In response to my question
------------------------------
> What is the algebraic difference between an equality operator
> and an equivalence operator, and in consequence between and
> equality set and an equivalence set?
Gil replied:
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Let me answer in the language of sets rather than algebra.
Think of the relevant operators as rules for assigning entities
to sets. An equivalence operator assigns entities to a set
based on their possession of a *particular* characteristic or
set of characteristics.
Two entities are *equal* iff they are equivalent in all
dimensions or characteristics allowed by the relevant axiom set.
In terms of algebra, here's an example: for any integer x, 2x
and 4x are equivalent in being even numbers, but they are not
equal.
In effect, although it's not so obvious, Marx is saying that two
integers are equal because they're both even.
Paul
----
I am not sure that this gets us much further. Given a set S, an
arbitary binary equivalence operator which we will write ~,and
an element e in S, then the triple {S, ~, e} defines the set of
elements in S equivalent to e under ~.
Let us define S to be the set of commodity bundles, such that
each element of the set is a vector of commodities. (This is a
generalisation of Marx's argument where he deals only with
quantities of individual commodities. Marx's quantities of
individual commodities constitute the basis vectors of the
commodity bundle space.) Let us further define ~ to be the
operator meaning, exchanges for. Given a point in commodity
bundle space, [ 1000kg flour, 800kg rice, 4000litre paraffin],
the ~ defines all bundles of commodities that will exchange with
this bundle.
Since ~ is an equivalence relation, it induces a partitioning of
the commodity bundle space.
Gil says that two entities are equal if they are equivalent in
all dimensions or characteristics alowed by the relevant axiom
set.
The question becomes what is the relevant axiom set. In
discussing equivalences between commodities, properties such as
weight are clearly irrelevant. The only thing that is relevant
is what they will exchange for, in these terms the elements of
the set induced by ~e are equal to one another. So the
equivalence relation is an equality relation.
But Marx's argument goes beyond this, he says that the elements
of the equivalence sets are equal because each element of the
equivalence set belongs to it by virtue of a relationship they
share to another space, a scalar dimension ( quantity of common
substance) which he identifies with labour time.
There are two questions here:
1) Is it valid to deduce that the partitioning of commodity
bundle space is associated with a distinct scalar dimension?
2) Is it empirically reasonable to conclude that this scalar
dimension is that of labour time.
I will argue that the answer to both questions is yes.
1. Partitioning or ordering
------------------------
Equivalence relations always induce a partitioning of the space
on which they operate. They partition the space into equivalence
classes. Let us consider the new space formed by these
equivalence classes, i.e., one in which each point in the space
is an equivalence class in commodity space. Call this induced
space exchange value space. The question is
(1.1) whether commodity exchange also induces an ordering, and
thus an ordering on exchange values
(1.2) whether this ordering makes exchange values space
homeomorphic to the real number line, and thus associates with
each point in exchange value space a real valued potential.
It is easy to demonstrate (1.1). We first define the relation <
for commodity bundle space. To do this we need only say that for
two points x,y in commodity bundle space, the relation x<y holds
if there exists a commodity dimension k such that x(k) < y(k)
and x(i) <= y(i) for all i not equal to k.
Now since ~ induces a partitioning on commodity bundle space to
create exchange value space, the < relation will be inherited by
the exchange value space.
We can demonstrate (1.2) by arguing that there exists at least
one commodity dimension to commodity bundle space which is
continuous and thus itself homeomorphic to the real line. Chose
that dimension to be k in the argument above, then it must
induce a real morphism on exchange value space.
There must thus exist a real valued potential associated with
each point in commodity bundle space.
This real valued potential must exist as a necessary logical
consequence of the exchange relation, whatever enters into the
exchange - including unimproved land or whatever. However, given
that exchange value space has this single scalar degree of
freedom, it implies that the information content of all of the
possible exchange ratios for commodities can be derived from a
single operator associated with each type of commodity. This
operator is its exchange-value, its type is then
V(q:real commodity -> real exchange-value )= vq
or, in words, it is a function mapping real numbered quantities
of a given commodity to points in an exchange value space that
is homeo-morphic to the real numbers, and this function can be
modeled by multiplying q, its input parameter by a number v.
2. What is the attractor of exchange value
---------------------------------------
Now consider the configuration space constituted by the set of
such exchange value numbers v. For N commodities this
consititues an N dimensional configuration space. At each time
step, the exchange system is located within distinct valid
region in this space ( a region not a point if we allow for
stochastic effects ). Over time, the valid region wanders
through configuration space, describing a trajectory.
The problem of the 'law of value' is to define a law of motion
for this trajectory. We have to discover what the attractor for
this motion is.
As I understand Marx's Capital I, he first restricts his
attention to the sub-manifold of produced exchangeable items
which he calls commodities. That excludes other exchangeable
things like titles to land or titles to ownership of corporate
entites. Within this submanifold, he describes a process by
which a trajectory through a dual space, that of values or
commodity labour contents, acts as the atractor for the
trajectory through the appropriate submanifold of the
configuration space of exchange values.
In Capital III he describes another dual space, that of prices
of production, which he suggests may be a better approximation
as an attractor for movement through configuration space.
In Capital III he also suggests how, with the addition of
another operator, the rate of interest, the movement of the
price of production system can be said to drive the trajectory
of the entire system of exchange value configuration space,
(including land etc).
It is only in comparatively recent years, that the problem
testing these hypotheses against real data has been addressed.
The evidence accumulated so far, on the first and second
hypotheses, shows them both to be remarkably well confirmed.
However, it is not yet clear whether both attractors are
simultaneously active, or, whether in practice the trajectories
that are followed in both dual spaces are so similar as to be
indistinguishable. These are open questions for research.