# [OPE-L:7125] [OPE-L:627] Re: Use and abuse of mathematics [OPE 574]

Gil Skillman (gskillman@mail.wesleyan.edu)
Mon, 08 Mar 1999 12:32:24 -0500

Alan raises a number of deep and interesting points in this post about
the difficulties involved in establishing a relationship of "equality."
However, to the extent they apply I think their logical force works
*against* Marx's Chapter 1 argument (and thus attempts to defend it)
rather than my critique of it. I'll try to demonstrate this below.

Along the way I'll try to explain why I haven't really imposed any of
those nasty things on Marx's argument that Alan keeps accusing me of
(trying to impute neoclassical or mathematical assumptions, etc.). I now
have a sense of why it might necessarily *appear* to Alan (and to those
who share his position) that I'm doing these things, while it seems to me
(and those who share my position) that Alan is continually evading the
central point. I'll speak to this at the end.

>I begin with an apparently minor point: as Gil points out (4B)
reflexivity can

>be deduced from symmetry and transitivity.

But I *don't* "point [this] out"; my point was rather that it is a
*confusion* to assert that reflexivity follows from symmetry and/or
transitivity. I'm pretty sure no mathematician would assert this. If
"E" stands for the relation "is exchanged for", then reflexivity, or aEa,
means "a is exchanged for itself," which of course never happens, by the
nature of exchange. In other words, mathematicians who have for all
years have defined an equivalence relation as including reflexivity as a
separate condition from symmetry and reflexivity haven't simply been
foolish or careless in doing so. Of course, one can simply stipulate
exchange reflexivity axiomatically, quite apart from what happens in
reality, and I'd be happy to do this for the sake of discussion.

>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

What? No I don't. In the argument Alan refers to, I say 1) the
assertion that exchange satisfies reflexivity, as the axiom is generally
understood, is problematic, but (2) *even given* reflexivity, symmetry,
transitivity, and composability thrown in for good measure, exchange does
not establish a relationship of equality in a sense sufficient to support
Marx's subsequent argument. How can Alan criticize my argument if he
doesn't even know what it is?

>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.

This seems to be a remarkable case of throwing stones from a glass house.
Alan makes a *much stronger* assertion of authority for Birkhoff and
MacLane than simply being "standard" when he wrote

>"That's what 'equal' means. Nothing more, and nothing less."

This sounds *exactly* like an attempt "to turn [mathematics] into a
source of authority."

> 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.

And of course, contrary to Alan's accusation, my point was not to use
mathematics as an authority, but rather to point out that, by his own
source,

Alan's conditions of RST plus substitution do not establish "equality"
but at most "congruence." Do we agree on this point now? And the issue
has never been whether mathematics " 'proves ' Marx right", the issue is
whether this interpretation is adequate to support Marx's use of the
term "equality" in his chapter 1 argument. And as I've argued in detail,
it isn't.

>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.

Excuse me, but this accusation is simply preposterous. As originally
stated, my critique of Marx's Chapter 1 argument did not "use
mathematics" at all. *Alan* introduced a particular mathematical
treatment of equality to illustrate a particular interpretation of the
notion, and I addressed this notion *on its own terms* to show it was
inadequate to support Marx's use of the term "inequality" in his chapter
1 argument. (And it also misrepresented Birkhoff and MacLane's definition
of the term "equality.")

>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.

Again, this accusation is ludicrous. I agreed with Alan that Birkhoff
and MacLane's discussion helped to illuminate various senses of the term
"equality", and then used that discussion to give some sense of the
difficulty in Marx's argument. See below for further discussion of the
term "equality."

>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.

More stone throwing from the glass house. Would a "true mathematician"
then say, as Alan does, "That's what 'equal' means. Nothing more, and
nothing less"?

>Actually in mathematics there *is* no standard definition of

>equality.

Compare this position to Alan's statement quoted just above.

> 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".

Yes, I've said all along that this is one possibility: one could
*define* equality in such a way that commodity exchange can be said to
establish an equality relation as a simple tautology. More on this
approach at the end of my post. But this does not appear to be what Alan
has been trying to do, which is to establish that exchange establishes a
relationship of equality in something like the sense we commonly
understand the term. In particular, as I've shown in detail, exchange,
even under the conditions of reflexivity, symmetry, transitivity, and

(de-)composability, cannot be understood to establish an "equality" in
the sense required for Marx's Chapter 1 argument.

>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.

What? No it doesn't. The equivalence relation is established in terms
of a particular dimension, in this case color. No claim *whatsoever* is
argument requires.

> 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.

Skillman, not really a mathematician at all, has no problem with it
either, but it doesn't support Marx's Chapter 1 argument.

> 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.

See above. How Alan arrives at this assessment is mysterious to me. I
see Carnap as making pretty much the same claim about equivalence
relations that Birkhoff and MacLane make, and the "kind of equality" thus
established exists only with reference to the *explicitly asserted*
relation R, not *any other* relation.

>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.

Here, I think, is the nub of the issue. Whether or not there is an
"absolute" distinction between equality and equivalence, it seems clear
that the existence of an equivalence relation, i.e. a binary relation E
satisfying reflexivity, symmetry, and transitivity is insufficient to
establish equivalence along *some other* dimension L, as Marx's Chapter 1
argument requires. And as for what is or isn't "employed in
mathematics", I wonder if Alan knows of any mathematician who would allow
an equation between entities measured in different units, such as Marx
asserts when he writes "1 quarter of corn = <italic>x </italic>cwt of
iron."

>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?

I'll accept this point, since it bolsters the critique of Marx's Chapter
1 rather than detracts from it.

>If not, what justification is there for treating

>Gil Skillman as an economic agent with reflexive preferences?

This is a non sequitur. Even if I'm not the same person, the old and new
me may still be held, unproblematically, to have reflexive preferences.
In other words, non-identity does not imply that there are *no* features
in common. In contrast, Marx's argument requires that, given an
equivalence relation established (in whatever sense) by exchange, there
is a *particular* other feature in common among exchangeable bundles.
This also does not follow.

> 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)

This passage is doubly a non sequitur, first for the reason given above,
and second because I have not relied on "bluff empiricist common sense"
for my critique. I find the Quine stuff quite interesting but not
applicable to this point.

>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.

That's fine. This does not speak to my critique of Marx's Chapter 1
argument. As I'll show below, it strengthens it, if anything.

>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.

I appreciate these comments. I'd add just for the intrinsic interest of
the subject that there's another way out of the difficulty Russell
addressed, provided by G. Spencer-Brown in his amazing book <italic>Laws
of Form. </italic>But so far as I can tell neither approach rescues
Marx's Chapter 1 argument.

>The whole approach being used in this discussion by participants,
particularly

>Gil, therefore misrepresents what is at issue.

I beg to differ.

>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.

I'm puzzled. Just a few posts ago, Alan was anything but agnostic on the
issue. Having posited the exchange conditions of reflexivity, symmetry,
and transitivity plus (de-)composability (RSTC) he stated "That's what
'equal' means. Nothing more, and nothing less" Note, not "what [Alan]
takes it to mean for the sake of this discussion" but "what 'equal'
means." And he accuses *me* of fence-straddling and trying to "impose"
mathematical definitions....

> They actually say 'look, you

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.

>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

I cited this, let us not forget, in response to Alan's blithe assertion
that Birkhoff and MacLane define equality as an equivalence relation
satisfying a substitution property, which they don't. Do we now agree
that Alan's representation of Birkhoff and MacLane's argument was
inaccurate?

>'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.

Excuse me, but this is the first I've heard that *infinite* sets were
even potentially at issue in this discussion. There are no infinite
commodity bundles I know of. There is not an infinite number of exchange
relations. I have not heard anyone assert that given commodities have an
infinite number of characteristics. [And if they did, wouldn't we need to
consume only one commodity?] So Alan, first let me ask if you think the
problem of infinite sets is relevant *in this case*. If not, I see no
problem with using B&M's definition, at least as a point of reference.

But suppose that the problem of infinite sets is relevant to this
discussion. The force of this point is that in that case it may be
impossible to define "equality" unambiguously. Fine; that's an
additional strike against Marx's Chapter 1 argument, because he states
that "valid exchange-values of a particular commodity express something
*equal*" and "[an exchange relation] can *always* be represented by an

Bottom line: if Alan's point here is relevant to the discussion, it
counts as an additional argument against Marx's *un*ambiguous claims in
Chapter 1 about the "equalizing" properties of exchange.

>Would Gil like to explain how one compares an infinite number of
objects?

Perhaps Gil can't. Would Alan like to explain how the problem of
infinity is relevant to a world of scarcity? And if it does, how the
ambiguity Alan refers to here doesn't count as an *additional* strike
against Marx's decidedly *un* ambiguous claims in Chapter 1?

>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.

that's only necessary if in fact we're dealing with infinite sets in a
finite economic world), it isn't sufficient to support Marx's Chapter 1
claims.

>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.

And what *I* find wearying about the discussion is that it keeps
meandering away from the central point: there is no interpretation of
the term "equality" that I know of, other than by simple tautology, that
is sufficient to support Marx's inference from the fact of exchange
(under whatever conditions) that there exists "a common element of
identical magnitude" in exchanged bundles. In particular, Marx's
inference does not follow from Alan's *particular* representation of
exchange based on RSTC. And at best it's unclear how adding "a couple of
extra axioms" would do the trick.

> The concepts 'equality' or 'property' are taken as given, as

>something we can borrow without question from the mathematicians.

This characterization is entirely false, and turns the discussion on its
head. It is *Marx* who insists on a *particular* "equalizing" property
of exchange, so the burden of proof is on those who agree with this
assertion. It doesn't follow from the commonplace notion of the term; it
doesn't follow from RSTC; if it can follow from Alan's "RSTC plus a
couple additional axioms" or Andrew's "alternative interpretation", it is
incumbent on them to demonstrate it. [For what it's worth, Andrew's
paper, for all of its other virtues, does not do so.]

> 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.

Again, this turns the discussion on its head, because my core point is
that *Marx* has not been careful in his use of the term "equality" in his
Chapter 1 analysis.

>

>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.

These accusations are both false. See above.

> So far, so good. But then the following

>creeps in: mathematics, or competitive general equilibrium *produces
results

Again, false. Nothing in my argument relies on a *particular*
mathematical formalization or the assumption of competitive general
equilibrium (indeed, earlier Alan was accusing me of not even allowing
the law of one price, so the present accusation actually contradicts his
former one).

> 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.

This has nothing to do with what I've been saying. Marx is apparently
wrong because there is no evident sense in which exchange establishes an
"equation" in a sense sufficient to support Marx's inference that "a
common element of identical magnitude exists in two different [exchanged]
things". Now of course, one could follow Alan and Rosenbloom and
*define* exchange (under a particular set of conditions, perhaps), to
establish a sort of equality, in which Marx's inferences follow as a
simple tautology. I'd be interested in seeing the properties of that
relation, which have not yet been established, either by Marx or by Alan.

>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.

Again, the person now urging caution and warning against strait-jackets
is the one who said a few posts ago, evidently throwing caution to the
winds and seeking to impose a strait-jacket on the discussion, "That's
what 'equal' means. Nothing more, and nothing less". In contrast, as
originally stated my critique makes no use of mathematics, and I only
responded to *Alan's* mathematization because it evidently misrepresented
Birkhoff and MacLane's understanding of the term "equality."

>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.

I accept this, which should be no surprise since I've never argued
against it. But theories must still be internally consistent according
to their *own* logic. Therefore I'll reiterate my (not mathematically
based, not neoclassically based, not general equilibrium based) point:
there is no evident sense, other than by definition (a definition we have
not yet seen, by the way) in which exchange, under whatever conditions,
establishes a relation of "equality" with respect to commodity bundles in
a sense sufficient to support Marx's argument in Chapter 1. A *minimal*
condition for such a theory, in order to avoid a proof by contradiction,
is that the asserted equalizing properties of exchange must only obtain
with respect to *commodities* but not other exchangeables.

>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.

Again, I won't argue this, and Alan may well be getting at what's been
troubling this discussion and causing so much frustration. It may well
be that Alan and those who agree with him are taking it as a *primitive
assumption* that commodity exchange establishes a form of equality,
rather than as an *inference* (as Marx strongly suggests in using terms
like "therefore" , "it follows from this that..." and "must
therefore...") while I, and those who agree more or less with me, are
taking it as an inference. But if this is the case we've been attempting
to debate across a paradigmatic divide, a necessarily fruitless
endeavor.

So let me ask Alan: do you take the claim that exchange (under some set
of conditions) establishes a relation of equality with respect to
commodity bundles as an *inference*, as Marx suggests in his Chapter 1
argument, or as an *assumption*? And if the former, what is the
definition of the "equality" purported to arise in exchange that is held
to entail Marx's subsequent inferences?

Gil