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 (, 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 concepts—it is rather a complete emancipation from such notions.
The evolution of the function concept can be seen as a tug-of-war between two elements, twomental images: the geometric (expressed in the form of a curve) and the algebraic (expressed as a formula—first finite and later allowing infinitely many terms, the so-called analytic expression). (See , 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.