Themathematical concept of afunction dates from the 17th century in connection with the development ofcalculus; for example, the slope${\displaystyle dy/dx}$ of agraph at a point was regarded as a function of thex-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such asOresme.

Mathematicians of the 18th century typically regarded a function as being defined by ananalytic expression. In the 19th century, the demands of the rigorous development ofanalysis byKarl Weierstrass and others, the reformulation ofgeometry in terms of analysis, and the invention ofset theory byGeorg Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from oneset to another.

In the 12th century, mathematicianSharaf al-Din al-Tusi analyzed the equationx³ +d =b ⋅ x² in the formx² ⋅ (b –x) =d, stating that the left hand side must at least equal the value ofd for the equation to have a solution. He then determined the maximum value of this expression. It is arguable that the isolation of this expression is an early approach to the notion of a "function". A value less thand means no positive solution; a value equal tod corresponds to one solution, while a value greater thand corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development inIslamic mathematics, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe.^[1]

According toJean Dieudonné^[2] and Ponte,^[3] the concept of a function emerged in the 17th century as a result of the development ofanalytic geometry and theinfinitesimal calculus. Nevertheless, Medvedev suggests that the implicit concept of a function is one with an ancient lineage.^[4] Ponte also sees more explicit approaches to the concept in theMiddle Ages:

Historically, some mathematicians can be regarded as having foreseen and come close to a modern formulation of the concept of function. Among them is[Nicole] Oresme (1323–1382). . . In his theory, some general ideas about independent and dependent variable quantities seem to be present.^[5]

The development of analytical geometry around 1640 allowed mathematicians to go between geometric problems about curves and algebraic relations between "variable coordinatesx andy."^[6] Calculus was developed using the notion of variables, with their associated geometric meaning, which persisted well into the eighteenth century.^[7] However, the terminology of "function" came to be used in interactions between Leibniz and Bernoulli towards the end of the 17th century.^[8]

The term "function" was literally introduced byGottfried Leibniz, in a 1673 letter, to describe a quantity related to points of acurve, such as acoordinate or curve'sslope.^[9][10]Johann Bernoulli started calling expressions made of a single variable "functions." In 1698, he agreed with Leibniz that any quantity formed "in an algebraic and transcendental manner" may be called a function ofx.^[11] By 1718, he came to regard as a function "any expression made up of a variable and some constants."^[12]Alexis Claude Clairaut (in approximately 1734) andLeonhard Euler introduced the familiar notation${\displaystyle f(x)}$ for the value of a function.^[13]

The functions considered in those times are called todaydifferentiable functions. For this type of function, one can talk aboutlimits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input. Such functions are the basis ofcalculus.

In the first volume of his fundamental textIntroductio in analysin infinitorum, published in 1748, Euler gave essentially the same definition of a function as his teacher Bernoulli, as anexpression orformula involving variables and constants e.g.,${\displaystyle x^2+3x+2}$.^[14] Euler's own definition reads:

A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.^[15]

Euler also allowed multi-valued functions whose values are determined by an implicit equation.

In 1755, however, in hisInstitutiones calculi differentialis, Euler gave a more general concept of a function:

When certain quantities depend on others in such a way that they undergo a change when the latter change, then the first are calledfunctions of the second. This name has an extremely broad character; it encompasses all the ways in which one quantity can be determined in terms of others.^[16]

Medvedev^[17] considers that "In essence this is the definition that became known as Dirichlet's definition." Edwards^[18] also credits Euler with a general concept of a function and says further that

The relations among these quantities are not thought of as being given by formulas, but on the other hand they are surely not thought of as being the sort of general set-theoretic, anything-goes subsets of product spaces that modern mathematicians mean when they use the word "function".

In hisThéorie Analytique de la Chaleur,^[19]Joseph Fourier claimed that an arbitrary function could be represented by aFourier series.^[20] Fourier had a general conception of a function, which included functions that were neithercontinuous nor defined by an analytical expression.^[21] Related questions on the nature and representation of functions, arising from the solution of thewave equation for a vibrating string, had already been the subject of dispute betweenJean le Rond d'Alembert and Euler, and they had a significant impact in generalizing the notion of a function.Luzin observes that:

The modern understanding of function and its definition, which seems correct to us, could arise only after Fourier's discovery. His discovery showed clearly that most of the misunderstandings that arose in the debate about the vibrating string were the result of confusing two seemingly identical but actually vastly different concepts, namely that of function and that of its analytic representation. Indeed, prior to Fourier's discovery no distinction was drawn between the concepts of "function" and of "analytic representation," and it was this discovery that brought about their disconnection.^[22]

During the 19th century, mathematicians started to formalize all the different branches of mathematics. One of the first to do so wasAugustin-Louis Cauchy; his somewhat imprecise results were later made completely rigorous by Weierstrass, who advocated building calculus onarithmetic rather than ongeometry, which favoured Euler's definition over Leibniz's (seearithmetization of analysis). According to Smithies, Cauchy thought of functions as being defined by equations involvingreal orcomplex numbers, and tacitly assumed they were continuous:

Cauchy makes some general remarks about functions in Chapter I, Section 1 of hisAnalyse algébrique (1821). From what he says there, it is clear that he normally regards a function as being defined by an analytic expression (if it is explicit) or by an equation or a system of equations (if it is implicit); where he differs from his predecessors is that he is prepared to consider the possibility that a function may be defined only for a restricted range of the independent variable.^[23]

Nikolai Lobachevsky^[24] andPeter Gustav Lejeune Dirichlet^[25] are traditionally credited with independently giving the modern "formal" definition of a function as arelation in which every first element has a unique second element.

Lobachevsky (1834) writes that

The general concept of a function requires that a function ofx be defined as a number given for eachx and varying gradually withx. The value of the function can be given either by an analytic expression, or by a condition that provides a means of examining all numbers and choosing one of them; or finally the dependence may exist but remain unknown.^[26]

while Dirichlet (1837) writes

If now a unique finitey corresponding to eachx, and moreover in such a way that whenx ranges continuously over the interval froma tob,${\displaystyle y=f(x)}$ also varies continuously, theny is called acontinuous function ofx for this interval. It is not at all necessary here thaty be given in terms ofx by one and the same law throughout the entire interval, and it is not necessary that it be regarded as a dependence expressed using mathematical operations.^[27]

Eves asserts that "the student of mathematics usually meets the Dirichlet definition of function in his introductory course in calculus.^[28]

Dirichlet's claim to this formalization has been disputed byImre Lakatos:

There is no such definition in Dirichlet's works at all. But there is ample evidence that he had no idea of this concept. In his [1837] paper for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the functionhas two values: ...^[29]

However, Gardiner says"...it seems to me that Lakatos goes too far, for example, when he asserts that 'there is ample evidence that [Dirichlet] had no idea of [the modern function] concept'."^[30]Moreover, as noted above, Dirichlet's paper does appear to include a definition along the lines of what is usually ascribed to him, even though (like Lobachevsky) he states it only for continuous functions of a real variable.

Similarly, Lavine observes that:

It is a matter of some dispute how much credit Dirichlet deserves for the modern definition of a function, in part because he restricted his definition to continuous functions....I believe Dirichlet defined the notion ofcontinuous function to make it clear that no rule or law is required even in the case of continuous functions, not just in general. This would have deserved special emphasis because of Euler'sdefinition of a continuous function as one given by single expression-or law. But I also doubt there is sufficient evidence to settle the dispute.^[31]

Because Lobachevsky and Dirichlet have been credited as among the first to introduce the notion of an arbitrary correspondence, this notion is sometimes referred to as the Dirichlet or Lobachevsky-Dirichlet definition of a function.^[32] A general version of this definition was later used byBourbaki (1939), and some in the education community refer to it as the "Dirichlet–Bourbaki" definition of a function.

Dieudonné, who was one of the founding members of the Bourbaki group, credits a precise and general modern definition of a function to Dedekind in his workWas sind und was sollen die Zahlen,^[33] which appeared in 1888 but had already been drafted in 1878. Dieudonné observes that instead of confining himself, as in previous conceptions, to real (or complex) functions, Dedekind defines a function as a single-valued mapping between any two sets:

What was new and what was to be essential for the whole of mathematics was the entirely general conception of afunction.^[34]

Hardy 1908, pp. 26–28 defined a function as a relation between two variablesx andy such that "to some values ofx at any rate correspond values ofy." He neither required the function to be defined for all values ofx nor to associate each value ofx to a single value of y. This broad definition of a function encompasses more relations than are ordinarily considered functions in contemporary mathematics. For example, Hardy's definition includesmultivalued functions and what incomputability theory are calledpartial functions.

Logicians of this time were primarily involved with analyzingsyllogisms (the 2000-year-old Aristotelian forms and otherwise), or asAugustus De Morgan (1847) stated it: "the examination of that part of reasoning which depends upon the manner in which inferences are formed,and the investigation of general maxims and rules for constructing arguments".^[35] At this time the notion of (logical) "function" is not explicit, but at least in the work of De Morgan andGeorge Boole it is implied: we see abstraction of the argument forms, the introduction of variables, the introduction of a symbolic algebra with respect to these variables, and some of the notions of set theory.

De Morgan's 1847 "FORMAL LOGIC OR, The Calculus of Inference, Necessary and Probable" observes that "[a]logical truth depends upon thestructure of the statement, and not upon the particular matters spoken of"; he wastes no time (preface page i) abstracting: "In the form of the proposition, the copula is made as abstract as the terms". He immediately (p. 1) casts what he calls "the proposition" (present-day propositionalfunction orrelation) into a form such as "X is Y", where the symbols X, "is", and Y represent, respectively, thesubject,copula, andpredicate. While the word "function" does not appear, the notion of "abstraction" is there, "variables" are there, the notion of inclusion in his symbolism "all of the Δ is in the О" (p. 9) is there, and lastly a new symbolism for logical analysis of the notion of "relation" (he uses the word with respect to this example " X)Y " (p. 75) ) is there:

" A₁ X)Y To take an X it is necessary to take a Y" [or To be an X it is necessary to be a Y]

" A¹ Y)X To take a Y it is sufficient to take a X" [or To be a Y it is sufficient to be an X], etc.

In his 1848The Nature of Logic Boole asserts that "logic . . . is in a more especial sense the science of reasoning by signs", and he briefly discusses the notions of "belonging to" and "class": "An individual may possess a great variety of attributes and thus belonging to a great variety of different classes".^[36] Like De Morgan he uses the notion of "variable" drawn from analysis; he gives an example of "represent[ing] the class oxen byx and that of horses byy and the conjunctionand by the sign + . . . we might represent the aggregate class oxen and horses byx + y".^[37]

In the context of "the Differential Calculus" Boole defined (circa 1849) the notion of a function as follows:

"That quantity whose variation is uniform . . . is called the independent variable. That quantity whose variation is referred to the variation of the former is said to be afunction of it. The Differential calculus enables us in every case to pass from the function to the limit. This it does by a certain Operation. But in the very Idea of an Operation is . . . the idea of an inverse operation. To effect that inverse operation in the present instance is the business of the Int[egral] Calculus."^[38]

Eves observes "that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory ofsets, orclasses, from a foundation in the logic of propositions and propositional functions".^[39] But by the late 19th century the logicians' research into the foundations of mathematics was undergoing a major split. The direction of the first group, theLogicists, can probably be summed up best by Bertrand Russell 1903 – "to fulfil two objects, first, to show that all mathematics follows from symbolic logic, and secondly to discover, as far as possible, what are the principles of symbolic logic itself."

The second group of logicians, the set-theorists, emerged withGeorg Cantor's "set theory" (1870–1890) but were driven forward partly as a result of Russell's discovery of a paradox that could be derived from Frege's conception of "function", but also as a reaction against Russell's proposed solution.^[40]Ernst Zermelo's set-theoretic response was his 1908Investigations in the foundations of set theory I – the firstaxiomatic set theory; here too the notion of "propositional function" plays a role.

In hisAn Investigation into the laws of thought Boole now defined a function in terms of a symbolx as follows:

"8. Definition. – Any algebraic expression involving symbolx is termed a function ofx, and may be represented by the abbreviated formf(x)"^[41]

Boole then usedalgebraic expressions to define both algebraic andlogical notions, e.g., 1 − x is logical NOT(x),xy is the logical AND(x,y),x + y is the logical OR(x,y),x(x + y) isxx + xy, and "the special law"xx =x² =x.^[42]

In his 1881Symbolic Logic Venn was using the words "logical function" and the contemporary symbolism (x =f(y),y =f⁻¹(x), cf page xxi) plus the circle-diagrams historically associated withVenn to describe "class relations",^[43] the notions "'quantifying' our predicate", "propositions in respect of their extension", "the relation of inclusion and exclusion of two classes to one another", and "propositional function" (all on p. 10), the bar over a variable to indicate not-x (page 43), etc. Indeed he equated unequivocally the notion of "logical function" with "class" [modern "set"]: "... on the view adopted in this book,f(x) never stands for anything but a logical class. It may be a compound class aggregated of many simple classes; it may be a class indicated by certain inverse logical operations, it may be composed of two groups of classes equal to one another, or what is the same thing, their difference declared equal to zero, that is, a logical equation. But however composed or derived,f(x) with us will never be anything else than a general expression for such logical classes of things as may fairly find a place in ordinary Logic".^[44]

Gottlob Frege'sBegriffsschrift (1879) precededGiuseppe Peano (1889), but Peano had no knowledge ofFrege 1879 until after he had published his 1889.^[45] Both writers strongly influencedRussell (1903). Russell in turn influenced much of 20th-century mathematics and logic through hisPrincipia Mathematica (1913) jointly authored withAlfred North Whitehead.

At the outset Frege abandons the traditional "conceptssubject andpredicate", replacing them withargument andfunction respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the wordsif, and, not, or, there is, some, all, and so forth, deserves attention".^[46]

Frege begins his discussion of "function" with an example: Begin with the expression^[47] "Hydrogen is lighter than carbon dioxide". Now remove the sign for hydrogen (i.e., the word "hydrogen") and replace it with the sign for oxygen (i.e., the word "oxygen"); this makes a second statement. Do this again (using either statement) and substitute the sign for nitrogen (i.e., the word "nitrogen") and note that "This changes the meaning in such a way that "oxygen" or "nitrogen" enters into the relations in which "hydrogen" stood before".^[48] There are three statements:

• "Hydrogen is lighter than carbon dioxide."
• "Oxygen is lighter than carbon dioxide."
• "Nitrogen is lighter than carbon dioxide."

Now observe in all three a "stable component, representing the totality of [the] relations";^[49] call thisthe function, i.e.,

"... is lighter than carbon dioxide", is the function.

Frege calls theargument of the function "[t]he sign [e.g., hydrogen, oxygen, or nitrogen], regarded as replaceable by others that denotes the object standing in these relations".^[50] He notes that we could have derived the function as "Hydrogen is lighter than . . .." as well, with an argument position on theright; the exact observation is made by Peano (see more below). Finally, Frege allows for the case of two (or more) arguments. For example, remove "carbon dioxide" to yield the invariant part (the function) as:

• "... is lighter than ... "

The one-argument function Frege generalizes into the form Φ(A) where A is the argument and Φ( ) represents the function, whereas the two-argument function he symbolizes as Ψ(A, B) with A and B the arguments and Ψ( , ) the function and cautions that "in general Ψ(A, B) differs from Ψ(B, A)". Using his unique symbolism he translates for the reader the following symbolism:

"We can read |--- Φ(A) as "A has the property Φ. |--- Ψ(A, B) can be translated by "B stands in the relation Ψ to A" or "B is a result of an application of the procedure Ψ to the object A".^[51]

Peano defined the notion of "function" in a manner somewhat similar to Frege, but without the precision.^[52] First Peano defines the sign "K meansclass, or aggregate of objects",^[53] the objects of which satisfy three simple equality-conditions,^[54]a =a, (a =b) = (b =a), IF ((a =b) AND (b =c)) THEN (a =c). He then introduces φ, "a sign or an aggregate of signs such that ifx is an object of the classs, the expression φx denotes a new object". Peano adds two conditions on these new objects: First, that the three equality-conditions hold for the objects φx; secondly, that "ifx andy are objects of classs and ifx =y, we assume it is possible to deduce φx = φy".^[55] Given all these conditions are met, φ is a "function presign". Likewise he identifies a "function postsign". For example ifφ is the function presigna+, then φx yieldsa+x, or if φ is the function postsign +a thenxφ yieldsx+a.^[54]

While the influence of Cantor and Peano was paramount,^[56] in Appendix A "The Logical and Arithmetical Doctrines of Frege" ofThe Principles of Mathematics, Russell arrives at a discussion of Frege's notion offunction, "...a point in which Frege's work is very important, and requires careful examination".^[57] In response to his 1902 exchange of letters with Frege about the contradiction he discovered in Frege'sBegriffsschrift Russell tacked this section on at the last moment.

For Russell the bedeviling notion is that ofvariable: "6. Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they containvariables. The notion of the variable is one of the most difficult with which logic has to deal. For the present, I openly wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. . . . We shall find always, in all mathematical propositions, that the wordsany orsome occur; and these words are the marks of a variable and a formal implication".^[58]

As expressed by Russell "the process of transforming constants in a proposition into variables leads to what is called generalization, and gives us, as it were, the formal essence of a proposition ... So long as any term in our proposition can be turned into a variable, our proposition can be generalized; and so long as this is possible, it is the business of mathematics to do it";^[59] these generalizations Russell namedpropositional functions.^[60] Indeed he cites and quotes from Frege'sBegriffsschrift and presents a vivid example from Frege's 1891Function und Begriff: That "the essence of the arithmetical function 2x³ + x is what is left when thex is taken away, i.e., in the above instance 2( )³ + ( ). The argumentx does not belong to the function but the two taken together make the whole".^[57] Russell agreed with Frege's notion of "function" in one sense: "He regards functions – and in this I agree with him – as more fundamental thanpredicates andrelations" but Russell rejected Frege's "theory of subject and assertion", in particular "he thinks that, if a terma occurs in a proposition, the proposition can always be analysed intoa and an assertion abouta".^[57]

Russell would carry his ideas forward in his 1908Mathematical logical as based on the theory of types and into his and Whitehead's 1910–1913Principia Mathematica. By the time ofPrincipia Mathematica Russell, like Frege, considered the propositional function fundamental: "Propositional functions are the fundamental kind from which the more usual kinds of function, such as "sinx" or logx or "the father ofx" are derived. These derivative functions . . . are called "descriptive functions". The functions of propositions . . . are a particular case of propositional functions".^[61]

Propositional functions: Because his terminology is different from the contemporary, the reader may be confused by Russell's "propositional function". An example may help. Russell writes apropositional function in its raw form, e.g., asφŷ: "ŷ is hurt". (Observe the circumflex or "hat" over the variabley). For our example, we will assign just 4 values to the variableŷ: "Bob", "This bird", "Emily the rabbit", and "y". Substitution of one of these values for variableŷ yields aproposition; this proposition is called a "value" of the propositional function. In our example there are four values of the propositional function, e.g., "Bob is hurt", "This bird is hurt", "Emily the rabbit is hurt" and "y is hurt." A proposition, if it issignificant—i.e., if its truth isdeterminate—has atruth-value oftruth orfalsity. If a proposition's truth value is "truth" then the variable's value is said tosatisfy the propositional function. Finally, per Russell's definition, "aclass [set] is all objects satisfying some propositional function" (p. 23). Note the word "all" – this is how the contemporary notions of "For all ∀" and "there exists at least one instance ∃" enter the treatment (p. 15).

To continue the example: Suppose (from outside the mathematics/logic) one determines that the propositions "Bob is hurt" has a truth value of "falsity", "This bird is hurt" has a truth value of "truth", "Emily the rabbit is hurt" has an indeterminate truth value because "Emily the rabbit" doesn't exist, and "y is hurt" is ambiguous as to its truth value because the argumenty itself is ambiguous. While the two propositions "Bob is hurt" and "This bird is hurt" aresignificant (both have truth values), only the value "This bird" of thevariableŷsatisfies the propositional functionφŷ: "ŷ is hurt". When one goes to form the class α:φŷ: "ŷ is hurt", only "This bird" is included, given the four values "Bob", "This bird", "Emily the rabbit" and "y" for variableŷ and their respective truth-values: falsity, truth, indeterminate, ambiguous.

Russell definesfunctions of propositions with arguments, andtruth-functionsf(p).^[62] For example, suppose one were to form the "function of propositions with arguments"p₁: "NOT(p) ANDq" and assign its variables the values ofp: "Bob is hurt" andq: "This bird is hurt". (We are restricted to the logical linkages NOT, AND, OR and IMPLIES, and we can only assign "significant" propositions to the variablesp andq). Then the "function of propositions with arguments" isp₁: NOT("Bob is hurt") AND "This bird is hurt". To determine the truth value of this "function of propositions with arguments" we submit it to a "truth function", e.g.,f(p₁):f( NOT("Bob is hurt") AND "This bird is hurt" ), which yields a truth value of "truth".

The notion of a "many-one" functional relation": Russell first discusses the notion of "identity", then defines adescriptive function (pages 30ff) as theunique valueιx that satisfies the (2-variable) propositional function (i.e., "relation")φŷ.

N.B. The reader should be warned here that the order of the variables are reversed!y is the independent variable andx is the dependent variable, e.g.,x = sin(y).^[63]

Russell symbolizes the descriptive function as "the object standing in relation toy":R'y =_DEF (ιx)(x R y). Russell repeats that "R'y is a function ofy, but not a propositional function [sic]; we shall call it adescriptive function. All the ordinary functions of mathematics are of this kind. Thus in our notation "sin y" would be written " sin 'y ", and "sin" would stand for the relation sin 'y has toy".^[64]

David Hilbert set himself the goal of "formalizing" classical mathematics "as a formal axiomatic theory, and this theory shall be proved to beconsistent, i.e., free from contradiction".^[65] InHilbert 1927The Foundations of Mathematics he frames the notion of function in terms of the existence of an "object":

13. A(a) --> A(ε(A)) Here ε(A) stands for an object of which the proposition A(a) certainly holds if it holds of any object at all; let us call ε the logical ε-function".^[66] [The arrow indicates "implies".]

Hilbert then illustrates the three ways how the ε-function is to be used, firstly as the "for all" and "there exists" notions, secondly to represent the "object of which [a proposition] holds", and lastly how to cast it into thechoice function.

Recursion theory and computability: But the unexpected outcome of Hilbert's and his studentBernays's effort was failure; seeGödel's incompleteness theorems of 1931. At about the same time, in an effort to solve Hilbert'sEntscheidungsproblem, mathematicians set about to define what was meant by an "effectively calculable function" (Alonzo Church 1936), i.e., "effective method" or "algorithm", that is, an explicit, step-by-step procedure that would succeed in computing a function. Various models for algorithms appeared, in rapid succession, including Church'slambda calculus (1936),Stephen Kleene'sμ-recursive functions(1936) andAlan Turing's (1936–7) notion of replacing human "computers" with utterly-mechanical "computing machines" (seeTuring machines). It was shown that all of these models could compute the same class ofcomputable functions.Church's thesis holds that this class of functions exhausts all thenumber-theoretic functions that can be calculated by an algorithm. The outcomes of these efforts were vivid demonstrations that, in Turing's words, "there can be no general process for determining whether a given formulaU of the functional calculusK [Principia Mathematica] is provable";^[67] see more atIndependence (mathematical logic) andComputability theory.

Set theory began with the work of the logicians with the notion of "class" (modern "set") for exampleDe Morgan (1847),Jevons (1880),Venn (1881),Frege (1879) andPeano (1889). It was given a push byGeorg Cantor's attempt to define the infinite in set-theoretic treatment (1870–1890) and a subsequent discovery of anantinomy (contradiction, paradox) in this treatment (Cantor's paradox), by Russell's discovery (1902) of an antinomy in Frege's 1879 (Russell's paradox), by the discovery of more antinomies in the early 20th century (e.g., the 1897Burali-Forti paradox and the 1905Richard paradox), and by resistance to Russell's complex treatment of logic^[68] and dislike of hisaxiom of reducibility^[69] (1908, 1910–1913) that he proposed as a means to evade the antinomies.

In 1902 Russell sent a letter to Frege pointing out that Frege's 1879Begriffsschrift allowed a function to be an argument of itself: "On the other hand, it may also be that the argument is determinate and the function indeterminate . . .."^[70] From this unconstrained situation Russell was able to form a paradox:

"You state ... that a function, too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Letw be the predicate: to be a predicate that cannot be predicated of itself. Canw be predicated of itself?"^[71]

Frege responded promptly that "Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic".^[72]

From this point forward development of the foundations of mathematics became an exercise in how to dodge "Russell's paradox", framed as it was in "the bare [set-theoretic] notions of set and element".^[73]

The notion of "function" appears as Zermelo's axiom III—the Axiom of Separation (Axiom der Aussonderung). This axiom constrains us to use a propositional function Φ(x) to "separate" asubsetM_Φ from a previously formed setM:

"AXIOM III. (Axiom of separation). Whenever the propositional function Φ(x) is definite for all elements of a setM,M possesses a subsetM_Φ containing as elements precisely those elementsx ofM for which Φ(x) is true".^[74]

As there is nouniversal set — sets originate by way of Axiom II from elements of (non-set)domain B – "...this disposes of the Russell antinomy so far as we are concerned".^[75] But Zermelo's "definite criterion" is imprecise, and is fixed byWeyl,Fraenkel,Skolem, andvon Neumann.^[76]

In fact Skolem in his 1922 referred to this "definite criterion" or "property" as a "definite proposition":

"... a finite expression constructed from elementary propositions of the forma εb ora =b by means of the five operations [logical conjunction, disjunction, negation, universal quantification, and existential quantification].^[77]

van Heijenoort summarizes:

"A property is definite in Skolem's sense if it is expressed . . . by awell-formed formula in the simplepredicate calculus of first order in which the sole predicate constants are ε and possibly, =. ... Today an axiomatization of set theory is usually embedded in a logical calculus, and it is Weyl's and Skolem's approach to the formulation of the axiom of separation that is generally adopted.^[78]

In this quote the reader may observe a shift in terminology: nowhere is mentioned the notion of "propositional function", but rather one sees the words "formula", "predicate calculus", "predicate", and "logical calculus." This shift in terminology is discussed more in the section that covers "function" in contemporary set theory.

The history of the notion of "ordered pair" is not clear. As noted above, Frege (1879) proposed an intuitive ordering in his definition of a two-argument function Ψ(A, B).Norbert Wiener in his 1914 (see below) observes that his own treatment essentially "revert(s) toSchröder's treatment of a relation as a class of ordered couples".^[79]Russell (1903) considered the definition of a relation (such as Ψ(A, B)) as a "class of couples" but rejected it:

"There is a temptation to regard a relation as definable in extension as a class of couples. This is the formal advantage that it avoids the necessity for the primitive proposition asserting that every couple has a relation holding between no other pairs of terms. But it is necessary to give sense to the couple, to distinguish the referent [domain] from the relatum [converse domain]: thus a couple becomes essentially distinct from a class of two terms, and must itself be introduced as a primitive idea. . . . It seems therefore more correct to take an intensional view of relations, and to identify them rather with class-concepts than with classes."^[80]

By 1910–1913 andPrincipia Mathematica Russell had given up on the requirement for anintensional definition of a relation, stating that "mathematics is always concerned with extensions rather than intensions" and "Relations, like classes, are to be taken inextension".^[81] To demonstrate the notion of a relation inextension Russell now embraced the notion ofordered couple: "We may regard a relation ... as a class of couples ... the relation determined by φ(x, y) is the class of couples (x, y) for which φ(x, y) is true".^[82] In a footnote he clarified his notion and arrived at this definition:

"Such a couple has asense, i.e., the couple (x, y) is different from the couple (y, x) unlessx = y. We shall call it a "couple with sense," ... it may also be called anordered couple.^[82]

But he goes on to say that he would not introduce the ordered couples further into his "symbolic treatment"; he proposes his "matrix" and his unpopular axiom of reducibility in their place.

An attempt to solve the problem of theantinomies led Russell to propose his "doctrine of types" in an appendix B of his 1903The Principles of Mathematics.^[83] In a few years he would refine this notion and propose in his 1908The Theory of Types twoaxioms of reducibility, the purpose of which were to reduce (single-variable) propositional functions and (dual-variable) relations to a "lower" form (and ultimately into a completelyextensional form); he andAlfred North Whitehead would carry this treatment over toPrincipia Mathematica 1910–1913 with a further refinement called "a matrix".^[84] The first axiom is *12.1; the second is *12.11. To quote Wiener the second axiom *12.11 "is involved only in the theory of relations".^[85] Both axioms, however, were met with skepticism and resistance; see more atAxiom of reducibility. By 1914 Norbert Wiener, using Whitehead and Russell's symbolism, eliminated axiom *12.11 (the "two-variable" (relational) version of the axiom of reducibility) by expressing a relation as an ordered pair using the null set. At approximately the same time,Hausdorff (1914, p. 32) gave the definition of the ordered pair (a,b) as {{a,1}, {b, 2}}. A few years laterKuratowski (1921) offered a definition that has been widely used ever since, namely {{a,b}, {a}}".^[86] As noted bySuppes (1960) "This definition . . . was historically important in reducing the theory of relations to the theory of sets.^[87]

Observe that while Wiener "reduced" the relational *12.11 form of the axiom of reducibility hedid not reduce nor otherwise change the propositional-function form *12.1; indeed he declared this "essential to the treatment of identity, descriptions, classes and relations".^[88]

Where exactly thegeneral notion of "function" as a many-one correspondence derives from is unclear. Russell in his 1920Introduction to Mathematical Philosophy states that "It should be observed that all mathematical functions result form one-many [sic – contemporary usage is many-one] relations . . . Functions in this sense aredescriptive functions".^[89] A reasonable possibility is thePrincipia Mathematica notion of "descriptive function" –R 'y =_DEF (ιx)(x R y): "the singular object that has a relationR toy". Whatever the case, by 1924,Moses Schönfinkel expressed the notion, claiming it to be "well known":

"As is well known, by function we mean in the simplest case a correspondence between the elements of some domain of quantities, the argument domain, and those of a domain of function values ... such that to each argument value there corresponds at most one function value".^[90]

According toWillard Quine,Schönfinkel 1924 "provide[s] for ... the whole sweep of abstract set theory. The crux of the matter is that Schönfinkel lets functions stand as arguments. For Schönfinkel, substantially as for Frege, classes are special sorts of functions. They are propositional functions, functions whose values are truth values. All functions, propositional and otherwise, are for Schönfinkel one-place functions".^[91] Remarkably, Schönfinkel reduces all mathematics to an extremely compactfunctional calculus consisting of only three functions: Constancy, fusion (i.e., composition), and mutual exclusivity. Quine notes thatHaskell Curry (1958) carried this work forward "under the head ofcombinatory logic".^[92]

By 1925Abraham Fraenkel (1922) andThoralf Skolem (1922) had amended Zermelo's set theory of 1908. But von Neumann was not convinced that this axiomatization could not lead to the antinomies.^[93] So he proposed his own theory, his 1925An axiomatization of set theory.^[94] It explicitly contains a "contemporary", set-theoretic version of the notion of "function":

"[Unlike Zermelo's set theory] [w]e prefer, however, to axiomatize not "set" but "function". The latter notion certainly includes the former. (More precisely, the two notions are completely equivalent, since a function can be regarded as a set of pairs, and a set as a function that can take two values.)".^[95]

At the outset he begins withI-objects andII-objects, two objectsA andB that are I-objects (first axiom), and two types of "operations" that assume ordering as a structural property^[96] obtained of the resulting objects [x,y] and (x,y). The two "domains of objects" are called "arguments" (I-objects) and "functions" (II-objects); where they overlap are the "argument functions" (he calls them I-II objects). He introduces two "universal two-variable operations" – (i) the operation [x,y]: ". . . read 'the value of the functionx for the argumenty . . . it itself is a type I object", and (ii) the operation (x,y): ". . . (read 'the ordered pairx,y') whose variablesx andy must both be arguments and that itself produces an argument (x,y). Its most important property is thatx₁ =x₂ andy₁ =y₂ follow from (x₁ =y₂) = (x₂ =y₂)". To clarify the function pair he notes that "Instead off(x) we write [f,x] to indicate thatf, just likex, is to be regarded as a variable in this procedure". To avoid the "antinomies of naive set theory, in Russell's first of all . . . we must forgo treating certain functions as arguments".^[97] He adopts a notion from Zermelo to restrict these "certain functions".^[98]

Suppes^[99] observes that von Neumann's axiomatization was modified by Bernays "in order to remain nearer to the original Zermelo system . . . He introduced two membership relations: one between sets, and one between sets and classes". Then Gödel [1940]^[100] further modified the theory: "his primitive notions are those of set, class and membership (although membership alone is sufficient)".^[101] This axiomatization is now known asvon Neumann–Bernays–Gödel set theory.

In 1939, the collaborationNicolas Bourbaki, in addition to giving the well-known ordered pair definition of a function as a certain subset of thecartesian productE ×F, gave the following:

"LetE andF be two sets, which may or may not be distinct. A relation between a variable elementx ofE and a variable elementy ofF is called a functional relation iny if, for allx ∈E, there exists a uniquey ∈F which is in the given relation withx.We give the name of function to the operation which in this way associates with every elementx ∈E the elementy ∈F which is in the given relation withx, and the function is said to be determined by the given functional relation. Two equivalent functional relations determine the same function."

Both axiomatic and naive forms of Zermelo's set theory as modified by Fraenkel (1922) and Skolem (1922)define "function" as a relation,define a relation as a set of ordered pairs, anddefine an ordered pair as a set of two "dissymetric" sets.

While the reader ofSuppes (1960)Axiomatic Set Theory orHalmos (1970)Naive Set Theory observes the use of function-symbolism in theaxiom of separation, e.g., φ(x) (in Suppes) and S(x) (in Halmos), they will see no mention of "proposition" or even "first order predicate calculus". In their place are "expressions of the object language", "atomic formulae", "primitive formulae", and "atomic sentences".

Kleene (1952) defines the words as follows: "In word languages, a proposition is expressed by a sentence. Then a 'predicate' is expressed by an incomplete sentence or sentence skeleton containing an open place. For example, "___ is a man" expresses a predicate ... The predicate is apropositional function of one variable. Predicates are often called 'properties' ... The predicate calculus will treat of the logic of predicates in this general sense of 'predicate', i.e., as propositional function".^[102]

In 1954, Bourbaki, on p. 76 in Chapitre II of Theorie des Ensembles (theory of sets), gave a definition of a function as a triplef = (F,A,B).^[103] HereF is afunctional graph, meaning a set of pairs where no two pairs have the same first member. On p. 77 (op. cit.) Bourbaki states (literal translation): "Often we shall use, in the remainder of this Treatise, the wordfunction instead offunctional graph."

Suppes (1960) inAxiomatic Set Theory, formally defines arelation (p. 57) as a set of pairs, and afunction (p. 86) as a relation where no two pairs have the same first member.

The reason for the disappearance of the words "propositional function" e.g., inSuppes (1960), andHalmos (1970), is explained byTarski (1946) together with further explanation of the terminology:

"An expression such asx is an integer, which contains variables and, on replacement of these variables by constants becomes a sentence, is called a SENTENTIAL [i.e., propositional cf his index] FUNCTION. But mathematicians, by the way, are not very fond of this expression, because they use the term "function" with a different meaning. ... sentential functions and sentences composed entirely of mathematical symbols (and not words of everyday language), such as:x + y = 5 are usually referred to by mathematicians as FORMULAE. In place of "sentential function" we shall sometimes simply say "sentence" – but only in cases where there is no danger of any misunderstanding".^[104]

For his partTarski calls the relational form of function a "FUNCTIONAL RELATION or simply a FUNCTION".^[105] After a discussion of this "functional relation" he asserts that:

"The concept of a function which we are considering now differs essentially from the concepts of a sentential [propositional] and of a designatory function .... Strictly speaking ... [these] do not belong to the domain of logic or mathematics; they denote certain categories of expressions which serve to compose logical and mathematical statements, but they do not denote things treated of in those statements... . The term "function" in its new sense, on the other hand, is an expression of a purely logical character; it designates a certain type of things dealt with in logic and mathematics."^[106]

See more about "truth under an interpretation" atAlfred Tarski.

• Boole, George (1854).An Investigation into the Laws of Thought on which are founded the Laws of Thought and Probabilities. Walton and Marberly.
• De Morgan, Augustus (1847).Formal Logic, or The Calculus of Inference, Necessary and Probable. Walton and Marberly.
• Dedekind, Richard; Pogorzelski, H.; Ryan, W.; Snyder, W. (1995).What are Numbers and What Should They Be?. Research Institute for Mathematics.
• Dieudonné, Jean (1992).Mathematics-The Music of Reason. Springer-Verlag.
• Dirichlet, G. P. Lejeune (1889).Gesammelte Werke, Bd. I. Berlin.ISBN 9780828402255.{{cite book}}:ISBN / Date incompatibility (help)CS1 maint: location missing publisher (link)
• Edwards, Harold M. (2007)."Euler's definition of the derivative".Bulletin of the American Mathematical Society.44 (4):575–580.doi:10.1090/s0273-0979-07-01174-3.MR 2338366.
• Euler, Leonhard (1988).Introduction to Analysis of the Infinite. Book I. Springer-Verlag.
• Euler, Leonhard (2000).Foundations of Differential Calculus. Springer-Verlag.
• Eves, Howard (1990).Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover.ISBN 0-486-69609-X.
• Fourier, Joseph (1822).Théorie analytique de la chaleur. Paris: Firmin Didot Père et Fils.
• Grattan-Guinness, Ivor; Bornet, Gérard (1997).George Boole: Selected Manuscripts on Logic and its Philosophy. Springer-Verlag.ISBN 3-7643-5456-9.
• Halmos, Paul (1970).Naive Set Theory. New York, Springer-Verlag.ISBN 9780387900926.
• Hardy, Godfrey Harold (1908).A Course of Pure Mathematics.Cambridge University Press (published 1993).ISBN 978-0-521-09227-2.{{cite book}}:ISBN / Date incompatibility (help)
• Kleene, Stephen Cole (1952).Introduction to Metamathematics. North-Holland (published 1971).ISBN 978-0-7204-2103-3.{{cite book}}:ISBN / Date incompatibility (help)
• Lavine, Shaughan (1994).Understanding the Infinite. Harvard University Press.
• Lobachevsky, Nikolai (1951).Works. Moscow-Leningrad.{{cite book}}: CS1 maint: location missing publisher (link)
• Luzin, N. (1998). "Function: Part II".The American Mathematical Monthly.105 (3):263–270.doi:10.2307/2589085.JSTOR 2589085.
• Medvedev, Fyodor A. (1991).Scenes from the History of Real Functions. Birkhauser.ISBN 9780817625726.
• Ponte, João Pedro (1992)."The history of the concept of function and some educational implications".The Mathematics Educator.3 (2):3–8.
• Russell, Bertrand (1903).The Principles of Mathematics. Cambridge University Press.
• Russell, Bertrand (1920).Introduction to Mathematical Philosophy (2nd ed.). Dover.ISBN 0-486-27724-0.{{cite book}}:ISBN / Date incompatibility (help)
• Smithies, Frank (1997).Cauchy and the Creation of Complex Function Theory. Cambridge University Press.
• Suppes, Patrick (1960).Axiomatic Set Theory (1972 ed.). Dover.ISBN 0-486-61630-4.{{cite book}}:ISBN / Date incompatibility (help) cf. hisChapter 1 Introduction.
• Tarski, Alfred (1946).Introduction to Logic and to the Methodology of Deductive Sciences (1995 ed.). Courier Dover.ISBN 0-486-28462-X.{{cite book}}:ISBN / Date incompatibility (help)
• Venn, John (1881).Symbolic Logic. Macmillan.
• van Heijenoort, Jean (1976) [1967].From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931 (3rd printing ed.). Harvard University Press.ISBN 0-674-32449-8.
  • ——;Frege, Gottlob (1967) [1879]. "Frege (1879)Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought".ibid. pp. 1–82. With commentary by van Heijenoort.
  • ——;Peano, Giuseppe (1967) [1889]. "Peano (1889)The principles of arithmetic, presented by a new method".ibid. pp. 83–97. With commentary by van Heijenoort.
  • ——;Russell, Bertrand (1967) [1902]. "Russell (1902)Letter to Frege".ibid. pp. 124–125. With commentary by van Heijenoort. Wherein Russell announces his discovery of a "paradox" in Frege's work.
  • ——;Frege, Gottlob (1967) [1902]. "Frege (1902)Letter to Russell".ibid. pp. 126–128. With commentary by van Heijenoort.
  • ——;Hilbert, David (1967) [1904]. "Hilbert (1904)On the foundations of logic and arithmetic".ibid. pp. 129–138. With commentary by van Heijenoort.
  • ——;Richard, Jules (1967) [1905]. "Richard (1905)The principles of mathematics and the problem of sets".ibid. pp. 142–144. With commentary by van Heijenoort. TheRichard paradox.
  • ——;Russell, Bertrand (1967) [1908a]. "Russell (1908a)Mathematical logic as based on the theory of types".ibid. pp. 150–182. With commentary byWillard Quine.
  • ——;Zermelo, Ernst (1967) [1908]. "Zermelo (1908)A new proof of the possibility of a well-ordering".ibid. pp. 183–198. With commentary by van Heijenoort. Wherein Zermelo rails against Poincaré's (and therefore Russell's) notion ofimpredicative definition.
  • ——;Zermelo, Ernst (1967) [1908a]. "Zermelo (1908a)Investigations in the foundations of set theory I".ibid. pp. 199–215. With commentary by van Heijenoort. Wherein Zermelo attempts to solve Russell's paradox by structuring his axioms to restrict the universal domain B (from which objects and sets are pulled bydefinite properties) so that it itself cannot be a set, i.e., his axioms disallow a universal set.
  • ——;Whitehead, Alfred North;Russell, Bertrand (1967) [1910]. "Whitehead and Russell (1910)Incomplete symbols: Descriptions".ibid. pp. 216–223. With commentary byW. V. Quine.
  • ——;Wiener, Norbert (1967) [1914]. "Wiener (1914)A simplification of the logic of relations".ibid. pp. 224–227. With commentary by van Heijenoort.
  • ——;Skolem, Thoralf (1967) [1922]. "Skolem (1922)Some remarks on axiomatized set theory".ibid. pp. 290–301. With commentary by van Heijenoort. Wherein Skolem defines Zermelo's vague "definite property".
  • ——;Schönfinkel, Moses (1967) [1924]. "Schönfinkel (1924)On the building blocks of mathematical logic".ibid. pp. 355–366. With commentary byWillard Quine. The start ofcombinatory logic.
  • ——;von Neumann, John (1967) [1925]. "von Neumann (1925)An axiomatization of set theory".ibid. pp. 393–413. With commentary by van Heijenoort. Wherein von Neumann creates "classes" as distinct from "sets" (the "classes" are Zermelo's "definite properties"), and now there is a universal set, etc.
  • ——;Hilbert, David (1967) [1927]. "Hilbert(1927)The foundations of mathematics".ibid. pp. 464–479. With commentary by van Heijenoort.
• Whitehead, Alfred North;Russell, Bertrand (1913).Principia Mathematica to *56 (1962 ed.). Cambridge University Press.ISBN 978-0-521-62606-4.{{cite book}}:ISBN / Date incompatibility (help)

• Dubinsky, Ed; Harel, Guershon (1992).The Concept of Function: Aspects of Epistemology and Pedagogy. Mathematical Association of America.ISBN 0-88385-081-8.
• Frege, Gottlob (1879).Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle.
• Kleiner, Israel (1989)."Evolution of the Function Concept: A Brief Survey".The College Mathematics Journal.20 (4). Mathematical Association of America:282–300.doi:10.2307/2686848.JSTOR 2686848.
• Lützen, Jesper (2003)."Between rigor and applications: Developments in the concept of function in mathematical analysis". In Roy Porter (ed.).The Cambridge History of Science: The modern physical and mathematical sciences. Cambridge University Press.ISBN 0521571995. An approachable and diverting historical presentation.
• Malik, M. A. (1980). "Historical and pedagogical aspects of the definition of function".International Journal of Mathematical Education in Science and Technology.11 (4):489–492.doi:10.1080/0020739800110404.
• Monna, A. F. (1972). "The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue".Archive for History of Exact Sciences.9 (1):57–84.doi:10.1007/BF00348540.S2CID 120506760.
• Reichenbach, Hans (1947)Elements of Symbolic Logic, Dover Publishing Inc., New York NY,ISBN 0-486-24004-5.
• Ruthing, D. (1984). "Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N.".Mathematical Intelligencer.6 (4):72–77.doi:10.1007/BF03026743.S2CID 189883712.
• Youschkevitch, A. P. (1976). "The concept of function up to the middle of the 19th century".Archive for History of Exact Sciences.16 (1):37–85.doi:10.1007/BF00348305.S2CID 121038818.

• Functions fromcut-the-knot.

• Functions and mappings
• Basic concepts in set theory
• History of mathematics
• History of logic

• Articles with short description
• Short description is different from Wikidata
• CS1 errors: ISBN date
• CS1 maint: location missing publisher