I hope to communicate today to my students an idea that they may find rather difficult to grasp, or so I fear. Perhaps if I have a go at explaining it here, the effort will help me to clarify my exposition. Let me say, by way of setting the scene, that today I shall try to bring together into a single integrated account the four strains of argument I have been developing in my course: Marx's economic analysis of capitalism, Marx's historical account of the development of capitalism, the modern mathematical formalization of the classical and Marxian tradition of Political Economy, and my literary critical explanation of the extraordinary language of the first six or seven chapters of Capital. The reading assigned for today is my 1981 essay, "A Critique and Reinterpretation of Marx's Labor Theory of Value." That essay, which so far as I can determine, in David Hume's words, fell stillborn from the presses, is a rather difficult read for philosophers, containing as it does a good deal of linear algebra. [Linear Algebra, which is mother's milk to economists, is actually quite elementary undergraduate math for a math major, but as C. P. Snow observed many decades ago, we live in a society of two cultures, and philosophers whose intellectual sophistication knows no bounds confront a page of mathematics as though it were Linear B.]
The idea, in a sentence, is this: The narrative accompanying a mathematical model frequently contains a good deal of information that finds no formal representation in the model, and it is a mistake to think that the deductions from the model constitute a proof or endorsement of those elements present only in the narrative. In the case of Marx's theory, the distinction between labor power and labor, which Marx claims is the key to understanding the origin of profit in a capitalist system, actually finds no representation in the formalism implicit in Marx's argument and made explicit by the modern mathematical formalization of that argument. The goal of my essay is two-fold: First, to demonstrate that the labor/labor-power distinction plays no formal role in Marx's argument, and that all of his results can be reproduced for any arbitrarily chosen commodity as "substance of value;" and Second, to find an alternative formalization of what I believe to be Marx's correct analysis and critique of capitalism.
There are a number of examples of this sort of mismatch between a formal argument and the accompanying narrative. My favorite is the narrative that has grown up around the so-called Prisoner's Dilemma. I shan't reproduce here my analysis of that familiar example, from Game Theory, of a two-person non-zero-sum game in which each player has two pure strategies. Those who are interested will find it in the book-length tutorial I wrote on The Use and Abuse of Formal Methods in Political Philosophy, archived at box.net. Other examples, considerably more important at one time in U. S. military affairs, are the mismatches between the narratives of deterrence and nuclear war and the game theoretic analyses accompanying them. [To be found in my unpublished book, The Rhetoric of Deterrence, also archived.]
If one prefers a more light-hearted example, one can consider the practice adopted by some Grade School teachers of drawing circles as happy faces and squares as Sponge Bob Square Pants when introducing little children to Geometry. Sooner or later, the children must learn that the mathematical properties of a circle are independent of whether a happy or sad face is drawn in it.