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In everyday plain English, the word “function” is understood to be roughly synonymous with words like “goal” or “purpose”. For example, the function of a fruit knife is to cut fruit.

In linguistics, “function” is usually understood to have this kind of meaning. For example, the primary function of the English determiner the, when used alongside a noun within a noun phrase like the cow, is to express deictic meaning (“that particular cow”), whilst the determiner a/an, as in a cow, has the function of expressing non-deictic (singular) quantification (“just one of the cows”).

In mathematics and computer science, the word “function” seems to have a somewhat different meaning. The introduction of the term “function” to mathematics is credited to Gottfried Wilhelm Leibniz, who first used it in 1673 [1]. In maths, a function is understood as a mapping or relationship between the elements of two sets. Some use the metaphor of a black-box machine to describe a function: it takes an element of one set as input, performs some transformation upon it, and gives as output an element of another set.

Let us consider an example from set theory. Given x ∈ ℝ (meaning x is a number in the set ℝ, where ℝ is the infinite set of all real numbers), the function f(x): x + 1, then, is a mapping from any number x within ℝ to the number that is greater than it by 1. Here, the function takes x as input, adds 1 to it, and then returns the result as output. In this case, both the input and output are members of the set ℝ, but it is possible that they could be members of different subsets of ℝ. For example, if x ∈ X where X ⊆ ℝ {x | x is an even number and x > 0} (meaning X is a subset of ℝ where for any member x of X, x is an even number greater than 0) and if y ∈ Y where Y ⊆ ℝ {y | y is an odd number and x > 1} (meaning Y is a subset of ℝ where for any member y of Y, y is an even number greater than 1), then the input for f(x): x + 1 is a member of the set X and the output (that is, “x + 1”) is a member of the set Y. Thus, f(x): x + 1 is a mapping from X to Y.

The Shared Meaning of “Function” in Maths and Linguistics

It seems (to me at least) that the word “function” means something different the way it is used in linguistics versus the way it is used in mathematics and computer science. The link between the two is not very clear, and they are even translated differently in Chinese (功能 [lit.”result ability”] for linguistic functions and 函數 [lit. “contained number”] for mathematical functions). It also seems that the linguistics/everyday sense of “function” predates the mathematical sense. In Latin, the language from which Leibniz derived the mathematical term, “fūnctiō” bears the meaning of “performance” or “execution of a task” [2]. Wiktionary further specifies “fūnctiō” as a nominalisation of the Latin verb “fungor”, meaning “to perform, execute, or complete something” [3]. Thus, it would seem that not much has changed, in terms of meaning, between the Latin “fūnctiō” and modern everyday English “function”. The function of a fruit knife is to cut fruit, and the act denoted by “cutting fruit” is an instance of “executing a task”.

So, the link between the Latin meaning of “fūnctiō” and the meaning of “function” as used in linguistics is straightforward. Functionalist linguistics is the school of thought within linguistics that takes the idea of linguistic functions as a central core concept. That is, bits and stretches of language should be described in terms of the functions they perform (typically, these are communicative or social functions, which include the deictic and quantification functions mentioned earlier). When analysing a stretch of language, the linguist asks: What functions do the components of this stretch execute?

The question, of course, is if this understanding of “function” has anything to do with the way “function” is understood in mathematics. To answer this, it helps to first understand why mappings in maths of the kind just discussed are called “functions” to begin with. Perhaps it might be embarrassing, but I struggled for a while to understand this.

Leibniz coined the term “function” for mathematics when he was developing calculus, studying quantities related to the points of a curve. For example, the slope of a line can be described by a quantity representing some change between two points on the line. Let’s say the line exists on a two-dimensional space, so there is an x-axis and a y-axis. Each point is defined by its value on the x-axis and its value on the y-axis. The slope for any two points A with coordinates (xA, yA) and B with coordinates (xB, yB) can be derived by the function m(A, B): (yA - yB)/(xA - xB) [4]. In 1718, Johann Bernoulli, a contemporary of Leibniz, came to regard a function as “any expression made up of a variable and some constants” [5, 6].

Using the slope example, we may say in plain English that “the slope denoted m(A, B) is a function of the points A and B”. What is meant by “function” in the phrase “is a function of”? Most obviously, the mathematical sense of “function” as a mapping or relationship between A and B is meant. Given the coordinates of A and the slope m(A, B), one can derive the coordinates of B. Thus, the slope m(A, B) defines a relationship that holds between A and B (though, crucially, it is not the only relationship).

We can begin to work towards an alignment between the mathematical and linguistic notions of “function” when we consider the following re-wording: “the points A and B function together to give the slope denoted m(A, B)”. Here, the word “function” much more clearly expresses its original sense “to perform” or “to execute a task”, where the task is to produce the slope. What we need to understand is that a function of something is the relation from that something to the result it can give. The function of a fruit knife gives us cut fruit, and we call this function “to cut fruit”.

This is the understanding that is common to both maths and linguistics. In language, we can consider the function(s) of the indefinite determiner a/an in analogous mathematical terms. One function of a/an is the indication of deictic meaning, specifically the semantic feature [-deictic] (in contrast to the, whose function is to indicate [+deictic]). This function can be written mathematically as deixis(“a/an”): [-deictic], which can be read as “the meaning [-deictic] is a function of the determiner a/an”.

To summarise, both maths and linguistics understand a function as a relation between something and a result that could come from it.