History of the function concept

The mathematical concept of a function dates from the 17th century in connection with the development of calculus; for example, the slope of a graph at a point was regarded as a function of the x-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 as Oresme.
Mathematicians of the 18th century typically regarded a function as being defined by an analytic expression. In the 19th century, the demands of the rigorous development of analysis by Karl Weierstrass and others, the reformulation of geometry in terms of analysis, and the invention of set theory by Georg Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set to another.

Functions before the 17th century

In the 12th century, mathematician Sharaf al-Din al-Tusi analyzed the equation in the form stating that the left hand side must at least equal the value of 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 than means no positive solution; a value equal to corresponds to one solution, while a value greater than corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in Islamic mathematics, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe.
According to Jean Dieudonné and Ponte, the concept of a function emerged in the 17th century as a result of the development of analytic geometry and the infinitesimal calculus. Nevertheless, Medvedev suggests that the implicit concept of a function is one with an ancient lineage. Ponte also sees more explicit approaches to the concept in the Middle Ages:
The development of analytical geometry around 1640 allowed mathematicians to go between geometric problems about curves and algebraic relations between "variable coordinates x and y." Calculus was developed using the notion of variables, with their associated geometric meaning, which persisted well into the eighteenth century. However, the terminology of "function" came to be used in interactions between Leibniz and Bernoulli towards the end of the 17th century.

Notion of function in analysis

The term "function" was literally introduced by Gottfried Leibniz, in a 1673 letter, to describe a quantity related to points of a curve, such as a coordinate or curve's slope. 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 of x. By 1718, he came to regard as a function "any expression made up of a variable and some constants." Alexis Claude Clairaut and Leonhard Euler introduced the familiar notation for the value of a function.
The functions considered in those times are called today differentiable functions. For this type of function, one can talk about limits 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 of calculus.

Euler

In the first volume of his fundamental text Introductio in analysin infinitorum, published in 1748, Euler gave essentially the same definition of a function as his teacher Bernoulli, as an expression or formula involving variables and constants e.g.,. Euler's own definition reads:
Euler also allowed multi-valued functions whose values are determined by an implicit equation.
In 1755, however, in his Institutiones calculi differentialis, Euler gave a more general concept of a function:
Medvedev considers that "In essence this is the definition that became known as Dirichlet's definition." Edwards also credits Euler with a general concept of a function and says further that

Fourier

In his Théorie Analytique de la Chaleur, Joseph Fourier claimed that an arbitrary function could be represented by a Fourier series. Fourier had a general conception of a function, which included functions that were neither continuous nor defined by an analytical expression. Related questions on the nature and representation of functions, arising from the solution of the wave equation for a vibrating string, had already been the subject of dispute between Jean le Rond d'Alembert and Euler, and they had a significant impact in generalizing the notion of a function. Luzin observes that:

Cauchy

During the 19th century, mathematicians started to formalize all the different branches of mathematics. One of the first to do so was Augustin-Louis Cauchy; his somewhat imprecise results were later made completely rigorous by Weierstrass, who advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's. According to Smithies, Cauchy thought of functions as being defined by equations involving real or complex numbers, and tacitly assumed they were continuous:

Lobachevsky and Dirichlet

and Peter Gustav Lejeune Dirichlet are traditionally credited with independently giving the modern "formal" definition of a function as a relation in which every first element has a unique second element.
Lobachevsky writes that
while Dirichlet writes
Eves asserts that "the student of mathematics usually meets the Dirichlet definition of function in his introductory course in calculus.
Dirichlet's claim to this formalization has been disputed by Imre Lakatos:
However, Gardiner says
"...it seems to me that Lakatos goes too far, for example, when he asserts that 'there is ample evidence that had no idea of concept'."
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 he states it only for continuous functions of a real variable.
Similarly, Lavine observes that:
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. A general version of this definition was later used by Bourbaki, and some in the education community refer to it as the "Dirichlet–Bourbaki" definition of a function.

Dedekind

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 work
Was sind und was sollen die Zahlen, which appeared in 1888 but had already been drafted in 1878. Dieudonné observes that instead of confining himself, as in previous conceptions, to real functions, Dedekind defines a function as a single-valued mapping between any two sets:

Hardy

defined a function as a relation between two variables x and y such that "to some values of x at any rate correspond values of y." He neither required the function to be defined for all values of x nor to associate each value of x 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 includes multivalued functions and what in computability theory are called partial functions.

Logicians' function

Prior to 1850

s of this time were primarily involved with analyzing syllogisms, or as Augustus De Morgan 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". At this time the notion of "function" is not explicit, but at least in the work of De Morgan and George 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 " logical truth depends upon the structure of the statement, and not upon the particular matters spoken of"; he wastes no time abstracting: "In the form of the proposition, the copula is made as abstract as the terms". He immediately casts what he calls "the proposition" into a form such as "X is Y", where the symbols X, "is", and Y represent, respectively, the subject, copula, and predicate. 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" is there, and lastly a new symbolism for logical analysis of the notion of "relation" Y " is there:
In his 1848 The 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". Like De Morgan he uses the notion of "variable" drawn from analysis; he gives an example of "represent the class oxen by x and that of horses by y and the conjunction and by the sign +... we might represent the aggregate class oxen and horses by x + y".
In the context of "the Differential Calculus" Boole defined the notion of a function as follows:

Logicians' function 1850–1950

Eves observes "that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory of sets, or classes, from a foundation in the logic of propositions and propositional functions". 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, the Logicists, can probably be summed up best by – "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 with Georg Cantor's "set theory" 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. Ernst Zermelo's set-theoretic response was his 1908 Investigations in the foundations of set theory I – the first axiomatic set theory; here too the notion of "propositional function" plays a role.