Editing Ambiguity (section)

== Mathematical notation ==
[[Mathematical notation]] is a helpful tool that eliminates a lot of misunderstandings associated with natural language in [[physics]] and other [[science]]s. Nonetheless, there are still some inherent ambiguities due to [[Lexical semantics|lexical]], [[syntactic]], and [[Semantics|semantic]] reasons that persist in mathematical notation.

=== Names of functions ===
The '''ambiguity''' in the style of writing a [[Function (mathematics)|function]] should not be confused with a [[multivalued function]], which can (and should) be defined in a deterministic and unambiguous way. Several [[special function]]s still do not have established notations. Usually, the conversion to another notation requires to scale the argument or the resulting value; sometimes, the same name of the function is used, causing confusions. Examples of such underestablished functions:
* [[Sinc function]]
* [[Elliptic integral#Complete elliptic integral of the third kind|Elliptic integral of the third kind]]; translating elliptic integral form [[MAPLE]] to [[Mathematica]], one should replace the second argument to its square; dealing with complex values, this may cause problems.
* [[Exponential integral]]<ref name="irene">{{cite book|url=http://www.math.sfu.ca/~cbm/aands/page_228.htm|title=Handbook on mathematical functions|last1=Abramovits|first1=M.|last2=Stegun|first2=I.|page=228}}</ref>
* [[Hermite polynomial]]<ref name="irene" />{{rp|775}}

=== Expressions ===
Ambiguous expressions often appear in physical and mathematical texts.
It is common practice to omit multiplication signs in mathematical expressions. Also, it is common to give the same name to a variable and a function, for example, {{nowrap|<math>f=f(x)</math>.}} Then, if one sees {{nowrap|<math>f=f(y+1)</math>,}} there is no way to distinguish whether it means <math>f=f(x)</math> '''multiplied''' by {{nowrap|<math>(y+1)</math>,}} or function <math>f</math> '''evaluated''' at argument equal to {{nowrap|<math>(y+1)</math>.}} In each case of use of such notations, the reader is supposed to be able to perform the deduction and reveal the true meaning.

Creators of algorithmic languages try to avoid ambiguities. Many algorithmic languages ([[C++]] and [[Fortran]]) require the character * as a symbol of multiplication. The [[Wolfram Language]] used in [[Mathematica]] allows the user to omit the multiplication symbol, but requires square brackets to indicate the argument of a function; square brackets are not allowed for grouping of expressions. Fortran, in addition, does not allow use of the same name (identifier) for different objects, for example, function and variable; in particular, the expression <math>f = f(x)</math> is qualified as an error.

The order of operations may depend on the context. In most [[programming language]]s, the operations of division and multiplication have equal priority and are executed from left to right. Until the last century, many editorials assumed that multiplication is performed first, for example, <math>a/bc</math> is interpreted as {{nowrap|<math>a/(bc)</math>;}} in this case, the insertion of parentheses is required when translating the formulas to an algorithmic language. In addition, it is common to write an argument of a function without parenthesis, which also may lead to ambiguity.
In the [[scientific journal]] style, one uses roman letters to denote elementary functions, whereas variables are written using italics.
For example, in mathematical journals the expression <math>s i n</math> does not denote the [[sine function]], but the product of the three variables {{nowrap|<math>s</math>,}} {{nowrap|<math>i</math>,}} {{nowrap|<math>n</math>,}} although in the informal notation of a slide presentation it may stand for {{nowrap|<math>\sin</math>.}}

Commas in multi-component subscripts and superscripts are sometimes omitted; this is also potentially ambiguous notation.
For example, in the notation {{nowrap|<math>T_{mnk}</math>,}} the reader can only infer from the context whether it means a single-index object, taken with the subscript equal to product of variables {{nowrap|<math>m</math>,}} <math>n</math> and {{nowrap|<math>k</math>,}} or it is an indication to a trivalent [[tensor]].

=== Examples of potentially confusing ambiguous mathematical expressions ===
An expression such as <math>\sin^2\alpha/2</math> can be understood to mean either <math>(\sin(\alpha/2))^2</math> or {{nowrap|<math>(\sin \alpha)^2/2</math>.}} Often the author's intention can be understood from the context, in cases where only one of the two makes sense, but an ambiguity like this should be avoided, for example by writing {{nowrap|<math>\sin^2(\alpha/2)</math> or <math display="inline">\frac{1}{2}\sin^2\alpha</math>.}}

The expression <math>\sin^{-1}\alpha</math> means <math>\arcsin(\alpha)</math> in several texts, though it might be thought to mean {{nowrap|<math>(\sin \alpha)^{-1}</math>,}} since <math>\sin^{n} \alpha</math> commonly means {{nowrap|<math>(\sin \alpha)^{n}</math>.}} Conversely, <math>\sin^2 \alpha</math> might seem to mean {{nowrap|<math>\sin(\sin \alpha)</math>,}} as this [[exponentiation]] notation usually denotes [[function iteration]]: in general, <math>f^2(x)</math> means {{nowrap|<math>f(f(x))</math>.}} However, for [[trigonometric]] and [[hyperbolic functions]], this notation conventionally means exponentiation of the result of function application.

The expression <math>a/2b</math> can be interpreted as meaning {{nowrap|<math>(a/2)b</math>;}} however, it is more commonly understood to mean {{nowrap|<math>a/(2b)</math>.}}

=== Notations in quantum optics and quantum mechanics ===
It is common to define the [[coherent states]] in [[quantum optics]] with <math>~|\alpha\rangle~ </math> and states with fixed number of photons with {{nowrap|<math>~|n\rangle~</math>.}} Then, there is an "unwritten rule": the state is coherent if there are more Greek characters than Latin characters in the argument, and <math>n</math>-photon state if the Latin characters dominate. The ambiguity becomes even worse, if <math>~|x\rangle~</math> is used for the states with certain value of the coordinate, and <math>~|p\rangle~</math> means the state with certain value of the momentum, which may be used in books on [[quantum mechanics]]. Such ambiguities easily lead to confusions, especially if some normalized adimensional, [[dimensionless]] variables are used. Expression <math> |1\rangle </math> may mean a state with single photon, or the coherent state with mean amplitude equal to 1, or state with momentum equal to unity, and so on. The reader is supposed to guess from the context.

=== Ambiguous terms in physics and mathematics ===
Some physical quantities do not yet have established notations; their value (and sometimes even [[dimension]], as in the case of the [[Einstein coefficients]]), depends on the system of notations. Many terms are ambiguous. Each use of an ambiguous term should be preceded by the definition, suitable for a specific case. Just like [[Ludwig Wittgenstein]] states in [[Tractatus Logico-Philosophicus]]: "...&nbsp;Only in the context of a proposition has a name meaning."<ref>{{cite book|title=Tractatus Logico-Philosophicus|last=Wittgenstein|first=Ludwig|publisher=Dover Publications Inc.|year=1999|isbn=978-0-486-40445-5|page=39}}</ref>

A highly confusing term is ''gain''. For example, the sentence "the gain of a system should be doubled", without context, means close to nothing.
* It may mean that the ratio of the output voltage of an electric circuit to the input voltage should be doubled.
* It may mean that the ratio of the output power of an electric or optical circuit to the input power should be doubled.
* It may mean that the gain of the laser medium should be doubled, for example, doubling the population of the upper laser level in a quasi-two level system (assuming negligible absorption of the ground-state).

The term ''intensity'' is ambiguous when applied to light. The term can refer to any of [[irradiance]], [[luminous intensity]], [[radiant intensity]], or [[radiance]], depending on the background of the person using the term.

Also, confusions may be related with the use of [[atomic percent]] as measure of concentration of a [[dopant]], or [[Optical resolution|resolution]] of an imaging system, as measure of the size of the smallest detail that still can be resolved at the background of statistical noise. See also ''[[Accuracy and precision]]''.

The [[Berry paradox]] arises as a result of systematic ambiguity in the meaning of terms such as "definable" or "nameable". Terms of this kind give rise to [[Virtuous circle and vicious circle|vicious circle]] fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.<ref>Russell/Whitehead, ''Principia Mathematica''</ref>