Marx's Mathematical Manuscripts 1881

# ‘Third Draft’

Written: August, 1881;
Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;
First published: in Russian translation, in Pod znamenem marksizma, 1933.

If we now consider the differential of y in its general form, dy = f’(x)dx, then we already have before us a purely symbolic operational equation, even in the case where f’(x) from the very beginning is a constant, as in dy = d(ax) = adx. This child of 0/0 or dy/dx = f’(x) looks suspiciously like its mother. For in dy/dx = 0/0 numerator and denominator are inseparably bound together; in dy = f’(x)dx they are obviously separated, so that one is forced to the conclusion: dy = f’(x)dx is only a masked expression for 0 = f’(x)ยท0, thus 0 = 0 with which ‘nothing’s to be done’ (‘nichts zu wolle’). Looking more closely, analysts in our century, such as, for example, the Frenchman Boucharlat, smell a rat here too. He says:*

In ‘dy/dx = 3x², for example, 0/0 alias dy/dx, or even more its value 3x², is the differential coefficient of the function y. Since dy/dx is thus the symbol which represents the limit 3x², dx must always stand under dy but, in order to facilitate algebraic operation we treat dy/dx as an ordinary fraction and dy/dx = 3x² as an ordinary equation, and thus by removing the denominator dx from the equation obtain the result dy = 3x²dx, which expression is called the differential of y.’43

In order to ‘facilitate algebraic operation’, we thus introduce a false formula.

In fact the thing (Sache) doesn’t behave that way. In 0/0 (usually written (0/0)), the ratio of the minimal expression (Minimalausdrucks) of y1 - y, or of f(x1) - f(x), or of the increment of f(x), to the minimal expression of x1 - x, or to the increment of the independent variable quantity x, possesses a form in which the numerator is inseparable from the denominator. But why? In order to retain 0/0 as the ratio of vanished differences. As soon, however, as x1 - x = 0 obtains in dx a form which manifests it as the vanished difference of x, and thus y1 - y = 0 appears as dy as well, the separation of numerator and denominator becomes a completely permissible operation. Where dx now stands its relationship with dy remains undisturbed by this change of position. dy = df(x), and thus = f’(x)dx, is only another expression for dy/dx [ = f’(x) ], which must lead to the conclusion that f’(x) is obtained independently. How useful this formula dy = df(x) immediately becomes as an operational formula (Operationsformel), however, is shown, for example, by:

y² = ax ,

d(y²) = d(ax) ,

so that

y² = ax, [thus] = 2ax/a = 2x ;

so that 2x, double the abscissa, is the value of the subtangent of the usual parabola.

However, if dy = df(x) serves as the first point of departure (Ausgangspunkt), which only later is developed into dy/dx itself, then, for this differential of dy to have any meaning at all, these differentials dy, dx must be assumed to be symbols with a defined meaning. Had such assumptions not originated from mathematical metaphysics but instead been derived quite directly from a function of the first degree, such as y = ax, then, as seen earlier, this leads to (y1 - y)/(x1 - x) = a, which is transformed to dy/dx = a. From here as well, however, nothing certain is to be got a priori. For since Δy/Δx is just as much = a as dy/dx = a, and the, Δx, Δy, although finite differences or increments, are yet finite differences or increments of unlimited capacity to contract (Kontraktionsfähigkeit), one then may just as well represent dx, dy as infinitely small quantities capable of arbitrarily approaching 0, as if they originate from actually setting the equality x1 - x = 0, and thus as well y1 - y = 0. The result remains identical on the right-hand side in both cases, because there in itself there is no x1 at all to set = x, and thus as well no x1 - x = 0. This substitution = 0 on the other side consequently appears just as arbitrary an hypothesis as the assumptions that dx, dy are arbitrarily small quantities. Under (sub) IV) I will briefly indicate the historical development through the example of d(uz), but yet prior to that will give an example under (sub) III)44 which is treated the first time on the ground of symbolic calculus, with a ready-made operational formula (fertigen Operationsformel), and is demonstrated a second time algebraically. Enough (soviel) has been shown under (sub) II), so that the latter method alone, by means of its application to so elementary a function as the product of two variables, using its own results, necessarily leads to starting points (die Ausgangspunkte) which are the opposite pole as far as operating method goes.