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.
Contents
- Structured mathematical objects
- Functional notation
- Equality vs isomorphism
- Equivalence classes
- Subjectivity
- References
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
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,
Similarly, a topological space consists of a set X (the underlying set ) and a topology
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 ↦ 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 x2 + x + 1..." In fact x2 + x + 1 is not a function. The function is the operation that associates x2 + x + 1 to x, often denoted as x ↦ x2 + 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:
But this is not strictly true: if
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.