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Unformatted text preview: Evolution of the Function Concept: A Brief Survey Israel Kleiner The College Mathematics Journal, September 1989, Volume 20, Number 4, pp. 282300. Israel Kleiner received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the history of mathematics and its use in the teaching 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 intimate connection with problems in calculus and analysis. (A one-sentence definition of analysis as the study of properties of various classes of functions would not be far off the 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 step further: The keynote of Western culture is the function concept, a notion not even remotely hinted at by any earlier culture. And the function concept is anything but an extension or elaboration of previous number conceptsit is rather a complete emancipation from such notions. T he evolution of the function concept can be seen as a tug of war between two elements, two mental images: the geometric (expressed in the form of a curve) and the algebraic (expressed as a formulafirst finite and later allowing infinitely many terms, the so-called analytic expression). (See [7, p. 256].) Subsequently, a third element enters, namely, the logical definition of function as a correspondence (with a mental image of an input-output machine). In the wake of this development, the geometric 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 the excitement and the challenge that some of the best mathematicians of all time confronted in trying to come to grips with the basic conception of function that we now accept as commonplace. 1. Precalculus Developments. The notion of function in explicit form did not emerge until the beginning of the 18th century, although implicit manifestations of the concept date back to about 2000 B . C . The main reasons that the function concept did not emerge earlier were: lack of algebraic prerequisitesthe coming to terms with the continuum of real numbers, and the development of symbolic notation; lack of motivation. Why define an abstract notion of function unless one had many examples from which to abstract? In the course of about two hundred years (ca. 14501650), there occurred a number ofIn the course of about two hundred years (ca....
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