[OPE-L:7141] [OPE-L:643] Re: Re: Use and abuse of mathematics [OPE 574]

Steve Cullenberg (stephen.cullenberg@ucr.edu)
Tue, 09 Mar 1999 10:35:45 -0800

Unfortunately, I have been too occupied with university things here to
respond as late but let me make a point in response to Andrew's post about

>Andrew writes about Gil:

>I have put forth an interpretation of the text according to which
>Marx DOES NOT infer the third thing from the *fact of exchange*.
>Unless you prove that my interpretation is false, you are not
>entitled to refer to "Marx's inference." That assumes what needs to
>be proved.

Marx writes on page 127 of the Penguin edition of Volume I the following:

"....1 quarter of corn = x cwt of iron. What does this equation signify?
It signifies that a common element of identical magnitude exists in two
different things, in 1 quarter of corn and similarly in x cwt of iron.
Both are therefore equal to a third thing, which in itself is neither the
one or the other."

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

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?"
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 equality. That adds more mathematical
structure and those restrictions need to be argued for. It seems to me
that too many people in this debate have considered the operators of
exchange and equality as self-obvious, and they aren't. I think Marx
clearly recognized the analytical burden his argument about equality
implied, namely that equality in this sense requires a "common element"
different from each, and he made that explicit, even though I don't think
he argued for it strongly.

3. Now, to Andrew's immediate point. I think there have been two senses
of equality in our discussion of this issue. One sense is what I would
like to call "substance equality", which is the sense in which Gil and I
have raised the question about equality, and I think the way expressed by
Marx on page 127. 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 classes.

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. I think the
numeraire equality is a mathematical or definitional notion of equality
useful for many analytical purposes and not unique certainly to marxian

5. If the idea of equality in question is the substance one then I think
it will be difficult to provide an argument that will lead to equality as
substance based on an "inference" as Gil wants and Andrew I think rightly
refuses. That would require an arbitrage argument of some sort, but that
is not really the relevant issue here, as I believe Marx was making a prior
ontological point, not a causal one. In fact, I don't think Andrew has
argued that substance equality should be viewed as an inference. Indeed,
quite a while ago 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?

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. 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.

I would be curious to Andrew's or Alan's or anyone's thoughts on this.

Stephen Cullenberg Office: 909-787-5037, ext. 1573
Department of Economics Fax: 909-787-5685
University of California Email: stephen.cullenberg@ucr.edu
Riverside, CA 92521 www.ucr.edu/CHSS/depts/econ/sc.htm