I was reading an online discussion about Karl Marx and his Maths knowledge. I asked various LLMs to explain and expand on this.
Person A: Marx wasn't skilled at math. He didn't contribute anything meaningful and struggled to understand what a limit is. He wrote 100s of pages going in circles. This isn't exactly controversial.
Person B: You may say to yourself this matters not, but Marx not believing in derivatives is why that model of an economy in volume two of Capital is so damn weird and incorrect.
ChatGPT
Got it — let’s break this down, expand it, and add the missing context.
1. What Person A is saying
Claim: Karl Marx was not mathematically skilled.
Evidence given: He struggled with basic concepts like limits (the foundation of calculus).
Consequence: His mathematical manuscripts (hundreds of pages of notes on differential calculus) are often seen as confused or repetitive, not really advancing mathematics.
Implied point: Marx’s economics may have been weakened because he lacked solid math tools.
👉 This refers to Marx’s Mathematical Manuscripts (written in the 1870s but published much later, in the 1960s). In them, Marx tried to grapple with 18th–19th century calculus, especially Newton and Leibniz. He disliked the way limits and infinitesimals were formulated, and he tried to rebuild calculus in terms of algebra. Historians of mathematics generally say his work wasn’t mathematically innovative — more of a philosophical critique.
2. What Person B is saying
Counterclaim: Marx’s rejection of derivatives (and limits) did matter.
Why? Because in Volume II of Capital, Marx builds models of economic circulation and reproduction (how capital flows through production, consumption, and reinvestment).
Problem: These models become static and “weird” because they don’t use calculus to model change dynamically.
Consequence: The mathematical structure of Marx’s economics ends up flawed — not just philosophically, but technically.
👉 What they’re pointing at is that modern economics (classical and neoclassical) relies heavily on calculus: rates of change, marginal productivity, marginal utility, dynamic equilibrium, etc. Marx instead constructed models more algebraically (balance sheets of inputs and outputs, “simple reproduction” vs. “expanded reproduction”). Without derivatives, he couldn’t formalize concepts like marginal cost, growth rates, or stability conditions.
This leaves Capital Vol. II full of long, wordy arguments about circulation that are mathematically cumbersome, and in places inconsistent.
3. Why this matters historically
Marx’s goal: He wanted to show that capitalism is internally unstable, prone to crisis.
What he lacked: The right math to model non-linear dynamics, growth rates, or feedback loops.
Result: He often worked with simplified, “snapshot” models of the economy. Later Marxist economists had to translate his ideas into dynamic mathematics (sometimes using input–output analysis, dynamic systems theory, or Sraffian models).
So Person B is saying: Marx’s discomfort with calculus wasn’t just an abstract philosophical quirk — it made his economic models less realistic.
4. Extra context
Why Marx distrusted calculus: He thought infinitesimals were “mystical” and logically unsound. In his manuscripts, he tries to “purge” calculus of limits and rewrite it as pure algebra. This is somewhat similar to Bishop Berkeley’s earlier critique (“the ghosts of departed quantities”), but Marx didn’t know about the 19th-century work of Cauchy, Weierstrass, and others who had just then made calculus rigorous.
Economics after Marx: By the late 19th century, the “marginal revolution” (Jevons, Walras, Menger) used calculus to model marginal utility and equilibrium. Marx never engaged with this because he rejected derivatives. This created a mathematical gulf between Marxist economics and mainstream economics.
✅ In summary:
Person A is right that Marx wasn’t strong in math and his manuscripts show real confusion.
Person B adds that this wasn’t trivial — Marx’s refusal to use derivatives crippled his economic models, especially in Capital Vol. II. Without calculus, his system looked static and sometimes unrealistic compared to the dynamic, derivative-driven models of mainstream economics.
Gemini
Of course. The conversation you've read touches on a well-documented aspect of Karl Marx's intellectual life: his intense but ultimately frustrating struggle with mathematics, particularly calculus, and how this influenced his economic work.
Here’s an expansion and explanation of the points made by Person A and Person B.
Marx's Obsession with Mathematics
Person A is correct that Marx was not a skilled mathematician and wrote hundreds of pages exploring mathematical concepts. These writings are known today as his "Mathematical Manuscripts."
1 However, his goal wasn't to contribute new theorems. Instead, he had two primary motivations:
Finding a Foundation for Dialectics: Marx believed that the process of change in society (the dialectic) was mirrored in the operations of calculus. He was particularly fascinated by the derivative, which captures instantaneous change. He thought that if he could understand the logical foundation of calculus, he could create a more rigorous, scientific foundation for his own theories of social change.
Critiquing Capitalism: He needed mathematics to analyze economic data and model the dynamics of capital, such as changes in the rate of profit and the circulation of commodities.
2 His struggle, as Person A notes, was with the fundamental concepts of calculus, specifically the limit and the derivative (dy/dx). During Marx's time, the foundations of calculus were still being solidified. He was deeply suspicious of the "metaphysical" or "mystical" explanations for how calculus worked and tried to derive its principles from basic algebra. This attempt to reinvent the wheel is what led him to write pages and pages of notes that often went in circles without reaching a satisfactory conclusion for him.
How This Affects "Capital, Volume Two" 🧐
This is where Person B's point comes in, and it's the crucial link. "Capital, Volume Two" focuses on the circulation of capital.
3 It analyzes processes that happen over time, like the turnover of money, production, and commodities. These are fundamentally dynamic processes involving rates of change.Today, an economist would model these processes using calculus and differential equations. It's the most efficient and powerful tool for describing systems that change over time.
However, because Marx never became comfortable with the methods of calculus (he couldn't get past what he saw as its logical inconsistencies), he couldn't use it in his analysis. Instead, he was forced to rely on what Person B calls a "weird and incorrect" model. More accurately, it was an incredibly cumbersome and archaic one.
To analyze the circulation of capital, Marx created complex arithmetical tables and schemes. He used discrete, step-by-step calculations to show how capital moves through the economy over a year. While not necessarily "incorrect" in their own logic, these methods were:
Clunky: They were far less elegant and powerful than using calculus.
Static: They could only show snapshots in time or average rates, struggling to capture the continuous, fluid motion of capital that derivatives describe so well.
Limited: They made it much harder to generalize his findings and create a flexible model of the economy.
So, Person B is right. Marx's "disbelief" in or struggle with derivatives forced him to use a much weaker mathematical toolkit. This is a primary reason why the models in "Capital, Volume Two" seem so strange and convoluted to anyone familiar with modern economics. He was essentially trying to describe the physics of motion using only basic arithmetic instead of the more appropriate tool of calculus.
Perplexity
This exchange is pointing toward a somewhat obscure but interesting intersection between Karl Marx, mathematics, and the economic models in Capital. Person A is pointing out Marx’s well‑documented mathematical struggles, while person B connects those struggles—especially Marx’s difficulties with calculus—to weaknesses in his economic system modeling. Let’s unpack this carefully.
Marx and Mathematics
Self‑study attempts: In his later years (especially the 1860s–1880s), Marx devoted considerable time to trying to teach himself higher mathematics. Surviving Mathematical Manuscripts show him working through algebra, calculus, and differential equations, often by paraphrasing textbook passages and re‑deriving results.
Difficulty with limits and derivatives: He struggled to grasp the limit concept, which was central to the then‑emerging rigorous formulation of calculus (by Cauchy, Weierstrass, etc.). Instead, he attacked the derivative mostly as an algebraic operation—what he sometimes called the “disappearance” of magnitudes—without fully internalizing the analytic foundation.
Long derivations in circles: The manuscripts often repeat the same problems, reinterpret them, or critique authors like Newton, Leibniz, and contemporary textbook writers. Historians of math generally conclude Marx didn’t advance mathematics at all, despite his efforts.
The Role of Calculus in Economics
Economic models need dynamical change: Many aspects of political economy involve rates of change, marginal analysis, and the relationship between infinitesimal shifts in quantities (capital, labor, profit, wages). By Marx’s time, economists like the marginalists (Jevons, Walras, Menger) were explicitly using calculus for marginal utility and optimization.
Marx without calculus: Marx rejected the marginalist framework and approached value theory through labor-time and reproduction schemes rather than marginal substitution. This was partly philosophical, but partly a methodological gap: without calculus, he had trouble building continuous, dynamic models.
Volume II of Capital: This volume deals with the circulation of capital and reproduction schemas. To model an entire economy reproducing itself year after year, you need a balance across sectors (production of means of production vs. consumer goods). Marx uses big tables (“departments I and II”) and proportionalities rather than differential equations. The absence of calculus here leads to rigid, somewhat unrealistic “proportional growth ratios” instead of continuous adjustment dynamics.
So when Person B says Marx’s disbelief in derivatives made the model in Capital, Vol. II weird and incorrect, they mean:
Instead of modeling how small changes in production, wages, or profits lead to smooth responses, Marx’s schemas deal in stepwise, fixed proportions.
This makes them mathematically clumsy and unable to capture the dynamism, instability, or equilibrating tendencies of an actual economy.
Critics argue this gap left Marx less able to counter the mathematical sophistication of the marginalist or later neoclassical economists.
Why This Was Important
Historical irony: Marx was contemporaneous with the formalization of differential calculus for economic problems, but he stood apart—philosophically mistrustful of infinitesimals and technically unable to use them effectively.
Consequences for Marxist economics: Later Marxist economists (e.g., von Bortkiewicz, Morishima, Sraffa) reformulated the reproduction schemes with linear algebra and differential equations to fix the inconsistencies in Marx’s numeric tables.
Philosophical consistency vs. mathematical sophistication: Marx preferred analyzing social relations qualitatively, avoiding “marginal calculus” because it embedded an individual utility framework he opposed. But his technical difficulties with math amplified that philosophical stance and left his economic models more rigid than intended.
In short:
Person A is correct that Marx struggled deeply with math and never made a meaningful contribution there. Person B is pointing to a real consequence of that weakness: Marx’s failure to embrace derivatives and limits—tools central to describing change—meant that his economic reproduction schemas in Capital, Vol. II ended up static, awkward, and, many economists argue, incorrect.
Let's also ask Chinese LLMs.
Deepseek
This is a fascinating and specific critique that touches on the history of economic thought, philosophy, and mathematics. Let's break down this conversation, expand on its claims, and add the necessary context. Summary of the Conversation
Person A makes a historical claim about Karl Marx's personal aptitude in mathematics, stating he was unskilled, failed to grasp fundamental concepts like calculus (specifically limits), and that his mathematical writings were voluminous but circular and unproductive.
Person B connects this alleged mathematical deficiency directly to Marx's economic work. They argue that Marx's philosophical rejection of the mathematical concept of a derivative (a core concept in calculus) led him to create an economic model in Capital, Volume II that is "weird and incorrect."
Now, let's expand and analyze these points in detail.
--- Expanding on Person A's Point: "Marx wasn't skilled at math..."
This claim is a mix of historical fact and contentious interpretation.
1. Marx's Late-Life Mathematical Journey: It is true that Marx intensely studied mathematics later in his life (roughly from the 1860s until his death in 1883). He filled hundreds of pages with notes, primarily working through textbooks on algebra, trigonometry, and most importantly, calculus—specifically, the works of mathematicians like Isaac Newton, Gottfried Leibniz, and later, Augustin-Louis Cauchy.
2. The Struggle with "The Limit": The concept of a limit is the fundamental bedrock of calculus. It's the idea of approaching a value infinitely closely without necessarily ever reaching it. For example, calculating the instantaneous velocity (a derivative) of a moving object involves taking the limit of its average velocity as the time interval shrinks to zero. Historical evidence from Marx's letters (particularly to Engels) shows he was indeed obsessed with the " metaphysical" nature of this concept. He wasn't just trying to learn how to calculate derivatives; he was trying to understand their philosophical foundation. He struggled with the "mystical" way early calculus (called "infinitesimal calculus") dealt with infinitely small quantities (infinitesimals) that seemed to magically appear and disappear in equations. He was more interested in the dialectical process of how a quantity changes into another, which he felt the standard mathematical notation obscured.
3. "Hundreds of pages going in circles": This is the contentious part. To a professional mathematician, Marx's manuscripts might seem circular because he was often re-deriving known results while trying to reformulate the logical steps in a way that satisfied his philosophical needs. He was attempting a critique of the foundations of mathematics from a dialectical materialist perspective, not simply trying to become a mathematician. Verdict: While Marx was certainly not a pioneering mathematician, calling him "unskilled" is an oversimplification. He was a deeply intelligent and rigorous thinker engaging with advanced mathematical concepts on a philosophical level, which is different from being unable to perform the calculations.
--- Expanding on Person B's Point: "Marx not believing in derivatives..."
This is the more substantive and interesting part of the critique, linking philosophy to economic modeling.
1. What does "not believing in derivatives" mean? Marx wasn't a mathematical crank who denied the utility of calculus. Instead, he took issue with its conceptual foundations. He believed the standard presentation (in his time) swept the inherent contradiction of movement and change (a key dialectical concept) under the rug with slick notation. He wanted to ground the process of differentiation in a logically rigorous, step-by-step process that exposed the "disappearance" of terms in a clear way.
2. The Connection to Volume II of Capital: Capital, Volume II is where Marx outlines his models of "simple" and "expanded" reproduction—essentially, models of how the entire economy circulates goods and capital between two main departments:
Department I: Produces means of production (machines, raw materials).
Department II: Produces means of consumption (goods for workers and capitalists).
The model is presented in terms of aggregate value categories (constant capital, variable capital, surplus value) and uses systems of equations to show the conditions under which equilibrium between these departments can be achieved.
3. Why is it "weird and incorrect"? This is where Person B's point becomes clear. Mainstream (neoclassical) economics, which developed after Marx, relies heavily on marginalist concepts—essentially, derivatives. Think of marginal utility, marginal cost, or marginal productivity. These concepts are about the rate of change of one economic variable with respect to another (e.g., how much more output you get from one more unit of input).
Marx's Static Model: Marx's reproduction schema is largely static and discrete. It describes economic circulation in terms of totals and sums at a given point in time or over a defined period. It's a snapshot, not a video. It excels at showing the necessary proportions for equilibrium but is clunky for analyzing dynamic changes, optimization, or the behavior of individual actors.
The Marginalist (Derivative-Based) Model: Modern economics uses the calculus-based language of margins and rates of change to create dynamic, optimizing models. This allows economists to model how individuals and firms make decisions at the margin (e.g., should I hire one more worker?) and how smooth, continuous changes ripple through the economy.
Critics argue: By building his model on aggregate value sums instead of marginal relationships, Marx created a framework that is ill-suited for analyzing the very things he wanted to analyze: dynamic change, crisis, and price formation. It feels "weird" because it doesn't use the mathematical language that became standard in economics. It's deemed "incorrect" by many modern economists because its underlying assumptions (the labor theory of value) and its lack of marginal analysis are rejected by the mainstream.
--- Conclusion and Added Perspective
The conversation highlights a fundamental clash between two different ways of understanding economics:
1. A Macro-Social Model of Reproduction (Marx): Focused on class relations, aggregate flows of value, and the structural conditions for crisis. Its math is based on algebra and systems of equations.
2. A Micro-Foundational Model of Optimization (Neoclassical): Focused on individual choice, scarcity, and equilibrium achieved through marginal decisions. Its native language is calculus.
Person B is correct that Marx's philosophical approach to mathematics influenced his economic modeling. His suspicion of the "mystical" derivative led him to develop a unique, non-marginalist framework. Whether this makes it "incorrect" depends entirely on your school of economic thought.
To a Marxist, the reproduction schema reveals the deep structure of capitalist exploitation that marginalist models ignore. To a neoclassical economist, it's an archaic and clumsy model that fails to explain how economies actually function on a day-to-day basis.
The conversation, therefore, isn't just about math skills; it's about how deeply our philosophical and mathematical tools shape the economic theories we create.
Qwen
This conversation touches on a niche but fascinating intersection of intellectual history, economics, and mathematics — specifically, Karl Marx’s engagement with calculus and how it may have influenced (or hindered) his economic theories, particularly in Volume II of Capital. Let’s unpack, expand, and contextualize what’s being said here.
Person A: “Marx wasn’t skilled at math. He didn’t contribute anything meaningful and struggled to understand what a limit is. He wrote 100s of pages going in circles. This isn’t exactly controversial.”
This is broadly accurate, though it requires nuance.
Marx’s Mathematical Manuscripts
In the 1870s and 1880s, Marx wrote a series of notebooks now known as the Mathematical Manuscripts of Karl Marx. These were not published in his lifetime and only became widely available in the 20th century (first in Russian in 1933, then in English in 1983). In them, Marx attempted to reconstruct the foundations of calculus — specifically, the concept of the derivative — from first principles, using an algebraic rather than analytical approach.
He was particularly interested in the “mystery” of the differential calculus — how you can cancel out infinitesimals (like dx) and yet still get meaningful results. He rejected the “mystical” infinitesimals of Newton and Leibniz and also disliked the “mystification” he saw in later limit-based approaches (à la Cauchy or Weierstrass). Instead, he tried to ground calculus in algebraic manipulation — for example, deriving the derivative of x² by expanding (x + h)², subtracting x², dividing by h, and then “canceling” h — but without fully embracing the concept of a limit as h → 0.
Why This Matters
Marx’s approach was philosophically motivated. He wanted to “demystify” calculus — to show that its operations were grounded in real, rational, dialectical processes, not in metaphysical infinitesimals or abstract limits. He saw calculus as a product of historical development, much like economic categories.
But mathematically? His work was not original or rigorous by the standards of 19th-century mathematics. He reinvented wheels that had already been refined by Cauchy, Riemann, and others. He didn’t grasp — or perhaps rejected — the formal epsilon-delta definition of limits, which was already emerging in his time. He wrote hundreds of pages trying to justify calculus through algebraic “cancellation” and “negation,” which modern mathematicians would see as hand-waving or circular.
So yes — it’s not controversial among historians of mathematics or even Marxist scholars that Marx’s mathematical work was idiosyncratic, non-rigorous, and ultimately a philosophical exercise rather than a mathematical contribution.
Person B: “You may say to yourself this matters not, but Marx not believing in derivatives is why that model of an economy in volume two of Capital is so damn weird and incorrect.”
This is a provocative and insightful claim — and it deserves serious unpacking.
Volume II of Capital: The Problem of Reproduction
Volume II of Capital is about the circulation of capital — how money turns into commodities, commodities into more money, and how this process reproduces itself across the entire economy. Marx constructs abstract “schemes of reproduction” — essentially, macroeconomic models showing how different sectors (Department I: means of production; Department II: consumer goods) must exchange outputs to sustain expanded reproduction (i.e., economic growth).
The problem? Marx’s models are static, discrete, and algebraic. He assumes fixed proportions, fixed turnover times, and no continuous adjustment. He doesn’t model rates of change, feedback loops, or dynamic equilibria — precisely the kinds of things that calculus (especially differential equations) is designed to handle.
The Connection to Calculus
Person B is suggesting that Marx’s discomfort with — or philosophical rejection of — the derivative as a mathematical object led him to avoid modeling economic processes as continuous, dynamic systems. Instead, he modeled the economy as a series of discrete “periods” or “turnovers,” where everything resets at the end of each cycle.
This results in models that feel “weird” or “mechanical” to modern economists. For example:
- Marx assumes that Department I must produce exactly enough constant capital for both departments in the next period — a rigid proportionality that doesn’t allow for gradual adjustment or disequilibrium dynamics.
- He doesn’t account for time lags, inventory fluctuations, or marginal adjustments — all of which require thinking in terms of rates of change (derivatives) or integrals over time.
- His “expanded reproduction” schemes are essentially solved algebraically, like balancing a ledger, rather than modeled dynamically, like a system of differential equations.
In short: Marx’s aversion to the derivative — or at least, his failure to internalize it as a tool for modeling continuous change — may have led him to construct economic models that are brittle, overly schematic, and unable to capture the fluid, evolving nature of capitalist economies.
Historical Context
It’s worth noting that Marx was writing before the mathematization of economics really took off. The marginalist revolution (Jevons, Menger, Walras) was just beginning, and economists like Léon Walras were starting to use calculus to model equilibrium. But Marx was ideologically opposed to marginalism — he saw it as bourgeois apologetics — and he was also philosophically committed to a dialectical, historical, and qualitative method.
So his avoidance of calculus wasn’t just a mathematical shortcoming — it was also a methodological choice. But as Person B implies, that choice had consequences. It made his macroeconomic models in Volume II feel “off” — too rigid, too schematic, too divorced from the messy, continuous churn of real capitalist economies.
Broader Implications
This conversation touches on a deeper issue: how the tools we use to think shape what we can think about.
Marx’s philosophical commitment to demystifying calculus led him to reject the very tools that might have helped him model economic dynamics more realistically. His economic system — brilliant in its historical and structural insights — suffers in its formal mechanics because he lacked (or refused) the mathematical language to express continuous, interdependent change.
Modern economists — even Marxist ones — have tried to “rescue” Marx’s schemes by reformulating them with differential equations, input-output matrices (à la Leontief), or even systems dynamics models. But these are retrofits. Marx himself didn’t — and perhaps couldn’t — think that way.
Conclusion
Person A is right: Marx’s mathematical work was unoriginal, philosophically driven, and mathematically flawed. He didn’t understand limits in the modern sense, and his hundreds of pages on calculus are more of a historical curiosity than a contribution.
Person B is also right — and more interesting: Marx’s failure to embrace the derivative as a modeling tool likely contributed to the “weirdness” and “incorrectness” of his macroeconomic models in Volume II. His economy is a machine with fixed gears, not a flowing river. And that limitation stems, at least in part, from his philosophical and mathematical blind spots.
This doesn’t invalidate Marx’s broader critique of capitalism — his insights into exploitation, alienation, accumulation, and crisis remain powerful. But it does show that even the greatest thinkers are constrained by the tools they’re willing — or able — to use.
Further Reading (if you’re curious):
- The Mathematical Manuscripts of Karl Marx (1983, New Park Publications)
- “Marx’s Theory of Differential Calculus” by Joseph M. Levy (in Science & Society)
- “Marx, Mathematics and Economics” by Paul Cockshott
- Volume II of Capital — especially Chapters 20–21 on simple and expanded reproduction
- Compare with Léon Walras’s Elements of Pure Economics (1874) — one of the first uses of calculus in economics
Marx may not have been a mathematician — but his struggle with math reveals a lot about how he thought, and where his economic models hit their limits.
🧭 PART I: Setting the Stage — The Conversation in Context
Person A: Marx wasn’t skilled at math. He didn’t contribute anything meaningful and struggled to understand what a limit is. He wrote 100s of pages going in circles. This isn’t exactly controversial.
Person B: You may say to yourself this matters not, but Marx not believing in derivatives is why that model of an economy in volume two of Capital is so damn weird and incorrect.
This isn’t just a snarky academic jab — it’s a doorway into a profound and underappreciated problem in the history of economic thought: how Marx’s philosophical and mathematical limitations shaped — and arguably crippled — his most ambitious attempt to model the capitalist economy as a whole system.
Let’s break this down carefully, expand it, and add relevant historical, mathematical, and economic context.
📚 PART II: Marx’s “Mathematical Manuscripts” — The 100s of Pages Going in Circles
Between 1873 and 1883 — while he was supposed to be finishing Capital Vol. II and III — Marx filled notebooks with explorations of calculus. These were published posthumously as the Mathematical Manuscripts of Karl Marx (first in Russian in 1933, then in English in 1983).
These manuscripts are fascinating — not because they advanced mathematics, but because they reveal Marx’s philosophical struggle with the foundations of calculus.
➤ What Was Marx Trying to Do?
He wasn’t trying to invent new math. He was trying to demystify calculus — to show that its core operations (like taking a derivative) could be understood through dialectical materialism: that is, through real, concrete, historical processes of negation and transformation — not through “mystical infinitesimals” or “metaphysical limits.”
He focused especially on the derivative — the idea that you can find the “rate of change” of a function by examining what happens when you make an infinitely small change in the input.
For example, for y = x²:
- Classical Newton/Leibniz approach: introduce an infinitesimal dx, compute dy = (x+dx)² - x² = 2x·dx + dx², then “cancel” dx to get dy/dx = 2x + dx → “let dx = 0” → dy/dx = 2x.
- Later “rigorous” approach (Cauchy/Weierstrass): define the derivative as the limit of Δy/Δx as Δx → 0.
- Marx’s approach: treat dx not as infinitesimal or limit, but as a real algebraic difference that gets negated through a dialectical process.
Instead, he tried to ground calculus in algebraic manipulation — for example, deriving the derivative of x² by expanding (x + h)², subtracting x², dividing by h, and then “canceling” h — but without fully embracing the concept of a limit as h → 0.
He called this method “the algebraic method” or sometimes “the historical method,” because he believed it mirrored how calculus actually developed — through practical manipulation before formal justification. Marx was fascinated by the fact that early practitioners like Newton and Leibniz got correct results even though their foundations were logically shaky. He saw this as evidence that mathematical truths emerge dialectically — through practice, contradiction, and negation — not from static axioms.
For instance, when deriving dy/dx for y = x², Marx would write:
Then — here’s the key move — he’d say: now let h be “negated,” not “approaching zero.” The term h doesn’t vanish via limit; it is dialectically sublated — canceled out because it belongs to a lower, preliminary stage of the operation. What remains — 2x — is the “real” derivative, born from the negation of the intermediary form.
This is philosophically rich. It’s also mathematically inadequate.
Modern analysis requires rigor: you must define what it means for h to “approach” zero, quantify error bounds (ε–δ), and prove convergence. Marx dismissed this as “mystification” — he thought mathematicians were hiding the real, concrete process of cancellation behind layers of abstraction. But in rejecting limits, he cut himself off from the tools needed to handle continuous change — which brings us directly to Person B’s point.
—
📉 PART III: Why Marx Avoided Calculus — It Wasn’t Just Ignorance (now properly sequenced)
Person A says Marx “struggled to understand what a limit is.” That’s true — but it wasn’t just a cognitive shortcoming. It was philosophical. And this matters immensely.
➤ Hegel vs. Newton: The Dialectical Suspicion of “Mystical Infinitesimals”
Marx was steeped in Hegelian dialectics. For Hegel (and Marx after him), reality unfolds through contradictions, negations, and qualitative leaps — not smooth, continuous functions. The derivative — especially as conceived in Newton’s “fluxions” or Leibniz’s “infinitesimals” — felt metaphysical, even mystical to them.
“The calculus… operates with ‘incomprehensible’ zeroes… cancels them as if they were nothing — yet out of this very nothing, something determinate emerges!”
— Marx, Mathematical ManuscriptsMarx saw calculus as a kind of alchemy: you start with something (dx), treat it as zero, cancel it — and yet produce real knowledge. He wanted to rationalize this process — to show that dx isn’t a ghostly infinitesimal, but the result of an algebraic process of negation.
He called this “the negation of the negation” — a core Hegelian-Marxist concept — applied to mathematics. In his view, the derivative wasn’t about limits approaching zero; it was about a qualitative transformation emerging from quantitative subtraction and cancellation.
➡️ So Marx didn’t “fail to understand limits” — he rejected them as mystifying. He preferred to think of dx as the “ghost of departed quantities” (a phrase from Berkeley he would’ve loved) made concrete through dialectical negation.
This is brilliant philosophy. Terrible mathematics.
—
📉 PART IV: How This Shaped Volume II of Capital — The Static Economy Problem
Let’s get concrete. What does it actually look like when someone tries to model capitalist reproduction without derivatives?
➤ Marx’s Two-Department Model — A Snapshot, Not a Movie
In Volume II, Marx divides the economy into:
- Department I: produces means of production (machines, steel, etc.)
- Department II: produces consumer goods (food, clothes, etc.)
He then constructs “reproduction schemes” — essentially balance sheets — showing how output from each department must be allocated to sustain simple reproduction (no growth) or expanded reproduction (with accumulation).
Here’s a simplified version of his expanded reproduction scheme:
Marx resolves this by assuming capitalists adjust production plans based on last period’s results — but discretely, not continuously. There’s no notion of:
- Gradual adjustment (e.g., d(output)/dt)
- Inventory buffers smoothing mismatches
- Prices signaling shortages/surpluses
- Time lags in investment
It’s all-or-nothing. Either the proportions are exact — or crisis ensues.
➡️ This is the “weirdness” Person B refers to. The model has no shock absorbers. No feedback loops. No dynamics. It’s like trying to drive a car by jumping between fixed gears instead of using a clutch and accelerator.
Modern economists (even non-Marxists) would model this with systems of differential equations — tracking flows, rates of change, disequilibrium adjustments. Think Goodwin’s growth cycles, Kalecki’s investment lags, or even basic Solow-Swan models.
Marx couldn’t — or wouldn’t — do that.
—
⚖️ PART V: The Tragedy — Marx Was So Close to Something Revolutionary
Ironically, Marx intuitively understood dynamic processes better than almost anyone.
- His concept of capital as “self-expanding value” (M–C–M’) is inherently temporal and processual.
- His theory of crisis hinges on disproportionalities developing over time.
- His historical materialism is all about change, transformation, emergence.
He had the substance of dynamics — but not the formal language to express it.
Imagine if Marx had embraced the mathematics of change — if he’d used differential equations to model:
- The rate of exploitation changing with technological innovation
- The organic composition of capital rising over time
- The falling rate of profit as a differential tendency, not a mechanical law
- Sectoral imbalances propagating through input-output networks
He might have built a truly dynamic theory of capitalism — one that could speak to business cycles, financialization, supply chain disruptions, algorithmic trading — not just 19th-century textile mills.
Instead, his formal models remained frozen in algebraic snapshots — and generations of Marxists inherited those frozen models, treating them as sacred texts rather than starting points.
—
🔁 PART VI: Modern Attempts to “Fix” Marx With Math
Many 20th- and 21st-century economists — Marxist and otherwise — have tried to “rescue” Marx by giving him the math he lacked.
➤ Michał Kalecki (1899–1970)
A Polish Marxist-Keynesian who modeled investment cycles, profit determination, and class conflict using differential equations and lag structures. He essentially did what Marx couldn’t — formalized dynamics without abandoning class analysis.
“The workers spend what they get, the capitalists get what they spend.”
— Kalecki (a dynamic identity with distributional consequences)➤ Richard Goodwin (1913–1996)
Built a predator-prey model (Lotka-Volterra equations) to describe cycles of employment and profit share — explicitly inspired by Marx’s crisis theory, but expressed in phase diagrams and differential equations.
(Employment rate vs. wage share — cyclical, not static)➤ Anwar Shaikh, Thomas Vietorisz, Paul Cockshott
Contemporary Marxist economists who use input-output tables, linear algebra, simulation models, even agent-based modeling to update Marx’s framework.
Shaikh’s Capitalism: Competition, Conflict, Crises (2016) is perhaps the most ambitious — he rebuilds classical/Marxian economics using modern tools while preserving Marx’s core insights.
➡️ The lesson? Marx’s economics doesn’t have to be “weird” or “incorrect.” But to fix it, you have to leave behind his 19th-century formalism — including his philosophical aversion to calculus.
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🧭 PART VII: Broader Philosophical Lesson — Tools Shape Thought
This isn’t just about Marx. It’s about all thinkers.
- Freud didn’t have neuroscience — so he invented metapsychology.
- Darwin didn’t have genetics — so natural selection remained abstract until the Modern Synthesis.
- Marx didn’t have dynamical systems — so his macroeconomics remained schematic.
We’re all prisoners of our tools. The danger comes when we mistake our tools for reality — or worse, when we reject better tools because they feel alien to our worldview.
Marx rejected calculus because it felt “bourgeois,” “mystical,” “ahistorical.” But in doing so, he crippled his ability to model the very system he sought to critique.
There’s a deep irony here: Marx, the theorist of historical development and transformation, failed to develop his own theoretical tools historically. He clung to algebra while the world was moving toward analysis.
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💡 Final Thought: What Would Marx Do Today?
If Marx were alive today — seeing algorithmic high-frequency trading, global supply chains, AI-driven productivity shifts, climate-driven commodity shocks — do you think he’d still be scribbling algebraic reproduction schemes?
Unlikely.
He’d probably be:
- Learning Python to scrape financial data
- Using system dynamics software to model sectoral crises
- Reading papers on nonlinear econophysics
- Arguing with post-Keynesians and complexity economists
Because Marx was, above all, a materialist. He believed theory must bend to reality — not the other way around.
The real betrayal of Marx isn’t updating his math — it’s failing to update it.
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✅ TL;DR — The Big Picture
- Marx’s struggle with calculus wasn’t incidental — it was philosophical, ideological, and deeply consequential.
- His rejection of limits and derivatives led to static, brittle economic models in Volume II — models that can’t handle real-world dynamics.
- Modern Marxists have largely moved beyond this — using differential equations, simulations, and systems theory to revive Marx’s insights in mathematically robust ways.
- The lesson: Don’t worship the form — preserve the substance. Marx’s critique of capitalism remains vital. His 19th-century math does not.
- Updating Marx isn’t betrayal — it’s fidelity to his method: follow the material reality, wherever it leads — even if it means learning calculus.
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