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1.1. A brief survey of its evolution and development
Israel Kleiner received his Ph.D. in ring theory at McGill University, and has been atYork University for over twenty years. He has been involved in teacher education at theundergraduate and graduate levels and has given numerous talks to high school studentsand teachers. One of his major interests is the history of mathematics and its use in theteaching of mathematics.Introduction. The evolution of the concept of function goes back 4000 years; 3700 of these consist of anticipations. The idea evolved for close to 300 years in intimateconnection with problems in calculus and analysis. (A one-sentence definition ofanalysis as the study of properties of various classes of functions would not be far offthe mark.) In fact, the concept of function is one of the distinguishing features of“modern” as against “classical” mathematics. W. L. Schaaf [24, p. 500] goes a stepfurther:The keynote of Western culture is the function concept, a notion not evenremotely hinted at by any earlier culture. And the function concept isanything but an extension or elaboration of previous number concepts—itis rather a complete emancipation from such notions.The evolution of the function concept can be seen as a tug of war between twoelements, two mental images: the geometric (expressed in the form of a curve)and the algebraic (expressed as a formula—first finite and later allowing infinitelymany terms, the so-called “analytic expression”). (See [7, p. 256].) Subsequently, a thirdelement enters, namely, the “logical” definition of function as a correspondence (with amental image of an input-output machine). In the wake of this development, thegeometric conception of function is gradually abandoned. A new tug of war soon ensues(and is, in one form or another, still with us today) between this novel “logical”(“abstract,” “synthetic,” “postulational”) conception of function and the old “algebraic”(“concrete,” “analytic,” “constructive”) conception.In this article, we will elaborate these points and try to give the reader a sense of theexcitement and the challenge that some of the best mathematicians of all timeconfronted in trying to come to grips with the basic conception of function that we nowaccept as commonplace.1. Precalculus Developments. The notion of function in explicit form did not emergeuntil the beginning of the 18th century, although implicit manifestations of the conceptdate back to about 2000 B.C. The main reasons that the function concept did not emergeearlier were:• lack of algebraic prerequisites—the coming to terms with the continuum of realnumbers, and the development of symbolic notation;• lack of motivation. Why define an abstract notion of function unless one had manyexamples from which to abstract?
In the course of about two hundred years (ca. 1450–1650), there occurred a number ofdevelopments that were fundamental to the rise of the function concept:• Extension of the concept of number to embrace real and (to some extent) evencomplex numbers (Bombelli, Stifel, et al.);• The creation of a symbolic algebra (Viète, Descartes, et al.);• The study of motion as a central problem of science (Kepler, Galileo, et al.);• The wedding of algebra and geometry (Fermat, Descartes, et al.).The 17th century witnessed the emergence of modern mathematized science and theinvention of analytic geometry. Both of these developments suggested a dynamic,continuous view of the functional relationship as against the static, discrete view held by the ancients.In the blending of algebra and geometry, the key elements were the introduction ofvariables and the expression of the relationship between variables by means ofequations. The latter provided a large number of examples of curves (potentialfunctions) for study and set the final stage for the introduction of the function concept.What was lacking was the identification of the independent and dependent variables inan equation:Variables are not functions. The concept of function implies a unidirectionalrelation between an “independent” and a “dependent” variable. But in the case ofvariables as they occur in mathematical or physical problems, there need not besuch a division of roles. And as long as no special independent role is given to oneof the variables involved, the variables are not functions but simply variables [2, p. 348].See [6], [15], [27] for details.The calculus developed by Newton and Leibniz had not the form that students see today.In particular, it was not a calculus of functions. The principal objects of study in 17th-century calculus were (geometric) curves. (For example, the cycloid was introducedgeometrically and studied extensively well before it was given as an equation.) In fact,17th-century analysis originated as a collection of methods for solving problems aboutcurves, such as finding tangents to curves, areas under curves, lengths of curves, andvelocities of points moving along curves. Since the problems that gave rise to thecalculus were geometric and kinematic in nature, and since Newton and Leibniz werepreoccupied with exploiting the marvelous tool that they had created, time and reflectionwould be required before the calculus could be recast in algebraic form.The variables associated with a curve were geometric—abscissas, ordinates, subtangents,subnormals, and the radii of curvature of a curve. In 1692, Leibniz introduced the word“function” (see [25, p. 272]) to designate a geometric object associated with a curve. For example, Leibniz asserted that “a tangent is a function of a curve” [12 p. 85].Newton’s “method of fluxions” applies to “fluents,” not functions. Newton calls hisvariables “fluents”—the image (as in Leibniz) is geometric, of a point “flowing” along a curve. Newton’s major contribution to the development of the function concept was his use of power series. These were important for the subsequent development of that concept.2
As increased emphasis came to be placed on the formulas and equations relating thefunctions associated with a curve, attention was focused on the role of the symbolsappearing in the formulas and equations and thus on the relations holding among thesesymbols, independent of the original curve. The correspondence (1694–1698) betweenLeibniz and Johann Bernoulli traces how the lack of a general term to representquantities dependent on other quantities in such formulas and equations brought aboutthe use of the term “function” as it appears in Bernoulli’s definition of 1718 (see[3, p. 9]and [27, p. 57] for details):One calls here Function of a variable a quantity composed in any mannerwhatever of this variable and of constants [23, p. 72].This was the first formal definition of function, although Bernoulli did not explain what“composed in any manner whatever” meant. See [3], [6], [12], [27] for details of thissection.2. Euler’s Introductio in Analysin Infinitorum.In the first half of the 18th century,we witness a gradual separation of 17th-century analysis from its geometric origin andbackground. This process of “degeometrization of analysis” [2, p. 345] saw thereplacement of the concept of variable, applied to geometric objects, with the concept offunction as an algebraic formula. This trend was embodied in Euler’s classic Introductioin Analysin Infinitorum of 1748, intended as a survey of the concepts and methods ofanalysis and analytic geometry needed for a study of the calculus.Euler’s Introductio was the first work in which the concept of function plays an explicitand central role. In the preface, Euler claims that mathematical analysis is the generalscience of variables and their functions. He begins by defining a function as an “analyticexpression” (that is, a “formula”):A function of a variable quantity is an analytical expression composed in anymanner from that variable quantity and numbers or constant quantities [23, p. 72].Euler does not define the term “analytic expression,” but tries to give it meaning byexplaining that admissible “analytic expressions” involve the four algebraic operations,roots, exponentials, logarithms, trigonometric functions, derivatives, and integrals. Heclassifies functions as being algebraic or transcendental; single-valued or multivalued;and implicit or explicit. The Introductio contains one of the earliest treatments oftrigonometric functions as numerical ratios (see [13]), as well as the earliest algorithmictreatment of logarithms as exponents. The entire approach is algebraic. Not a singlepicture or drawing appears (in v. 1).Expansions of functions in power series play a central role in this treatise. In fact,Euler claims that any function can be expanded in a power series: “If anyone doubtsthis, this doubt will be removed by the expansion of every function” [3, p. 10]. Thisremark was certainly in keeping with the spirit of mathematics in the 18th century. Hawkins [10, p. 3] summarizes Euler’s contribution to the emergence of function as an important concept:This term, which will appear often throughout this paper, was formally defined only in the late19th century (see sec. 7).Youschkevitch [27, p. 54] claims that “because of power series the concept of function as analytic expression occupied the central place in mathematical analysis.”