Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion). However, the concept of formal correctness depends on time and on the context. Therefore, many notations in mathematics are qualified as abuse of notation in some context and are formally correct in other contexts; as many notations were introduced a long time before any formalization of the theory in which they are used, the qualification of abuse of notation is strongly time dependent. Moreover, many abuses of notation may be made formally correct by improving the theory. Abuse of notation should be contrasted with misuse of notation, which should be avoided.

A related concept is abuse of language or abuse of terminology, when not notation but a term is misused. Abuse of language is an almost synonymous expression that is usually used for non-notational abuses. For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V "a representation of G". A common abuse of language consists in identifying two mathematical objects that are different but canonically isomorphic. Examples include identifying a constant function and its value or identifying to $\mathbb {R} ^{3}$ the Euclidean space of dimension three equipped with a Cartesian coordinate system.

Examples

Structured mathematical objects

Many mathematical objects consist of a set, often called the underlying set, equipped with some additional structure, typically a mathematical operation or a topology. It is a common abuse of notation to use the same notation for the underlying set and the structured object. For example, ${\mathbb {Z}}$ may denote the set of the integers, the group of integers together with addition, or the ring of integers with addition and multiplication. In general, there is no problem with this, and avoiding such an abuse of notation would make mathematical texts pedantic and difficult to read. When this abuse of notation may be confusing, one may distinguish between these structures by denoting $({\mathbb {Z}},+)$ the group of integers with addition, and $({\mathbb {Z}},+,\cdot )$ the ring of integers.

Similarly, a topological space consists of a set $X$ (the underlying set ) and a topology ${\mathcal {T}},$ which is characterized by a set of subsets of $X$ (the open sets). Most frequently, one considers only one topology on $X$ , and there is no problem to denote by $X$ both the underlying set, and the pair consisting of $X$ and its topology ${\mathcal {T}},$ although they are different mathematical objects. Nevertheless, it occurs sometimes that two different topologies are considered simultaneously on the same set; for distinguishing the corresponding topological spaces, one must use notation such as $(X,{\mathcal {T}})$ and $(X,{\mathcal {T}}').$

Functional notation

One encounters, 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 $f (x)$ denotes normally 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 $x \mapsto f (x)$ be a function ..." This abuse of notation is widely used, as it simplifies the formulation, and the systematic use of a correct notation becomes quickly pedantic.

A similar abuse of notation occurs in sentences such as "Let us consider the function $x 2 + x + 1$ ..." In fact $x 2 + x + 1$ is not a function. The function is the operation that associates $x 2 + x + 1$ to $x$ , often denoted as $x \mapsto x 2 + x + 1$ . Nevertheless, this abuse of notation is widely used since it is generally not confusing.

Equality vs. isomorphism

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 – they are 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.

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

(E \times F) \times G = E \times (F \times G) = E \times F \times G

But this is not strictly true: if $x \in E$ , $y \in F$ and $z \in G$ , the identity $((x, y), z) = (x, (y, z))$ would imply that $(x, y) = x$ and $z = (y, z)$ , and so $((x, y), z) = (x, y, z)$ would mean nothing.

This notion can be made rigorous in category theory, using the idea of a natural isomorphism.

Equivalence classes

Referring to an equivalence class of an equivalence relation by x instead of [x] is an abuse of notation. Formally, if a set X is partitioned by an equivalence relation ~, then for each x ∈ X, the equivalence class {y ∈ X | y ~ x} is denoted [x]. But in practice, if the remainder of the discussion is focused on equivalence classes rather than individual elements of the underlying set, it is common to drop the square brackets in the discussion.

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 x ≡ y (mod n). The elements of that group would then be [0], [1], …, [n − 1], but in practice they are usually just denoted 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".

Subjectivity

The terms "abuse of language" and "abuse of notation" depend on context. Writing "f: A → B" 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.

References

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