I think the part of discussion which involves myself and Gil is at last
getting down to the core of the matter.
I'm not one to stand by forms of words that get in the way of communication.
If the phrase "That's what 'equal' means. Nothing more, and nothing less"
causes difficulties then I retract it, because it's not necessary to my
The key statement in Gil's post is the following:
>It may well be that Alan and those who agree with him are taking it as a
>*primitive assumption* that commodity exchange establishes a form of equality,
>rather than as an *inference* (as Marx strongly suggests in using terms like
>"therefore" , "it follows from this that..." and "must therefore...") while I,
>and those who agree more or less with me, are taking it as an inference. But
>if this is the case *we've been attempting to debate across a paradigmatic
>divide*, a necessarily fruitless endeavor. [my emphasis -AF]"
My response to this is straightforward: yes, we are arguing across a
paradigmatic divide, no, it's not fruitless. On the contrary, that's how
science progresses. That's the whole issue, the central point which I think
practically every TSS contributor has been trying to make in the debate about
Marx, from day one. If even the Marxists, let alone the economics profession,
were to understand this single point today, the case against Marx would
Precisely because we are arguing across a paradigmatic divide, two entirely
different sets of meanings can be attributed to the terms in virtually every
substantial debate. The problem is that the dominant side in the debate
*systematically and unconsciously argues as if its own set of meanings were
the only ones possible*. I think that's what has to stop, if communication
is to be established, and if the debate about Marx is to take place on a
level playing ground.
I *don't* agree that arguing across a paradigmatic divide is fruitless and I
think Gil should consider the implications of what I hope was a throwaway
statement. If this was really true, progress in science would be impossible.
It is both possible and necessary to argue across a paradigmatic divide on the
basis of a thoroughgoing commitment to pluralism, in which each participant in
the debate *admits the possibility of a different assigment of meaning* to the
terms in the debate, undertakes to understand the meaning which other
participants are employing in their own terms, and does not impose her or his
meaning on the basis of any a priori assertion that no other meaning is
It is also possible to submit *evidence* in arguments across a paradigmatic
divide; to submit grounds for supposing that one attribution of meaning is
superior. Great care, however, has to be exercised in defining what is
acceptable as evidence. My point about authority is that it does not count as
evidence; it is a basic logical fallacy, proof by acclaim, to say that
anything is true because a lot of people believe it. I do think, however, that
when proof by authority is offered by one side in a debate, it is legitimate
to challenge this by citing an alternative authority. There is a narrow line
which it is easy to cross, and it may be that I've done that in the heat of
the moment, which leads one to forms of words that suggest the alternative
authority is *superior*. I retract any such implication in any of my
arguments. My purpose in citing non-equilibrium Walrasians, and in citing
foundational mathematicians, is to demonstrate that there is no *sole*
authority for the meanings which Gil attaches to equality.
To be precise,
(a) the concept of equality in exchange does *not* necessarily presuppose
(b) the concept of equality is *not* necessarily distinguishable absolutely
from equivalence or congruence;
(c) the inference that a quantititive relation (such as, but not confined to,
equality) presupposes a common predicate is *not* a necessarily illegitimate
Once one accepts these three propositions, I think the bulk of Gil's case
against Marx's chapter 1 (and chapter 5, don't forget) argument, disappears.
I'll deal with (c) in more detail since I agree there is ground still to
cover, but I do think that points (a) and (b) have been more or less
established by my contributions to the debate.
As a preliminary to dealing with (c), I think it is relevant that the paradigm
in which I am situated is in general a temporal paradigm; in relation to Marx,
specifically, it is the temporal single-system paradigm. I believe that the
paradigm in which Gil is situated is a form of equilibrium paradigm, and
though I don't want to deny Gil the right to explain his paradigm in his own
terms, I don't think it's unreasonable of me to point out the many places
where he implicitly makes assumptions that are valid only in an equilibrium
paradigm, wherever it turns out that his argument against Marx depends on
attributing these assumptions to Marx.
Also, there is a certain amount of cross-fire. Some of the points that have
cropped up in posts responding to Gil apply more to contributions from others
-- I think Steve tends in this direction but we haven't really cleared this up
yet -- who read into Marx's use of 'equality' an assertion that goods actually
exchange in proportion to their values. I think this is a directly equilibrium
reading of Marx; to assert such a form of equality Marx would have to reason
like Ricardo that value was the proportion in which goods exchanged in a
special, hypothetical society where, inter alia, supply matched demand. I
think Gil does not now attribute that argument to Marx and in this interchange
at least, we can perhaps leave that particular issue behind.
That allows me to address what I think is the core of Gil's objection:
>Here, I think, is the nub of the issue. Whether or not there is an "absolute"
>distinction between equality and equivalence, it seems clear that the
>existence of an equivalence relation, i.e. a binary relation E satisfying
>reflexivity, symmetry, and transitivity is insufficient to establish
>equivalence along *some other* dimension L, as Marx's Chapter 1 argument
>requires. And as for what is or isn't "employed in mathematics", I wonder if
>Alan knows of any mathematician who would allow an equation between entities
>measured in different units, such as Marx asserts when he writes "1 quarter of
>corn = x cwt of iron."
If Gil was really willing to abandon the idea that Marx makes an absolute
distinction between equality and equivalence, the discussion could progress
considerably. Later I'll cite from his own post many cases in which his
argument against Marx does indeed attribute to Marx an absolute distinction
between equality and equivalence, and moreover his charges against Marx depend
crucially on this attribution.
My response is that I don't think the word 'equality' plays any special role
beyond pointing out that exchange-equivalence establishes aquantitative
relation between commodities. For what it's worth, that was what I intended
with the words 'nothing more and nothing less'; 'equality' asserts, and only
asserts, that objects enter into some quantitative relation with each other,
the quantitative relation of belonging to equivalence-classes with a linear
relation between them.
Let's set this aside however and deal with the above paragraph as it stands.
The issue is that the binary relation E is not just any old equivalence
relation. It is a quantitative relation. It says that if 1 ton of coal
exchanges for 1 ounce of gold, then 2 tons of coal exchanges for 2 ounces of
gold. That's why the other 'dimension' is needed.
I see the substance of Marx's argument as follows: In order for a quantitative
relation to exist, both sides in the quantitative relation must possess some
predicate that is quantitatively related, just as, for example, the
quantitative relation established by weighing requires objects to have a
Gil objects, I think, that equivalence establishes nothing more than the
predicate of 'being exchangeable for a definite quantity of something' and
that we can leave things there, perhaps by singling out some special
something, money for example, which can be made the measure of this predicate.
The predicate that corresponds to being in the same price-class is then
'exchanging for the same quantity of money' or some such definition.
If this was sufficient, why has physics developed the category of mass? Why
has it found it necessary to define precisely 'another dimension' than weight?
Why do we find, in practically every sphere in science or mathematics where a
quantitative relation between objects is specified, the assertion of some
independent property of the objects that enter into this quantitative relation
that does not depend on the particular properties of any of these objects, be
it distance, area, electrical charge, or whatever?
The whole problem with defining the quantitative relation of exchange in terms
of one particular exchanged object is that it turns the discussion of this
relation into a discussion of the particular properties of the object used as
a measure. Exchange does *not* depend on the properties of money. Money arises
from the activity of exchange.
That is why whenever science really gets down to the business of defining
quantitative relations between objects, it always seeks to define a predicate
of the objects prior to and independent of the relation; their mass, their
charge, their temperature, and so on. Once we do this, we can undertake the
real business of science which is not just to explain each phenomenon
separately, but to relate together apparently unrelated phenomena.
I develop this point more thoroughly in a second post;now I want to return to
the nature of Gil's argument. I think the essence of the matter is this: I
don't think Marx's inference depends on some special meaning
of the word 'equality' that singles it out, raises it to a level distinct from
equivalence. I think that Gil's argument against Marx does depend on saying
that Marx is attributing some special such meaning to 'equality'.
That's why I wrote, as Gil cites me:
>Indeed it's quite hard to see how equality *could* be rigorously distinguished
By this I meant, since it is quite hard in any case, independent of Marx, to
distinguish equality from equivalence, why attribute to Marx the crime of
trying to do so?
This is eactly, however, what Gil accuses Marx of. He writes:
>But suppose that the problem of infinite sets is relevant to this discussion.
>The force of this point is that in that case it may be impossible to define
>"equality" unambiguously. Fine; that's an additional strike against Marx's
>Chapter 1 argument, because he states that "valid exchange-values of a
>particular commodity express something *equal*" and "[an exchange relation]
>can *always* be represented by an *equation*" [emphases added].
>And what *I* find wearying about the discussion is that it keeps meandering
>away from the central point: there is no interpretation of the term "equality"
>that I know of, other than by simple tautology, that is sufficient to support
>Marx's inference from the fact of exchange (under whatever conditions) that
>there exists "a common element of identical magnitude" in exchanged bundles.
>It is *Marx* who insists on a *particular* "equalizing" property of exchange,
>so the burden of proof is on those who agree with this assertion.
But my point is that Marx doesn't require some wonderful unambiguous
definition of equality in order to make his inference. The argument would for
example still apply if everywhere Marx wrote the word 'equality' we were to
strike it out and substitute the words 'quantitatively equivalent' or even
'quantitatively related'. tjhe decisive thing about exchange is that it places
disparate use-values in a quantitative relation with one another, and that's
all that really matters. Indeed in many places where the translators
distinguish 'equality' from 'equivalence', Marx in German uses the same word.
The argument simply doesn't depend on a special meaning of the word
My response finally, therefore, to Gil's final question:
>So let me ask Alan: do you take the claim that exchange (under some set of
>conditions) establishes a relation of equality with respect to commodity
>bundles as an *inference*, as Marx suggests in his Chapter 1 argument, or as
>an *assumption*? And if the former, what is the definition of the "equality"
>purported to arise in exchange that is held to entail Marx's subsequent
Commodity exchange establishes a *quantified* relation of equivalence between
commodities. You may call this an assumption if you like, but it is an
assumption that follows from direct observation of the way that a commodity
society both uses, and speaks of, commodities. This quantified relation of
equivalence is, I think, what Marx *means* by equality: dare I say it, nothing
more, and nothing less. I see Marx's inference as this: observing society we
note that it establishes, both in practice and in its manner of conceiving of
this practice, a quantified equivalence relation between disparate use-values.
To be able to do this -- Marx 'infers' -- society must also in practice,
whether or not it conceives
of it openly, furnish these disparate use-values with a quantifiable predicate
which is qualitatively the same for all of them, and which is defined prior to
and independent of the relation of exchange. That is, use-values functioning
as commodities must also have values. The exchange-relation then expresses the
ratio in which these values exchange against one another.
PS I'm off to Boston tomorrow, so if I don't reply to anything over the next
week it isn't because I don;t want to but becase I can't.