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 (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 
, and two topological spaces
and 
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 
as simply G when the group operation is clear from context.
The new use may achieve clarity in the new area in an unexpected way, but it may borrow arguments from the old area that do not carry over, creating a false analogy.
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."
Examples
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".
The computation of the vector product as the determinant of the matrix
is a significant abuse of notation as 
are treated as scalars but are in fact
vectors.
With Big O notation, we say that some function f "is"
O(g(x)) (given some function g, where x is one of f's parameters). Intuitively this
notation groups functions according to their growth respective to some parameter; as such, it would be appropriate to use the set
membership notation and say that 
. However, the usual notation is f = O(g(x)), despite
the fact that the implied relationship is not symmetric (which the symbol "=" would imply). One reason for this is that, as an
extension of the abuse, it is useful to overload relation symbols such as < and ≤, such that, for example, f <
O(g(x)) means that f's real growth is less than g(x). But this further abuse is not
necessary if, following Knuth one distinguishes between O and the closely related
o and Θ notations.
Another common abuse of notation is to blur the distinction between equality and isomorphism.
Quotation
- "We will occasionally use this arrow notation unless there is no danger of confusion."
(Ronald L. Graham, Rudiments of Ramsey Theory)
See also
External links
- 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
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