# [OPE-L:7138] [OPE-L:639] Exchange and infinite sets

Alan Freeman (a.freeman@greenwich.ac.uk)
Tue, 09 Mar 1999 12:14:57 +0000

I've separated this point from my main response to Gil

Gil writes
==========

Excuse me, but this is the first I've heard that *infinite* sets were even
potentially at issue in this discussion. There are no infinite commodity
bundles I know of. There is not an infinite number of exchange relations. I
have not heard anyone assert that given commodities have an infinite number of
characteristics. [And if they did, wouldn't we need to consume only one
commodity?] So Alan, first let me ask if you think the problem of infinite
sets is relevant *in this case*. If not, I see no problem with using B&M's
definition, at least as a point of reference.

Alan responds
=============

All the exchange-classes we are discussing are infinite, that is, contain an
infinite number of mutually-exchangeable baskets. EG if one ton of coal
exchanges for one ounce of gold per ton and iron is exchanges for one ounce
of gold per ton then the following baskets are all equivalent to two ounces
of gold:

1 ton of iron and 1 ton of coal
1.1 tons of iron and 0.9 tons of coal
1.2 tons of iron and 0.8 tons of coal

etc., in general uncountably infinite.

This is one of the reasons I find it problematic to say that two complete
exchange classes are the 'same' because they contain the same elements. One
cannot in general enumerate all the elements. That's why I've always preferred
(irrespective of any attachment to Marx) to enquire into the predicates of the
elements themselves. I have always entertained dark suspicions about
enumerating infinitely many things. Not least, the relation of
exchange-equivalence does in fact involve many exchanges that never actually
take place so that the enumeration does not actually happen in society; this
does not stop us saying that the baskets concerned are exchange-equivalent.
The relation of exchange-equivalence is not limited to the exchanges we
actually observe, and this is one of the strongest reasons, for me, for saying
that it is defined deductively in terms of predicates, not inductively in
terms of set membership. It is society that does it this way, not just me or
Marx; any shopkeeper will give you a price for an arbitrary basket of goods
regardless of whether you buy them.

The fact that exchange-classes are infinite follows from a point that I still
think hasn't been adequately absorbed in the discussion, namely, the objects
that enter into relations of commodity exchange are not devoid of properties
prior to exchange; they come with a structure, they are quantified before they
enter exchange, and exchange is a relation not just between individual objects
but between combinations of objects defined by this structure (if one ton of
coal exchanges for one ounce of gold then two tons of coal exchanges for two
ounces of gold).

This is illustrated by a remark of Ian's that things mentioned by him on
5/3/99

>therefore both belong to the equivalence class of things
>mentioned by me on 5/3/99, but they will not share a property in any
>substantive (causally explanatory sense)

There is no quantitative relation between things mentioned by me on 5/3/99. A
quantitative relation arises if, for example, I want to say 'I said twice as
many things on 5/3/99 as on 6/3/99'. In that case one needs a measure of the
number of things said which is independent of their content; precisely
because, if we don't, we have to adopt a very tortuous way of talking,
speaking of the equivalence classes between baskets of things said on 5/3/99
and baskets of things said on 6/3/99, and so on. The 'predicate' that becomes
necessary to quantify things said, so that they may be compared -- 'equated'
-- is that of *number of things*. If then one wants to say 'I said twice as
much on 5/3/99 as I wrote' one needs a measure that does not depend on the
mode of delivery, that is independent of whether I wrote it, said it, or
e-mailed it. One needs a measure of information, and the category of
'information' presents itself immediately as a third category, a substance, a
category we require to make quantitative comparisons between disparate
communications.

The requirement for an independent predicate arises *specifically* from the
fact of a quantitative relation. This point is continuously overlooked.

But once one introduces quantity, one always introduces infinity.

Fundamentally, though I think I showed that one can pose the issue without
beginning with linearity (by using decomposition instead), this is why I feel
it is important to the structure of Marx's argument that exchange is a
linear relation; that is, a quantified relation.

One way to escape the problem that exchange is a relation between infinite
classes is to say that two exchange-classes are the 'same' if they contain any
pair of elements that are exchange-equivalent to each other. One can then
infer, using transitivity, that any element in one class is
exchange-equivalent to any element in the other class and that the two
classes are the same. But I don't think this gets to the bottom of it. One
might for example define the relation of having the same mass by saying that a
pound of coal has the same mass as a pound of iron if they balance each other
in a pair of scales. This doesn't help us understand why two pounds of iron
balances two pounds of coal.

That's why I think its is preferable to proceed differently, as Marx does, and
associate a magnitude with each class, and say that the two classes are the
same if this magnitude is the same. The reason I think this is better, as I've
said in the discussion, is that the classes have a quantifiable relation to
each other: if A is exchange-equivalent to B, then 2A is exchange-equivalent
to 2B.

I think Gil's argument about predicates in two dimensions might be phrased
this: in exchange, values do not 'balance', so this reasoning is
inappropriate; all we can say is that the two exchangeable objects have the
same price, so that it is their price, their measure in terms of another
use-value, which gives us the 'common property' that defines the
equivalence-classes. This is indeed the way in which the standard of weight
was itself defined for a long time; the standard British pound was defined as
the quantity of a standard iron bar, kept at Greenwich, which would balance
something. Weight was thus 'the number of standard iron bars' that something
would balance with.

The question that has to be asked, however, is why this definition was
abandoned. Why did the French revolution find it necessary to define the
Kilogram in terms of something *else*, a volume of water? Why were physicists
compelled to begin, not from the weight of an object which depends on the act
of weighing, but from its mass, which doesn't? In my view, because defining
weight in terms of weighing involved circular reasoning. Physicists wanted to
quantifying objects in terms of it. They concluded that there was a
fundamental property of matter, mass, which all massive objects had in common,
which could be defined independent of the act of weighing. I think this was a
sound instinct because with the category of mass, one may explain all sorts of
other phenomena -- such as gravity -- in which there is no relation of simple
equality.

Actually, therefore, the need to define something independent of weight --
mass -- arises once we investigate relations between objects, such as
gravitational force, in which they don't balance, but do enter into a
quantitative relation. The need for an additional predicate presents itself
with full force when there is a variable quantitative relation between
objects. We then require a quantitative measure of these objects that does not
depend on any quantitative equality between them. I reason, therefore, in
exactly the opposite way to Gil; it seems to me that it is *only* in
equilibrium, *only* when goods exchange at values, that one may get away with
not using the category of value. It is precisely because value is not equated
in exchange, that one needs the category of value.

My reading of Marx's derivation places much less emphasis on the word
'equality' than Gil; I think, for example, it could be replaced by words such
as 'enter into a quantitative relation with' and the inference would stand.
The essence of Marx's inference (which follows Hegel's 'Law of Appearance'
quite closely) is that in order for objects to enter into a lawlike
quantitative relation with one another of any kind, these objects must already
partake of a common substance.

If one proceeds by merely observing the exchange-relation and taking it as
defining the predicate which makes objects exchangeable, one can, it is true,
remain at the level of use-value. This magnitude can be defined by singling
out one particular commodity, gold, say, as the use-value in terms of which
all values are expressed. One then defines price in use-value terms, as the
amount of gold to which a basket is equivalent.

I find this deeply unsatisfactory. What's so special about gold? This idea
would mean that, for example, if we remove gold from circulation, our
predicate is no longer defined. The same applies for any *particular*
use-value that we choose to use as the measure of price.

I want to show that the relation of exchange-equivalence can also be defined
in terms of value. This argument
does not prove that it is necessary to do it in terms of value; at this stage
I only want to show how it can be done, without supposing that goods exchange
at their value.

Suppose that each commodity in the basket has an independently-defined
predicate, its value, and that this predicate is qualitatively the same for
all commodities. One can then reason as follows:

(a) observe the ratio in which 1 unit of value, embodied in iron, exchanges
for value embodied in gold. Call this I.
(b) observe the ratio in which 1 unit of value, embodied in coal, exchanges
for value embodied in gold. Call this C.
(c) for any basket composed of iron and coal, one may then calculate the
amount of value to which this basket is equivalent, when it exchanges for
gold. This is equal to B=I*iron + C*coal.
(d) All baskets whose value, so calculated, is equal to B, are
exchange-equivalent.

It might be argued that this definition still singles out a special commodity,
gold, and merely substitutes the value in the gold for the quantity of the
gold, as the predicate in terms of which we estimate price.

However, and this is the 'SS' part of 'TSS', I would argue that one can go a
step farther -- and that this farther step is a key step in Marx's concept --
and measure price not in terms of the value in the gold, but the value that
the gold represents in exchange. This is done by calculating the MEL as we
propose; by asking how much value a given quantity of gold represents in
exchange, not how much value is in the gold. To discover how much value the
gold represents in exchange, one adds up the prices and the values of all
goods (expressed in hours using the approach above), divides one by the other,
and deduces that in exchange, each hour of gold represents a given proportion
of the total value of all the goods in society. From this one may translate
any price into an absolute number of hours, independent of which commodity
serves as the money commodity. This number of hours will be different, in
general, from the value of the commodity. Commodities whose price so defined
is above their value, will in exchange transfer value to the seller;
commodities whose price so defined is lower than their value, will in exchange
transfer value to the buyer. The sum of all transfers is zero.

This representation completely absolves us from any mention of the ratios in
which physical magnitudes exchange; all exchanges are now represented in terms
of a single magnitude. Physical magnitudes enter only in the calculation of B.

It specifies, deductively, a means of finding out whether two baskets are part
of the same exchange-class which does not demand comparing them with any
particulat commodity and indeed, does not demand observing them in the act of
exchange.

It completely solves the problem of finding out if two exchange-classes are
equal (the same class) without having to enumerate each of the elements in
each of the classes, and compare them.

It permits us to map, uniquely, any system of prices into a system of
transfers of value and thus attain something which Post-Keynesians at least
regard as a kind of holy grail, and not without reason; it lets us define the
relation between price and distribution in terms of a distributed substance
which is given independent of and prior to the formation of price. It lets us
*connect two apparently unconnected phenomena*, price formation and
distribution; I think the object of all scientific enquiry is to connect
together phenomena which appear on the surface to be unconnected, and it
should at least be acknowledged that Marx's category of value, interpreted as
I have done, permits us to do this; if we define our equivalence classes in
use-value terms, we cannot do this.

Put another way, I think the value-category has a superior explanatory power,
and that ultimately is the best test of any concept.

Alan