defined through a characterizing property (often a universal property). Once this desired property is defined, there may be various ways to construct the structure, and the corresponding results are formally different objects, but which have exactly the same properties (i.e., isomorphic). As there is no way to distinguish these isomorphic objects through their properties, it is standard to consider them as equal, even if this is formally wrong.[3]

One example of this is the Cartesian product, which is often seen as associative:

Another example of similar abuses occurs in statements such as "there are two non-Abelian groups of order 8", which more strictly stated means "there are two isomorphism classes of non-Abelian groups of order 8".

For example, in modular arithmetic, a finite group of order n can be formed by partitioning the integers via the equivalence relation "x ~ y if and only if xy (mod n)". The elements of that group would then be [0], [1], …, [n − 1], but in practice they are usually denoted simply as 0, 1, …, n − 1.

Another example is the space of (classes of) measurable functions over a measure space, or classes of Lebesgue integrable functions, where the equivalence relation is equality "almost everywhere".

The terms "abuse of language" and "abuse of notation" depend on context. Writing "f: AB" for a partial function from A to B is almost always an abuse of notation, but not in a category theoretic context, where f can be seen as a morphism in the category of sets and partial functions.