I’ve seen a screenshot floating around on twitter many times and it always bothers me when I see it.

Personally, I think it’s a poor representation of Marx’s abilities in mathematics and is often used to demean Marxists as an attempt to smugly assert Marx as an idiot. It ignores the historical nuance of when the manuscripts were written and leaves half the story out.

On "On the Concept of the Derived Function"

The most pressing issue is the conclusion most people point out in the picture, is that $0/0 = a = dy/dx$. While yes, Marx did write this, the context gives a more reasonable interpretation of his thinking. In particular, Marx was critiquing the use of mystical infinitesimals used often in Leibniz’s notation. These infinitesimals are defined as a number smaller than any other real number, but not zero; however, it could be treated as zero when needed during computation. It’s this ambiguous nature that led many people to be skeptical of infinitesimal calculus, most notably Bishop Berkeley. In his book, The Analyst, Berkeley called infinitesimals (and could be interpreted to extend to differentials) “ghosts of departed quantities,” and went as far as to say:

It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

Here, he questions the nature of Fluxions (Newton’s derivatives) and “evanescent increments” (infinitesimals), calling them ill-defined quantities. He did acknowledge that Fluxions and increments as tools to get a final (correct) value, but due to the ambiguous nature, felt they were unstable. Marx took a similar approach, however less mathematical and perhaps more metaphysical. He thought that infinitesimals’ ambiguous nature caused a natural contradiction and it played into his ideas of thesis, antithesis, and synthesis. Footnote 5 from the manuscript’s upload on marxists.org puts it plainly. $0/0$ is handled by predefinition by continuity. $\Delta y / \Delta x$ equals the preliminary derivative for all $\Delta x \neq 0$, and $0/0$ is predefined by continuity. Therefore, there is no actual division of zero by zero to be used as an argument.

This philosophical lens is the key to understanding Marx’s critique. The infinitesimal embodies a fundamental dialectical contradiction. Marx’s method applies thesis, antithesis, and synthesis to this process. The ratio of finite differences $\Delta y / \Delta x$ serves as the thesis. The ambiguous transition where the increment vanishes creates the anthesis, the internal contradiction of a quantity that must be treated as two different things at once. The stable, resulting derived function $dy/dx$, which Marx referred to as the "absolute minimal expression", is the synthesis, resolving the ambiguity by providing the real algebraic value. This focus on motion and contradiction is what made the problem of calculus a "touchstone" for applying his materialist dialectics to mathematics.

We now know that treating $dy/dx$ as an algebraic equation is wrong due to a variety of reasons. For one, $dy/dx$ is the derivative operator being applied to a function $y$. The symbols $d$ and $dx$ have no independent meaning. Another reason is that the derivative is now defined as the limit of finite differences $\Delta y / \Delta x$ as $\Delta x$ approaches zero.

You cannot split the limit of quotients into the quotient of limits when the limit of the denominator is zero. Since the denominator approaches zero, the expression must be treated as the result of the limit process. The problem becomes more apparent in derivatives of the second or third order. Again, however, Marx was unaware of this because the limit definition was not formalized until Weierstrass.

It’s especially important to keep in mind Marx’s own command of mathematical literature. The epsilon-delta definition was first introduced in 1821 by Cauchy and would be known among many mathematicians of the era, but Marx was not a mathematician and had better expertise in the domain of philosophy and economics. Additionally, it was around the time period of Marx writing his manuscripts that Weierstrass had formalized the limit definition of the derivative, solving the issue Marx had stumbled upon 100 years after Berkeley had. However, Marx was not made aware of that material and thus did not know of a better alternative to infinitesimals. He was, however, correct in his conclusion that the definition of derivatives and thus much of calculus was ambiguous and shaky, certainly infinitesimals were a chimera.

In short, calculus was understood in terms of infinitesimals, numbers which were non-zero but infinitely small so that, while performing calculations, they could be considered to be effectively zero. With Berkeley (and by extension Marx), it became clear that the formulation of infinitesimals led to a paradox and the foundations of calculus itself were at risk. Weierstrass fixed these issues with his definition of a limit, removing the notion of an infinitesimal completely, at least until the introduction of hyperreals and surreals in non-standard analysis.

What were the paradoxes/issues? Because infinitesimals were so informally defined by Leibniz, there was no concrete definition for them and the ambiguity takes out a lot of rigor in any proof that used them prior to non-standard analysis. The other pressing issue can be summarized as follows:

Consider the function $f(x)=x^2$. Using an infinitesimal $h$ to find the rate of change, observe:

What follows is:

Anyone familiar with calculus would notice that the left hand side is quite similar to the current limit definition taught to students in high school.

Using the power rule, we’d simply say that the derivative of $x^2$ would be $2x$. However, where is the $h$ present before? We were treating $h$ as a non-zero infinitesimal while we’re trying to solve for the derivative of $f$, then subsequently treating it as zero. However, if $h$ is $0$, we can’t divide by $h$.

In non-standard analysis, this problem no longer exists.

Noting these two issues, it is easy to see how someone can come to Marx’s conclusion: that by using the definitions being used in his own textbooks and by other calculus students at the time, $dy/dx = 0/0$, an indeterminate number.

Those that don’t understand the historical context and nuance often point at the original picture at the top of the page and attack Marx’s later work: Capital. Marx saw his mathematical work as part of a broader philosophical project to incorporate mathematics into a scientific worldview. He sought an algebraic rigor in calculus that mirrored his analysis of economic concepts, aiming to establish rigorous foundations for his study of value and change. This mathematical focus informed his use of algebraic formulas to analyze and derive economic laws. For instance, he derived the dependence of the rate of profit on the organic composition of capital. Furthermore, he utilized his critique of the dialectical contradiction inherent in infinitesimals as a “touchstone” for analyzing economic dualities within capitalism, such as the relationship between use-value and exchange-value.