Karl Marx’s Mathematical Manuscripts are a collection of over 1,000 pages of his mathematical notes, drafts, and reflections, primarily focused on the foundations and history of differential calculus. He wrote these manuscripts during the final decade of his life, from roughly 1873 to 1883, and they were first published in the Soviet Union in 1968.
Content and analysis
The manuscripts contain four independent treatises, along with supplementary materials:
- On the Concept of the Derived Function: Written around 1881, this treatise demonstrates how to calculate derivatives for several basic functions using an algebraic approach rather than the more common geometric arguments of the time.
- On the Differential: In this work, Marx attempts to define a derivative from first principles without relying on the concept of a limit, which he was suspicious of. He was heavily influenced by the work of French mathematician Jean-Louis Boucharlat but intentionally steered away from his use of limits.
- On the History of Differential Calculus: Marx applies a dialectical historical analysis to the development of calculus, dividing it into three phases:
- The “mystical” differential calculus of Isaac Newton and Gottfried Leibniz.
- The “rational” differential calculus of Jean-Baptiste d’Alembert.
- The “purely algebraic” differential calculus of Joseph-Louis Lagrange.
- Taylor’s Theorem, MacLaurin’s Theorem, and Lagrange’s Theory of Derived Functions: This section includes notes and an unfinished thesis on these foundational topics of calculus.
The purpose behind Marx’s mathematical studies
Marx’s work on the mathematical manuscripts was motivated by both intellectual curiosity and a desire to better ground his economic theories. His goal was to achieve a greater understanding of variable economic relationships, which he believed would benefit from a deeper mathematical foundation. He saw parallels between his theories on the history of economics and the development of calculus.
Legacy and significance
- Anticipation of non-standard analysis: Though working in isolation and unaware of many contemporary advancements in mathematics, Marx’s approach to infinitesimals and differentials is now seen as anticipating aspects of 20th-century non-standard analysis. His work embraced infinitesimals as genuine mathematical entities, a concept that was out of favor among mathematicians of his time.
- Limited contemporary impact: Because the manuscripts were not published until many decades after his death, they had no direct influence on the historical development of mathematics.
- A “dialectical” approach: The work provides a rare example of Marx applying his dialectical method not to social theory, but to a field of pure science.
- Continued relevance: While not a professional mathematician, Marx’s mathematical writings continue to be a subject of analysis, particularly for those interested in the history of mathematics, Marxist theory, and the philosophy of science.