Marx left over 1000 manuscript pages of mathematical notes on his attempts at discovering the foundations of calculus. The majority of these manuscript pages have been collected into four papers, along with drafts and supplementary notes in the published editions of his collected works. In these works, Marx attempted to draw analogies between his theories of the history of economics and the development of calculus by constructing differential calculus in terms of mathematical symbols altered by an upheaval that would reveal their meaning.
On the Concept of the Derived Function Marx wrote
On the Concept of the Derived Function in 1881, just two years before his death. In this work, he demonstrates the mechanical steps needed to calculate a derivative for several basic functions from first principles. Despite the fact that Marx's principal sources primarily relied on geometric arguments for the
definition of the derivative, Marx's explanations rely much more strongly on algebraic explanations than geometric ones, suggesting he likely preferred to think of things algebraically. Fahey et al. state that although "We might be alarmed to find a student writing 0/0 ... [Marx] was well aware of what he was doing when he wrote '0/0'." However, Marx was evidently disturbed by the implications of this, stating that "The closely held belief of some rationalising mathematicians that
dy and
dx are quantitatively actually only infinitely small, only approaching 0/0, is a chimera".
On the Differential In
On the Differential, Marx tries to construct the definition of a derivative d
y/d
x from first principles, without using the definition of a
limit. He appears to have primarily used an elementary textbook written by the French mathematician Boucharlat, who had primarily used the traditional
limit definition of the derivative, but Marx appears to have intentionally avoided doing so in his definition of the derivative. Fahey et al. state that, as evidenced by the four separate drafts of this paper, Marx wrote it with considerable care.
On the History of Differential Calculus Fahey et al. state that although Marx never used this term in his mathematical papers, his history of calculus can be understood in terms of
thesis, antithesis, synthesis. Marx identified three historical phases of development - the "mystical" differential calculus of
Newton and
Leibniz, the "rational" differential calculus of
d'Alembert, and the "purely algebraic" differential calculus of
Lagrange. However, as Marx was not aware of the work of
Cauchy, he did not carry his historical development any further. ==Legacy==