# [OPE-L:7039] [OPE-L:534] Re: How mathematicians think about equality [OPE

Steve Cullenberg (stephen.cullenberg@ucr.edu)
Fri, 26 Feb 1999 13:28:10 -0800

Alan referred to (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.
>

Alan,

OK, we have the algebraic definition of equality out in the open now and it
helps I think see where our differences lie. Let me make a couple of points.

1. Your axiom of basket decomposition rules out complementarities between
exchanged commodities.

Let b = 2 books
x = 3 bread
c = 1 socks
y = 3 jam
R = the exchange operator

Thus, following your axiom, bRx and cRy then (bUc)R(xUy) my example would
read:

if 2 books exchange for 3 bread and 1 socks exchange for 3 jam, then 2
books and 1 socks would exchange for 3 bread and 3 jam. This is unlikely
as bread and jam are complementary and therefore together would demand more
books and socks in exchange than they would in the antecedent exchanges.

Of course, if elements b, x, c, y were units of foreign currency say, then
this axiom makes sense. Or, if these elements were different anounts of
some monetary unit, or some measure of abstract labor, or any other
numeraire measure of value then your definition makes sense.

But this suggests two issues to me:

a. As a fundamental description of exchange, thinking about exchange as an
equivalence or equality relation is problematic. As Gil and I argued
before, exchange does not obey reflexivity (apples do not exchange for
apples), which already problematizes your example of equality relations
because they are also equivalence relations. Second, equality as you
define it, rules out a large class of commodities because of
complementarities as the example I gave illustrates.

b. But maybe you are not thinking about direct exchange of commodity
against commodity, but are thinking about exchange in an already monetized
or "priced" world, where the question of value has already been settled in
some sense. Then, I agree it does make sense to think about commodity
exchange as an exchange of equality. Two dollars of oranges will exactly
equal two dollars of bread. OK....even those vile neoclassicals would
agree that exchange at this level of abstraction is a relation of equality.
But isn't this like wondering which weighs more, one hundred pounds of
feathers or one hundred pounds of steel?

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.

Let me put the question slightly differently. Does it make sense to think
about the process of exchange as one where some substance is "conserved" in
and through exchange? That's the notion of equality I've been driving at,
and I thought we were, but I see from this post on the mathematicians
notion of equality that you have something different in mind, something
that I don't necessarily disagree with, but which I think is much more
restrictive question than the one I, or I believe Marx, raised.

Steve
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