[OPE-L:7108] [OPE-L:607] Re: Use and abuse of mathematics [OPE 574]

Ajit Sinha (sinha@cdedse.ernet.in)
06 Mar 99 13:13:28 IST (+0530)

I found Alan's post extremely interesting. Alan, could you now go
back to Marx's chapter one and explain to us how Marx's reasoning
of deriving the equality of labor from commodity exchange relation
is valid, given Carnap and Rosenbloom's ideas of equality and
equivalence? I'm sorry I have been away from ope-l for a long time
and have not followed this current debate. For those who care, my
new (and always temporary) address is given below.
Cheers, ajit sinha

Dr. Ajit Sinha
Visiting Fellow
Centre for Development Economics
Delhi School of Economics
University of Delhi, Delhi 110007
> ...
> 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