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 (while being unlikely to introduce errors or cause confusion). Abusing notation should be contrasted with "misusing" notation which should be avoided.

Common examples occur when speaking of compound mathematical objects. For example, a topological space consists of a set $T$ and a topology $\backslash mathcal\{T\}$, and two topological spaces $(T,\; \backslash mathcal\{T\})$ and $(T,\; \backslash mathcal\{T\text{'}\})$ can be quite different if they have different topologies. Nevertheless, it is common to refer to such a space simply as $T$ when there is no danger of confusion or when it is implicitly clear what topology is being considered. Similarly, one often refers to a group $(G,\; \backslash star)$ as simply $G$ when the group operation is clear from context.

Advantages

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The new use may achieve clarity in the new area in an unexpected way.

Disadvantages

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The new use may borrow arguments from the old area that do not carry over, creating a false analogy.

Examples

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John Harrison cites "the use of f(x) to represent both application of a function f to an argument x, and the image under f of a subset, x, of f's domain".

Quotation

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"We will occasionally use this arrow notation unless there is no danger of confusion."

(Ronald L. Graham, Rudiments of Ramsey Theory)

See also

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• Mathematical notation

External links

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• Section 2.2: Criticism and reconstruction from "Formalized Mathematics", by John Harrison
• "Strong Symbols", by Henning Thielemann (PDF Slides) Section 5: Common abuse of notation

Mathematical notation | Mathematical terminology

Abuso di notazione