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 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.

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 $\mathbb {R} ^{3}$ the Euclidean space of dimension three equipped with a Cartesian coordinate system.

Examples

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, ${\displaystyle {\mathbb {Z}}}$ may denote the set of the integers, the group of integers together with addition, or the ring of integers with addition and multiplication. In general, there is no problem with this, and avoiding such an abuse of notation would make mathematical texts pedantic and difficult to read. When this abuse of notation may be confusing, one may distinguish between these structures by denoting ${\displaystyle ({\mathbb {Z}},+)}$ the group of integers with addition, and ${\displaystyle ({\mathbb {Z}},+,\cdot )}$ the ring of integers.

Similarly, a topological space consists of a set X (the underlying set ) and a topology ${\displaystyle {\mathcal {T}},}$ which is characterized by a set of subsets of X (the open sets). Most frequently, one considers only one topology on X, and there is no problem to denote by X both the underlying set, and the pair consisting of X and its topology ${\displaystyle {\mathcal {T}},}$ although they are different mathematical objects. Nevertheless, it occurs sometimes that two different topologies are considered simultaneously on the same set; for distinguishing the corresponding topological spaces, one must use notations such as $(X,{\mathcal {T}})$ and ${\displaystyle (X,{\mathcal {T}}').}$

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 as, generally, it is not confusing. However, in most computer algebra systems, expressions are distinct from functions, and the habit of this abuse of notation leads many beginners in computer algebra to make erroneous computations.

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, which have exactly the same properties: they are isomorphic). As there is no way for distinguishing these isomorphic object through their properties, it is standard to consider them as equal, even if this is formally wrong.

Some of the most common examples are detailed in next subsections.

Integers

The set of integers is generally defined as a set containing the natural numbers, in which subtraction is defined for every pair of elements, and such that every integer is the difference of two natural numbers. Two sets that have these properties are isomorphic in the sense that there is a unique bijection between them, which fixes the natural numbers and is compatible with addition (and subtraction).

There are two standard ways for constructing integers. An integer may be defined being either 0 or a pair (s, n), where n is a nonzero natural number, and s is + or −; this pair is standardly denoted +n or −n. Integers may also be defined as equivalence classes of pairs of natural numbers, for the equivalence relation

${\displaystyle (a,b)\sim (c,d)\Longleftrightarrow a+d=b+c}$

With these definitions, the integer +3 (for example) is a pair, and is therefore different from the equivalence class of (6, 3), which is an infinite set, and from the natural number 3. Thus, writing that these three objects are equal is an abuse of notation, which it is so common that there is not any standard notation for the equivalence class, and that writing

3 = +3

is rarely identified as an abuse of notation.

However, in computer programming, natural numbers, generally called unsigned integers, must be distinguished from the nonnegative integers, also called nonnegative signed integers, as passing from one to the other requires a type conversion.

Rational numbers

Commonly, Rational numbers are formally defined as equivalence classes of pairs (a, b) of integers (such that b ≠ 0) for the equivalence relation

${\displaystyle (a,b)\sim (c,d)\Longleftrightarrow ad=bc.}$

The equivalence class of the pair (a, b) is denoted a/b. It follows that writing, for example, 3/2 = 6/4 is not an abuse of notation, as the two members of this equation, although written differently, denote the same equivalence class.

On the other hand, 3/1 is not formally an integer, as it is a set of pairs of integers. It follows that the commonly written equality

${\displaystyle {\frac {3}{1}}=3}$

is an abuse of notation, consisting in identifying integers with fractions having 1 as a denominator. There is generally no problem with this identification and the resulting abuse of notation, as the map ${\displaystyle n\to {\frac {n}{1}}}$ is a ring isomorphism of the integers onto the image of this map, and is the unique ring homomorphism of the integers into the rational numbers.

Real numbers

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.

Cartesian product as associative

The Cartesian product is often seen as associative, with:

$(E \times F) \times G = E \times (F \times G) = E \times F \times G$

This of course cannot be rigorously true: if $x \in E$, $y \in F$ and $z \in G$, the identity $((x, y), z) = (x, (y, z))$ would imply that $(x, y) = x$ and $z = (y, z)$, and so $((x, y), z) = (x, y, z)$ would mean nothing.

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".

Del operator

The del operator, denoted by $\nabla$, is a tuple of partial derivative operators posing as a vector. This suggests notations such as $\nabla f$ for gradient, $\nabla \cdot \vec v$ for divergence, and $\nabla \times \vec v$ for curl. The notation is extremely convenient because $\nabla$ does behave like a vector most of the time. But it can be regarded as an abuse because $\nabla$ doesn't commute with vectors, and so doesn't satisfy all properties of vectors.

(A contrary view is that notation is not abused if one does not think of $\nabla$ as a vector. The vector-like notations are simply specially defined uses of the dot and cross.)

Trigonometric functions

In some countries it is common to denote the square of the value of $\sin(x)$ as $\sin^2(x)$, and the inverse function as $\sin^{-1}(x)$. In his article on notation in the Edinburgh Encyclopedia Charles Babbage complains at length of this abuse of notation, and suggests two alternatives for the notation $f^{n}(x)$

• Function composition, i.e. $f^2(x) = f(f(x))$ and $f^{-1}(x)$ is the inverse.
• Powers of the value, i.e. ${\displaystyle f^{2}(x)=f(x)^{2}}$ and ${\displaystyle f(x)^{-1}={\frac {1}{f(x)}}}$ is the reciprocal.

Babbage argues strongly for the former, and also that the square of the value should be notated as ${\displaystyle \sin x^{2}}$, but beware: Babbage intends ${\displaystyle (\sin x)^{2}}$ even though what he wrote is easily confused with ${\displaystyle \sin(x^{2})}$ (the only non-confusing way to avoid this abuse of notation is to always include the parentheses).

To press his example further, Babbage investigates what the function $\sin^2(x) = \sin(\sin x)$ is like, and also ${\displaystyle \sin ^{\frac {1}{2}}(x)}$, which is the function that, when composed with itself, equals $\sin(x)$, the functional square root.

Dirac delta function

The Dirac delta function cannot be interpreted as a function in classical analysis. However it is often treated as one, for example when calculating convolutions. Treating the Dirac delta "function" as a function lets the user save traditional limit notation, saving its visual clutter.

Values of a random variable

In probability theory, conventional methods of indicating the probability of a value of a random variable are abuse of notation in two ways: Writing $P(x)$ instead of $P(X = x)$ leaves out the identity of the random variable (here $X$), which can be confusing out of context. However, even when writing $P(X = x)$, there is a mismatch of types: the expression $X=x$ is an equation and from a type theory point of view has type boolean; that is, it evaluates to either "true" $T$ or "false" $F$. The domain of the $P$ function here is not $\{T,F\}$, though; instead $P$ should be logically thought of as taking two arguments: a random variable $X$ and a subset of that random variable $X$'s sample space $\Omega$. This is important: if one were to implement $P$ in a computer algebra system one would need to give it two arguments (and not only one boolean one), just like an implementation of the summation symbol $\sum_{i=min}^{max}f(i)$ is really a function of the form $F(f,i,min,max)$, not $F(f,truthvalue,max)$. So a logically more appropriate notation could be $P(X, \{x\})$ (the second argument here is the set of values we consider for $X$) or (borrowing from analysis, since the value set contains only the single element $x$ in this case) $P(\left.X\right|_{x})$, but everybody writes $P(X = x)$ or (abbreviated) $P(x)$.

There is a good reason for such widespread so-called abuse: Notational abuse is a matter of perspective. Despite the arguably suggestive manner in which it is written, the notation $P(\phi)$ does not (and is not meant to) mean applying some function to some value. Instead, the meaning is that $P(\phi)$ takes the entire expression $\phi$ as input --- not evaluated --- and expands into a particular, longer, expression in a (nominally) simpler language. Specifically, the notation can be defined by expanding to measure theory and set-builder notation as in (roughly):

$P(\phi) = P_\mu(\phi) = P_{\Omega,\mu}(\phi) = \frac{\mu(\{w \mid \phi(w) \text{ and } w\in\Omega\})}{\mu(\{w \mid w\in\Omega\})}$

In words: To compute the probability of a formula being true, build the set of all possible worlds in which the formula is true, measure that set, and finally divide that by the measure of the set of all possible worlds. There are, naturally, a number of other, better, ways to define the notation. That which matters here is just to recognize that the notation is no more abusive than some abbreviation ultimately resting on top of set-builder notation. (Whether we consider set-builder notation to be rigorous is another matter entirely.)

Regarding the computer science perspective: $P(\phi)$ can be --- directly --- implemented on a computer as a macro. (The abbreviations can be supported by default parameters, fields, closures, environments, global variables, and so forth.) That implementation is awkward in applicative-order evaluation, as initially sketched, but simple in normal-order evaluation, as just sketched, directly indicates that the concept is primarily about syntax.

So regarding $P(X=x)$, while it can be called abusive, it can also be said to exemplify proper use of notation: it is a primitive of the language of probability theory (so is "notation"), that has been shown to rigorously reduce to the language of set theory (so is "proper").

A perhaps uncontroversial example of abuse in probability theory is to take $P(X)$ as meaning the marginal distribution of random variable $X$, and, at the same time, to declare that $P(X=x)$ means a number. At face value this seems legitimate, and it could perhaps be kept that way, but for the fact that probability theorists permit any sort of expression inside the $P()$. So, what would $P(Z)$ mean, where $Z$ is a non-basic random variable (deterministically) defined by $Z=(X=x)$? That is, $Z$ is true when random variable $X$ equals our favorite value, $x$, and in all other cases is false.

Given that $Z=(X=x)$ then one concludes that $P(Z)=P(X=x)$ ought to hold. However, the left-hand side is supposed to mean a distribution, while the right hand side is supposed to mean a number. Distributions and numbers are not, of course, equal to one another, so contradiction ensues if we try to rigorously support both conventions at the same time.

The resolution is to call one convention the definition and the other the abuse. If we take $P(X=x)$ meaning a number as the abuse, then the abuse is more specifically that we implicitly typecast a marginal distribution over a Boolean random variable down to its probability of being true. If we take $P(X)$ meaning an entire distribution as the abuse, then the abuse is more specifically that we implicitly surround the expression with quantifiers ranging over all possible values of $X$ (so as to form its entire marginal distribution one entry at a time).

Bourbaki

The term "abuse of language" frequently appears in the writings of Nicolas Bourbaki:^[1]

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.

If one were to be extremely pedantic, one could say that even the term "partial function" could be called an abuse of language, because a partial function is not a function. (Whereas a continuous function is a function that is continuous.) But the use of adjectives (and adverbs) in this way is standard English 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 might be 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.

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.

See also

• Mathematical notation

References

1. Bourbaki, Nicolas (1988). Algebra I: Chapters 1-3. Elements of Mathematics. Springer.

External links

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