[OPE-L:7117] [OPE-L:619] Re: Use and abuse of mathematics [OPE 574]

Ian Hunt (Ian.Hunt@flinders.edu.au)
Mon, 8 Mar 1999 11:28:31 +1030

I largely agree with Alan's argument here. Equivalence relations are
relations of identity (however, they are relations of identity that are
compatible with difference). In set theory, or any restricted domain for
that matter, we can define identity as the equivalence relation between x
and y in that domain that implies that x and y satisfy all equivalence
relations in that domain. So "has the same elements as" can be taken as
identity in set theory. (ie for sets A and B, A=B iff (z)(z is a
member(element) of A iff z is a member(element) of B).

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
>(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
>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
>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
>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.
>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".
>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