What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§4. LEIBNIZ’ NOTATION AND THE “INFINITELY SMALL”

Newton and Leibniz knew how to obtain the integral and the derivative as limits. But the very foundations of the calculus were long obscured by an unwillingness to recognize the exclusive right of the limit concept as the source of the new methods. Neither Newton nor Leibniz could bring himself to such a clear-cut attitude, simple as it appears to us now that the limit concept has been completely clarified. Their example dominated more than a century of mathematical development during which the subject was shrouded by talk of “infinitely small quantities,” “differentials,” “ultimate ratios,” etc. The reluctance with which these concepts were finally abandoned was deeply rooted in the philosophical attitude of the time and in the very nature of the human mind. One might have argued: “Of course integral and derivative can be and are calculated as limits. But what, after all, are these objects in themselves, irrespective of the particular way they are described by limiting processes? It seems obvious that intuitive concepts such as area or slope of a curve have an absolute meaning in themselves without any need for the auxiliary concepts of inscribed polygons or secants and their limits.” Indeed, it is psychologically natural to search for adequate definitions of area and slope as “things in themselves.” But to renounce this desire and rather to see in limiting processes their only scientifically relevant definitions, is in line with the mature attitude that has so often cleared the way for progress. In the seventeenth century there was no intellectual tradition to permit such philosophical radicalism.

Leibniz’ attempt to “explain” the derivative started in a perfectly correct way with the difference quotient of a function y = f(x),

For the limit, the derivative, which we called f′(x) (following the usage introduced later by Lagrange), Leibniz wrote

replacing the difference symbol Δ by the “differential symbol” d. Provided we understand that this symbol is solely an indication that the limiting process Δx → 0 and consequently Δ y→ 0 is to be carried out, there is no difficulty and no mystery. Before passing to the limit, the denominator Δx in the quotient Δy/Δx is cancelled out or transformed in such a way that the limiting process can be completed smoothly. This is always the crucial point in the actual process of differentiation. Had we tried to pass to the limit without such a previous reduction, we should have obtained the meaningless relation Δy/Δx= 0/0, in which we are not at all interested. Mystery and confusion only enter if we follow Leibniz and many of his successors by saying something like this:

“Δx does not approach zero. Instead, the ’last value’ of Δx is not zero but an ’infinitely small quantity,’ a ’differential’ called dx; and similarly Δy has a ’last’ infinitely small value dy. The actual quotient of these infinitely small differentials is again an ordinary number, f′(x) = dy/dx.” Leibniz accordingly called the derivative the “differential quotient.” Such infinitely small quantities were considered a new kind of number, not zero but smaller than any positive number of the real number system. Only those with a real mathematical sense could grasp this concept, and the calculus was thought to be genuinely difficult because not everybody has, or can develop, this sense. In the same way, the integral was considered to be a sum of infinitely many “infinitely small quantities” f(x) dx. Such a sum, people seemed to feel, is the integral or area, while the calculation of its value as the limit of a finite sum of ordinary numbers f(x_i)Δx was regarded as something accessory. Today we simply discard the desire for a “direct” explanation and define the integral as the limit of a finite sum. In this way the difficulties are dispelled and everything of value in the calculus is secured on a sound basis.

In spite of this later development Leibniz’ notation, dy/dx for f′(x) and ∫ f(x) dx for the integral, was retained and has proved extremely useful. There is no harm in it if we consider the symbols d only as symbols for a passage to the limit. Leibniz’ notation has the advantage that limits of quotients and sums can in some ways be handled “as if” they were actual quotients or sums. The suggestive power of this symbolism has always tempted people to impute to these symbols some entirely unmathematical meaning. If we resist this temptation, then Leibniz’ notation is at least an excellent abbreviation for the more cumbersome explicit notation of the limit process; as a matter of fact, it is almost indispensable in the more advanced parts of the theory.

For example, rule (d) of page 429 for differentiating the inverse function x = g(y) of y = f(x) was that g′(y)f′(x) = 1. In Leibniz’ notation it reads simply

“as if” the “differentials” may be cancelled out from something like an ordinary fraction. Likewise, rule (e) of page 431 for differentiating a compound function z = k(x), where

z = g(y), y = f(x),

now reads

Leibniz’ notation has the further advantage of emphasizing the quantities x, y, z rather than their explicit functional connection. The latter expresses a procedure, an operation producing one quantity y from another x, e.g. the function y = f(x) = x² produces a quantity y equal to the square of the quantity x. The operation (squaring) is the object of the mathematician’s attention. But physicists and engineers are on the whole primarily interested in the quantities themselves. Hence the emphasis on quantities in Leibniz’ notation has a particular appeal to people engaged in applied mathematics.

Another remark may be added. While “differentials” as infinitely small quantities are now definitely and dishonorably discarded, the same word “differential” has slipped in again through the back door— this time to denote a perfectly legitimate and useful concept. It now means simply a difference Δx when Δx is small in relation to the other quantities occurring. We cannot here go into a discussion of the value of this concept for approximate calculations. Nor can we discuss other legitimate mathematical notions for which the name “differential” has been adopted, some of which have proved quite useful in the calculus and in its applications to geometry.