Just a quick response to a couple of points.
Alan writes:
I'm very rushed but I think I'm quite happy with Brendan's =
axiomatisation in
which money is assumed from the outset.=20
Brendan: Actually my subject heading was wrong. Money is NOT assumed =
from the outset - what is assumed is the general form of value not the =
money form, a universal equivalent rather than money.
Alan: [...] However, I think
it is useful to begin without supposing money. There are two reasons for =
this;
analytically, I don't like to begin by supposing something whose =
properties
one seeks to derive. I think this was Marx's reasoning and I think his
correspondence, writings and remarks on this question establish that he
considered important to *derive* money from the commodity and not to
presuppose it
Brendan: I agree we should not suppose money, but I think we have to =
assume a unversal equivalent if we want to follow Marx's own approach, =
because it is only with the general (universal equivalent) form of value =
that he introduces transitivity, a crucial property to establish =
equivalence.=20
And although I personally am prepared to go along with reflexive direct =
exchange, it's fair to say that Marx's argument does not directly use or =
require it. Exchange via a universal equivalent is unproblematically =
reflexive.
So I think it's futile to attempt an axiomatisation of Marx's chapter 1 =
argument using only direct exchange.=20
Alan: The second reason is a practical one; there is not, in most =
markets, a single
money. There are at least three competing world-currencies, setting =
aside the
many national monies, and each of these monies has many different forms. =
The
category of 'money', supposed as if there was only one, is therefore a =
bit
problematic.
Brendan: The axiomatisation I gave would also apply to several =
"universal equivalent" commodities, provided they were each universal.=20
And I think you could axiomatise a more complex relation of "exchange =
via (potentially several) 'universal' equivalent commodities" that would =
allow for the quasi-money commodities each being exchangeable for only a =
subset of commodities, so long as they were exchangeable for each other. =
It wouldn't add anything useful in the current context though.
Brendan
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