Marx, Scruton and Equivalence Classes

In reading Roger Scruton's Fools, Frauds and Firebrands: Thinkers of the New Left, I encountered a furious attack against the beginning of Marx's Das Kapital:

`Das Kapital opens with a disastrous argument, to the effect that if two commodities exchange against each other their ‘exchange-value’ must be ‘the mode of expression, the phenomenal form, of something contained in [them], yet distinguishable from them’. The remark, phrased already in the tendentious idiom of ‘classical German philosophy’, is justified by an important fallacy:

Let us take two commodities, e.g., corn and iron. The proportions in which they are exchangeable, whatever those proportions may be, can always be represented by an equation in which a given quantity of corn is equated to some quantity of iron: e.g., 1 quarter corn = x cwt. iron. What does this equation tell us? It tells us that in two different things – in 1 quarter of corn and x cwt. of iron, there exists in equal quantities something common to both. The two things must therefore be equal to a third, which in itself is neither the one nor the other. Each of them, so far as it is exchange value, must therefore be reducible to this third. [ref]'

Scruton continues 

Now the only logical conclusion to be drawn from the fact that two commodities exchange at a given rate is that they exchange at that rate. If a money value is assigned to the given equation then this is simply another fact of the same kind. The value of any commodity can be seen as an ‘equivalence class’. Just as a geometer would define the direction of a line as the set of all lines that are identically directed, and just as Frege and Russell defined the number of a class as the class of all classes that are equinumerous with it, so might the economist define the value of a commodity as the class of all commodities that exchange equally against it. The assumption of a ghostly ‘third’ item, in terms of which this equivalence is to be defined, is strictly redundant – a purely metaphysical commentary upon facts that provide no independent support for it. [Page 121]

As it is assumed in reading every book, I was reading it `in good will' and given the confidence of Scruton in `Marx’s fallacy' took this a real objection. The other night, I took advantage of mania's insomnia to reflect on this `fallacy'. Scruton claims that this is simply an equivalence class: all commodities with price $x. It is true that this is an equivalence class --which I shall not explain here, see Wikipedia--, but it does not tell us what a dollar is, which is what Marx is trying to find out. 

Applying Scruton's argument to some typical examples of equivalence classes makes it clear that there is no fallacy here, testifying only Scruton's poor understanding of mathematics (and its philosophy). I take examples from Wikipedia,

If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class would consist of all green cars, and X/~ could be naturally identified with the set of all car colors. Fine, but this does not tell us what color itself is! (let alone its qualia);

"Is similar to" on the set of all triangles. Fine, but does this tell us what a triangle is in the first place? We have to know what a triangle is in order to begin checking whether this is an equivalence relation or not, by referring to our intuition of a triangle and subsequently measuring and comparing the corresponding rations;

"Has the same absolute value as" on the set of real numbers. Okay, but this, by itself, cannot define the notion of absolute value to begin with. Absolute value is to be defined separately, beforehand.

Scruton's mistake is a fatal --yet common-- one: Associating to real world a pure axiomatic `empty' theory, devoid of any `phenomenal' (real word) significance. It is just as wrong as taking axiomatic Euclidean (or Minkowski for that matter) geometry to define what a meter or a second is. Defining length, upon which the whole geometry rests, requires a common basis to compare lengths with [see Euclid's Elements, book VII], for example a meter. A meter cannot be defined merely by geometry, it is a real external (phenomenal) world concept: the most recent definition of meter is done using vibrations of the Caesium atom. It is in this very act of defining a meter that `bare' geometry is connected to the physical world; geometry by itself does not and cannot talk about world, unless such a connection is established via a real object. The best example is one from physics: Consider the equivalence class of all objects that weigh x kilogram; indeed it is an equivalence relation:

1. A~A: A has the same weight as itself;
2. A~B iff B~A, trivially;
3. If A~B ∧ B~C then A~C: If A and B weigh the same (x), and B and C also weigh the same, then A and C would also weigh the same;

but this says nothing whatsoever about what a kilogram itself is: we can imagine a group of 50 Bic (Cristal model) pens to be equivalent to Scruton's book; but does this suffice to define a kilogram?! Of course not. And it is indeed this very example that Scruton is debunked: in comparing 50 Bics and Scruton's book, there indeed is a `ghostly ``third'' item', namely a kilogram! An operationalist might say `but in order to see whether 50 Bics and the book are equivalent, i.e. weigh the same, we must first weigh them by a balance'. Indeed this is the point: the very mechanism by which a balance works is based on a universal kilogram scale. 

Similarly for Scruton's fallacy (!), equivalence of all commodities that worth $x, has nothing to do with real-world transactions unless a dollar is defined in advance.