Marx's Mathematical Manuscripts 1881

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 *y _{1} - y*, or of

*y² = ax ,*

*d(y²) = d(ax) ,*

*2ydy = adx ;*

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 *(y _{1} - y)/(x_{1} - x) = a*, which is transformed to

To *(ad)* IV.

Finally *(following Lagrange)* it is to be noted that the *limit* or the *limit value*, which is already occasionally found in Newton for the differential coefficients and which he still derives from purely geometric considerations *(Vorstellungen)*, still to this day always plays a predominant role, whether the symbolic expressions appear *(figurieren)* as the limit of *f’(x)* or conversely *f’(x)* appears as the limit of the symbol or the two appear together as limits. This category, which Lacroix in particular analytically broadened, only becomes important as a substitute for the category ‘minimal expression’, whether it is of the derivative as opposed to the ‘preliminary derivative’, or of the ratio *(y _{1} - y)/(x_{1} - x)*, when the application of calculus to curves is treated. It is

^{*}
This is a translation of Marx’s German translation of a passage originally in French - *Trans*.

^{43}
The citation is from the book of Boucharlat (see, for example, J.-L. Boucharlat, 7th edition, Paris, 1858, pp.3-4).

^{44}
Here Marx projects a somewhat different enumeration of the sections of his work from that which he had followed earlier. In Section III he plans to locate materials which in the second draft were located in Section II; in Section IV, to comment on the historical development of differential calculus by means of the example of the history of the theorem on the differential of a product.

^{45}
In connection with this paragraph see note 5 as well as Appendix I, ‘On the Concept of “Limit” in the Sources Cited by Marx’, p.151 (where there is a discussion of how in Boucharlat’s textbook both sides of the equation *dy/dx = f’(x)* are treated as limits) and pp.152-153 (where the discussion is about the concept of limit in Lacroix’s long Traité and Marx’s related concept of the word in this paragraph). Exactly what Marx had in mind in his treatment of the symbolic expression as the limit of *f’(x)* remains unclear. (Perhaps he simply had in mind the fact that the derivative was obtained as a result of the supposition that *x _{1} = x*, that is, when the numerator and denominator of the ratio