I very much appreciate Alan's reference to the "mathematical treatment"
of equivalence and equality. This should promote clarity in subsequent
discussions, much as did Paul's treatment of exchange in terms of metric
spaces. However, it needs to be made clear up front that the passage
from Birkhoff and MacLane, and the inferences Alan draws from it, are
*utterly* non-responsive to the point I am making (along with Steve) with
respect to Marx's Chapter 1 argument.
In brief, I *grant* that equivalence relations, such as those established
by systematic exchange under LOOP, establish a "kind of equality." But
to quote any number of my previous posts on the subject, the equivalence
relations established by systematic exchange are not sufficient to
establish "equality" *in the sense required by Marx's Chapter 1
argument*. As you can see from the passage cited by Alan below,
exchange can only establish equality with respect to the exchange
relation R. *But Marx asserts that equivalence in exchange relations
establishes an equivalence in *another* dimension:
"[The exchange relation between corn and iron] signifies that a common
element of identical magnitude exists in two different things, in one
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 nor the other. Each
of them, so far as it is exchange-value, must therefore be reducible to
this third thing." [I, p 127]
But this inference is *nowhere* supported in the passage cited by Alan,
and furthermore it is fallacious in general. I've illustrated this with
the counter-example involving exchange of, say, <italic>x
</italic>boot-polish with <italic>z </italic>acres of unimproved land.
If the mere fact of systematic exchange (with or without LOOP) were
sufficient to establish "equality" in the sense required by Marx, it must
follow
that boot-polish and unimproved land share a "common element of identical
magnitude," which according to Marx's subsequent argument can
<italic>only</italic> be abstract labor. But unimproved land is not a
product of labor. Contradiction.
One could easily multiply the counter-examples. Consider exchanges in
airwave frequency rights, such as have been established by auction in the
U.S. According to Marx's argument in Chapter 1, the equivalence
conditions established by systematic exchange imply a "common element of
identical magnitude" in different sets of air wave rights. Well, what
can that common element be in this case? It can't possibly be abstract
labor; neither the air waves nor claim rights to them are products of
labor.
Or consider markets in financial instruments, where one is much much more
likely than in commodity markets to see LOOP obtain, due to the relative
ease of arbitrage. What is the "common element of identical magnitude"
in different bundles of securities that are exchangeable for each
other?
Note that nothing in the preceding critique depends on the assertion that
equal values are exchanged, or that markets are in equilibrium.
Furthermore, the critique holds even given the unlikely suppositions that
bundles can exchange for themselves (although this bit of reflexivity
virtually never happens, given the logic of exchange) and LOOP (which, in
terms of underlying microeconomic logic, is virtually as restrictive as
assuming market equilibrium; see my previous post).
Thus, Alan's concluding sermon--
>Of course if the 'vast majority of economists' wish to think
differently
>they are entitled to do so. They are not, however, entitled to tell us
that
>this is the only way to think, and they are not entitled to tell us
that
>Marx must have thought like they do.
--is necessarily beside the point.
Gil
>Here's what Birkhoff and MacLane (1963:155), in one of the standard
>undergraduate works on algebra, say about equality and equivalence.
>
>"We have asserted that any reflexive, symmetric and transitive
relation
>might be regarded as a kind of equality. We shall now formulate the
>significance of this assertion. For convenience, a relation R which has
the
>reflexive, symmetric and transitive properties,
>
> aRb, aRb->bRa, aRb and bRc -> aRc
>
>for all members of a set S, will be called an *equivalence* relation.
>If...we are willing to treat suitable *subsets* of S as elements, such
an
>equivalence relation becomes an ordinary equality"
>
>The way this works is as follows: we define an equivalence class R(a)
as
>the set of all elements R-equivalent to a. (eg everything that
exchanges
>for a coat). We then find that
>
> aRb -> R(a)=R(b)
>
>that is, equivalence between the original elements a,b translates into
>equality between the subsets.
>
>If, now, the original elements a, b etc have a structure, the
following
>applies (Birkhoff and MacLane p157)
>
>"In discussing the requisites for an admissible equality relation we
also
>demanded a certain 'substitution property' relative to binary
operations.
>In terms of the equivalence relation R and the binary operation a*b = c
on
>the set S, this property takes the form
>
> aRa' and bRb' imply (a*b)R(a'*b')"
>
>Let me repeat my axiom of basket decomposition [OPE 516] to compare it
>with the above:
>
>"(4 )if bRx and cRy then (b U c) R (x U y), where U is set union."
>
>The operation * in the standard algebraic formulation is thus the set
union
>operation in my formulation. This thus conforms to the textbook
definition
>of equality.
>
>"Equality" in Marx, as among the mathematicians, means the following:
we
>can partition the space of baskets of use-values into subsets of
baskets
>that exchange for each other. Two baskets are 'equal' under this
relation
>if they belong to the same subset. We can then identify the price of a
>basket with the subset to which it belongs. In common language, two
baskets
>are equal if they have the same price.
>
>That's what 'equal' means. Nothing more, and nothing less. It has no
>implication that their values are equal, it is completely independent
of
>the supposition of equilibrium, and it is a coherent, scholarly
definition.
>As far as I can see Marx's analysis conforms to it with almost
textbook
>clarity.
>
>Of course if the 'vast majority of economists' wish to think
differently
>they are entitled to do so. They are not, however, entitled to tell us
that
>this is the only way to think, and they are not entitled to tell us
that
>Marx must have thought like they do.
>
>References
>==========
>
>Birkhoff and MacLane (1963) 'A Survey of Modern Algebra', New
>York:MacMillan
>
>
>