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 to simplify the exposition or to suggest the correct intuition, while possibly minimizing errors and confusion at the same time.

A closely related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is used informally. 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 itself a "representation of G."

Since both mathematical notation and terminology vary across time and context, what is flagged as abuse in one context could be formally correct in another. Furthermore, abuse of notation should not be conflated 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.

ℝ3 is often used to mean 3D Euclidean space taken as a vector space, metric space, or topological space, depending on the properties relevant in context. This is because, for example, the Euclidean distance is understood to be the 'default' metric on X and a topology (ℤ,+,⋅) the ring of integers; by explicitly defining the topology, metric structure, etc. of interest on (X,𝒯′) to distinguish between different topological spaces with the same underlying set.

One may encounter, in many textbooks, sentences such as "Let f for the element 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 quickly becomes pedantic.

Another abuse of notation occurs in sentences such as "Let us consider the function x2+x+1 could formally be denoted f=c, when it would be more correct to write e.g. " since they are more concise and 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.

One example of this is the Cartesian product, which is often seen as associative: y∈F and (x,y)=x and x instead of ∼, then for each [x]. But in practice, if the remainder of the discussion is focused on the equivalence classes rather than the individual elements of the underlying set, then it is common to drop the square brackets in the discussion.

For example, in modular arithmetic, a finite group of order [0],[1],…,[n−1], but in practice they are usually denoted simply as B is almost always an abuse of notation, but not in a category theoretic context, where "More about the languages of math — Suppression of parameters". abstractmath.org.

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