Abuse of notation
Encyclopedia
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). 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. By an abuse of language, this itself is often referred to as "abuse of notation".
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."
consists of a set
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 
when there is no danger of confusion—that is, when it is implicitly clear what topology is being considered. Similarly, one often refers to a group
as simply 
when the group operation is clear from context.
, algebraic manipulations of the Leibniz notation
for the derivative
is commonly thought to be an abuse of notation. It is frequently convenient to treat the expression 
as a fraction. For example, it leads to the correct formula for differentiation of the composition of functions (commonly called the "chain rule
")
. Another example is the concept of separation of variables
in solving differential equations, in which one can rewrite the equation
as 
and then integrate. However, this treatment does not have to be viewed as an "abuse of notation", because it can be fully justified by taking the "differentials" 
and 
as simply being two numbers in the ratio 
: 1. If this is done, then the methods are completely rigorous, with no "abuse of notation" involved. In terms of a graph of 
against 
, this can be thought of as taking d
and d
as the horizontal and vertical components of a vector along a segment of a tangent to the graph. Leibniz's own interpretation was of a ratio of infinitesimal changes in 
and 
, and a similar interpretation occurs in non-standard analysis
, but in standard analysis no such "infinitesimals" are needed, as finite real numbers provide a fully rigorous justification.
operator, denoted by
, is a tuple of partial derivative operators posing as a vector. This suggests notations such as 
for gradient
,
for divergence
and
for curl. The notation is extremely convenient because 
does behave like a vector most of the time. But it can be regarded as an abuse because 
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
as a vector. The vector-like notations are simply specially defined uses of the dot and cross.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, abuse of notation occurs when an author uses a mathematical notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
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). 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. By an abuse of language, this itself is often referred to as "abuse of notation".
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
False analogy
-The Argument from Analogy:The process of analogical inference involves noting the shared properties of two or more things, and from this basis infering that they also share some further property...
.
Abuse of language is an almost synonymous expression that is usually used for non-notational abuses. For example, while the word representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
properly designates a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
from a group G to GL(V) where V is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, it is common to call V "a representation of G."
Examples
Common examples occur when speaking of compound mathematical objects. For example, a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
consists of a set
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 when there is no danger of confusion—that is, when it is implicitly clear what topology is being considered. Similarly, one often refers to a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
as simply when the group operation is clear from context.Derivative
In standard analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, algebraic manipulations of the Leibniz notation
Leibniz notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" increments of x and y, just as Δx and Δy represent finite increments of x and y...
for the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
is commonly thought to be an abuse of notation. It is frequently convenient to treat the expression as a fraction. For example, it leads to the correct formula for differentiation of the composition of functions (commonly called the "chain ruleChain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
")
. Another example is the concept of separation of variablesSeparation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
in solving differential equations, in which one can rewrite the equation
as and then integrate. However, this treatment does not have to be viewed as an "abuse of notation", because it can be fully justified by taking the "differentials" and as simply being two numbers in the ratio : 1. If this is done, then the methods are completely rigorous, with no "abuse of notation" involved. In terms of a graph of against , this can be thought of as taking d and d as the horizontal and vertical components of a vector along a segment of a tangent to the graph. Leibniz's own interpretation was of a ratio of infinitesimal changes in and , and a similar interpretation occurs in non-standard analysisNon-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, but in standard analysis no such "infinitesimals" are needed, as finite real numbers provide a fully rigorous justification.
Del operator
The delDel
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
operator, denoted by
, is a tuple of partial derivative operators posing as a vector. This suggests notations such as for gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
,
for divergenceDivergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
and
for curl. The notation is extremely convenient because does behave like a vector most of the time. But it can be regarded as an abuse because 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
as a vector. The vector-like notations are simply specially defined uses of the dot and cross.)Cross product
The determinant of a 3×3 matrix may be computed by "expanding along the first row" as follows:-
-
The cross productCross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
of the vectors (a1, a2, a3) and (b1, b2, b3) is given similarly by-
-
Thus the cross product may be computed by writing-
-
and expanding along the first row by rote, ignoring the fact that the matrix is not truly a matrix over the real or complex numbers (or whatever fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
the matrix entries belong to), and that thus the resulting computation does not compute an ordinary determinant. This is technically an abuse of notation, but is useful as a mnemonic to remember the formula for cross product and is very helpful in calculations.
Function application over set
John Harrison (1996) cites "the use of f(x) to represent both application of a functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f to an argument x, and the image under f of a subset, x, of fs domain". (Note that the last x is usually written differently, e.g. as X, in order to distinguish an element x of the domain from a subset X.)
On the other hand, it is questionable that this is an abuse, since the definition of a function as an operator on subsets of the domain is widely known and is often stated explicitly in articles and books.
Exponentiation as repetition
ExponentiationExponentiationExponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
is repeated multiplication, and multiplication is frequently denoted by juxtaposition of operands, with no operator at all. The suggested superscript notation for other associative operations denoted by juxtaposition follows:
- Function applicationFunction applicationIn mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range.-Representation:...
is sometimes denoted without parentheses:
. This suggests the functional powers notation: 
. This also generalizes nicely to represent function inverse for a power of -1 and functional square rootFunctional square rootIn mathematics, a half iterate is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function g is a function f satisfying f = g for all x...
for a power of 1/2.
- Exponentiation over sets.
- String repetition: "ab3c" = "abbbc".
Trigonometric functions
In some countries it is common to denote the square of the value of
as 
, and the inverse function as 
. In his article on notation in the Edinburgh EncyclopediaEdinburgh EncyclopediaThe Edinburgh Encyclopedia was an encyclopedia in 18 volumes, printed and published by William Blackwood and edited by David Brewster between 1808 and 1830...
Charles BabbageCharles BabbageCharles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...
complains at length of this abuse of notation and suggests two alternatives for the notation
- Function composition, i.e.
and 
is the inverse. - Powers of the value, i.e.
and 
is the reciprocal.
Babbage argues strongly for the former, and also that the square of the value should be notated as
, but beware: Babbage intends 
even though what he wrote is easily confused with 
(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
is like, and also 
which is the function which, when composed with itself, equals 
, the functional square rootFunctional square rootIn mathematics, a half iterate is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function g is a function f satisfying f = g for all x...
.
Big O notation
With Big O notationBig O notationIn mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
, we say that some term
"is" 
(given some function g, where x is one of f' s parameters).
Example: "Runtime of the algorithm is
" or in symbols "
".
Intuitively this notation groups functions according to their growth respective to some parameter; as such, the notation is abusive in two respects:
It abuses "=", and it invokes terms that are real numbers instead of function terms.
It would be appropriate to use the set membership notation and thus write
instead of 
.
This would allow for common set operations like
, 
,
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,
means that fs real growth is less than 
.
But this further abuse is not necessary if, following KnuthDonald KnuthDonald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
one distinguishes between O and
the closely related o and Θ notations.
Concerning the use of terms for scalar numbers instead of functions, one encounters the following troubles.
- You cannot cleanly define what
may mean, due to the fact 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. - The abused O notation is bound to one variable, and the identity of that variable can be ambiguous: for instance, in
one of the variables might be a parameter which is not in scope of the 
.
That is, you might mean
, since 
was the parameter that you assigned 2, or you might mean 
, since 
was the parameter substituted by 3 here.
Even
might be the same as 
, since 
might be a parameter, not the concerned function variable.
The carelessness regarding the use of function terms might be caused by the rarely-used functional notations, like Lambda notationLambda calculusIn mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
.
You would have to write
and 
.
The correct O notation can be easily extended to complexity functions which map tuples to complexities; this 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
,
which is not possible with the abused notation.
You can also state theorems like
is a convex coneConvex coneIn linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...
, and use that for formal reasoning.
Equality vs. isomorphism
Another common abuse of notation is to blur the distinction between equality and isomorphismIsomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
. For instance, in the construction of the real numbers from Dedekind cuts of rational numbers, the rational number
is identified with the set of all rational numbers less than 
, 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
Sound pressure level
Sound pressure level measurements are commonly in dB(A) where the suffix "A" denotes A-weightingA-weightingA Weighting curve is a graph of a set of factors, that are used to 'weight' measured values of a variable according to their importance in relation to some outcome. The most commonly known example is frequency weighting in sound level measurement where a specific set of weighting curves known as A,...
. There is ubiquitous misuse of "dB" in recording sound level measurements although a dB (decibel) is only a numerical ratio of two quantities.
Bourbaki
The term "abuse of language" frequently appears in the writings of Nicolas BourbakiNicolas BourbakiNicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
:
- 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 functionPartial functionIn mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
s 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 functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f: A' → B, whereA' is a subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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 functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
is a function that is continuous.) But the use of adjectiveAdjectiveIn grammar, an adjective is a 'describing' word; the main syntactic role of which is to qualify a noun or noun phrase, giving more information about the object signified....
s (and adverbAdverbAn adverb is a part of speech that modifies verbs or any part of speech other than a noun . Adverbs can modify verbs, adjectives , clauses, sentences, and other adverbs....
s) 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 magmaMagma (algebra)In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
is a generalized groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
.)
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.
Abuse of language or notation?
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 theoreticCategory (mathematics)In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
context, where f can be seen as a morphism in the category of partial functions.
External links
- "Strong Symbols", by Henning Thielemann (PDF Slides) Section 5: Common abuse of notation
- Function application
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