[OPE-L:7056] [OPE-L:553] Re: Re: Re: How mathematicians think about

Gil Skillman (gskillman@mail.wesleyan.edu)
Sun, 28 Feb 1999 17:10:06 -0500

Andrew writes:

>A reply to part of ope-l 534.
>Steve wrote
>>"2. The question I am getting at, and I believe Gil is too, is a
>>prior notion of equality in exchange, direct exchange, where such
>>exchange of two heterogeneous use values implies a third element,
>>something common to each, a substance....as discussed by Marx in
>>Chapter One."
>I think this is the crux of the matter, and I think this is a
>misreading -- unfortunately, a common misreading -- of Marx's
>argument. He is not saying that exchange implies the existence of a
>third thing. Indeed, in Ch. 2, he says that, at first
>(historically), "commodities ... become exchangeable through the
>mutual desire of their owners to alienate them" (Vintage, p. 182),
>not due to the existence of any third thing, and their exchange
>ratio is at first purely a matter of chance.

I don't understand Andrew's argument here. First, what Marx says in
Chapter 2 does not prove or disprove what Marx in fact says in Chapter 1.
Second, the cited passage is not logically inconsistent with the passages
in Chapter 1 that are under fire. The claim that "commodities...become
exchaneable through the mutual desire of their owners to alienate them"
does not speak one way or the other to the claim that systematic exchange
establishes relationships of equality.

Now, in Chapter 1, Marx does in fact say that 1) exchange establishes
"something equal" among commodities and that 2) one can infer from the
equality thus established the existence of a "third thing" that is distinct
from the exchanged commodities themselves.

On Point 1: "Therefore x boot-polish, y silk, z gold, etc., must, as
exchange values, be mutually replaceable or of identical magnitude. *It
follows from this* that, firstly, the valid exchange-values of a particular
commodity *express something equal*..." [emphasis added]

On Point 2: "What does this *equation* (established by an exchange
relation between corn and iron] signify? It signifies that a *common
element of identical magnitude exists* in two different things....Both are
*therefore equal to a third thing*, which in itself is neither the one nor
the other. Each of them, so far as it is exchange-value, *must therefore
be reducible to this third thing.*[emphasis added]

>It also seems to me that the point is just too obvious. Marx would
>have had to have been REALLY STUPID to thing that you have to have
>a common element in order to have an exchange.

Well, Andrew, you said it, I didn't. The extended discussion we've been
having with Alan on this issue suggests that the point is not so obvious.
In any case, I don't think Marx was stupid; just wrong. But it is
undeniable, given the passages cited above, that he asserts that it's the
*exchange* of commodities that establishes an equality relation from which
one can infer the existence of a "common element."

> In the exchange of
>Christmas gifts, or favors, or prisoners, there's no third thing
>(unless the relationships become *very* commodified), yet the
>exchange does take place.

That would be my point, too. See below.

>In Ch. 1, Marx instead proceeds from the equality of *commodities*
>(not exchange). The equality of commodities, as I think Steve
>agrees, is a fact.

Well, *I* don't agree (and I'm pretty sure Steve doesn't either). Andrew's
assertion begs the key issue: what exactly is it that is held to "equate"
commodities? The answer can only be exchange (with some set of conditions
attached, perhaps). But this property of exchange, if it exists, is not
ever shown, by Marx or by Andrew, to depend on the particular precondition
that only *commodities* are exchanged. But this demonstration is
*critical* for the distinction Andrew wants to draw above.

Furthermore, I don't agree that exchange "equates" commodities. It
establishes an *equivalence relation* among commodities. Thus, when Andrew

> Marx points out that their equality to each
>other implies an equality to a third thing, which is neither one
>of them, but which is the substance of both.

this begs the key question at hand, since this claim presumes that
exchange establishes an "equality" between any 2 pairs of commodities. It
doesn't. Marx doesn't explain how it does. Neither does Andrew. The
point of our ongoing discussion with Alan is that *even if* one asserts a
system of exchange satisfying reflexivity, symmetry, transitivity, and
linearity, it *still* doesn't follow that exchange establishes a relation
of *equality* among commodities.

Andrew thus asserts exactly that which must first be proved.