[OPE-L:7177] [OPE-L:694] Re: Use and abuse of mathematics [OPE 574]

Andrew Kliman (Andrew_Kliman@email.msn.com)
Tue, 16 Mar 1999 16:40:05 -0500

A reply to Steve's welcome and very thoughtful OPE-L 643.

He quotes the ""....1 quarter of corn = x cwt of iron" passage (p.
127, Penguin ed., Capital I) and comments:

"This is the crux of the issue on equality that Gil and I have been
concerned with, not the later passages in Marx where he talks about
a universal equivalent, etc."

Yes, I realize this, and this is also the matter with which my
comments have been concerned.

Steve: "Marx here is talking about what the fundamental nature of
the exchange is when two commodities trade hands."

I interpret the argument differently. I do not think it is
important to his argument that the commodities exchange, just that
they could exchange. That's a minor point. My major difference is
that I don't think he is concerned with the "fundamental nature of
the exchange." I don't think I know what this means. I do not
agree, and don't think he argued, that there is any fundamental
nature to exchange *as such*. Now, of course, your statement is
qualified by the use of the term "commodities," so perhaps you are
not thinking of exchange as such. Yet I also do not know what is
meant by "the fundamental nature of the exchange" of commodities.

I think Marx is instead concerned with the nature of *the
commodity*. He has already identified one aspect. It is a
use-value. Now he is trying to show that there's another aspect,
and that this second aspect is NOT exchange-value, but rather
(intrinsic) value. "... an intrinsic value, i.e. an exchange-value
that is inseparably connected with the commodity, inherent in it,
seems a contradiction in terms. Let us examine the matter more
closely" (p. 126).

In this, the 2nd paragraph on p. 127, he is taking a conclusion that
he derived in the immediately prior paragraph, with respect to the
expanded form of value, and applying it to the accidental form.
Specifically, he concludes that, although the *exchange-value* of
the corn is the iron, and v.v., each of the commodities is, in its
own nature, a third thing -- *value* -- and that this shared
property "in itself is neither the one nor the other." In other
words, he's saying that it is NOT the case that the corn has value
by virtue of its exchangeability with the equivalent (iron) and NOT
the case that its character as value is created by, established by
means of (etc.), the exchange.

What's clearly underlying this demonstration are the notions that
value arises in exchange and that commodities are valuable because
they are exchangeable against money. Marx is opposed to these

Steve: "2. Note: Marx clearly chose to write this exchange as an
equality, using the equality operator "=". He could have said the
following: "1 quarter wheat <---> x cwt of iron. What does this
exchange signify?"

I would say, instead, that he wrote the *exchange relation* as an
equality; whether the commodities exchange or not is unimportant to
his purpose. Now, the question is whether he is right is claiming
that "Whatever their exchange relation may be, it can always be
represented by an equation in which a given quantity of corn is
equated to some quantity of iron." In other words, does there
always exist an x such that

1 quarter of corn = x cwt of iron

given his premise that corn and iron are commodities? I think the
claim is correct.

Moreover, as you rightly note, he does not ask what the *exchange*
signifies, but rather what the *equality* signifies. This supports
my reading, namely that he is concerned with the nature of the
commodity, not the nature of exchange. I.e., the nature of the
commodity is such that its "physical body" is but one aspect of it,
the other aspect being that it itself is a value.

Steve: "The issue Gil and I have been pressing concerns the fact
that when two commodities exchange that in no way implies
necessarily that the act of exchange is to be perceived as an

I think I agree with this. If you mean that one cannot validly
deduce from the exchange of commodities that the exchange is a
relation of equality, then I agree. I don't think Marx argues

Steve: "The other notion of equality is one I would like to call
"numeraire equality" and is the one where one values in some way
(labor, steel, dollars, etc.) the distinct use values in exchange
and then groups equal amounts of value in equivalence or equality

Well, I'm sorry to say that I don't understand the math. Yet the
following may nonetheless be relevant. If I understand "numeraire
equality," then I think there's a problem with it. Imagine that you
value all the distinct use-values in terms of steel. In terms of
what do you value steel itself? The value of a thing and the thing
itself cannot be the same. (This is why Marx notes that, precisely
because labor is the substance of value, it itself cannot have a

Steve: "4. I think there is a deep fundamental theoretical
difference between substance equality and numeraire equality. I
think substance equality is an ontological primitive, supposed by
Marx, and as Gil and I have been saying in different ways, not
argued for very persuasively."

If I understand what you mean by "substance equality," then I would
disagree that it concerns ontology. I think it concerns the very
meaning of equality. As you write in OPE-L 661: "The equality sign
only implies that there is some common denominator and along that
dimension the common denominator for both things compared is equal."
I think this is EXACTLY what Marx is saying when he writes in this
paragraph that "this equation ... signifies that a common element of
identical magnitude exists in two different things." What you call
"common denominator," he calls common element. You say the "common
denominator ... is equal"; he says that the common element is of
"identical magnitude." So I don't see that he needs to argue for
this. He's simply telling us that equality presupposes a third

Steve: "... at the beginning of one of the rounds of this debate
(if I remember correctly), Andrew argued rather that Marx merely
had asserted that an exchange ratio between two commodities
"could always be found" so that there would be a substance
equality of sort. Again, on page 127, Marx wrote "Let us now
take two commodities, for example corn and iron. Whatever their
exchange relation may be, it can always be represented by an
equation in which a given quantity of corn is equated to some
quantity of iron..." He then goes on and asks, as I recounted
above, what does this equation mean?"

Yeah, exactly. I see I have repeated myself above.

Steve: "6. I agree with Marx's "possibility theorem" that we can
always find a ratio of two commodities where a well defined
substance is made equal, not as an inference from the laws of
exchange, but rather as an analytical starting point."

Good. Please also note that this is an *existence* theorem. The
substance *exists* and it has been shown to exist. This is no small
matter. I think it is exactly what Marx was trying to establish in
this paragraph and the one above.

Steve: "But the question remains, as I raised it long ago: why
make this strong ontological assumption? what analytical results
depend on this peculiar notion of exchange? and what how robust are
the major analytical results of Marxian econmics if this singular
assumption is dropped? These are questions that go to the heart of
marxian economics, ones like the conservation of value, no value
being created in exchange, the creation of surplus value, etc."

Good questions. But please note that we are now on a very different
terrain. It is no longer at issue whether Marx's argument is
logically valid, or whether it is true! That has been settled in
his favor!

Let me take a stab at answering these questions -- I will not
attempt a definitive answer:

"why make this strong ontological assumption?"

I don't think it is a strong assumption to say that "The equality
sign only implies that there is some common denominator and along
that dimension the common denominator for both things compared is
equal." This is "ontological" in the sense that you (and Marx)
hold that "there is" a common denominator. But is it an
assumption, much less a strong one? I do not see that.

Why state it? In order to show that the commodity is a value, that
value is intrinsic. This blows away the opposite perspective, that
of Bailey and people like Ajit Sinha, that value is purely a
relation between commodities.

"what analytical results depend on this peculiar notion of

I do not think Marx is advancing a "notion of exchange,"
but a notion of equality in which it implies the existence of a
common element. Many results depend on the notion that value is not
simply a relation between commodities, which is why Bailey was so
concerned to try to establish that value is simply this relation.
It is also why Ajit invariably asserts the same thing when he tries
to contest and to suppress the TSS interpretation of Marx's value
theory. He has to deny the very basis of Marx's theory in order to
declare our interpretation illegitimate.

Let me just indicate one crucial result: If value were reducible to
exchange-value, then a commodity could always be used as the measure
of value (numeraire). The "value" of the numeraire is always
supposed to be equal to itself.

I have argued that THIS is an ontological primitive, "... a
metaphysical materialist primitive: value is a veil, only relative
prices (ratios of things) matter. A thing's 'value' *is* the
quantity of another thing it commands. A unit of corn commands a
unit of corn, so it is always selfsame economically as well as
physically." P. 211, "A Value-theoretic Critique of the Okishio
Theorem," _Marx and Non-equilibrium Economics_.

In any case, note that numeraire metaphysic does not permit the
value of the numeraire to fall -- no matter how much the labor-time
needed to produce it declines! Thus, what is denied is the law of
value and all that flows from it.

To see the consequences of this, take a one-sector (corn) model,
with continuously rising productivity of living labor, a constant
real wage rate, and a constant "capital/output" ratio. Value
becomes irrelevant under the numeraire metaphysic -- all ratios of
quantities of value are identically ratios of corn itself. So,
under the conditions stipulated, since the corn output continually
rises relative to the corn input, the "profit" rate continuously
rises as well.

What we have, then, is a capital-productivity theory of the profit
rate. This is antithetical to Marx, as you know. He argued that
the profit rate FALLS BECAUSE labor becomes more productive. Once
we give up the numeraire metaphysic, then this becomes possible. If
the *value* of the corn output falls relative to the *value* of the
corn input when productivity rises, then a continually falling
profit rate can result.

"how robust are the major analytical results of Marxian
econmics if this singular assumption is dropped?"

Well, academic "Marxian economics" is rooted firmly in numeraire
theory (or in simultaneous valuation, which likewise precludes a
change in the value and/or prices of commodities between time of
input and time of output -- so its analytical role as a destroyer of
Marx's theory is much the same). *Its results thus do not depend on
the third thing, but rather deny it. I think it is, however,
crucial to *Marx's own* results (which is why he is at pains to
establish it at the outset). As you yourself indicate quite
rightly, nothing less is involved than "the conservation of value,
no value being created in exchange, the creation of surplus value,


Andrew Kliman