Abuse of notation

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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). Abuse of notation should be contrasted with misuse of 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 $\mathcal{T}$ , and two topological spaces $(T, \mathcal{T})$ and $(T, \mathcal{T'})$ 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 $(G, \star)$ as simply $G$ when the group operation is clear from context.

In standard analysis, another example is in the Leibniz notation for the derivative $\frac{dy}{dx}$ . Although the derivative is not strictly a fraction, abusing this notation leads to the correct chain rule $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ . (This is valid in non-standard analysis however.) Often good notation is judged by whether or not its abuses will lead to correct interpretations.

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."

1 Examples
2 Quotation
3 See also
4 References
5 External links

[edit] Examples

John Harrison (1996) 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

$\mathbf{a}\times\mathbf{b}=\det \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{bmatrix}$

is a significant abuse of notation as $\mathbf{i},\mathbf{j},\mathbf{k}$ are treated as scalars but are in fact vectors.

With Big O notation, we say that some term $f (x)$ "is" $O (g (x))$ (given some function g, where x is one of f's parameters). Example: "Runtime of the algorithm is $O (n 2)$ " or in symbols " $T (n) = O (n 2)$ ". Intuitively this notation groups functions according to their growth respective to some parameter; as such, the notation is abuse with respect to two aspects: It abuses "=" and it invokes terms of real numbers instead of function terms. It would be appropriate to use the set membership notation and thus write $f(n)\in O(g(n))$ instead of $f (n) = O (g (n))$ . This would allow for common set operations like $O(n\cdot\log n) \subset O(n^2)$ , $O(2^n) \bigcup O(n^2)$ , and it would make clear, that the relation is not symmetric in contrast to what the "=" symbol suggests. Some argue for "=", because as an extension of the abuse, it could be useful to overload relation symbols such as < and ≤, such that, for example, $f < O (g)$ means that f's real growth is less than $g$ . But this further abuse is not necessary if, following Knuth one distinguishes between O and the closely related o and Θ notations. Concerning the use of terms for scalar numbers instead of functions you run into the following troubles: First, you cannot cleanly define, what $f(n)\in O(g(n))$ may mean. The O notation is about growth of functions, but to the left hand and the right hand side of the relation there are scalar values, and you cannot decide, whether the relation holds if you look at particular function values. Second, the abused O notation is bound to one variable, and you have problems identifying the right one. E.g. in $O (n m)$ one of the variables might be a parameter, that is not in scope of the $O$ . That is you might mean $O (2 m)$ , since $n$ was the parameter that you assigned 2, or you might mean $O (n 3)$ , since $m$ was the parameter substituted by 3 here. Even $O (c)$ might be the same as $O (1)$ , since $c$ might be a parameter, not the concerned function variable. The carelessness regarding the use of function terms might be caused by the yet rarely used functional notations, like Lambda notation. You would have to write $(n\mapsto n\cdot\log n) \in O(n\mapsto n^2)$ and $O(n\mapsto n\cdot\log n) \subset O(n\mapsto n^2)$ . The correct O notation can be easily extended to complexity functions which map tuples to complexities and that lets you formulate a statement like "the graph algorithm needs time proportional to the logarithm of the number of edges and to the number of vertices" by $T_{\mbox{graph}}\in O((v,e)\mapsto v\cdot\log e)$ , which is not possible with the abused notation. You can also state theorems like $O (f)$ is a convex cone and use that for formal reasoning.

Another common abuse of notation is to blur the distinction between equality and isomorphism. For instance, in the construction of the real numbers from Dedekind cuts of rational numbers, the rational number r is identified with the set of all rational numbers less than r, even though the two are obviously not the same thing (as one is a rational number and the other is a set of rational numbers). However, this ambiguity is tolerated, because the set of rational numbers and the set of Dedekind cuts of the form {x: x<r} have the same structure. It is through this abuse of notation that Q is regarded as a subset of R.

The term "abuse of language" frequently appears in the writings of Nicolas Bourbaki:

We have made a particular effort always to use rigorously correct language, without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses of language, without which any mathematical text runs the risk of pedantry, not to say unreadability. Bourbaki (1988).

For example:

Let E be a set. A mapping f of E × E into E is called a law of composition on E. [...] By an abuse of language, a mapping of a subset of E × E into E is sometimes called a law of composition not everywhere defined on E. Bourbaki (1988).

In other words, it is an abuse of language to refer to partial functions from E × E to E as "functions from E × E to E that are not everywhere defined." To clarify this, it makes sense to compare the following two sentences.

1. A partial function from A to B is a function f: A' → B, where A' is a subset of A.

2. A function not everywhere defined from A to B is a function f: A' → B, where A' is a subset of A.

Strictly speaking, even the term "partial function" could arguably be called an abuse of language, because a partial function is not a function that is partial. (Whereas a continuous function is a function that is continuous.) But the use of adjectives (and adverbs) in this way is common practice, although it can occasionally be confusing. Some adjectives, such as "generalized", can only be used in this way. (e.g., a magma is a generalized group.)

The words "not everywhere defined", however, form a relative clause. Since in mathematics relative clauses are rarely used to generalize a noun, this is considered an abuse of language. As mentioned above, this does not imply that such a term should not be used; although in this case perhaps "function not necessarily everywhere defined" would give a better idea of what is meant, and "partial function" is clearly the best option in most contexts.

Using the term "continuous function not everywhere defined" after having defined only "continuous function" and "function not everywhere defined" is not an example of abuse of language. In fact, as there are several reasonable definitions for this term, this would be an example of woolly thinking or a cryptic writing style.

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 partial functions.

[edit] Quotation

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

(Ronald L. Graham, Rudiments of Ramsey Theory)

[edit] See also

Mathematical notation

[edit] References

Bourbaki, Nicolas (1988). Algebra I: Chapters 1-3, Elements of Mathematics. Springer.
Harrison, John (1996). "2.2 Criticism and reconstruction", Formalized Mathematics, Technical Reports 36. Turku Centre for Computer Science. ISBN 951-650-813-8.

[edit] External links

"Strong Symbols", by Henning Thielemann (PDF Slides) Section 5: Common abuse of notation

Categories: Mathematical notation | Mathematical terminology

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Abuse of notation

From Wikipedia, the free encyclopedia

Contents

[edit] Examples

[edit] Quotation

[edit] See also

[edit] References

[edit] External links

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