[OPE-L:7086] [OPE-L:584] Re: How mathematicians think about equality [OPE 523]

Andrew Kliman (Andrew_Kliman@email.msn.com)
Wed, 3 Mar 1999 12:24:26 -0500

A reply to OPE-L 571.

Gil wrote: "I do not mean to sound peremptory in posing such

No problem. I don't mind answering the questions. I'm a firm
believer in the Socratic method as a road to truth.

Gil: "In fact I am quite curious, even more so after you answered
my questions, to hear what you think Marx *was* saying, especially
concerning the first two paragraphs on p. 127 of the Penguin or
Vintage edition of V. I."

A section of my recent paper, "Marx's Concept of Intrinsic Value,"
is devoted to precisely this. (The paper is a revision of one I
presented at the 1995 EEA conference.) I have copied this section

The rest of the paper is also relevant, but not as directly. Other
issues I take up are

(1) misunderstandings concerning the concept of intrinsic value and
the distinction between value and exchange-value, and how the usual
claims of errors or internal inconsistencies in Marx's theory stem
from these misunderstandings;

(2) the process of development of the concept in Marx's work (up to
the publication of the Contribution (1859), he didn't posit the
distinction clearly, but largely in response to Bailey, whose
_Critical Dissertation_ Marx excerpted only after that, he began to
work out the distinction and increasingly made it a focal point of
his work through 1872); and

(3) the significance of the concept of intrinsic value in his work;
especially, why Marx bothered to analyze the nature of the commodity

If anyone wants the whole paper, I'll be happy to send it.





In this section, I will argue that the primary purpose of Marx’s
analysis at the beginning of _Capital_ was to establish a clear
distinction between value and exchange-value, *to break from the
conception of value as a ratio in exchange*. This interpretation
differs sharply from the common view that in the opening pages he
was instead advancing a “labor theory of value” (i.e., a theory that
exchange ratios are governed by relative quantities of labor), at
least as a “first approximation” to reality. (note 1)

In response to Adolph Wagner’s critique, Marx (1975:183, 198)
emphasized that “neither `value’, nor `exchange-value’ are my
subjects, but *the commodity*. ... What I start out from is the
simplest social form in which the labour-product is presented in
contemporary society, and this is the `*commodity*’.” My thesis,
that the distinction between value and exchange-value is the focal
point of his initial analysis, may seem to contradict this.
Actually, however, it reinforces Marx’s point, since his analysis
discloses that value, unlike exchange-value, is an intrinsic
property of the commodity itself. Moreover, a key reason
commentators have read a labor theory of exchange ratios into his
argument is that they seem to have misconstrued the object of
analysis, taking it to be exchange and the determination of exchange
ratios. Thus, to establish my contrary thesis, I will frequently
have reason to emphasize that Marx’s object of analysis is the
commodity itself.

Marx does, of course, examine the “exchange relation” (Marx,
1977:127), the expression of one commodity’s equality with another.
But as we shall see, he does so in order to establish that value is
intrinsic to the commodity. It is only in the second chapter,
entitled “The Process of Exchange” that Marx begins to investigate
the act of exchange; the title of Chapter 1 is “The Commodity.”

Although the title of the first section makes clear that the two
“factors” of the commodity are use-value and value, Marx first
states that commodities are use-values and “material bearers” of
exchange-value (Marx, 1977:125, 126). Rather than this indicating
that value and exchange-value are the same for him, Marx is simply
adopting the standpoint of the economists, but only provisionally.
As he writes later in the chapter (Marx, 1977:152, emphasis added):

“When at the beginning of this chapter, we said in the customary
manner that a commodity is both a use-value and an exchange-value,
this was, strictly speaking, wrong. A commodity is a use-value or
object of utility, and a `value’. ... [Its] form of manifestation
is exchange-value, and the commodity never has this form when
*looked at in isolation*, but only when it is in a value-relation or
an exchange relation with a second commodity.”

Marx thus begins from the “form of manifestation” as part of an
analytical strategy meant to enable the reader to see “beyond” the
exchange relation of commodities, to focus on the commodity itself,
in isolation. Indeed, he moves immediately to distinguish the
content, value, from its form of manifestation. Because
exchange-value “appears” as the ratio in which one thing exchanges
for another, and because this ratio is constantly changing,

“exchange-value appears to be something accidental and purely
relative, and consequently an intrinsic value, i.e. an
exchange-value that is inseparably connected with the commodity,
inherent in it, seems a contradiction in terms. Let us consider the
matter more closely.” [Marx, 1977:126]

The “matter” Marx wishes to consider is not whether the
quantitative exchange ratio is accidental or determinate. Instead,
it is whether value is an accidental phenomenon — one that arises
only contingently, in and through the act of exchange, a phenomenon
that is nothing other than this relation between the things,
“something ... purely relative.” Or is the opposite possible — that
value is intrinsic, “inherent in” the commodity itself?

Thus, it is in order to demonstrate that value is indeed inherent
in the commodity that Marx turns to the exchange relation; the
subject-matter is not exchange, but the commodity. He notes that,
because the commodity (a quarter of wheat) is exchanged for a
variety of other things “in the most diverse proportions,” it has
“many exchange values instead of one” (Marx, 1977:127). Its
exchange-values are the *other things* for which it exchanges; if 5
tins of boot polish, 10 yards of silk, or 1/35 ounce of gold, etc.
are exchanged for a quarter of wheat, the latter’s exchange-values
*are* the 5 tins of boot polish, the 10 yards of silk, or the 1/35
ounce of gold. Marx is showing that, when value is taken to be
exchange-value or relative price, value is not a “property” of the
commodity at all, but is another commodity, the *physical body* of
the other commodity, itself.

Yet although the various exchange-values of wheat are physically
different, each is equally the exchange-value of the quarter of
wheat. As exchange-values, they must be “of equal magnitude” and,
therefore, they “express something equal” (Marx, 1977:127). They
are all, in other words, “equally” the expression of *something
else*. (note 2) Marx (1977:127) thus concludes: “exchange-value
cannot be anything other than the mode of expression, the `form of
appearance’, of a content distinguishable from it.” (It is only on
the next page that this content is finally denoted as “value.”) He
is not concerned here with causality or magnitude, but is drawing
out the relationship between, and the distinction between, the
*form* and the *content* of this equality. (note 3)

The next paragraph seems to derive the same result by looking at
just two commodities exchanging with one another. This apparent
repetition can be understood in light of the subsequent development
of exchange-value in the third section of the chapter. As we have
seen, Marx is seeking to dispel the semblance that exchange-value is
accidental. This semblance arises when the exchange relation is
considered as the exchange of two commodities alone; Marx (1977:139)
will later call this the “accidental form” of value . To dispel the
semblance, Marx therefore first considers what he will call the
“total or expanded form” of value (Marx, 1977:154), before returning
to the accidental form. Once the equal content of each of a
*series* of commodities is established, in other words, the equal
content of *two* commodities can then be recognized more easily.

Beginning with the total form of value also dispels the opposite
illusion, an illusion which appears most strikingly in the “money
form” (Marx, 1977:162) of value — namely, that the second commodity,
with which the first exchanges, is value itself, is “endowed with
the form of value by nature itself” (Marx, 1977:149), and that the
first is therefore valuable by virtue of its exchange with the
second. Thus, an examination of the total form helps demonstrate
both that there is an identical content to each commodity, and that
this content is distinct from any of the commodities themselves.

Having first established this content, common to, but
distinguishable from, all commodities, Marx (1977:127) can then turn
back to the relation of two commodities and draw the conclusion that
“a common element of identical magnitude exists in two different
things .... Both are therefore equal to a third thing, which in
itself is neither the one or the other.”

This “third thing” argument has frequently been misunderstood.
Marx is not asking what allows commodities (much less use-values as
such) to exchange, as Böhm-Bawerk (1984:68-69) believed. Calling
Marx’s conclusion an “assumption,” Böhm-Bawerk argued that exchange,
“change of ownership,” involves “inequality” rather than equality —
evidently because each owner wants the *other* commodity.
Ironically, when Marx does finally turn to the process of exchange
in Chapter 2, he states the same thing. (note 4)

Again, however, Marx’s first object of investigation is not
exchange, but the nature of the commodity itself. He is not asking
why the commodities are exchanged (instead of being hoarded or
consumed), or what about them enables them to be exchanged — he
argues later in the text that products at first “bec[a]me
exchangeable through the mutual desire of their owners to alienate
them” (Marx, 1977:182). Instead, Marx is asking *as what* do the
commodities exchange. (note 5)

Put differently, he derives the existence of intrinsic value from a
postulated exchange of equivalents, *not the converse*. In the
previous paragraph, he proceeded from the equal magnitudes of the
exchange-values to derive a content common to them all. Similarly,
he here proceeds from the exchange of two equivalent commodities to
derive their equality to a third thing: *if* “1 quarter of corn = x
cwt of iron” (Marx, 1977:127), *then* a common element of “identical
magnitude” exists in each. “If A, then B” does not imply “if B,
then A.”

Moreover, Marx is still dealing with form and content. The causal
determination and magnitude of the exchange ratio are not at issue
here. Böhm-Bawerk (1984:69), among many others, apparently believed
the opposite, when he objected that chemical elements do not unite
“because they possess an exactly equal degree of chemical affinity.”
Yet had Marx wished to state, either as “theory” or as “first
approximation,” that 1 quarter of corn exchanges for x cwt of iron
because in that precise ratio the two are of equal value, he was
capable of doing so in clear and unambiguous terms. (note 6) Again,
however, the “third thing” argument answers a very different
question — *as what* do commodities exchange?

To understand Marx’s subsequent argument, it is crucial to
recognize that he has now indeed established that commodities
exchange as bearers of an intrinsic value, a “third thing,” present
in each. He now turns to a different question: “what is this third
thing?” It is also crucial to recognize that, since the common
element has been shown to “exist in” (Marx, 1977:127) each
commodity, the search is for a “property” (Marx, 1977:127, 128) of
the commodity *itself*. Marx thus *discontinues the examination of
the exchange relation*, which he pursued precisely in order to
establish that this third thing exists, and returns to an
investigation of the commodity.

It seems that failure to understand these points has led many
critics to charge that Marx asserts, rather than proves, what the
common element is, or that his proof is faulty, since some possible
candidates — e.g., utility, scarcity, the commodities’ existence as
appropriated things (Böhm-Bawerk, 1984:74-75) — are not considered.
It is quite true that the things could not exchange as commodities
unless they were scarce, owned, and useful. But none of these is a
property of the things *themselves*; all are *relations between* the
things and people. (Although the usefulness of things is dependent
on their physical properties, usefulness itself is not such a
property.) The basis of the criticisms is a misconception of the
object under investigation at this point.

This does not mean that Marx proved that the common element is
labor. First, he does not even state that it is, despite a popular
belief to the contrary. (note 7) Marx (1977:128, emphasis added)
actually writes: “only one property remains, that of being
*products of* labour.” Here again, the error results from
inattention to the fact that the commodity, not exchange or what
regulates exchange, is the object of analysis. Second, once one
recognizes that the object of analysis is the commodity itself, and
that what Marx *means* by “commodity” is (a) a useful thing that is
also (b) the product of labor, (note 8) there is no need for proof.
Once all physical properties of the commodity that make it useful
are rejected as the common property — they are qualitative
properties, but the exchange relation, as a quantitative relation,
abstracts from the qualities of commodities (Marx, 1977:127-28) — it
is then *self-evident* that “only one property remains, that of
being products of labor” (Marx, 1977:128).

What is not self-evident, what no one before Marx had identified,
is the dual character of this labor. The commodities are different
not only as useful, concrete things, but (for the same reason) also
as the products of the different sorts of useful, concrete laboring
activities. Only as products of “human labour in the abstract”
(Marx, 1977:128) are they the same.

Viewing commodities from the standpoint of what they have in
common, then, what remains, according to Marx (1977:128), is only a
“residue.” Nothing physical, concrete, or useful — about them or
the labor that produces them — is left. All that is left is a mere
abstraction, a “phantom-like objectivity; they are merely congealed
quantities of homogeneous human labour. ... As crystals of this
social substance, which is common to them all, they are values —
commodity values” (Marx, 1977:128).


1. Marx surely did hold that value is determined by labor-time, but
I question the view that he attempted to explain *relative* prices
on the basis of labor-time. In any case, my argument here is that
the opening section makes no such attempt.

2. The equality is thus a qualitative one. The various commodities
would still all “express something equal” were they to exchange for
7 or 9 bushels of wheat instead of a quarter (8 bushels).

3. Cf. Marx (1975:198): “`*commodity*’ is, on the one hand,
use-value, and on the other hand, `value’, not exchange-value, since
the mere form of appearance is not its proper *content*.”

4. “All commodities are non-use-values for their owners, and
use-values for their non-owners. Consequently, they must all change
hands” (Marx, 1997:179).

5. Cf. Marx (1971:144): “what is this unity of objects exchanged
against each other? ... As what do they become *exchangeable*?”

6. Contrast the “third thing” argument with the passage at the end
of Chapter 5 (pp. 268-69 and 269n), in which Marx first does state —
but only as an *assumption* — that equal values exchange: “The
money owner ... must buy his commodities at their value, sell them
at their value .... If prices actually differ from values, we must
first reduce the former to the latter, i.e. disregard this situation
as an accidental one.”

7. Böhm-Bawerk (1984:77): “labor is shown to be the sought-for
common factor”; Kay (1979:51): “Marx’s argument is ... that in
exchange, labour is the common property that regulates the terms of

8. Böhm-Bawerk (1984:71) was perhaps justified in complaining that
Marx had omitted to mention this from the start. Yet Marx was
following Ricardo’s (1982:12) well known delimitation of the term


Böhm-Bawerk, E. von. 1984. Karl Marx and the Close of His System
(Philadelphia: Orion Editions).

Kay, G. 1979. Why Labour is the Starting Point of Capital, in D.
Elson (ed.), Value: The representation of labour in capitalism
(London: CSE Books).

Marx, K. 1971. Theories of Surplus-Value, Part III (Moscow:
Progress Publishers).

__________. 1975. Notes on Adolph Wagner, in K. Marx, Texts on
Method (Oxford: Basil Blackwell).

__________. 1977. Capital: A critique of political economy, Vol. I
(New York: Vintage Books).

Ricardo, D. 1982. On the Principles of Political Economy and
Taxation (Cambridge: Cambridge Univ. Press).