# [OPE-L:1170] LVB6:What's in a definition?

Alan Freeman (100042.617@compuserve.com)
Wed, 21 Feb 1996 01:56:57 -0800

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Is the expression total values = total prices merely a
convenient or arbitrary definition? I'll finish on this one
apart from a summary in the next and last post.

The first point which has to be made is that there is nothing
wrong with definitions. I have in front of me a list of
definitions (I got them from a library) including:

the number 1
the number 2
a set
equivalence
Latitude 60 North
mass
the centimetre
an isosceles triangle
a theorem
a proof
isomorphism

Without them, neither human thought nor Gil's attacks on Marx
would get very far.

These definitions are all landmarks in human thought. People
worked for them. They were hard to come by. They may lack something
but they are nothing to be ashamed of. In short, they're bloody
good definitions.

Not all definitions, even apparently obvious definitions, are good
ones. Frege's (contradictory) definition of a set brought an
era of philosophy to an end, and Russell's started a new one.
The search for a definition of proof at the heart of a hundred and
fifty years of studies in the foundations of modern mathematics.

The really difficult thing in constructing a theory is not to
avoid definitions but to get the right ones. Therefore, I don't think
that it tells us much to learn that key results are derived from key
definitions. Something more is required.

Behind the notion that a definition is in some sense 'trivial'
or cheating is, I think, one of three things: either,

(1) the definition is not explicit or,
(2) the definition is inconsistent, or
(3) the definition is circular(see point 3)

I think there are two further reasons to criticise a definition
which don't give rise to the same feeling of being cheated, but
which should not be neglected:

(4) the definition does not fit reality, or
(5) the definition is not independent.

If Marx, in Chapter 5, employs a 'hidden' definition, so that the
identity of the sum of values and the sum of prices is assumed but
not stated, then I take this for a genuine (but not intractable)
difficulty. I think I have shown there is no hidden 'extra' equality
but a consistent application of the definition of money.

I certainly think Marx's claim to consistency has been re-
established, which Gil I think accepts. And I have argued,
though briefly, that the interpretation which restores this
definition matches observed reality well.

That leaves two questions: are there circular definitions in
Marx, and are there superfluous definitions?

The argument that Marx contains circular definitions has not
been raised in the debate, so I'll just mention (i) that
Mino has dealt with this quite well in couple of papers
(ii)that what is often taken for circularity is in fact nothing
but mathematical induction. Mathematical induction is an
argument that says, if a property of (t+1) can be deduced from
a property of (t), and if the property is true of any given t(0),
then it is true of all (t) after that. This feature of Marx's
reasoning is completely lost in simultaneist presentations.

It is useful to ask further, whether the concept of the value of
money, or which is the same thing, the equality of total values
and total prices, is derived or independent (an allegedly
'arbitrary' axiom is in fact an *independent* axiom). This is
certainly one of the questions the debate has posed for me.

What is a redundant, and what is an independent, definition?
============================================================

The most difficult, and in general intractable problem, in
analysing any formal axiomatic structure, is the dreaded
question: what are the minimum necessary assumptions?

Gil often writes as if it is possible to trace backwards,
through an automatic process, from the conclusions an author
draws and the arguments s/he uses, to the assumptions s/he must
have made. I think this is more problematic than he may have
realised.

One can always establish by some sort of textual analysis the
assumptions an author *says* s/he made. This doesn't get you
far unless your only aim is to destroy them.

In general it is very difficult to prove that an argument, which
does not flow from given premises, will not flow from other
premises, or to establish the minimum necessary premises for the
conclusion to be deducible. One cannot 'run the film of logic
backwards' and even when one can, it is very difficult to get

Indeed in a certain sense the whole endeavour of Speculative
Philosophy, in which Marx was steeped, was to establish the
minimum necessary postulates for the whole of human thought.

What is found, when this is done, is that the issue: is an
axiom independent? (or as Gil would have it 'arbitrary') is
deeply connected to the question: is an axiom redundant? For,
if an axiom is redundant, then it is not independent, and
vice versa. But it is in general very difficult to establish
that an axiom is definitely not redundant. That is, it is
extremely difficult to prove it is genuinely independent.

The case of geometry illustrates the difficulty.

Theorems based on the axiom that parallel lines do not meet
are vital to the whole of engineering. Is this a postulate or
a fact of reality? Geometry approached this question by asking
whether this axiom could be *deduced* from the other axioms.

This debate dogged it for centuries. Were all these vital theorems
generally true, generally false, based on arbitrary definitions,
true for particular cases only? Geometers struggled to prove that
the axiom was redundant - that all theorems of geometry could be
proved without it.

We now know the axiom is not redundant, but only because
Riemann and Lobachewsky exhibited geometries in which it does
not hold. If they had not done so, we still would not know
whether it was genuinely independent, or merely a rather
obscure deduction.

More interesting still, it is only when it was proved to be definitely
an independent axiom that it was ejected into the realm of the
empirically testable. It wasn't until we knew that (and how) our
universe might be curved, that we could test whether it actually was.

So the status of 'total value = total price' is clearly quite
important.

You can show a postulate is redundant by deducing it from other
postulates. But a failure to prove a postulate is redundant does
*not* prove it is not redundant.

The only way you can actually show a postulate is not redundant -
that is, independent - is to demonstrate a system which does
without it. And I think that is what Gil should concentrate on.

*But* this proof of independence is not achieved until that system
has been subjected to all the usual tests; consistency, generality,
noncircularity, match with reality, and so on. *Until* that has been
done, there is no basis for a claim it is arbitrary and such
arguments actually lead up a blind alley.

Until we have seen the whole of Gil's system and are sure it works,
there is no basis for accepting his criticisms of an alternative
system which definitely does work. This is yet another reason why,
in my view, *starting* from the 'errors' in Marx is a no-hoper and
indeed positively dangerous.

Euclid did not commit any error by writing *as if* his geometry
were the only one possible. He did not make a 'fundamental logical
error' by neglecting non-Euclidean geometry. On the contrary, his
genius consists in the fact that, even given the limitations of
his time, his profound knowledge of the structure of reality, and
of geometrical thought, prevented him from stating parallelism to
be self-evident and led him instead to pronounce it a separate axiom.

I cling to the obstinate belief that value theory is a class of
axiomatic systems and that, just as one can with various axioms
define various geometries or various algebras (groups, modules,
rings, fields etc), one can with various axioms define various
economics, and study their mutual interrelation. Probably even
deduce their class basis.

Is 'total values equals total prices' an axiom or a theorem?
============================================================

This , I think, is a real question and the debate with Gil has
helped bring it to light. But if it turns out to be an axiom I
would probably think more, rather than less of it, though I
would want it cast in what, for me, is its correct form, namely
that the monetary expression of labour is equal to the total
price of all goods in circulation, divided by their total value
(and that, hence, each pound represents in exchange a
corresponding portion of this total value)

Indeed the major question for investigation, for me, is the
whole issue of exactly *which* set of goods, in exchange with
money, constitutes the social labour which money represents? In
simple circulation without stocks I would argue this is just all
goods that are brought to market. But once stocks are introduced
we get the meat of the whole issue concerning moral depreciation;
and once world trade is introduced we come up against the origin
and magnitude of unequal exchange. So it is a very important
question.

In exactly the same way that it is a great geometrical
discovery to recognise the parallelism axiom as a separate axiom
and thus prove its independence, if it emerged that " total prices =
total values" is a genuinely independent axiom, then this would
seem to me rather an important discovery, not at all a slur on
the system.

However, my view it is not an independent axiom.

Why is 'total value equal to total prices'? That is, in terms
of the thought experiment, why is it that nothing that happens
on January 2nd 1990 can create or destroy value already produced in
1989, even in its price representation?

We have already dealt with this directly, by analysing the
monetary expression of value. But an alternative argument,
already hinted at, may illustrate why any other conclusion

Consider the change which takes place between 1st January and
2nd January, when the same set of goods is redistributed at new
prices. We can prove the invariance of the value sum without
referring to the sum of values directly, as follows.

Once sale at any given prices has occurred (1st January
prices), we can assess the effect of a new set of prices (2nd
January prices) independently of the amount of value that the
original sum represents, by valuing the new money in terms of
the old. Indeed, this is actually what is done by the exchange
markets, when one currency is valued in terms of another.

For, given that the goods concerned sell for given prices on
1st January and given that a new set of prices is reached on
2nd January, we can directly compare each 2nd January price
with the correspnding January 1st price; just as we could, for
example, compare a set of dollar prices with a set of pound
prices.

Suppose that the total of all prices for which goods sell on
2nd January is \$3000bn, twice as much as the total price which
the same goods fetched a day ago. Clearly, the new currency by
any reasonable standard is worth half the old one. If this is
disputed by Gil, then it seems to me the onus is on him to
explain what alternative definition of the magnitude of
inflation should be adopted under these circumstances.

But this means we can establish a set of 'deflated' 2 January
prices, by dividing every individual price by the sum of 2
January prices and multiplying by the sum of 1 January prices.
Thus if wine and corn cost \$40 and \$60 on 1 January, and were
found to cost \$120 and \$80 on 2 January, their deflated 2
January prices would be \$120/2=\$60 and \$80/2=\$40. Thus in
*relative* price terms, wine would gain \$20 and corn would lose
\$20, but due to inflation wine would rise by \$60 and corn by
\$40.

We can thus establish the relative change in price regardless
of the value it represents. We can, in particular, establish
the ratio between the original monetary expression of value and
the modified monetary expression of value, without knowing what
the underlying values actually are. We may therefore express
the modified relative price of two commodities in terms of the
original money. All that is required for this calculation, is
that the two sets of prices should refer to the same two sets
of goods.

In the successive passage from one such set of prices to
another such set of prices to a third such set of prices, the
terms representing monetary inflation in each transition are
simply additive, being in total no other than the differences
between successive price sums.

Since we can pass from any set of prices to any other set of
prices by means of a continuous transformation, we therefore
find that the sum of prices decomposes uniquely into two sums,
a fixed 'initial' magnitude and a term representing monetary
inflation. This fixed 'initial' magnitude is in effect a
constant of integration and does not vary as we move from price
set to price set. It is analogous to the fact that potential
energy is defined only to within an arbitrary constant.

But one particular case of prices is given by sale at values.
The 'definition' involved in equating total prices to total
values, therefore consists only in stating that the monetary
inflation involved in sale at values, is zero, or to put it the
other way around, that the arbitrary constant is equal to the
sum of values.

Moreover, since the non-inflationary component of the sum of
prices is constant, the invariance of 1990 surplus value with
respect to price changes on 2nd January 1990 can be established
from this result alone.

We can conceptualise the argument as follows: how do I compare
the distance of Greenwich, US from the centre of the earth with
the distance from Greenwich, UK from the centre of the earth?

Method 1(normalisation): go to Greenwich, UK and walk to the
centre of the earth, carrying a large measuring tape,
preferably calibrated in standard commodity units. Record the
result. Go to Greenwich, US and repeat the experiment. Compare
the results

Method 2(easier): travel from Greenwich, UK to Greenwich, US by
boat, noting the rises and falls. Total up the rises and falls.
The result is the difference in the distance from the centre of
the earth.

Moreover since we know that both Greenwiches are at sea level
to eight decimal places, without conducting the actual
experiment we can conclude they are at the same distance from
the centre of the earth.

Alan