Gil writes [1439 of 11/3]
"Imagine my reaction, then, when innumerable posts of mine
have failed to elicit general agreement about the manifest
invalidity of Marx's conclusion to Ch. 5, which for example
has nothing to do with Marx's notion of the value of money
or whether new value can be created solely in the process of
exchange (of course it can't, by Marx's definitions)."
I think this is a fair point, but one which illustrates my
point.
To be totally blunt about it, my view is that it is worse
to fail to answer a numerical example than a proposal framed in
the language of the propositional calculus.
I am not talking about how insulting, frustrating or demeaning
it is. I am talking about the proof of truth. Again, to be
blunt; from the fact that you have not had a satisfactory reply
to your arguments in logic, I deduce that there is a problem of
mutual understanding and respect. But I do not deduce that you
are right, or that those who have failed to respond to you are
wrong.
>From the fact that a theory cannot respond to a set of numbers
which do not violate its base assumptions but deduce a
contradiction, I deduce that this theory is wrong.
That's the difference.
I agree it is not good that Gil has not had a reply to his
arguments. I have tried to answer as much as I can and my view
is the following: although you may feel the answer I have given
is not satisfactory, I don't think you can say with a straight
face that I haven't given an answer.
But be that as it may, I don't think that from the exchange
between us, any major point in truth is settled or can be. I
think the people who agree with Gil will go on agreeing with
Gil, and the people who agree with me will go on agreeing with
me, and the people who don't care one way or the other will go
on not caring one way or the other. My only aim in such a
discussion is that all three groups will understand better than
before what each of us is arguing. In short, I don't expect a
debate of this character - an argument in formal logic -
actually to settle anything.
An argument in formal logic can settle something between people
who agree a common basis, but I really don't think it can ever
settle anything between people who don't agree a common basis.
I think if you try to achieve this you will get very very
frustrated. I sympathise with you, but there is nothing I can
do. Because if you want to settle the issues involved by means
of formal logic, you won't. Modern model theory has actually
proved this.
What we are really doing, is arguing across paradigms. And
between paradigms, there is no proof in logic that will ever
settle it. No proof in logic establishes that the earth goes
round the sun. It is a matter of observation, of fact, of data,
of numbers.
When Aristarchus established that the earth going round the sun
was the only way to explain the observed phenomena, Cleanthes
of Samos argued that he should be tried for impiety as follows:
"Cleanthes used to say that it behoved the Greeks to bring
[charges] against Aristarchus of Samos for moving the
Hearth of the Universe, because he tried to save the
phenomena by the assumption that the heaven is at rest, but
that the earth revolves in an oblique orbit, while also
rotating about its own axis"
Well, what can I say? In *logic* Cleanthes' argument is
actually impeccable. If one assumes that the earth is the
Hearth of the Universe, *then* Aristarchus is impious. There is
nothing more to be said. For that matter if one studies the
accusation against Galileo it is astonishing how *logically*
incorrect he was and how right his accusers were.
There is only one minor problem: by the standpoint of the
facts, he was right.
You have different prior assumptions from me. How can I show
that your assumptions are wrong? How can you possibly show that
mine are wrong?
I think I have shown that on the basis of my prior assumptions,
chapter 5 can be interpreted completely logically and
consistently and without any arbitrary presuppositions.
On the basis of your prior assumptions, my argument includes an
arbitrary axiom.
My disproof of your disproof consists in rejecting your axiom,
your disproof of my disproof consist in rejecting my axiom. I
don't know how much further we can go.
You think it is arbitrary to assume that the value of money is
equal to the total value of all goods in circulation. I think
it is arbitrary to assume that relative prices are given prior
to the emergence of a universal equivalent.
How do we settle this? My conclusion from the debate is that it
*cannot* be settled. There is no way, from within formal
mathematical logic, that it can ever be decided whether Gil
Skillman or Alan Freeman has got the better of it. The best we
can hope for is mutual understanding.
The proof of this is itself empirical: I don't know of any
major scientific issue that has ever been settled in logic.
The most we can hope for within mathematical logic is that both
Alan Freeman and Gil Skillman have put forward different
arguments on different prior assumptions and that each think
they are internally consistent (BTW I am sure your arguments
*are* internally consistent. Thus the argument is slightly
asymmetric in that I allow you your arguments, but you deny me,
and Marx, our arguments).
I remain extremely sceptical that it is possible to progress
beyond mutual understanding in cross-paradigm discussion. So
what can we do in cross-paradigm discussion? Well, we can agree
that we are both talking about the same external world.
That means, I can learn from you my looking at the phenomena
that you want to understand, and you can learn from me by
looking at the phenomena that I want to understand.
I can try and look at the way in which usury and mercantile
capital give rise to phenomena of exploitation, and you can
look at how moral depreciation gives rise to disparities of
income and endowment.
And we can illustrate these phenomena with numbers. True, in
doing so we add a few assumptions (not to the numbers
themselves but to the explanation of the numbers). But it is
always open to the other person to keep the same numbers and
change the assumptions.
In my view one of the most fruitful exchanges we had was when
we tried to clarify why I thought Marx's '$50 of corn exchanges
with $40 of wine but overall no-one gains or loses' argument
could be resolved with a price of $45, and you explained your
objections. I don't think anything was settled, but I think
each of us learned a lot about each other's way of approaching
the issue. I don't think that would have happened if we had not
discussed definite numbers, from within a very different
paradigm. The numbers illustrated what the differences were.
They allowed each of us to explain in our own language, on our
own ground, what we thought of these numbers.
That's why I think Marx was absolutely right, in Capital, to
use numbers instead of models. His numbers still form the basis
of almost every argument today. The very fact that so mahy
*different* arguments from so many *different* paradigms, can
all discuss the *same* numbers, proves why their usefulness is
much more universal.
I think we can both present ideas across paradigms by simply
spelling out either 'example' numbers as Einstein did - which
he used to call a 'thought-experiment' - or even by looking at
actual numbers in an actual economy. Generally speaking I would
rather do the latter, but sometimes the difficulty can be that
it is not possible to isolate the particular phenomena which
one wishes to discuss because in real data there are always a
very large number of determinations and one may wish to study
one of these in isolation from one of the others. In this case,
one has to choose a set of numbers that one hopes can be
accepted by both paradigms as 'reasonable'.
But then we are involved in a different level of discussion and
a different means of settling arguments.
To take an extreme case: no matter how internally consistent,
no-one could possibly accept an argument that predicted the
world had ended on March 13th, and still be there to argue the
case on March 14th. It is refuted not by logic but by the
*data*.
To take things one step forward. No-one could argue, no matter
how consistently, that all numbers add up to three, if anyone
can produce even two numbers that do not add up to three. This
argument is refuted not by logic but by *data*.
This distinction is made very clearly in model theory.
An argument in logic is a formal deduction from linguistic
statements, possibly involving logical variables, on the basis
of formal rules of deduction. An example of this type of
reasoning is 'all B implies A, C implies B, therefore C implies
A'. Impeccable.
But a second type of independent verification of such a
statement is possible. One can put actual values for the
variables A, B, C and see what happens. One can say A is 'I am
wet', B is 'it raining' and C is 'we are in Manchester'.
Then one can check the truth or falsity of the statement by
exhaustively enumerating all possible truth-values for the
deductions and establishing in all such cases, the deduction is
valid.
In model theory, there is a very important type of theorem that
establishes that any deduction is necessarily true for all sets
of truth values, and that anything that is true for all
possible sets of truth values, must be deducible in logic. It
is called a completeness theorem.
Unfortunately, it is only true for the very simple and trivial
logic called the propositional calculus.For the kind of logics
that I and you use in our debates, this type of theorem does
not hold.
Nevertheless, it is the case that if even *one* set of truth-
values can be discovered for which a formal deduction does not
hold, then this deduction *must* be false.
This remains true, even if we cannot discover a proof, within
formal logic, that the deduction is false!
In short, if your theorems don't work on the data, your
theorems are false. But if you cannot deduce a contradiction
from your theorems, you cannot conclude these theorems are
true.
And I think this is as far as it is possible to go without
discussing actual examples.
Alan