I begin with an apparently minor point: as Gil points out (4B) reflexivity can
be deduced from symmetry and transitivity. (proof: suppose aRb, then bRa by
symmetry, hence aRa by transitivity). Steve makes the same point.
Only one conclusion follows from the above result, namely, we can reduce the
axiom set by one axiom.
This is an excellent result. It shows we don't need to imagine things
exchanging with themselves, to reproduce Marx's argument. Consequently, this
argument doesn't depend logically on something that can't happen. Excellent.
Wish I could say the same for neoclassical general equilibrium.
The question for me is: Why does Gil have a problem with that? He reasons
thus:
(a) the 'standard' definition of equality demands reflexivity
(b) exchange isn't reflexive
(c) Marx invokes equality to discuss exchange
(d) therefore, Marx's reasoning is false
Behind this lies an attempt which is alien to mathematics: to turn it into a
source of authority. I was very careful, in citing Birkhoff and MacLane, not
to speak of a 'standard' definition of equality, the word that Gil uses. My
aim was the opposite of Gil's. I sought to show only that Marx's argument is
*possible*, provided one attaches to the word equality a reasonable meaning
which is compatible with Marx's usage. Since this was attacked as an
unreasonable use of the word equality, I pointed out that mathematicians
regularly use the word in a very similar way without qualms. In no sense did I
intend this to say that mathematics 'proves' Marx right.
Gil's purpose, as far as I can make out,is to use mathematics to prove Marx
wrong: to set up mathematics as a superior standard by which to judge the
validity of Marx's concepts. His method in essence is to prove that Marx's
definition of equality (and for that matter, exchange) does not conform, does
not comply with approved mathematical standards.
This is a forlorn enterprise, which no true mathematician would undertake. The
function of mathematics is not to tell people how to think, but to help
clarify what they actually do think.
Actually in mathematics there *is* no standard definition of
equality. In the Penguin Dictionary of mathematics, there's no entry for it.
In two of the standard works on logic, Carnap(1958) and Rosenbloom(1950), it
isn't indexed. It is not a mathematical, but a metamathematical concept; it is
one of the things one 'takes as known'; one supposes the enquirer has a valid
concept of equality, whatever that might be, and tries to specify its
properties.
As Rosenbloom states (p9): "the relation '=' is taken to be part of the known
syntax language. The only properties of this relation which will be used are
[R,S,T] and their consequences...Hence, we could alternatively take '=' as an
undefined term, and postulate [R,S,T]. A relation satisfying the latter
conditions is called an *equivalence* relation.
Let's just re-phrase that because it's in very condensed language and it's
easy to miss what's going on. Rosenbloom says "look, actually, I as a
mathematician cannot tell you what equality is. It's up to you. You can give
it to me as part of your syntax or as part of your semantics, I don't care.
*My* job is to tell you what properties your 'equality' must have, if it's
going to work for you. And what I have to tell you is this: it works like
equivalence".
Let's go into more detail. I'm going to cite a passage from Carnap which I
think throws considerable light on the 'third property' argument and supports
Marx rather strongly. It's rather worth reading:
"Suppose R is a relation which expresses likeness (or equality, or agreement)
in some particular respect, e.g. color. Then obviously R is an equivalence
relation; the equivalence classes with respect to R are the maximal classes of
individuals having the same color; and each equivalence class corresponds to a
particular color. This approach presupposes the separate colors as primitive
concepts. If, however, the relation Having-the-same-color is taken as a
primitive concept, then the several colors can be defined as the equivalence
classes of that relation"
First off note that this more or less *exactly* reproduces Marx's 'third
property' argument. The equivalence relation is directly explained as arising
from possessing a property in common, namely, color. Carnap, a reasonably
eminent mathematician, seems to have no problem with this idea. This doesn't
mean that Carnap is necessarily right, but it knocks a rather big dent in the
idea that Marx is necessarily wrong, or that his 'third property' argument is
in some sense mathematically illegitimate.
Second, Carnap, like most mathematicians, does not employ any 'absolute'
concept of equality. Equality is always equality 'in some respect'. 'Having
the same color' may equally be considered an equivalence or an equality. The
absolute distinction between equality and equivalence which Gil and others
seek to make, is not employed in mathematics.
Indeed it's quite hard to see how equality *could* be rigorously distinguished
from equivalence: the nearest one might get is to say that equality is in some
sense 'identity'; well, if you can give me a precise and uncontroversial
definition of identity, I'd really like to hear it. To take only one non-minor
issue, is Gil Skillman at the end of reading this post identical to Gil
Skillman at the beginning? If not, what justification is there for treating
Gil Skillman as an economic agent with reflexive preferences? If Gil Skillman
is not equal to Gil Skillman, how can he figure as a variable in an equation,
pray? If you think this is an unproblematic question to be settled with bluff
empiricist commonsense, just check out a few writers like Quine(1953)
Third, and this is the crux, Carnap states above that the idea of defining
equivalence by means of equivalence classes is *just another way of talking*
about equivalence defined, in everyday language, as having a property in
common. His actual words are 'can be defined' as the equivalence classes of
that relation. Note that, Gil. Not 'must be defined' or 'can only be defined'
but 'CAN be defined'. It's a choice; moreover it's our choice, not the
mathematician's choice. There is no argument in mathematics that says it's
better to start from the property and deduce the equivalence class, or start
from the equivalence class and use that to define the property. One must seek
an argument from outside mathematics, from philosophy or from the nature of
the subject matter, or wherever.
Carnap himself goes on, following the passage I cited, to trace the history of
the modern concept which has been unconsciously (and uncritically) absorbed
and reproduced by the participants in this discussion. This concept, which
chooses to define equivalence in terms of equivalence classes instead of
common properties, did not descend from the skies or the mind of God; it was
initiated by Frege [1884:73] and systematised by Russell [1903: 166] and goes
by the name of 'definition by abstraction'.
The idea that one may speak of equivalence classes, forgetting the
properties that they come from, is neither divinely ordained nor necessarily
true. It's a reasoning tool, a method of approaching the rather difficult idea
of equality, which was devised not because it was found to be mathematically
necessary but as the outcome of an intense *philosophical* debate which began
with Frege's attempt to escape Aristotle's distinctions between subject and
predicate. Frege set out to define predicates in terms of sets; this was his
path-breaking contribution to logic. He said 'instead of using the predicate
"red", we can *define* this predicate as the common property of all red
objects.' He then demonstrated mathematically how this could be done in terms
of set theory. Ironically his attempt to do so fell down because it was
internally contradictory as Russell showed. Russell then produced an escape
route by distinguishing sets from classes, and his approach has from that time
more or less dominated foundational studies in mathematical logic.
The whole approach being used in this discussion by participants, particularly
Gil, therefore misrepresents what is at issue.
First, it fails to realise that the 'equivalence class' approach is not a
mathematical result, but the mathematical formalisation of a philosophical
discussion. Mathematics cannot itself supply the authority for speaking of
classes instead of predicates. That authority has to come from observation and
philosophical analysis. If you speak to real mathematicians about it, what you
find is that they are *agnostic* on the question. They actually say 'look, you
can start with predicates, or you can start with sets. You choose. I can do it
either way. Each is equally valid.'
Second, it is by no means unproblematic to do things Frege's way, to define
predicates in terms of sets, instead of defining sets in terms of predicates.
The whole field is fraught with contradiction, paradox and concealed
assumptions.
Just to give one: everyone supposes that we can define the equality of sets as
if it was no problem. Gil (2) blithely cites Birkhoff and MacLean's definition
'A=B if they consist of the same elements' as if it was completely
unproblematic. Excuse me; this definition is unambiguously valid *only* for
finite sets, as any competent logician will tell you. But there are an
infinite number of possible baskets that can be composed from any finite
number of use-values.
Would Gil like to explain how one compares an infinite number of objects? If
so, he will have achieved in one short post what mathematics has been
struggling with for a hundred and twenty years. There are an infinite number
of equivalence classes defined by the exchange-relation. So far, mathematics
has not *agreed* on a method of enumerating infinite classes or testing for
their equality. It simply adopts an extra axiom to say that it can be done,
because without this axiom, nothing works. This is literally the only reason
offered for this axiom. An entire branch of foundational logic, Intuitionism,
simply refuses to accept it.
What I find very wearying about much of the discussion is that it hardly if
ever enquires into the origins, weaknesses, or limits of the concept of
property/predicate, or of equality, which it seeks to impose on Marx or use to
understand Marx. The concepts 'equality' or 'property' are taken as given, as
something we can borrow without question from the mathematicians. The
discussion doesn't even borrow carefully, with due attention to the origin and
meaning of the borrowed concepts they borrow; worse still, it entertains no
doubt that the concepts *work*; even though any practicing mathematicians will
warn you ceaselessly against the use that we are trying to make of them, and
vigorously debate such uses among themselves.
The discussion takes the following form, therefore: we want to try and
understand Marx. Marx is difficult to understand. Let's re-formulate Marx,
therefore, in terms of something we think we do understand: mathematics, or
competitive general equilibrium. So far, so good. But then the following
creeps in: mathematics, or competitive general equilibrium *produces results
that seem to contradict Marx*. Mathematics appears to deny that one needs a
third property. CGE appears to establish that there can be forms of exchange
other than those discussed by Marx. Therefore Marx must be wrong.
No: 'mathematics' can be wrong. I put 'mathematics' in scare-quotes because
the mathematicians themselves are infinitely more cautious, and would not
impose on the structure of enquiry, the straight-jacket that its users seek to
place around it.
We must do is drop, once for all, the notion that there is some arbiter
of logic, some *deductive* (dare I say Cartesian) process that will settle
disputes between theories that attach different meanings to the terms they
contain. We have to proceed in two stages:
(1) we should enquire in the most *sympathetic way possible* as to the
possible meaning of the theories we wish to compare, using mathematics only to
interrogate their structure in their own terms, and in this way try to get
clear what the theory actually says, in its own language, with its own logic.
(2) we should then compare all such theories, not against some canon of
authority such as Palgrave or mathematics, but against the commonly-observed
phenomena of the world.
The test of a theory is whether it *best explains what we see*; all attempts
to interpose an authority between interpreting and testing a theory, to
rule a theory out of court *before* it is tested against reality, are
ultimately attempts to suppress the use of science.
References
==========
Daintith, John and R.D. Nelson (eds) (1989), "The Penguin Dictionary of
Mathematics", Harmondsworth:Penguin.
Rosenbloom, Paul(1950) "The elements of mathematical logic". Dover
Carnap, Rudolf(1958) "Introduction to Symbolic Logic and its Applications".
Dover.
Frege, Gottlob (1884) "Die Grundlagen der Arithmetik", Halle 1884 (English:
Oxford 1953)
Russell, Bertrand (1903) "The Principles of Mathematics", 2nd ed London (1937)
and New York (1938): Cambridge
Quine, Willard van Orman (1953) 'Identity, ostension and hypostasis' in "From
a Logical Point of View", New York: Harper