Groucho Marxism
Questions and answers on socialism, Marxism, and related topics
Category: Mathematics
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A central question of Marx’s Das Capital is: why is capitalism profitable and productive? Marx’s answer to this question boils down to: because capitalists exploit workers. But why does worker exploitation result in positive profits and positive growth? To tackle this problem scientifically, Marx had to go it alone as much of the mathematical apparatus…
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In 1937 the Hungarian mathematician and physicist Jon von Neumann published a paper entitled A Model of General Economic Equilibrium, in which he set out an abstract mathematical model of an economy. Von Neumann’s model is still used widely today, particularly by Marxist economists. In this blog post I will provide a brief summary. The…
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Consider an economy with m sectors that each specialize in the production of one commodity type. The economy is determined by an m×m input-output matrix of inter-sector coefficients A = (aij), where aij ≥ 0 is the quantity of commodity i required to produce one unit of commodity j; a 1xm vector of direct labour coefficients…
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Value, Price and Profit is a transcript of a lecture series delivered in 1865 by Karl Marx. Having just finished reading it I thought I would provide a short summary. In this text, Marx sought to refute the theoretical basis for the economic policy of his contemporary John Weston, who argued: “(1) that a general…
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In a previous blog post I derived discrete versions of two key equations in quantum mechanics: the canonical commutation relation between position and momentum, and the Schrödinger equation. I did this by considering a particle moving in discrete time and space where time was taken to be imaginary. The eagle-eyed reader would have noticed a…
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Mathematical optimization is the selection of a best element, with regard to some criterion, from some set of available alternatives. The criterion is specified using an objective function on a given domain and the aim is to find the variable that maximizes or minimizes this function within this domain. Mathematical optimization is generally divided into…
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Calculus is the mathematical study of change. It has two major branches: differential calculus and integral calculus. The former concerns rates of change and the slopes of curves, whereas the latter concerns accumulation of quantities and areas under or between curves. Calculus was formulated separately in the late 17th century by Isaac Newton and Gottfried…
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In linguistics, the term ‘phoneme’ refers to any of the perceptually distinct units of sound in a specified language that distinguish one word from another; for example p, b, d, and t in the English words pad, pat, bad, and bat. Languages vary considerably in the number of phonemes they have, from as few as…
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The principle of stationary action is a great unifying principle of physics, and is usually presented in a continuous-time framework. In this blog post I will attempt to present a discrete-time formulation of the principle in accordance with the materialist conception of physics I put forward in a previous blog post. Let x(t) denote the…
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Consider an economy which produces m commodities using n types of labour. For such an economy we can define a commodity vector as a vector with positive elements of length m, and a labour vector as a vector with positive elements of length n. The economy is defined by an activity set, with the interpretation…
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In a previous blog post I suggested that some ideas in physics that seem strange when formulated in a continuous framework based on infinite sets make more sense in a discrete framework based on finite sets. In this blog post I will attempt to flesh this out in more detail. Consider a particle moving in…
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An axiomatic system is a set of formal statements (axioms) used to logically derive other statements. Zermelo-Fraenkel set theory, henceforth ‘ZF’, is an axiomatic system proposed by mathematicians Ernst Zermelo and Abraham Fraenkel in the early twentieth century to formulate a theory of sets. Today, ZF is the standard form of axiomatic set theory and…
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