# [OPE-L:7094] [OPE-L:593] Linearity, basket decomposition, etc@

Alan Freeman (a.freeman@greenwich.ac.uk)
Thu, 04 Mar 1999 09:36:25 +0000

Like Gil I am now finding it hard to keep up.

One minor clarification however, and some thoughts about how to open up a
more coherent procedure for addressing the questions at issue:

Gil writes(referring to my post below):
=======================================
>Actually, as (4) is stated, the two *aren't* the same. Axiom 4 only
>indicates that you can go from components to aggregate baskets, not
>vice-versa. And as someone has already noted, the converse is problematic:

>saying that you can exchange a pair of gloves for 4 apples does not say
>you can exchange, say, the left glove alone for an apple.

=============
>>(4 )if bRx and cRy then (b U c) R (x U y), where U is set union.
>>
>>Axiom 4 says we can decompose two baskets, or make them up from
>>components (the two are the same)

I don't think so, Gil.

Either you can exchange left gloves on their own, or you can't. If you
can't then there is no such commodity as a left glove; a pair of gloves
isn't really a basket but an indissoluble entity that happens to have two
physical parts, just as a human has arms and legs that are not generally
available separately.

If a left glove sells separately then it must be in the exchange-class of a
definite number of apples.

Consider now any basket A made up of a left-glove and something else -- B,
say. This can be represented as A={g,B} where g is the glove.

Axiom 4 then tells us that the exchange-class of A is the union of the
exchange-classes of g and B. That is, provided the basket A decomposes at
all, its price is equal to the sum of the prices of its parts.

I can think of only two ways out of this. The first might be that the B
doesn't sell apart from the g, even though the g sells apart from the B.
There may be an instance of this but I can't think of it.

The other would be, I guess, if gloves and apples were in some sense
'absolutely non-exchangeable', that is, there is no way by repeated
exchanges to get from a glove to an apple. In that case the entire market
would be partitioned into two (or more) completely non-exchangable classes,
one containing apples and the other containing gloves, with no way to get
from one to the other: that is, two markets. There couldn't even be a money
in such a system. Perhaps one needs an extra axiom to assert this, but I
don't think it's an important case, because it's really just two markets
instead of one; there can't be any cross-over at all.

However this particular exchange illustrates a more general point,
namely,why does it matter anyhow?

Such fine-grain analysis of the basic axiomatisation can at most lead to
the conclusion that we need a couple extra axioms. So what? If we need an
extra axiom or two to fully mathematise Marx's commodity-concept, then
fine; be constructive, let's add them. The constructive way to approach
this would be to try and devise the minimum such axiom set and then study
the conclusions that can be deduced from this axiom-set; then we can study,
for example, whether the axiom-set can be relaxed to cover less developed
forms of distribution such as arbitrage

[BTW the most obvious case of this in the real world comes not from the
rather strained examples presented on the list, but from the existence of
distinct national markets and the restricted existence of world values.
That is something I consider a *serious* theoretical issue and one Marx was
perfectly aware of, which constitutes IMO one of the few productive lines
of real theoretical development in Marxist thinking since Marx]

It is also perfectly simple to address the question of the 'third thing',
of the requirement for a substance, and so on. But first, we have to know
the basis on which the discussion begins.

I'm perfectly prepared to have a proper discussion of the issues you raise,
but the ground rules currently being followed in this debate do not permit
it. In order to have a discussion, *certain* things have to be agreed.
Otherwise, it isn't a discussion but a war of attrition. What I have asked
for agreement on is something very modest: grant me that my axiomatisation
of Marx is a *possible* interpretation. What's so difficult about that? You
don't have to agree that your own axiomatisation is wrong, merely that it
isn't the only one.

If we approach things in this way, as any mathematician would, I don't
think we'll find what you claim: that Marx's commodity-concept is 'either
tautological or false'. But at least we can discuss the question in a
logical framework in which the initial premises are agreed, instead
of constantly revisiting the premises.

proper mathematical form. But you don't seem to be doing this. You seem to
be attempting to establish that whatever axiomatisation of Marx is
attempted (if any axiomatisation at all is possible) a contradiction must
result. This prejudges the issue. All you do, therefore, is take pot-shots
at attempts to do so, all the while grizzling because your own issues are
is some agreed initial premise.

At the end of the day what you seem to do is to attribute your own axioms,
which, I hope you'll excuse me for pointing out, are generally derived in a
more or less unreconstructed and rarely very critical manner from
neoclassical microeconomic suppositions (existence of neoclassical general
equilibrium, consumer rationality, etc) to Marx, and deduce your
conclusions about Marx from this axiomatisation.

But this doesn't prove anything. You must demonstrate that there is *no*
possible axiomatisation of Marx without either a contradiction or a failure
to reproduce Marx's stated conclusions. It proves nothing to establish that
one particular axiomatisation fails, especially your own and least of all a
neoclassical axiomatisation, which is what most of the discussion seems
to be trying to impose on Marx (neoclassical general equilibrium,
indifference, etc)

I've addressed your claim by producing axioms that make sense of Marx in
his *own* terms, lead to the conclusions which you say cannot be deduced
from his argument, and are internally coherent. To refute this you must
either show that

(a) my axioms are not a possible axiomatisation of Marx's concept, or:

(b) Marx's conclusions do not follow from my axioms.

So far, I don't even see an attempt to do either of those things, though
it's possible I've missed it in the flurry of exchanges. If you have
done either of these things I'd be grateful if you could collate the
relevant passages as succinctly as possible and without extraneous
material, and put it into a new post.

I think it is because you don't attempt to do this, that some of the
discussion becomes frustrating. No doubt some of the things I do are
frustrating also; however I do think there is a logical order to things.

First we have to establish some measure of agreement or clear
disagreement over what Marx was saying. Then we can discuss whether it
was wrong.

How can we discuss whether it is wrong, when we don't agree what it is?

Alan