Leibniz though that strict identity was simply a matter of satisfying all
equivalence relations (ie identity of indiscernibles).
However, satisfying an equivalence relation implies that a property is
shared in common only if we take a nominalist definition of property that
equates it with an equivalence class. Two things can belong to an
equivalence class without sharing a property in any substantive sense eg.
two things can be equivalent in that they are both mentioned by me on
5/3/99 and theerefore both belong to the equivalence class of things
mentioned by me on 5/3/99, but they will not share a property in any
substantive (causally explanatory sense).
It is better to take the example of weight and mass used in earlier posts,
I think. It is an open queston whehter marx is right to say that there is a
comon property analogous to mass that explains price equivalence in some
way comparable to the way that mass explains weighing the same.
>Thanks for Gil's reasonably comprehensive summary of his argument, to which
>I'm now responding in part, in particular the 'third property' argument.
>
>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
Dr Ian Hunt,
Associate Professor in Philosophy,
Director, Centre for Applied Philosophy,
Philosophy Dept,
Flinders University of SA,
Humanities Building,
Bedford Park, SA, 5042,
Ph: (08) 8201 2054 Fax: (08) 8201 2556