Dr. Ajit Sinha
Visiting Fellow
Centre for Development Economics
Delhi School of Economics
University of Delhi, Delhi 110007
India
sinha@cdedse.ernet.in
___________
> ...
> 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
>