[OPE-L] equality versus equivalence

From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Fri Jul 13 2007 - 05:19:11 EDT

" What is the difference between equality and equivalence?
Does equivalence require commensurability?
Commensurability is the crucial point in Marx's argument."

Equivalence is a more general concept than equality and basically means
that if A and B are equivalent under some criterion P then wherever A
exists in some formula Y to be judged by criterion P, then we may
substitute B for the occurrence of A without changing our judgement of Y
by this criterion P.

Equivalence can be a weaker concept because it is qualified by the
criterion. Suppose we have two collections of triangles. One criterion P
might be equality of area, another criterion Q might be similarity,
another might be
Congruence. Given a triangle B defined by its coordinates in a Cartesian
frame of reference, we would have several different equality sets {B|
A=B} depending whether we took the = sign to mean area equivalence,
similarity, or congruence.

You are right that Marx actually demands commensurability which is
stronger than just an equivalence relation in
That it involves the need for the operator < as well as = . Thus to go
back to the example of triangles, if we consider the equivalence
relation of symmetry, this does not impose an order on the induced
equivalence sets, whereas, area equality does.
By itself therefore an equivalence relation is weaker than what we want
for commodities, because we also have the notion of commodities being
more valuable than one another, this is may be what you mean by

This point about < is not brought out particularly clearly in Capital,
it is assumed but not explicitly stated.

But we can go further than just asserting that the relations = and <
apply ( with appropriate interpretation ) to commodities. What Marx
analyses is just the properties of individual commodities and how these
relate in exchange, 1 coat exchanges with 12 boots etc. If one looks at
the equality sets that apply not to commodities but to vectors of
commodities then one can see that the conserved property must be a
scalar, by looking at the properties of the metric space that is
involved with commodity relations.

Consider a two dimensional Cartesian space. All points in this space can
be partitioned into equivalence sets   by their distance from the origin
- and these sets would have the operator < defined on them as well. In
fact though we find that the properties of of such a vector space are
not met by commodities. Their space is not a vector space, and the
reasons for this can be deduced from the separability of commodities in
commodity bundles. If one looks at the properties of the actual space
formed by commodity bundles, the argument for there being a conserved
scalar can be made logically stronger than the argument Marx presented
in Capital.

I published an article in Politica e Cultura a few years ago trying to
demonstrate this:


-----Original Message-----
From: OPE-L [mailto:OPE-L@SUS.CSUCHICO.EDU] On Behalf Of Fred Moseley
Sent: 13 July 2007 02:54
Subject: Re: [OPE-L] Ajit Sinha and equality versus equivalence

Quoting Paul Cockshott <wpc@DCS.GLA.AC.UK>:
> I think he is right to argue that it is this structure that allows
> the existence of money as a dimension reduction device. The assertion
> that the common scalar is labour, is only an assertion, it is not
> proved in the argument. There are obviously other possible common
> scalars - weight, volume, carbon content etc, but none of these had
> the same immediate empirical plausibility as labour.
> I would argue that the logic of commodity exchange implies a common
> conserved scalar value, and that hypothetically systems of commodity
> exchange could exist in which something other than labour provided
> this conserved scalar, ( some system of robot production for instance
> ), but that in contemporary human society it actually is labour that
> is the underlying scalar. But this is something to be empirically
> tested, not something to be logically deduced.

Hi Paul,

I agree with what I understand to be the general thrust of your argument
Marx makes a strong argument for the plausibility of labor as the
common property of commodities that determines their exchange values,
but a logical proof of this hypothesis is not possible.  The ultimate
test of this hypothesis is its empirical explanatory power.  And on
this ground, I think that labor as the common property has much more
explanatory power than any other possible hypothesis.  Starting with
the necessity of money and the value of money in SEction 3 of Chapter 1.

What is the difference between equality and equivalence?
Does equivalence require commensurability?
Commensurability is the crucial point in Marx's argument.


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