Re: [OPE-L] equality versus equivalence

From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Sat Jul 14 2007 - 17:13:39 EDT



Paul, you mentioned in an earlier post that Marx's choice of labor
objectified labor) as a scalar was an assertion and was not logically
proved.  And Fred you responded (to Andy and Paul, I think) by saying
Marx's argument did not constitute a logical proof, but rested instead
its empirical explanatory adequacy.  I read your arguments in this post
the strength of Marx's argument, but first I want to understand what it
you have in mind by insisting that Marx has not offered a logical proof.
Equally (or is it equivalently?), Paul, I want to understand what you
that Marx's argument about labor is only an assertion.

Neither of you may intend this, but the argument always seems inevitably
sound as if this were somehow a failing of Marx's argument:  that he
have offered a logical proof, but failed to do so.

I am not criticising what marx did here, given the time he was working,
and the much lower development of logic an algebra in the mid 19th
century, he presents a remarkably good formal argument. I think that
using more modern formalisms the formal argument can be pushed further,
but it would be anachronistic to expect this in the 1860s. Concepts like
equivalence sets and metric spaces were not well developed then.

However, as a general methodological point a formal argument can only
establish that the properties of the exchange relation imply a conserved
scalar - value. We can not from this formal argument identify it with

To do this one needs non-formal arguments - arguments about the nature
of society and the central role of labour in production are one avenue,
empirical evidence that actual exchange values appear to be
proportionate to labour used is another line of argument. It is a
willingness to put the hypothesis to empirical test that marks it out as

Suppose, for instance, we found that in practice the correlation between
prices and labour content was only about 50%, whereas the correlation
between prices and energy content was 98%, we would then have to reject
the hypothesis that labour was the conserved scalar in exchange, and
replace it with the hypothesis that energy was being conserved in

What we actually find is the reverse, the correlation between exchange
value and labour is much higher than the correlation between energy and
exchange value, so we can say that Marx's hypothesis is well supported.

For example, Paul, is bare assertion the only alternative available to
logical proof in science?


Clearly not. 

The important alternatives involve things like the explanatory power of
a theory, its conciseness, and the degree to which its predictions are
born out by observation.


----- Original Message -----
From: "Fred Moseley" <fmoseley@MTHOLYOKE.EDU>
Sent: Friday, July 13, 2007 10:11 PM
Subject: Re: [OPE-L] equality versus equivalence

Hi Paul,

Thanks very much for your explanation.

Would it be accurate to say that equality is QUANTITATIVE and
equivalence may not be quantitative?

Why do you think that your argument is stronger than Marx's?  Because
it is based on the weaker assumption of equivalence?

I think Marx's argument is quite strong.  Marx analylzes the process of
exchange from an OBJECTIVE point of view - i.e. he considers only the
objective properties of commodities - as opposed to the neoclassical
SUBJECTIVE point of view, which focuses on the choices of individuals.
 From this objective point of view, a mutually consistent (i.e.
transitive) system of exchange is necessarily the exchange of equals
(strong meaning), which in turn requires that commodities must be
commensurable, i.e. must possess some common property in terms of which
they are commensurable.  It seems to me that the only way to reject
this conclusion is to reject the objective method on which it is based.

What I find especially interesting about your argument is that it also
seems to be based on a similar objective method - "looking at the
properties of the vector space".  And your objective analysis of the
properties of vector spaces leads to the same conclusion - that
commodities must possess a common property that determines their


Quoting Paul Cockshott <wpc@DCS.GLA.AC.UK>:

> Fred:
> " 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
> 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
> 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
> 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
> 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
> for commodities, because we also have the notion of commodities being
> more valuable than one another, this is may be what you mean by
> commensurability.
> 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
> relate in exchange, 1 coat exchanges with 12 boots etc. If one looks
> 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
> be partitioned into equivalence sets   by their distance from the
> - 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
> 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
> -
> 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
> What is the difference between equality and equivalence?
> Does equivalence require commensurability?
> Commensurability is the crucial point in Marx's argument.
> Comradely,
> Fred
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