As I've noted, I do not understand the math, nor do I think the
opening argument in _Capital_ is deducing anything from the
properties of exchange per se. But I understand Alan's
methodological comments, and I think they are quite right.
Alan: "If we need an extra axiom or two to fully mathematise Marx's
commodity-concept, then fine; be constructive, let's add them."
Of course.
Alan: [To prove Marx was wrong or inconsistent,] "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, ..."
Right again. It is interesting to note that this means that NONE of
the attempts to prove errors or internal inconsistencies in Marx's
work are successful, i.e. valid. (They are quite successful
ideologically, i.e., quite successful in suppressing Marx's theory
and current work rooted in that theory.) None of them prove that
the interpretation used to contradict Marx is the *only possible*
one, or that a contradiction must arise under *all* possible
interpretations.
But actually, these failures do prove *something*. Given that some
interpretation does succeed in replicating Marx's conclusions, the
failure of another interpretation to do so proves that it is a
flawed interpretation.
Andrew Kliman