THE LOGICAL FOUNDATIONSOF THE

MARXIAN THEORY OF VALUE

SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Managing Editor:

JAAKKO HINTIKKA, Boston University

Editors:

DONALD DAVIDSON, University ofCalifornia. BerkeleyGABRIEL NUCHELMANS, University ofLeydenWESLEY C. SALMON, University ofPittsburgh

VOLUME 223

THE LOGICAL FOUNDATIONS OF THE

MARXIAN THEORY OF VALUE

ADOLFO GARCIA DE LA SIENRA Institute of Philosophical Research.

Universidad Nacional Aut6noma de Mexico

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

Garcia de la Sienra, Adolfo The logical foundations of the Marxian theory of value I Adolfo

Garcia de la Sienra. p. cm. -- (Synthese library; v. 223)

Includes bibliographical references and index. ISBN 978-94-010-5193-4 ISBN 978-94-011-2694-6 (eBook)

1. Surplus value. 2. Labor theory of value. 3. Marxlan economlCS. I. Title. 11. Series. HE!20S.G34 1992 335.4' 12--dc20 92-14028

ISBN 978-94-010-5193-4

Printed on acid-free paper

All Rights Reserved © 1992 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1992 Softcover reprint of the hardcover 1 st edition 1992

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DOI 10.1007/978-94-011-2694-6

To Carolina

TABLE OF CONTENTS

INTRODUCTION

1 THE PROTOTYPE OF MARX'S LABOR THEORY OF VALUE 71.1 Marx's concept of value 81.2 Description of the prototype 181.3 Mathematical modeling of the prototype 20

2 THE PROBLEM OF FOUNDATIONS 412.1 The sense and import of the Law of value 422.2 The problem of generalizing the prototype 532.3 A concise history of MTV 54

3 STRUCTURES AND REPRESENTATIONS 633.1 Structures 643.2 Representations 76

3.2.1 The ontological framework 763.2.2 A case of representation 80

4 THE DIALECTICAL METHOD 914.1 Hegel's dialectical method 924.2 The Marxian "inversion" of Hegel's dialectic 994.3 A new formulation of dialectic 104

4.3.1 The theological framework 1044.3.2 Dialectic revisited 1104.3.3 The dialectic of the theory of value 115

4.3.3.1 Dialectic as model construction 1174.3.3.2 Dialectic as theory construction 1224.3.3.3 The dialectical method in axiomatic systems 125

5 ABSTRACT LABOR 1295.1 Krause's treatment of abstract labor 1335.2 The concept of abstract labor 1375.3 The existence of abstract labor 1425.4 The representation of abstract labor 149

viii TABLE OF CONTENTS

6 THE GENERAL AXIOMS OF THE THEORY6.1 A general market economy6.2 The Fundamental Marxian Theorem

7 GENERAL REPRODUCIBILITY

8 THE PROTOTYPE REVISITED8.1 The Leontiefeconomy8.2 Mathematical modeling of the Leontiefeconomy

NOTES

BIBLIOGRAPHY

NAME INDEX

SUBJECT INDEX

159161175

179

191192193

207

213

219

221

INTRODUCTION

Mter the impressive collapse ofthe Soviet Union and the EasternEuropean socialist countries, it is more pertinent than ever torecover the scientific legacy of Karl Marx. This legacy is mainly(if not exclusively) constituted by his work in the field of eco­nomic theory. Marx's economic theory was intended by his au­thor as a scientific objective theory about the nature of capitalisteconomies, a theory that was going to serve as the foundationof the critique of bourgeois political economy. His "laws" aboutthe demise of capitalism, like the tendency of the profit rate tofall or the law ofthe cyclical crises, have been shown to hold un­der certain conditions but not in general. At any rate, it is likelythat had not the industrialized countries changed the situationof the working class, and allowed some intervention of the Statein the economy (especially after the Great Depression), capital­ism would have hardly survived, even though it is impossible toguess what kind of regime would have been instaurated in itsplace.

The present book is concerned with the very foundations ofMarx's economics, hence with the very foundations of his scien­tific legacy. I hope that after reading the book the reader willbe convinced that Marx's scientific work was indeed serious andthat this is the time to recover it as an important paradigm in

2 INTRODUCTION

scientific research. I think that the reader will be convinced thatMarx's economic theory is no less serious and mathematicallytractable than, say, general equilibrium theory. Quite anotherquestion is whether Marx's economic theory constitutes by it­self a critique of capitalism. Some authors would say yes since,for one thing, the Fundamental Marxian Theorem is one ofits central results. This theorem asserts that the exploitation ofthe workers is both necessary and sufficient for capitalist profit.These authors would claim that -in consequence- capitalism isstripping the worker part of his labor, but this view -which I amnot despising at all- can be adopted in a very naive way, with­out taking into account many other factors like the risk taken byentrepeneurs, the cost of creating jobs, the need to have an in­centive for investment, the creativity required of entrepeneurs,and so on. In the present book I do not consider these ques­tions, but only apply myself to put a solid logical foundation forfurther development of the labor theory of value.

The labor theory of value as presented by Marx in Capital be­gan to receive mathematical treatment toward the end of thenineteenth century, and was reformulated with some completiontoward the beginning ofthe fifties, thanks mainly to the work ofLeontief. The mathematical reconstruction -or formulation­of Marx's original theory I shall call 'the prototype'. In the firstpart of the book I present a rather detailed formulation of theprototype, proving the most basic theorems. The chapter beginswith a detailed discussion of the concept ofvalue in Marx's Capi­tal. It is shown thereby that there is an ambiguity in Marx's con­ception of value. In some passages, Marx insists that the mag­nitude of value must be determined independently of marketrelations, solely in the sphere of production, whereas in othersMarx writes as if the market were the factor reducing heteroge­neous to abstract labor. In Capital, as well as in the usual Marxistliterature, the first interpretation has prevailed over the otherand, accordingly, the prototype is built upon this interpretation.

One of the most outstanding exponents of the prototype hasbeen Morishima (1973). My own formulation of the prototype

INTRODUCTION 3

owes a lot to the one provided by Morishima, but there are somedifferences that have mainly to do with the methods of proof.The idealized economy described by the prototype I call a 'sim­ple Marxian economy'. The main result of the chapter is theproof that the Law of Value, namely that prices are proportionalto values, holds only if all industries in the economy have thesame value composition of capital.

In the second chapter I proceed to discuss in some detail thefoundational problems of the theory of value. These problemsarise from the difficulties faced by Marx to clarify in what sense"value regulates prices". Mter reviewing all the attempts madeby Marx to solve this problem (throughout volume 3 ofCapital),I discuss the criticisms advanced by Bohm-Bawerk against suchattempts, in order to conclude that these criticisms are sound.Indeed, this book can be seen as an attempt to provide new foun­dations for the theory of value, foundations that are no longersubject to Bohm-Bawerk devastating criticisms. This attempt isdeeply related to the task of generalizing the prototype. In thesame chapter, I present eight restrictive assumptions built intothe prototype that are candidates to be eliminated. The chapterends presenting a concise history of the efforts made to get rid ofsuch assumptions. It is shown that the main problem is to allowheterogeneous labor in a rather general setting of productionsets.

Chapter 3 is a revision of formal techniques. I present theoutlines ofa system of mathematical logic, up to the concept ofamodel, in order to discuss the need of introducing another con­cept of model more akin to Bourbaki's concept of structure. Inthe second part I discuss the problem of the relation betweenabstract mathematical models and scientific objects. As a wayof approaching this problem, I present the concept of funda­mental measurement together with an example that illustratesclearly -or so I hope- the idea ofa fundamental measurement.The measurement I provide afterward -the measurement ofabstract labor- is not fundamental, because the structure to bemeasured already is a mathematical one, of course having a very

4 INTRODUCTION

specific economic meaning. But the techniques explained andexemplified in the chapter hold also for this case, except thatthe structure to be measured is not "ontological".

Chapter four is a rather long chapter in which I discussHegel's dialectical method and its relation to Marx's dialecticallogic. After presenting in some detail Hegel's method, I discussMarx's "inversion" of Hegel in order to show that his misappli­cation of Hegel's method might be responsible for his severingthe determination of value from the market relations. In orderto proceed to a new formulation of dialectic, I analyze the theo­logical framework of Hegel's dialectic in order to conclude thatHegel's concept of Spirit (Geist) is not acceptable on theologicaland metaphysical grounds, since it implies a particularly strongform ofAverroism. After discussing Hegel's theory of universals,I discuss a little the root of the so-called "problem of incommen­surability", which indeed is nothing but a pseudoproblem fromthis point of view. In the last section I develop in some detailthis modified Hegelian view as a theory of science with specialreference to the axiomatic method.

In chapter 5 I claim that Marx's decision to sever value fromexchange is a dialectical mistake. This is established as a corollaryof the main result of chapter 1, because the proposition that thetransformation problem is solvable is logically inconsistent withthe proposition that the organic composition of capital is not thesame for all industries. After reviewing other passages of Marx'swork, especially in the A Contribution to a Critique ofPolitical Econ­omy, where Marx suggests that value is a result of the exchangeprocess and not a ready made prerequisite, I adopt this differentview. This view had been held only by Isaak Illich Rubin in thetwenties and, more recently and using mathematical techniques,by Professor Ulrich Krause (University of Bremen). After con­sidering the important work of Krause's, I introduce a formaldefinition of abstract labor and then proceed to provide a moregeneral proof of the existence of a representation for abstractlabor. This representation or measurement of abstract labor iswhat we call 'value'. The Law of Value results as a corollary ofthe representation theorem for abstract labor.

INTRODUCTION 5

The general axioms of the labor theory of value are intro­d uced in chapter six. This theory is thereby axiomatized throughthe definition of the set-theoretical predicate 'Marxian capitalisteconomy'. The fundamental law of the theory is not the Lawof Value, but one asserting that the behavior of the capitalistfirms consists of maximizing their profit. Nevertheless, the Lawof Value is a consequence of this axiom. Another consequenceof the same is the Fundamental Marxian Theorem, accordingto which the exploitation of the workers is both necessary andsufficient for capitalist profit.

Chapter 7 is by far the most technical of the book. In thischapter I define the concept of a reproducible global decision,i.e. a global production process chosen by the firms which canreprod uce itself in the sense of being able to reprod uce all thewage and capital goods its operation requires. The existence ofa reproducible global decision is actually equivalent to the exis­tence of a special kind of competitive equilibrium -that I labela 'Marxian competitive equilibrium'. Almost all of the chapter isdevoted to prove the existence of this equilibrium, as a means ofproving the existence of the reproducible global process.

In the last chapter of the book I return to the prototype to seehow the foundational problems of the theory have been solved.In fact, the Law of Value holds in Leontief economies in thestrongest possible form: if the equilibrium price system in thesense of chapter 7 is profitable for all the firms, then this pricesystem is actually positive and equal to the the unique price sys­tem at which the profit rate is positive; moreover, the labor­values are equal to this price system. In other words, in chap­ter 8 the Law of Value means that labor-values are the (unique)equilibrium prices. In this form, the "dialectical contradiction"caused by Marx's severing of value from exchange is solved andthe theory of value is seen to have a solid scientific foundation.

Chapter 1

THE PROTOTYPE OF MARX'S

LABOR THEORY OF VALUE

In the first two volumes of Capital,) Karl Marx developed in arather informal fashion a theory of value. It is known that thistheory ofvalue owes a lot to Ricardo's labor theory of value, butI shall not be concerned in the present book with establishing inwhat respects Marx's theory is similar or different from that ofRicardo's. Instead, I shall take Marx's theory as point of depar­ture of the history ofa theory ofvalue that everybody shall agreein calling 'the Marxian theory of value' (MTV, for short).

One ofthe theoretical aims pursued by Marx in C was to provethat the source of the capitalist's profits is nothing but that un­paid part of the worker's labor which he calls surplus-value. Inorder to prove this, it is obviously necessary to define first a quan­titative concept ofvalue, since surplus-value is just the differencein value between the goods the worker can buy with his salaryand those he produced to earn that salary. Hence, any seriousattempt to make sense of Marx's MTV as formulated in the firsttwo volumes ofC must show in the first place that value, namelyas a quantitative concept, is indeed well defined. This is also re­quired to formulate the fundamental law of MTV; the Law ofValue, according to which the magnitude of value of commodi­ties regulates the proportions in which they exchange. In orderto address this problem, I shall make use of techniques pertain­ing to contemporary logic and the theory of science to analyze

7

8 CHAPTER ONE

and reformulate MTV as given by Marx in the first two volumesof C. The leading thread shall be the requirement of producinga quantitative concept of value useful to formulate the Law ofValue and to define the concept of surplus-value. In §1 I will in­troduce and discuss Marx's concept of value as it was presentedby the author in the first chapter of C. In §2 I will describe anidealized fictitious market economy that I have labeled 'simpleMarxian economy'; roughly speaking, this is the economy de­scribed by Marx himself mainly in the first two volumes of C.The task of producing a quantitative concept of value for thissimple economy shall be tackled in §3.

Once we reach a clear cogent restatement of MTV, we will bein an advantageous position to formulate the foundational prob­lems of the theory. We shall see throughout the exposition thatall these foundational problems of MTV are clustered aroundthe problems of providing a fairly general quantitative conceptof value useful to give a correct general formulation of the Lawof Value.

1.1 MARX'S CONCEPT OF VALUE

As it is well known, the starting point of Marx in C is the con­cept of commodity. A commodity is a use-value, a useful thingwhich is produced by a capitalist firm with the purpose of seIl­ing it in the market. Assuming that use-values are produced indifferent kinds and that there is a standard or unit of measure­ment for these objects, Marx proceeds to analyze the exchangeprocess, in which commodities of different kinds exchange inthe market in certain proportions. Introducing the concept ofexchange-value, as "the quantitative relation, the proportion inwhich use-values ofone kind exchange for use-values ofanotherkind",2 Marx proceeds to consider whether exchange-value issomething merely relative and accidental, or whether, on thecontrary, there is an intrinsic value, inherent to the commodity.From the fact that the exchange value of one kind of commod­ity is represented by specified amounts of goods of other kinds,

THE PROTOTYPE 9

Marx infers that any of these two amounts must be mutually re­placeable or "of identical magnitude", and then claims:

It follows from this that, firstly, the valid exchange-values fgiiltigenTauschwerte] of a particular commodity express something equal,and secondly, exchange-value cannot be anything other than themode of expression, the 'form of appearance', of a content distin­guishable from it.3

Considering the "equation'" 1 quarter ofcorn = x cwt ofiron',Marx asks: What does this equation signify?; and responds thefollowing:

It signifies that a common element of identical magnitude exists intwo different things, in 1 quarter of corn and similarly in x cwt ofiron. Both are therefore equal to a third thing, which in itself is neither theone nor the other. Each ofthem, so far as it is exchange value, must thereforebe reducible to this third thing. [...] the exchange values ofcommoditiesmust be reduced to a common element, of which they represent agreater or a lesser quantity.4

Since the exchange relation of commodities is characterizedprecisely by its abstraction from their use-values, and hence fromthe natural properties of the goods involved, Marx infers thatthis common element cannot be a natural property of them.Then, after seemingly reflecting about which properties are left,he claims that if we "disregard the use-value of commodities,only one property remains, that of being products of labor". Inthis way, through a sort of abstractive process, Marx introducesfor the first time in C his concept of value as labor. It is a plausi­ble abductive reasoning by which he intends to establish that thelabor required to produce the commodities is the only factor bywhich they are treated as equivalents in the exchange process,provided that the exchange-values of the commodities are valid.Methodologically, this procedure suggests that value is arrivedat a posteriori, that value is somehow discovered in the exchangerelation, that it is manifested in valid exchange-value.

Mter the former revelation has taken place, Marx goes onto consider the nature of value "independently of its form

10 CHAPTER ONE

of appearance". This methodological turn indicates that Marxthinks that this object (that is, value) so discovered can be deter­mined independently of its form of manifestation (which is validexchange-value). This methodological decision -as we shallsee- has enormous implications for the understanding ofMTV,for the way Marx treats value along the remaining part of sec­tion 1, and throughout section 2, presupposes that the value ofcommodities -both the substance and magnitude of its value­not only can, but also must be determined independently of themarket. This is apparent from the outset, since Marx is clearlybringing forth what he calls valid exchange-value. What coulddistinguish valid from non-valid exchange-value? It will be plainthat, once we adopt the above mentioned methodological deci­sion of severing the determination of value from the exchangeprocess, the ground for the distinction between valid and non­valid exchange-value cannot be but the value of commodities asdetermined independently oj the market: a valid exchange-value isone by which commodities exchange by their values, where thesevalues are determined solely in the sphere ojproduction. We will showthat this is indeed so in C, and that it shall give rise to a very im­portant contradiction between the first two volumes ofC and thethird, a contradiction pointed out in due time by Bohm-Bawerkin his "The Conclusion of the Marxian System" (1896), as wellas to many other foundational difficulties that will appear later.Before considering all this, I would like to convey to the readerwhat Marx "really meant" by 'value' -at least in C- and to pro­vide, out of Marx's writings, a new systematic formulation of histheory of value, a formulation which -being closer to our stan­dards ofclarity- will entitle us to dissect the foundational prob­lems that it posed for posterity. We shall begin to consider thesein the next chapter, starting with Bohm-Bawerk's criticisms.

Right after finishing his argument from the equivalence ofex­changeable commodities, Marx asserts that "a use-value, or use­ful article, therefore, has value only because abstract human la­bor is objectified or materialized in it". This claim, again, soundsas if Marx had made an empirical discovery: he seems to have

THE PROTOTYPE 11

found that, as a matter of fact, commodities can exchange as val­ues in certain proportions due exclusively to a "property" theypossess, namely, that ofbeing incarnations of objectified abstracthuman labor. From this point of view, commodities are nothingbut

merely congealed quantities of homogeneous human labor, i.e. ofhuman labor-power expended without regard to the form of its ex­penditure. All these things now tell us is that human labor-powerhas been expended to produce them, human labor is accumulatedin them. As crystals of this social substance, which is common tothem all, they are values -commodity values [Warenwerte].5

We shall refer to this characterization of value as congealedquantities of homogeneous human labor as 'the first definitionof value'. The first question that Marx rises in connection withvalue thus defined concerns the way in which it can be measured:

How, then, is the magnitude of this value to be measured? By meansofthe quantity ofthe 'value-forming substance', the labor, containedin the article. This quantity is measured by its duration, and thelabor-time is itself measured on the particular scale of hours, days,etc. 6

Marx conceived the total labor-power ofsociety as one homo­geneous mass of human labor-power, composed of individualunits oflabor-power, claiming that each of these units

is the same as any other, to the extent that has the character of a so­cially average unit of labor-power and acts as such, i.e. only needs,in order to produce a commodity, the labor-time which is necessaryon average, or in others words is socially necessary. Socially neces­sary labor-time is the labor-time required to produce any use-value under theconditions of production normal for a given society and with the averagedegree of skill and intensity of labor prevalent in that society [... ] what ex­clusively determines the magnitude ofthe value ofany article is therefore thearrwunt oflabor socially necessary, or the labor-time socially necessary for itsproduction.?

This characterization of the magnitude of value, as the a­mount of socially necessary labor-time required to produce the

12 CHAPTER ONE

commodity, introduces in fact a second characterization ofvalue,that we shall label 'the second definition of value'. On the otherhand, the claim that each individual unit of labor-power is thesame as any other, is one that needs justification. As Marx him­self acknowledges a few paragraphs later, the different units oflabor-power are not homogeneous:

The totality of heterogeneous use-values or physical commoditiesreflects a totality of similarly heterogeneous forms of useful labor,which differ in order, genus, species and variety: in short, a socialdivision of labor. This division of labor is a necessary condition forcommodity production [...tSince Marx insists that "the value of a commodity represents

labor pure and simple, the expenditure of human labor in gen­eral", he must needs solve the problem posed by the heterogene­ity of labor. In order to do so, Marx seems to perform an intel­lectual operation, pointing out that "if we leave aside the deter­minate quality of productive activity, and therefore the usefulcharacter of the labor, what remains is its quality of being an ex­penditure of human labor-power".9 After this operation, Marxproceeds to introd uce the concept ofsimple average labor, whichtakes as something given in any society, and goes on to assert thatin experience we find that a reduction of more complex labor tosimple labor is effected, since in fact a commodity which is theoutcome of more complex labor is posited "through its value" asequal to the product ofsimple labor. It is not clear, unfortunately,what Marx could have meant by saying that complex labor is re­duced to simple labor "through the value" of the correspond­ing produced commodities. He is trying to characterize value interms ofsimple average labor, and then goes on to point out thatthe reduction ofcomplex labor to this one is effected by value! Itseems to me that Marx should have said here 'exchange-value'instead ofjust 'value'. Had he done so, the passage could be readas claiming that the market, that is the exchange process, is the socialprocess -"that goes behind the backs of the producers"- thatestablishes "the various proportions in which different kinds oflabor are reduced to simple labor as their unit of measuremem".

THE PROTOTYPE 13

Nevertheless, Marx does not say so, leaving in obscurity the in­ner workings ofsuch process, because-I think- to say it wouldhave been inconsistent with his declared purpose of consideringthe nature of value "independently of its form of appearance",that is of exchange-value. Rather, trying to be consequent withthat purpose, Marx solves the problem by means of an act offaith, saying that

In the interests of simplification, we shall henceforth view everyform oflabor-power directly as simple labor-power; by this we shallsimply be saving ourselves the trouble of making the reduction. [... ]Just as, in viewing the coat and the linen as values, we abstract fromtheir different use-values, so, in the case of the labor representedby those values, do we disregard the difference between its usefulforms, tailoring and weaving. 10

Thus, Marx is in fact assuming that the reduction of complexto simple labor, or, more generally, the reduction of the differ­ent heterogeneous labor to a common unit of measurement, issomething which could be done independently of the market; itis only for theoretical purposes that we save ourselves, i.e. theeconometricians or the theoreticians, the trouble of making thereduction. In this form Marx finally arrives at his notion of ho­mogeneous labor, which he also calls 'abstract labor', claimingthat

all labor is an expenditure of human labor-power, in the physiolog­ical sense, and it is in this quality of being equal, or abstract, humanlabor that it forms the value ofcommodities.) I

Therefore, it is fairly clear that the impression Marx gives tothe reader in the first two sections of C is that through validexchange-value -which is the "form" of value even thoughsomeone may be in darkness concerning whether a particularexchange-value is valid or not- we become aware of the exis­tence of value as homogeneous labor -which is the "substance"ofvalue. The reader is also given the impression that in order tomeasure this substance in terms of socially necessary labor-time-which is the "magnitude" of value- we have to perform first

14 CHAPTER ONE

the operation ofreducing complex and heterogeneous labor to acommon unit (simple average labor) and then to perform directtime measurements over all the production processes. In otherwords, Marx instills in these sections the methodological maximthat the magnitude of value is one that has to be determinedindependently of the market. If Marx would have claimed thatthis was not his intention, he should have granted at least thathis exposition failed to convey what he really meant to say, beinghis second crucial section particularly obscure and sloppy.

In section 3, devoted to the different value-forms, Marx makesa claim that seems to be incompatible to what we saw he had saidin the previous section. Even though Marx in fact assumed thatthe reduction ofcomplex to simple labor, or, more generally, thereduction of the different heterogeneous labors to a commonunit of measurement, is something which could be done inde­pendently of the market (say by a scientific observer endowedwith the relevant required information), in section 3 Marx claimsthat this reduction is actually effected by the market:

By equating, for example, the coat as a thing of value to the linen,we equate the labor embedded in the coat with the labor embeddedin the linen. Now it is true that the tailoring which makes the coat isconcrete labor of a different sort from the weaving that makes thelinen. But the act of equating tailO1ing with weaving reduces the former infact to what is really equal in the two kinds oflabor, to the characteristic theyhave in comrrwn of being human labor [... ] It is only the expression ofequivalence between different sorts of commodities which brings toview the specific character ofvalue-creating labor, by actually reducingthe different kinds of labor embedded in the different kinds of comrrwdity totheir comrrwn quality of being human labor in general. 12

If the reduction of the different labors is effected by the mar­ket, the doubt about the coherence of Marx account arises be­cause in such a case it does not seem necessary to theoreticallymake that reduction: in such a case it is the market what savesourselves the trouble of making the reduction; all we have to doas theoreticians or econometricians is to find out in the economicworld just how the market made the reduction. This view brings

THE PROTOTYPE 15

important implications for the concept of value. We saw howMarx found the objective manifestation of value in exchange­value, presenting his unveiling ofvalue through exchange-valueas a sort of empirical discovery. Now, one thing is to say thatvalue is discovered or manifested through exchange-value, andquite another to say that the value commodities have happensto be precisely the one manifested by any given system of ex­change proportions. For instance, suppose that the price of allcommodities is set with respect to gold. Then the labor embed­ded in all commodities is reduced in fact, by the market, to thelabor embedded in gold. What this means is that we can takegold mining as the standard unit to compare all labors, and soall concrete labors can be expressed in terms of units of sociallynecessary mining time. As Marx puts it:

The body of the commodity, which serves as the equivalent, alwaysfigures as the embodiment of abstract human labor, and is alwaysthe prod uct ofsome specific useful and concrete labor. This concretelabor therefore becomes the exp ression ofabstract human labor. [... ]The equivalent form therefore possesses a second peculiarity: in itconcrete labor becomes the form of manifestation of its opposite,abstract human labor. [...And] possesses the characteristic of beingidentical with other kinds of labor. [... ] The equivalent form has athird peculiarity: private labor takes the form of its opposite, namelylabor in its direct social form. 13

More specifically and for the sake of illustration: ifwe say thatone coat is worth two ounces of gold, that it takes one hour ofmining to produce one ounce of gold and three hours of tailor­ing to produce one coat, then we would have to say that threehours of tailoring are equivalent to two hours of gold mining.If the market reduces all concrete labors in this way to miningtime, then the door is open to define the measure of the magni­tude of value of any good as the quantity of socially necessarymining-time to which it is equivalent. Clearly, this definition ofthe measure of value seems to involve an essential appeal to theactual exchange proportions in the market, making essentiallydependent any measurement or determination of value on the

16 CHAPTER ONE

proportions in which goods are actually exchanged in the mar­ket. One consequence of this is, of course, the Law of Value inits strongest form: the price of any bundle of goods is propor­tional to its value. In this case, the claim that commodities areexchanged by their values would seem to be no longer a dar­ing synthetic proposition, but a consequence of the definitionof 'value'. Clearly, all these theses are inconsistent with the ideathat value determination is a business that has to be carried onwithout taking the market into account, independently of theexchange process.

Yet the idea that values have some autonomy with respect tothe motions taking place in the market pops up again in section3, just in the following passage:

The equation 20 yards of linen = 1 coat, or 20 yards of linen areworth one coat, presupposes the presence in 1 coat of exactly asmuch of the substance of value as there is in 20 yards of linen, im­plies therefore that the quantities in which the two commodities arepresent have cost the same amount of labor or the same quantity of labor­time. 14

Clearly, this idea-which is further confirmed by Marx's anal­yses of the influence of prod uctivity on the relative expression ofthe magnitude ofvalue- sends us back to the idea that somehowthe substance ofvalue does not depend on the exchange propor­tions; for otherwise it would not make sense the proviso that theformer equation "presupposes" the presence of the same sub­stance of value in both bundles of commodities. In the market­dependent view, the equation does not presuppose anything, itrather establishes the values. Moreover, after all this logical swing­ing, Marx seems to make his mind up and finally decides towardthe end of section 3 to sustain the market-independent view,when he asserts the Law of Value in the following form:

It becomes plain that it is not the exchange of commodities whichregulates the magnitude of their values, but rather the reverse, themagnitude of the value ofcommodities which regulates the propor­tions in which they exchange. 15

THE PROTOTYPE 17

Clearly, there appears to be certain inconsistency in Marxtreatment along sections 1, 2 and 3 of the first chapter of C.There appears to be a contradiction between the idea that valueis something determined in the sphere of prod uction, measuredin terms ofthe socially necessary labor-time required for the pro­duction of the different commodities, by means ofa reduction ofheterogeneous to simple homogeneous labor, which would haveto be made in a way Marx never explains, and the idea that valueis the outcome of exchange, which actually reduces all concreteprivate labors to a common unit. This contradiction is indeedthe first foundational problem posed by MTV: depending onwhich horn of the dilemma we choose to adopt, we can reachtwo very different versions of MTY. It turns out that most in­terpreters and critics of Marx have chosen the one that Marxseems to support most, namely the market-independent view.This is the interpretation followed by Bohm-Bawerk and moremodern (and less destructive) readers of Marx, although at leastone of his interpreters, Isaak Illich Rubin, tried to follow themarket-dependent view. I submit that it does not make muchsense for foundational work in MTV to decide which was, in theend, Marx's "ultimate position" on the matter. Whatever it was,he took it with him to the grave. My methodological stance to­ward this dilemma is the following, to wit, that the final decisionbetween these two interpretations (if it is true that they cannotbe made compatible to some extent) must be taken on the basisof which one solves more logical problems of MTV; keeping inmind that the leading thread of the present foundational workis the requirement of prod ucing a quantitative concept of valueuseful to define the concept of surplus-value and establish theLaw of Value.

As a methodological decision, I shall adopt here the market­independent view of value as the core of the prototype of MTV;and proceed accordingly to provide a logical reconstruction ofthis theory based upon that view. Trying to stick as much as pos­sible to the description of the valorization process as given by

18 CHAPTER ONE

Marx in C, I will try to make explicit, in a way which is canon­ical according to current logical standards, the minimum struc­ture or set of axioms which guarantee the existence of market­independent values for an economy of the sort described byMarx in his formulation of the theory of value.

1.2 DESCRIPTION OF THE PROTOTYPE

Throughout C, but mainly in the first two volumes, Marx de­scribes a fictitious idealized market economy -which I shall label'simple Marxian economy'- in order to discuss the laws that ex­plain the functioning of the capitalist mode of production. Thiseconomy is characterized by a set ofm formally independent pro­ducers, each one producing a particular kind of good, so thatm different kinds of goods are produced altogether in the econ­omy. The theory represents the workings ofthe economy duringone certain interval of time (say one year). At the beginning ofthe year all factories (one for each producer) are endowed withbrand new equipment, all the prime materials they will need,and hired personnel representing simple homogeneous labor­power. The capital goods in the factories have the property thatcan only be operated at a certain rythm, so that the same time ofuse always yields the same amounts of goods. Also, the workersnever get too tired, and so their rythm of production is con­stant all of the time. Hence, if the output of production wereincreased or decreased by any fixed amount (if the owners de­cided to produce, for instance, twice as many goods in the sameamount of time) then the amount of workers and equipmentwould have to be increased or decreased by the same amount(these technologies are described in the economic literature asyielding 'constant returns to scale'). Each factory operates everyday during the same number of hours, the workers are neverabsent and there never are strikes, power failures or any prob­lem preventing the normal operation of the factories. Moreover,all products have the same period of production, say the wholeyear, and the period of rotation of capital goods is such that at

THE PROTOTYPE 19

the end of the period, at the end of the year, they all are entirelyworn out and have to be replaced anew. Hence, the last day ofthe year appear all of a sudden all final net outputs (one unitfor each factory) together with the inputs produced along theperiod for replacement purposes, all original prime materialshave been entirely consumed (as planned) and all the originalmachines simultaneously break down, beyond any possibility ofrepair, exhalating their last breath after an exemplarly produc­tive life (thus, in contemporary terminology, these processes areof the point-input-point-output type).

Other traits characterizing the former, rather regular world,are that there is no choice of techniques in the economy, i.e. onlyone production procedure is used for each kind of commodity.There is no joint production, i.e. each factory produces only onekind of good as output. Also, every kind of good is producedunder oligopolic conditions, i.e. each kind is produced by oneproducer only, although different kinds are produced by differ­ent producers. Supposing that the reduction of heterogeneouslabor to a common standard can be effected as Marx suggested,all labors in the different factories are taken as simple homoge­neous labor and so there are no heterogeneous concrete labors.It is also assumed that every industry requires a positive amountoflabor. In order to reprod uce the labor-power expended in onehour ofwork, it is necessary the consumption ofcertain amountsofwage goods which taken together will constitute the consump­tion basket of the working class. We shall suppose that the wageof the workers is just enough to acquire this basket. We supposealso that the economy is closed, which means that all kinds ofmeans of production are produced in the economy. The wageand capital goods factories are semiproductive, i.e. they are atleast able to reproduce themselves, in the sense that they con­sume no more than they produce. Finally, all the wage and cap­ital goods factories are interconnected, i.e. there is no indepen­dent subgroup of capital or wage good industries, that is to say,no subgroup offactories producing capital or wage goods, whichdoes not require to employ outputs produced by another sub­group.

20 CHAPTER ONE

The previous economic fiction was not described by Marx ina complete manner, but was sketched by him along the first twovolumes ofC (although in the second volume he makes room forcapital goods having different rotation periods). The additionaltraits of the economy, those not explicitely given by Marx, wereintroduced to guarantee the existence of a unique quantity ofunskilled labor associated to each unit of good produced in theeconomy, i.e. to guarantee the existence of a unique numeri­cal labor-value for each unit of commodity produced within thesystem. This fiction can be mathematically modeled, a necessarystep in order to prove formally the existence of such numer­ical magnitudes. The mathematical modeling of this economyand of some others that will be introd uced later, as well as theformulation of the foundational problems of MTV, requires theintroduction of some basic notation at this point.

1.3 MATHEMATICAL MODELING OF THE PROTOTYPE

I ask the reader to grant the adoption now of certain notationalconventions. In C Marx distinguished in the labor process threemain constituents: (1) the labor-power, (2) the instruments ofla­bor, and (3) the objects oflabor. 16 The instruments and objects oflabor are called generically by Marx 'means of production', butusually we will call them 'inputs', following a widespread currentpractice. These constituents, as exem plified in a particular reallyexisting production process (say in one journey of a Volkswa­gen factory), can certainly be measured in terms of suitable units.In the case of labor-power, suppose that there are in the econ­omy n different trades, i.e. n different kinds of concrete labors(weaver, tailor, mason, carpenter, civil engineer, etcetera). Everyexpenditure of any of these kinds is to be measured -as Marxsuggested- in terms oflabor-time: XI hours ofweaving, Xz hoursoftayloring, and so on, where Xi (1 ::; i ::; n) is a nonnegative realnumber. Thus, the labor-power actually applied in the processcan be represented mathematically as an n-dimensional vectorx:

THE PROTOTYPE 21

Ifm kinds of goods are produced in the economy, we can mea­sure these goods in terms of fixed physical units of measure­ment; for instance, 20 yards of linen, 1 coat, half a ton of iron,and so on. We shall adopt from now on the convention that thefirst k components of vector

x = [x ···x x ···x)- -I -k-k+1 -m

represent amounts ofcapital goods, whereas the remaining m-krepresent amounts of luxury or wage consumption goods; weassume, moreover, that each position of such vectors is conven­tionally associated with one kind ofgood. This means that goodsbelonging to the kinds represented by positions k + 1 to marenever used as production means in any production process ofthe economy, whereas those represented by the first entries, 1to k, appear in such processes. We adopt the convention thatvectors of the form ~ represent in puts of some identifiable pro­duction process. Obviously, this entails that in vectors ~ the lastm - k entries are zero. In a similar way, vectors of the form

will represent outputs of the corresponding production process;in these vectors, any entry can be positive, since the economymay produce both capital as well as wage and luxury goods. Usu­ally, we shall gather vectors x, ~ and x into a single vector x, inthe following way:

x= [x,x,x).

Hence, vectors of this form represent production processes. Thenet output of production process x we shall represent by meansof the symbol X, which is defined as the difference x - ~. Thegeneric entries of vectors x, ~, x and xwill be written in a nat­ural way as Xi, ~i' Xi and Xi, respectively, with i running in eachcase over the appropriate set of indexes. In particular, if thesecomponents are zero, the vectors will be written as 0, Q, 0 andorespectively; whereas the whole production processes will be

22 CHAPTER ONE

written as O. Given any two vectors of the same dimension, sayW = [WI" .wn ] and x = [XI" ,xn ], their inner product

l:WiXii=1

shall be denoted by the juxtaposition wx of vectors wand x.The symbol::; among numerals shall mean that the first num­

ber is less than or equal to the second; among vector symbolsit means that some entry of the first vector is strictly less thanthe corresponding entry of the second, and that no entry of thesecond is strictly less than the corresponding entry of the first;~ is written only among vector symbols and it means that thefirst vector is equal to the second or that the relation expressedby::; holds. The symbol < among numerals means that the firstnumber is strictly less than the second; among vector symbols itmeans that all entries of the first vector are strictly less than theircorresponding entries in the second. The symbols ;:::, ~ and>express, respectively, the converses of the relations expressed by::;, ~ and <.

After this brief notational digression, let us return to our pre­vious concern, the mathematical modeling of our simple Marx­ian economy. In this economy there are m producers, each oneproduces only one kind of good (no joint production), no twoproducers produce the same kind of good (oligopoly) and thereis no choice oftecniques, so that every good is produced by oneproducer using one fixed tecnology. Hence, at the beginning ofthe year producer i (I ::; i ::; m) has production means repre­sented by vector ~i = [~Ii'" ~mJ, which are entirely consumedalong the year in order to produce one unit of good of kind i,represented by vector Xi = [Xli'" Xmi], where Xij = 0 for everyj 1- i, and Xii = 1. Since labor is unskilled, there are no hetero­geneous concrete labors and so the vector of labor inputs Xi ofprocess i is unidimensional, i.e. it is a scalar, that shall be rep­resented by Xi. Thus, following our previous conventions, theindustries producing capital goods are represented by vectors

THE PROTOITPE 23

AI = [xT••• xT

]-J -k ,

XI> X2' ... , Xk; the industries producing wage and luxury goodsby Xk+l> Xk+2, ... , xm • Now, let~;, ... , ~r be the transposes ofvec­tors ~!' ••• , ~k after their last m - k components (which happento be all zero) have been dropped. Analogously, let the vectors~r+1' ... ,~;:. be the transposes of vectors ~k+1' ••• , ~m after theirlast m - k components (which also happen to be all zero) havebeen eliminated. Then the matrix of capital good industries, AI,is defined by

whereas the matrix All of consumption goods industries is de­fined by

All = [~;+I ... ~;:'].

Clearly, A, is a square k x k matrix, while All is rectangular k x(m - k).

The labor inputs of processes XI to xm are collected in thematrices

Notice that since every industry requires a positive amount oflabor, L, and LII are both positive.

Now, according to Marx's first definition of value, as crystalsof homogeneous human labor commodities are values. Let Ai bethe amount ofvalue congealed in one unit ofcommodity of kindi (i = 1, ... , k). Since, according to Marx, the value ofa product iscomposed by the value ofthe production means expended in itsprod uction process, plus the amount ofdirect live labor investedin the same, the magnitude of value of one unit of capital goodi must satisfy equation

(1)

Analogously, the magnitude Aj (j = k + 1, ... , m) of value of aunit of non-capital good of kind j, satisfies equation

(2)

24 CHAPTER ONE

(Notice that we are writing the components of a vector ~I as ~IP

••• , ~kl for I = 1, ... , m). All formulas of type (1) and (2) can beput in concise form as

(3) and

provided that

These are the equations for value implied by the first definitionof value.

According to the second definition, only the amount ofsociallynecessary labor-time required for the prod uction of a commod­ity determines its magnitude of value. In order to compute thismagnitude, let us consider the total amounts of capital goodsYli, ···,Yki which are required for the production of a unit of cap­ital good i, after taking into account all the repercussions. Inorder to determine these amounts, the following equation mustbe solved:

(4)

where xi is the traspose of output vector Xi, after its last I - kcoordinates have been dropped, and

Assuming that equation (4) has a unique solution yi, theamount of socially necessary labor-time J-li required to produceone unit of capital good i is given by

(5)

On the other hand, since capital goods ~IJ' ••• '~kj (which arethe entries of ~T) are required for the production of one unit ofconsumption goodj (j = k + 1, ... , m), for replacement purposes

THE PROTOTYPE 25

goods must be produced in the amounts Ylj, •• • ,Ykj determinedby equation

(6) A T_IYj + ~j - Yj,

(Yj = [ylj ...Ykj ]T) in order to provide the wage or luxury goodsindustry with net outputs of capital goods in the amounts whichare just enough for the replacement.

Given that in the production of the capital goods LIYj units oflabor-time are expended, and the production of the consump­tion good j adds Xj units of direct labor to its product, the to­tal socially necessary labor-time required to produce one unit ofwage or luxury good j (k + 1 S: j S: rn) is given by

Hence, if we set M I = [J1]'" J1k)' MIl = [J1k+I'" J1m], YI =[YI ... Yk) and YII = [Yk+l ... Ym], we get the equations

and

(7)

Notice that I = [xi··· xk] is the identity matrix.The former are the equations for value according to Marx's

second definition. The first question that arises is whether thevalue equations (3) and (7) have unique positive solutions and, ifthe answer to this question is affirmative, the second question iswhether values as determined according to the first definition ofvalue coincide with values as determined according to the sec­ond one, i.e. whether AI = M 1 and All = MIl. The answer isaffirmative in both cases, but in order to prove it, it will be neces­sary to express in mathematical terms the semiproductivity andinterconnectedness of the wage and capital goods industries ofour simple economy.

26 CHAPTER ONE

In order to give mathematical expression to the reproducibil­ity or semiprod uctivity and the interconnectedness of the wageand capital goods industries of a simple economy, let us repre­sent first the consumption basket of the working class, namely asthe column vector b:

_ [b k

.+1

]b - : .bi

The amounts of goods bk + l , ••• , bl (I ::; m) are those that are nec­essary to reproduce the labor-power expended in one hour oflabor. It is also convenient to introduce the matrix

~Il ~Jk ~lk+1 ~ll

A= ~kl ~kk ~kk+l ~kl

0k+1l Ok+lk Ok+lk+1 Ok+ll

Oil Olk Olk+1 Oil

where ~Ij' .. ·~kj (1 ::; j ::; I) are the inputs of process Xi> theother entries of the matrix being 0; processes Xl> ... Xk are (aswe had said) those of the capital good industries, and processesXk+l> ... XI are those of the industries producing wage goods, i.e.goods of the types that constitute vector b. Consider now the I X Imatrices

01 01

B=Ok Ok

bk +! bk + 1

bi bi

and

L= n 1]

THE PROTOTYPE 27

B contains only zeros above the k + lth row and the last compo­nents of the columns are nothing but those of the consumptionvector b. All off-diagonal entries of L are zero and the diagonalelement of row i (i = 1, ... , l) is the labor input of process pro­ducing wage or capital good of kind i. These matrices are usefulto construct the product

01 OJ

BL = Ok Ok

bk+IXI bk+1XI

b/XI blXI

It can be seen that the ith column of this matrix (i = 1, ... , l)represents the amounts of wage goods bk+IXi' ••• , b/Xi requiredto reproduce the labor-power expended in processes Xi. There­fore, matrix

C = A + BL

~ll ~11

= ~kJ ~kl

bk+IXj bk+IXI

b/Xl blXI

CII CII

CkJ Ckl=

Ck+ll Ck+1I

C/I Cll

represents the amounts of production means and wage goodsrequired to operate the processes producing capital and wagegoods. Now we are in position to express in mathematical terms

28 CHAPTER ONE

the reproducibility and interconnectedness of the wage and cap­ital goods industries. We say that the capital and wage industriesare semiproductive or reproducible iff there is a positive (column)vector y such that Cy ;;; y. Also, we say that the wage and capitalgoods industries are interconnected iffthe matrix C is indecompos­able, i.e. ifit is not the case that Cij = 0 for all indices j belongingto some proper nonempty subset] of {I, .. " I} and indices i notbelonging to the same subset. What this means is that industriesrepresented by the indexes in] do not require as inputs-meansof production produced by the industries not represented by in­dexes in j. This implies that if the matrix C is indecomposablethen any increase in the final output ofany capital or wage goodsindustry necessarily increases the requirement of inputs for ev­ery other such industry.

Hence, the assumptions that the capital and wage goods in­dustries in the economy are both semiproductive and intercon­nected are expressed by saying that the matrix C is semiproduc­tive and indecomposable. As a matter offact, the reproducibilityand indecomposability of C imply that AI, the matrix of coef­ficients of capital goods industries, is also indecomposable and-what is stronger than mere semiproductivity- also quasipro­ductive, i.e. there is a positive vector z such that AIz :::; z. This canbe seen as follows. Since C is semiproductive, Cy ;;; Y for somepositive vector y. Drop the last I - k components of y, obtaininga vector z of dimension k. Then, given that AI is the left uppercorner of C and wage good industries require positive inputsfrom the capital good industries (because C is indecomposable),it is easy to see that AIz :::; z; in other words, some of the firstk components of Cy are strictly greater than the correspondingcomponents of AIz. That AI is indecomposable follows from thefact that any increase in the final output of any capital or wageindustry, in particular of any capital good industry, necessarilyincreases the requirement of inputs for all the wage and capitalgoods industries and so, in particular, for all of the second. Inorder to establish many of the results we are concerned with, itwill be necessary to prove the following

THE PROTOTYPE 29

LEMMA: Let E be a semipositive K x K square matrix that is quasipro­ductive and indecomposable. Then (I - E)-\ where 1 is the identitymatrix, is a positive matrix.

Proof Since E is quasiproductive, there is a y > 0 such that Ey ::;y. We are going to show that 1 - E possesses a semidominantdiagonal, i.e. that there exist positive YI, .. "YK such that

yd 1 - eid 2:: LYjl - eiJIjii

(i = 1, ... , K)

with at least one strict inequality.In fact, for every i = 1, ... , K, 0 < eilYI + ... + eiKyK ::; yi, where

this inequality is strict for at least one i, and so

Y· -x··y· > ~x··y > 0lilt _ L...t I) ) _ •

jii

Hence,y;jl- xid 2:: LYj!-Xijl2:: 0

Jii

with at least one strict inequality.Since E is indecomposable, so is 1 - E, and so it follows that

1 - E is nonsingular. Moreover, since all off-diagonal elements of1 - E are nonpositive, it follows that (I - E)-I> 0. 17 0

We are now in position to establish the first main result of thischapter. This is

THEOREM 1: In a simple Marxian economy, there exist unique systemsofpositive values AI and All in the sense of the first definition of value.Also, there exist unique systems ofpositive values M, and Mil in the senseof the second definition of value.

Proof Since AI is quasiproductive and indecomposable, by theLemma (I - A,)-J exists and is positive. Hence, setting A, =LI(I-A,)-J, YI = (I-A,)-J and YII = (I-A,)-lAII , we see at oncethat the matrices A" All, M I and Mil are unique and positive. 0

It can be shown that, in fact, in a simple Marxian economythe first definition of value is equivalent to the second. The first

30 CHAPTER ONE

one in having noticed that value can be characterized in a dualway was Michio Morishima (1973). He was also the first one inbringing to the light the hidden assumptions required to proveboth the previous theorem and the next one.

THEOREM 2: In a simple Marxian economy, values as determined ac­cording to the first definition coincide with values as determined accord­ing to the second one. That is to say, equations AI = M I and All = Milhold.

AI = AlAI + LI => AIYI = AIAlYI + LlYI

=> AI(YI - AIYI) = LIYI

=> All = LIYI

=> AI = MI.

Proof" It suffices to see that the following chain of implicationsholds:

This establishes the first identity. For the second one, using thefact that All = AIAII + LII , we have:

AI = AlAI + LI => All + AIYII = (AIAII + L II)

+ (AlAI YII + L lYII)

=> All + AIYII - AI(AIYII + All)

= LIYII + LII

=> All + AIYII - AlYII = Mil

=> All = Mil. 0

Thus, it turns out that in a simple economy the quantitativeconcept of value is well defined. I said that one of the lead­ing threads of our research was going to be the production ofa quantitative concept of value useful to define the concept ofsurplus-value. Hence, we must show that the just defined con­cept of value does permit to define that crucial notion. I shallconclude the present chapter by introducing the concepts ofsys­tern of hourly wages and system of prices, in order to prove theFundamental Marxian Theorem (FMT) for our simple economy.This theorem asserts that the rate of profit in terms of money of

THE PROTOTYPE 31

any process is positive iff the rate of exploitation, in terms ofvalue, is also positive. The FMT is important not merely as a"denunciation" of capitalism (as if exploitation were by itself thecause of evil in the capitalist societies) but first and foremost asthe establishment of a condition that prices and wages in a mar­ket economy must satisfy in order for the economy to be feasibleand able to reproduce itself. In other words, exploitation is ascientific concept required to formulate an important constrainton prices in a market economy.

In order to introduce the argument, let us adopt some addi­tional notational conventions. Let us suppose that any unit ofeach kind i (i = 1, ... , m) of commodities in the economy hasa single price Pi. Then the prices of all commodities can be puttogether in a positive vector [PI ... pm]. Nevertheless, it will beconvenient, for the purposes of treating the case of the simpleeconomy, to split this vector into two: the vector p = [Pl" 'Pd(l ::; m) of wage or capital goods, and the vector pi = [P1+1" ,pm]of luxury goods.

Let us denote the hourly wage paid to a worker belonging totrade i (i = 1, ... , n) as Wi. Then we can write the wage systemas a vector [WI' .. w n ] and so the total amount of salaries paid inprocess x = [x, K, x] is the inner product wx. In order to discussthe particular case of the simple economy, it will be convenient todenote with w the initial segment of vector [WI' .. wn ] containingthe hourly wages of workers of capital and wage good industries1, ... , l, and with Wi the remaining segment. Since labor in oursimple economy has been taken as simple and homogeneous, thecomponents of both vectors are all equal to the scalar w, denot­ing in each case the hourly wage of simple homogeneous labor.The assumption that workers do not save, i.e. that the wage ofthe workers is just enough to acquire the consumption basket b,implies that pb = w.

In a competitive market economy, no firm can survive unless itobtains profits by selling its prod uct in the market. A necessarycondition for this is that the production price be less than theselling price of the product. This can be expressed in general by

32

means of the formulas

(8)

CHAPTER ONE

pC < p

for wage and capital goods, and

(9)

for luxury goods, where

~Jl+1

pC' < p'

~Im

C' = ~km

bk+Jmxm

Depending on the price and the wage systems, the profityielded by the same production processes may be large or small.A measure of the size of the profit yielded by each unit of moneyinvested in the process is given by the profit rate of the process.The profit rate of process Xi under price system [p, p'], 7r(Xi), isthe ratio of the benefit to the prod uction cost:

(_ ) _ PXi - WXj

7r Xi -WXj + P~i

i = 1, .. . ,m.

Clearly, when we multiply the profit rate of process Xi by its pro­duction costs we obtain the net profit yielded by that process.

The only reason why a capitalist invests in order to producesome kind of use-value, is the desire for profits. On the otherhand, the reproduction of the workers requires, for each hour oflabor, the minimum consumption basket b of wage goods, whichare necessary to reproduce one hour of homogeneous labor­power. Hence, there are two basic constraints that a system ofwages and prices must satisfy in order to set the industries intomotion. The first is that the system must guarantee a profit rate

THE PROTOTYPE 33

greater than one for all industries; the second is that the salaryobtained by any worker for each hour oflabor be sufficient to buythe consumption basket b, i.e. WXj = pbx; for every i = 1, ... , m.It turns out that in our simple economy prices and salaries canbe established that satisfy these constraints. I will show that this istrue, first, with respect to the capital and wage industries; it willfollow easily that it holds also for the luxury goods industries. Asa matter oHact, an even stronger result can be obtained. We shallsee that it is possible to find a unique profit rate greater that oneand a (unique up to multiplication by a positive scalar) price vec­tor for all production processes in the economy. Before this, wewill prove that a price system p for wage and capital goods canbe found that at least permits to operate the industries of thesegoods without any losses (even though perhaps also without anyprofits).

THEOREM 3: There is a unique nonnegative profit rate r to whichthere correspond price systems p and pi such that

p = (1 + r)pC and pi = (l + r)pC'.

Moreover, these price systems are unique up to scale transformations.

Proof Since the matrix C is indecomposable, by the Perron­Frobenius theorem there is a unique positive real eigenvalue,(the Frobenius root) to which there corresponds a unique (up tomultiplication by a positive scalar) positive left eigenvector p:

pC = p,.

Let1-,

r=--.,Since C is semiproductive, there is a positive vector y such thatCy ~ y. Hence, pCy :S py and so , :S 1. It follows that r is auniquely determined nonnegative number and that 1 + r = ,-I.Therefore,

p = (l + r)pC.

34 CHAPTER ONE

We just set now the prices of luxury goods as required:

p' = (1 + r)pC'. 0

The number r shall be called also the equilibrium profit rate. Anyprice systems p and p' satisfying equations

p = (l + r)pC and p' = (1 + r)pC'

will be called equilibrium price systems. Clearly, if the equilibriumprofit rate r is greater than zero, then the price systems corre­sponding to it satisfy equations (8) and (9).

The existence of a positive uniform profit rate is not neces­sary, but it can be shown that a necessary condition for its exis­tence is the positivity of the exploitation rate. It can be shownthat the positivity of the exploitation rate, together with the as­sumption that all ind ustries in the economy have the same value­composition of capital, i.e. that the ratio of constant to variablecapital is the same in every production process, implies that theequilibrium profit rate is positive and -what is much more­that equilibrium prices turn out to be proportional to values.Nevertheless, it is possible to have in a simple economy both apositive uniform profit rate and a positive uniform rate of profitwithout all industries having the same value-composition of cap­ital. In such a case, the equilibrium price systems deviate fromthe value systems, in the sense that they are found to be non­proportional to the latter.

We have not introduced yet the important concept ofexplota­tion rate. In order to introduce it, it is important to define thesurplus-value of a production process. This is the difference be­tween the amount of live labor expended in the process andthe value of the goods the workers can afford with their salary.The value represented by the salary paid in prod uction processi (i = 1, ... , l) (operated at unitary level), the so-called necessarylabor, is just Vi = (Ak+lbk+l + ... A/b/)xj. On the other hand, thetotal amount of live labor expended in this process is Xj. Hence,

THE PROTOTYPE 35

the surplus-value of the process is Xi - Vi and so the rate of ex­ploitation, as defined by Marx in C,18 is the ratio ofsurplus-valueto necessary labor:

Xi -ViEi = ---.

Vi

Marx referred to number Ei as "an exact expression for the de­gree of exploitation of labor-power by capital, or of the workerby the capitalist". J8 For this reason, the rate of surplus-value isalso called 'the rate of exploitation'. In general, the rate of ex­ploitation may vary from one production process to another, butin the special case of a simple economy this rate turns out to beuniform for all production processes. This is the content of thenext theorem.

THEOREM 4: In a simple economy the exploitation rate is uniform.

Proof Consider any two production processes Xi and Xj (i,j =1, ... m) and let a be the number X)Xi' It is obvious that a > 0and that V) = aVi. Therefore,

X -vE =) )

) V)

a(xi - Vi)

aVi

= Ei. 0

Marx defined the constant capital of a labor process as thevalue of the means of production, 19 i.e. the constant capital ofproduction process Xi (i = 1, ... , m) is the number Ci as deter­mined by

Ci = ~1)1J + ... + ~kiAk'

On the other hand, the surplus-value is the number Si given by

Si = Xi - Vi,

which represents the amount of new value added to the capitalinvested in the process. Clearly, Si = EVi, and so the followingproposition is seen to be true.

36 CHAPTER ONE

THEOREM 5: The value Ai ofthe product ofprocess Xi (i = 1, ... , m)can be expressed as

By virtue of Theorem 5 we can write the value equations forcapital and wage goods as

(10)

where

A = AA + ABL + €ABL

A = [AI'" Ad

is the vector ofvalues of capital and wage goods industries. Anal­ogously, the value equations for luxury goods can be written as

(11)

where

A' = AA' + ABL' + €ABL'

A' = [AI+l ... Am]

is the vector of values of luxury goods,

[~ll+)

A ' - .- .

~kt+)

... x ]-1m

~km

is the matrix of inputs of luxury goods industries, and

is the vector oflabor inputs of the same industries.Now we are in position to establish a condition that is both

necessary and sufficient to guarantee profitable wages and pricesfor all the industries in a simple economy. As I said above, this isprecisely the FMT

THEOREM 6: (The Fundamental Marxian Theorem for a sim­ple economy). In a simple economy, there exists a price system at whichthe rate ofprofit of every process is positive iff the rate of exploitation €

is positive.

THE PROTOTYPE 37

Proof" Assume first that the rate of profit is positive for everyproduction process, i.e. assume that equations (8) and (9) hold.Since p is positive,

pCy < py

for any positive column vector y. This would not be possible ifCwere not productive, for the following reasons. Since C is semire­producible, there is a positive vector z such that Cz ~ z, i.e. suchthat either Cz = z or Cz ::; z. But the first case is impossiblebecause it implies

pCz = pz,

contradicting what was established above. Hence Cz ::; z. Thusthere is a positive vector y such that (see equation 10)

ACy + c:ABLy = Ay > ACy

and so c:ABLy > O. It follows that c: > O.Assume now that c: > 0 and let p = A, pi = N. Since pB = w,

AL = wL > 0 and AL' = wL' > O. Therefore, equations (10)and (11) yield

p = pC + c:wL > pC

andp' = pC' + c:wL' > pC'.

This establishes that the profit rate ofeach process is positive. 0The FMT establishes that a positive rate ofexplotation is both

necessary and sufficient for the existence of a price system atwhich all production processes in the economy are profitable.It does not establish, however, that this price system has to beexactly the one corresponding to the unique equilibrium rate ofprofit whose existence was established in Theorem 3. In otherwords, the profit rate ofone prod uction process is not necessarilyequaltothatofanothe~

The transformation problem is the problem of establishing a lawrelating labor-values and prices in such a way that the latter canbe derived out of the former. The transformation problem is

38 CHAPTER ONE

the problem of obtaining the equilibrium prices out of the valuesystems by means ofa regular nomological pattern. In the specialcase in which the constant composition of capital is the same inall the economy, the transformation problem can be solved ina straightforward way: as a matter of fact, it can be shown thatequilibrium prices are proportional to values iff all industriesin the economy have the same value-composition of capital, i.e.if the ratio of constant to variable capital is the same in everyproduction process. This is the gist of this, the last theorem ofthe chapter.

THEOREM 7: In a simple economy, equilibrium prices are propor­tional to values iff all industries in the economy have the same value­composition ofcapital.

Proof Assume first that equilibrium prices are proportional tovalues. That is to say, there is a positive 0: such that Pi = O:Aifor i = 1, ... , m. Let us write Ci for the sum in terms of equi­librium prices: Pl:Kli + ...Pk:Kki and Vi for the amount WXi, alsodetermined in terms of equilibrium prices. Then we have

PXi - WXir = L- _

WXi + P~i

_ Pi - (Ci + Vi)

Ci + Vi

_ Ai - (Ci + Vi)

Ci +ViVi

=E;---.

Ci + Vi

Therefore, C;fVi is constant and equal to r-1E; - 1.Assume now that C;fVi is constant for i = 1, ... , m, and set

Cir = E;-.

Vi

THE PROTOTYPE

Then we have

(1 + r)(Ci + Vi) = Ci + Vi + r(Ci + Vi)

= Ai - €Vi + r(Ci + Vi)

= Ai - €Vi + €Vi

= Ai

Hence, we have just obtained the equations

(Theorem 5)

39

A = (1 + r)AC and A' = (1 + r)ApC'.

Theorem 3 implies then that A is proportional to p and A' isproportional (by the same factor) to p'. 0

Morishima has shown that important results established byMarx in the third volume of C hold with certain modificationsand additional restrictive assumptions, and so Morishima thinksthat Marx was "succesful" in the transformation problem. Mor­ishima thinks that Marx motivation to tackle this problem was toshow that "individual exploitation and individual profit are dis­proportional unless some restrictive conditions are imposed".20According to Morishima,

it is clear that the transformation problem has the aim of showinghow 'the aggregate exploitation of labor on the part of the total so­cial capital' is, in a capitalist economy, obscured by the distortion ofprices from values; the other aim is to show how living labor canbe the sole source of profit [... ] Marx [... ] was very succesful in thetransformation problem.21

Nevertheless, leaving aside the fact that such additional as­sumptions are rather unrealistic, the gist of the labor theory ofvalue was in the first place to show that the Law of Value holdstrue. This law asserts that the magnitude ofvalue ofcommoditiesregulates (in a rather interesting, nontrivial sense) the propor­tions in which they exchange. Now, it is true that many inter­preters of Marx have taken this "regulating" as demanding theproportionality ofequilibrium prices and values. Bohm-Bawerk,one of the most outstanding critics of Marx in the nineteenth

40 CHAPTER ONE

century, claimed that the Law of Value asserts "and for all thatprecedes cannot assert, but that commodities exchange amongthemselves in proportion to the average socially necessary labor­time incorporated to them".22 Thus far we have seen that if thisproportionality is understood as the proportionality of valuesand equilibrium prices, within the conceptual framework of theprototype of MTV, then the Law of Value is seen to fail, exceptin the uninteresting case of equal value-composition of capital,according to Theorem 7. We shall see if it is possible to provide aformulation ofthe Law ofValue that circumvents this fundamen­tal difficulty. In this connection, the problem is to make sense ofthe idea that value "regulates" prices. This is one of the founda­tional problems ofMTY: We shall formulate it and all the othersin the following chapter.

These results are sufficient for our purposes ofexamining thefoundational problems of MTY: It can be seen that MTV is a se­rious important theory that deserves attention by scientists andphilosophers ofscience. Now I shall proceed to the next chapter,where the foundational problems ofthe theory will be presentedand analyzed in detail. We shall see that these problems arisemainly out of the effort to provide a general quantitative con­cept of value that makes possible a correct general formulationof the Law of Value, as well as from the attempt to generalize theassumptions that enabled us to prove the interesting theoremsof MTV for the prototype.

Chapter 2

THE PROBLEM OF FOUNDATIONS

What is the content of the problem offoundations and its impor­tance? We have seen that even the prototype of MTV -whichis a quite idealized version of a capitalist economy- is inter­esting as a particular nonexistent artificial system in which thestructure of the capitalist economy can be studied. I will discussthe scientific relevance of such systems in the chapter on the di­alectical method. In the meantime, we can say that the simpleeconomy studied in the previous chapter is useful because it setsthe stage for our raising of certain questions which must be ofinterest to anybody interested in the real workings of modernmarket economies. The two most important of these are, clearly,whether the idealizing assumptions that define simple systemscan be generalized in such a way that general laws can be formu­lated that obtain in those economies. In particular, is it possibleto prove that a quantitative concept of value can be defined inmore general structures? And what about the Law of Value? Isit possible to provide a more precise formulation of it, a formu­lation that can be shown to hold not only in the case of simplecommodity production with equal composition of constant cap­ital, but also in modern full-blown capitalist economies? Theseare the main foundational problems ofMTV. The present chap­ter intends to give a detailed and rigorous formulation of them.

41

42 CHAPTER TWO

2.1 THE SENSE AND IMPORT OF THE LAW OF VALUE

The aim and goal ofa theory ofvalue is to give a satisfactory ex­planation of the fact that, in the market, commodities gravitatearound certain prices. Why these commodities gravitate around theseprices and cannot go much below or much above them? This is one wayofasking the fundamental problem ofa theory ofvalue. Roughlyspeaking, the value a commodity has is "what it costs to get it".Common sense has a good appraisal of the value of commodi­ties in daily life, as it is evinced by the fact that nobody expectsto obtain in the market a Mercedes Benz for ten American dol­lars. This strong sense concerning the value of things suggeststhat there must be something objective, some reason or reasonswhy things happen to be as costly as they are. The labor theoryof value intends to explain the cost of things in terms of sociallabor. Essentially, a labor theory of value must explain the move­ment of prices in terms of social labor, and that is why the Lawof Value -which asserts that, in the very least, labor "regulates"the cost or value of commodities- constitutes in itself the verycore of the labor theory of value. As the Spanish philosopherFrancisco AIvarez (1986) has pointed out, the Law of Value isactually the law that defines MTV, very much as Newton's Sec­ond Law defines his mechanics. Hence, in a sense, Marx's Lawof Value is his labor theory ofvalue. This is the reason why we, asphilosophers of science, must take seriously the claims made byMarx concerning the regulative role of value in price formation,and try to solve the conceptual difficulties that it poses. Accord­ingly, I shall take here at its face value Marx's formulation ofthe Law of Value, in the sense that the magnitude of value ofcommodities regulates the proportions in which they exchange. Ishall discuss Bbhm-Bawerk's criticism of this law in order to con­clude that this criticism is right but not quite right. The blamefor the blast Marx gets from this critic lies no doubt on Marx ownhesitations, obscurities, inconsistencies and final methodologicaldecision (all of which we followed carefully in the first chapter)that led him to claim that values are to be determined indepen­dently of the market. This turned out to be a blind alley that

THE PROBLEM OF FOUNDATIONS 43

got Marx -despite Morishima's sympathetic interpretation­into the contradictions crudely pointed out by Bohm-Bawerk.We shall see that these contradictions are the result of the basicopposition introduced by Marx between the sphere ofcommod­ity production and the sphere of distribution, an opposition im­plicitly contained in the opposition between value as determinedsolely in the sphereofproduction and value as determined solelyin the sphere of distribution (in the market). We shall see thatthese are two arrested moments of a dialectical unity, which isthe concept of abstract labor taken in a sense rather different tothe one Marx gave to it in C.

Even within a simple economic system (in the sense ofchapter1), there seems to be a contradiction between the account Marxgives of value -especially of the Law of Value- in the first vol­ume of C and his acknowledging that if commodities were soldby their values then every firm or industry would tend to havea different profit rate, which contradicts his own remark that ina real capitalist economy the capitals move from one sphere ofproduction to another, looking for the highest return rate, whichinduces a generalized tendency within the economy toward theequalization of the profit rate. Clearly, if the Law of Value istaken to mean that commodities are (tendentially) sold at theirvalues, then this contradiction is nothing but a good Popperianrefutation of the Law of Value. The way out of this dilemma forMarx and his sympathetic interpreters has been twofold: (1) Onone hand, when Marx saw that his early formulation of the Lawof Value contradicted the results he had arrived at in volume 3of C, he began to water-down the import of the Law of Value,reducing it to the claim that value "regulates the exchange rela­tions" in a rather imprecise sense. (2) On the second, Marx andhis followers -notably Engels in the Preface and Supplementto volume 3 of C- have interpreted the distressing results ob­tained by Marx in volume 3 as showing that the Law of Valueas formulated in volume 1 is valid only in a society of "simplecommodity production":

44 CHAPTER TWO

The exchange of commodities at their values, or at approximatelythese values, thus corresponds to a much lower stage of develop­ment than the exchange at prices ofproduction, for which a definitedegree of capitalist development is needed.)

(The reader must notice that what Engels caHs 'simple commod­ity production' is not what I caHed 'simple Marxian economy'.Our simple economy is fuHy capitalist, even if rather idealizedand simplified in this sense. Engels' simple commodity produc­tion economy is supposed to be a stage historicaHy previous todeveloped capitalism, but in fact we have seen (see Theorem 7of chapter 1) that the Law of Value holds exactly in our capital­ist simple economy, under the assumption of an equal organiccomposition of capital.)

Before considering Bbhm-Bawerk's criticisms, we shall see thedifferent characterizations of the Law ofValue provided by Marxalong the third volume of C. Noticing that the equalization ofprofit rates contradicted his original formulation of that law,Marx suggested that even if prices are not proportional to val­ues, yet

the sum ofprices ofproduction for the commodities produced in so­ciety as a whole -taking the totality ofall branches of production­is equal to the sum of their values. 2

This is Marx's first attempt to cope with the problem. A cou­ple of pages later, complaining that the number of hours that theworker must work in order to prod uce his means of subsistence"is distorted by the fact that the prod uction prices of the neces­sary means of subsistence diverge from their values", Marx hadasserted that this "is always reducible to the situation" that

whenever too much surplus-value goes into one commodity, too lit­tle goes into another, and that the divergences from value that obtainin the production prices ofcommodities therefore cancel each otherout.3

And after this he immediately adds as a conclusion the foHow­ing statement:

THE PROBLEM OF FOUNDATIONS 45

With the whole of capitalist production, it is always only in a veryintricate and approximate way, as an average of perpetual fluctua­tions which can never be firmly fixed, that the general law prevailsas the dominant tendency.4

Unfortunately, if this conclusion was intended by Marx tosummarize his first attempt to deal with the transformationproblem, it is not well established, because -as Morishima hasshown- the sum of prices is equal to the sum of values only inthe uninteresting case in which both the rate of exploitation andthe rate of profit are zero.5

In the following chapter, Marx seems to try yet another wayof formulating the Law of Value:

Whatever be the ways in which the prices of different commodi­ties are first established or fixed in relation to one another, the Lawof Value governs their movement. When the labor-time requiredfor their production falls, prices fall; and where it rises, prices rise,as long as other circumstances remain equal. [... ] In whatever wayprices are determined, the following is the result: [...] The Law ofValue governs their movement in so far as reduction or increasein the labor-time needed for their production makes the price ofproduction rise or fall. [... ] The average profit, which determinesthe prices ofprod uction, must always be approximately equal to theamount of surplus-value that accrues to a given social capital as analiquot part of the total social capital. [...] Since it is the total valueof the commodities that governs the total surplus-value, while thisin turn governs the level of average profit and hence the generalrate of profit -as a general law or as governing the fluctuations­it follows that the Law of Value regulates the prices ofproduction.6

Unfortunately, the part of the former proposition which as-serts that prices move as the labor-time required for their pro­duction moves is trivially true: If the labor-time required for theproduction of commodities augments (diminishes), the salariesalso are increased (decreased) and this in turn implies a priceincrement (decrement) (use formula (5) of chapter 1). We shalldeal below with the clause that follows thereafter.

In the same chapter Marx proposes still another way of for­mulating the Law of Value when he says that

46 CHAPTER TWO

the relationship between demand and supply does not explain mar­ket value, but it is the latter, rather, that explains fluctuations in de­mand and supply.?

Here the formulation is sufficiently general as to elude crit­icism. But in chapter 18 Marx specifies the claim, denying themarket any role in the determination of equilibrium prices.There he sees the apparent determination of price by the turn­over of commercial capital as a distorting effect that obliteratesthe inner connection of the rate of profit with the formation ofsurplus-value, creating the "illusion" that

the circulation process as such determines the prices of commodi­ties, and that this is within certain limits independent of the processofproduction.8

Clearly, this is very suggestive of the role Marx attributes tothe Law of Value even at this very advanced stage of C, sincethe former complaint clearly implies that any determination ofprices independently of the sphere of production, even withincertain limits, is "mere illusion", that is, (equilibrium) prices mustbe determined by value. And this is indeed what Marx had saida few lines above, where he claimed that

while a closer consideration of the influence of turnover time onvalue formation in the case of the individual capital leads back tothe general law and the basis of political economy, viz. that commodityvalues are determined by the labor-time they contain, the influenceof the turnover of commercial capital on commercial prices exhibitsphenomena which, in the absence of a very far-reaching analysis ofthe intermediate stages of the process, seem to presuppose a purelyarbitrary determination ofprices.9

Hence, the Law ofValue appears at this point-despite all thedifficulties pointed out by Marx in previous chapters- as "thegeneral law and basis of political economy", as a law assertingthat values or prices are determined by labor-time.

On chapter 37, Marx provides yet another characterization ofthe Law of Value, this time as a regularity obtaining between val­ues and production prices whenever the social division oflabor is

THE PROBLEM OF FOUNDATIONS 47

proportionate to the social needs, i.e. whenever society allocatesthe required amounts of social labor to the different branches ofproduction, so that the different social needs are satisfied:

If [the social division oflabor] is in due proportion, products ofvar­ious types will be sold at their values (at a further stage of develop­ment, at their prices of production), or at least at prices which aremodifications of these values or production prices as determined bygeneral laws. This is in fact the Law of Value as it makes itselffelt, not inrelation to the individual commodities or articles, but rather to the total prod­ucts at a given time ofparticular spheres of social production autonomizedby the division oflabor; so that not only is no more labor-time devotedto each individual commodity than necessary, but out of the totalsocial labor-time only the proportionate quantity needed is devotedto the various types of commod ity. 10

Morishima followed the suggestion contained in the italicizedfragment of this quotation, finding a more general law of trans­formation of values into prices proportional to the equilibriumprices. Under these prices, the sum of prices of production forthe commodities produced in society as a whole is equal to thesum of their values and, moreover, the total surplus-value equalsthe total profits. Unfortunately, this holds under the rather re­strictive assumption that industries are linearly dependent (i.e.that the matrix

[AI, All ]

BLI , BLI

is singular) and so it hardly constitutes the desired general for­mulation of the Law of Value.

In chapter 49 Marx confronts the additional "confusion" de­rived from the transformation of surplus-value into profit andrent, but he insists that the relations the different agents of pro­duction have to these particular components "in no way alter thevalue determination and its law":

Just as little is the Law of Value affected by the fact that the equaliza­tion of profit, i.e. the distribution of the total surplus-value amongthe various capitals and the obstacles that landed property places

48 CHAPTER TWO

in the way of this (in absolute rent), gives rise to governing aver­age prices for commodities that diverge from their individual val­ues. This again affects only the addition ofsurplus-value to the var­ious commodity prices; it does not abolish surplus-value itself, northe total value of commodities as the source of these various price­components. II

Finally, in chapter 51 of volume 3, the next to last of C, Marxthinks of the Law of Value as asserting the regulation of "thetotal production by value":

the [...] characters of the product as commodity and the commodityas capitalistically produced commodity give rise to the entire de­termination of value and the regulation of the total production byvalue. 12

Hence, we can see that throughout volume 3 Marx providesseveral characterizations of the Law of Value, being the constantthat value determines or regulates the exchange relations. De­spite the serious difficulties encountered by Marx in his quest fora more definite formulation of the Law of Value, he neverthelesswas convinced, up to the last chapter of C, that value somehowregulated the production prices of commodities. Despite the in­sights provided by the former remarks on the Law of Value, theproblem of specifying the way in which value regulates prices isstill open. It seems to me that Morishima's sympathetic claim thatMarx was "very succesful" in the transformation problem is ex­cessively optimistic. For even Morishima's revised formulation ofMarx's solution to that problem depends on unduly and unnatu­rally restrictive assumptions. Rather, it seems that Marx thought,while he was writing the first volume of C, that he had unveiledthe secret of the value of capitalistically produced commoditiesonly to face sometime later -while he was writing volume 3­the distressing appearence of serious difficulties to sustain theLaw of Value in the way he seems to have thought it held whenhe wrote volume 1. 1 challenge the Marxian scholars to provethat Marx was aware, when he formulated his Law of Value involume 1, that it was valid only for a supposed "simple commod­ity production" economy. Quite on the contrary, I am inclined

THE PROBLEM OF FOUNDATIONS 49

to think that there is a real contradiction between Marx's for­mulation of the Law of Value in volume 1, and the hard factshe discovered later while writing volume 3. Nowhere in C wasMarx able to provide a solution to this contradiction; we shallsee that his remarks on the Law of Value along the third volumeof C do not really solve the problem. As Bohm-Bawerk stressed,if in reality we observe that there is a tendency toward equaliza­tion of profit rates, and that the composition of capitals is quitediverse, then Marx's formulation of the Law of Value appears tobe incorrect.

Bohm-Bawerk analyzed the main arguments advanced byMarx along volume 3 in order to solve the former difficulty.These are four (see the quotations above):

(1) The sum of prices of production for the commodities isequal to the sum of their values.

(2) When the labor-time required for the production ofcom­modities falls, prices fall; and where it rises, prices rise,as long as other circumstances remain equal.

(3) The Law of Value as given in volume 1 of C is ex­actly true in the case of "simple commodity production"economIes.

(4) In an advanced capitalist economy, the Law of Value reg­ulates at least indirectly and ultimately the productionprices, since it is the total value of the commodities thatgoverns the total surplus-value, while this in turn governsthe level of average profit and hence the general rate ofprofit.

In connection with the first argument, we have said that Mor­ishima has established that the sum of prices of production forthe commodities is equal to the sum of their values only un­der the arbitrary assumption that industries are "linearly depen­dent". Hence, contra Bohm-Bawerk, this proposition is far frombeing a mere tautology. I think that nevertheless this authorraises two serious criticisms to it. I think that Bohm-Bawerk'scontention that Marx confuses a mean among fluctuations and

50 CHAPTER TWO

a mean between constant and fundamentally unequal magni­tudes is right. Marx's fiesta of averages does not solve at all theproblem, because one thing is to establish that prices fluctuatearound values determined as in chapter 1, and quite another tosay that they fluctuate around a certain mean. As Bohm-Bawerkputs it, "that mean has a completely different meaning or, moreexactly, it is entirely meaningless for our law". 13 No matter howtough may this criticism sound, the main criticism is yet moresubtle and corrosive. Mter asking "What is the mission of the"Law of Value"?", Bohm-Bawerk replies -quite correctly: "Ev­idently, only that of clarifying the exchange relation ofgoods asit is observed in reality" .14 This is clearly seen in the original for­mulation of the law, according to which equilibrium prices areproportional to values (and we observe in reality a tendency to­ward equilibrium prices, i.e. to prices determined according to auniform rate of profit). Hence, Bohm-Bawerk is right in point­ing out that even if the claim that globally prices are equal tovalues were true, it would be fully irrelevant, since "at any rate,certainly you do not answer in political economy the question ofwhich is the exchange relation of commodities by indicating thesum of prices they get altogether" .15 It seems that in this pointBohm-Bawerk is right, and Marx's answer appears as a desper­ate effort to tackle with the problems the Law ofValue had begunto confront, since his answer is not even relevant to the point.

As I said above, proposition (2) is trivially true. If we keep allcircumstances equal, then it is obvious that an increase (decrease)of labor-time implies and increase (decrease) in the productionprice ofthe commodity. For in such a case it is necessary to investmore in salaries. Clearly, Bohm-Bawerk is quite right when heclaims that this argument is therefore useless to show that whatexclusively determines the magnitude of the value ofany article isthe amount of labor socially necessary, or the labor-time sociallynecessary for its production. For the argument shows only thatlabor-time is but one determinant cause of prices:

Evidently, one could affirm that this law dominates the movementsof prices only if a (permanent) change of prices could not be op-

THE PROBLEM OF FOUNDATIONS 51

erated or conditioned by any other cause but the variation of themagnitude of labor-time. 16

Bbhm-Bawerk attacks argument (3) by showing that in a "sim­ple commodity production" economy workers (who own theproduction means) would require different production times,and so each must wait for different periods of time to receivecompensation. Since it is a defining assumption of our simpleMarxian economy that all products have the same period of pro­duction, that assumption could be extended also to character­ize in its ideal purity a simple commodity production system.Hence, Bbhm-Bawerk's criticism can be raised also against theprototype of MTY. We shall deal with this type of criticisms,directed against idealizing assumptions, in the following sec­tion. At this point the interesting question is whether it can behistorically proven the existence of such systems, whether it ispossible to find traces of societies of that kind. Bohm-Bawerk'sconclusion is that "in reality, it is not possible to discover thistraces in any place whatsoever, neither in the historical pastnor in the present".]? According to Bohm-Bawerk, the Law ofValue has never exerted, and could not have exerted, a realsupremacy even in primitive conditions, due to the fact that thephenomenon of concurrency appeared already at the earlieststages of capitalist production (contrary to what Engels claims,the simple commodity production economies are capitalist).

Bohm-Bawerk considers that his rebuttal of arguments (1)-(3)establishes that three formulations of the Law of Value, intendedto show that that law had validity under certain restrictions, aremistaken and miss the target: The Law ofValue fails to hold evenunder such conditions. The fourth argument is of a differentkind, since it does not intend to show that the law holds underspecial circumstances, but rather to establish that the law is oneof unrestricted validity, only that its form is not like the one Marxhad enunciated in the first volume ofC. Bohm-Bawerk finds themost precise formulation of the generalized Law of Value in analready quoted passage which reads:

52 CHAPTER TWO

The average profit, which determines the prices ofproduction, mustalways be approximately equal to the amount of surplus-value thataccrues to a given social capital as an aliquot part of the total socialcapital. [...] Since it is the total value of the commodities that gov­erns the total surplus-value, while this in turn governs the level ofaverage profit and hence the general rate of profit -as a generallaw or as governing the fluctuations- it follows that the Law of Valueregulates the prices ofpmduction.

Bohm-Bawerk analyzes the links of the former chain of rea­sonings, showing that in each one of these links -total value ­total surplus-value, total surplus-value - average profit, averageprofit - general rate of profit, general rate of profit - prices ofprod uetion- Marx fails to see that one factor add itional to labor­time is a concomitant determinant of the following element inthe sequence. Hence, Bohm-Bawerk concludes that there is areal contradiction between all the formulations of the Law ofValue provided by Marx and the facts:

the Law of Value pretends that only the amount oflabor determinesthe exchange relations; the facts prove, on the contrary, that it is notonly the amount of labor, or factors homogenous to the same, whichdetermine the exchange relations. 18

It seems to me that Bohm-Bawerk's criticism of the formula­tions of the Law of Value given by Marx is solid and cogent. It isbeyond doubt that, at the very best, Marx only was able to showthat labor-value is but one of the multiple concomitant factorsdetermining the equilibrium or "production" prices. I shall takethese criticisms seriously, as constituting a harsh attack on thevery foundations of MTY. We shall see that Bohm-Bawerk wasright in claiming that the formulations of the Law of Value pro­vided by Marx conflict with the facts, but not quite right in thesense of not being able to notice that perhaps other formulationsare in agreement with them.

In the second section we shall complicate the panorama evenmore. The former criticisms of the Law of Value hold evenwithin the context of a simple Marxian economy. We shall askwhether it is possible to drop at least some of the assumptions,

THE PROBLEM OF FOUNDATIONS 53

or even better, ifit is possible to obtain a completely general for­mulation of MTV and, within this new formulation, whether itis possible to provide a new formulation of the Law of Value thatoverrides the correct criticisms advanced by Bohm-Bawerk.

2.2 THE PROBLEM OF GENERALIZING THE PROTOTYPE

In spite of the essentially failed attempts to provide a cogent for­mulation of the Law of Value within our simple Marxian econ­omy, we were nevertheless able to obtain a number ofsubstantialresults. The main of these is perhaps the Fundamental Marx­ian Theorem for such a kind of economies. Indeed, these re­sults were obtained under the assumption that certain condi­tions -some of which are quite restrictive- hold. Our task inthe present section will be to identify all of these conditions andto explore the question whether it is possible to obtain the sameresults -or even better results, since we also want an acceptableformulation of the Law of Value- by canceling those assump­tions which are restrictive. We shall see that the problem of pro­viding an acceptable formulation of the Law of Value is deeplyconnected with the problem of finding an adequate set of non­restrictive assumptions that imply the desired results.

The prototype of MTV is built upon assumptions which aregeneral and others that are restrictive. The restrictive assump­tions are the following:

(51) All capital goods have the same period of rotation

(52) There are no heterogeneous concrete labors, i.e. all labor-poweris homogeneous

(53) There is no choice of techniques

(54) There is no joint production

(55) The technology yields constant returns to scale, and so goods andproduction processes are infinitely divisible

(56) There is a fixed consumption bundle for all workers, and so thedemand structure of the proletariat is very rigid

54 CHAPTER TWO

(57) Workers do not save and so their salary is exactly suficient toacquire the consumption bundle

(58) The economy is closed

Having clearly before us the former assumptions, the problemof generalizing the protoype can be precisely formulated nowas the problem of providing an adequate characterization of acapitalist market economy, a characterization powerful enoughto entitle us to prove the existence of numerical labor-values,to prove the important theorems of the MTV, and above all toprovide a plausible formulation of the Law of Value. We shallpresent now an historical overview of the efforts that have beenmade to get rid of the former assumptions.

2.3 A CONCISE HISTORY OF MTV

The first step toward a mathematical formulation of MTV wasgiven by Wassily W. Leontief in Part II of The Structure ofAmeri­can Economy 1919-1929, published for the first time in 1941. Inthat book Leontief introd uced a "theoretical scheme" based onwhat later came to be known as "Leontiefmatrices", precisely be­cause of their appearing in this work of Leontief's. For the sakeofhistorical accuracy, however, it is fair to say that the first one inintroducing such matrices to theorize in the field of economicswas Vladimir Karpovich Dmitriev (1868-1913) in his essay "TheTheory of Value of David Ricardo", published originally in 1898.At the beginning of this essay Dmitriev considered the followingquestion: "How it is possible to calculate the amount of labourexpended for the production of a given economic good fromthe very beginning of history, when man managed without capi­tal, down to present time?,,19 After pointing out that the amountof labor expended on the production of a given product maybe determined "without such historical digressions",2° Dmitrievproduced a full-fledged input-output system which is identical toLeontief's labor input-output system. According to Nuti (1974),

THE PROBLEM OF FOUNDATIONS 55

the analytical apparatus provided by Leontief four decades lateradds two things: (i) a method for the actual computation of the so­lution, namely the inversion of the matrix (I - AT), where 1 is theidentity matrix and AT is the transpose ofthe matrix oftechnical co­efficients; and (ii) the generalization of the notion of full input (i.e.direct and indirect input requirements) from labor to other produc­tion inputs. 21

It is very difficult to know whether Leontief actually knewDmitriev's work before the publication of his famous book. Bethat as it may, the objective identity of Leontief's labor input­output system with that of Dmitriev's proves that Leontief wasnot the first one to devise it, although it also points out to some­thing which is more important, to wit, the Ricardian lineage ofLeontief's theoretical apparatus.

The starting point of the mathematical formulation ofthe pro­totype of MTV is Leontief's theoretical apparatus as presentedin his book of 1941, but -as we saw in chapter 1- this pro­totype is already sketched in the first two volumes of C. Oneof the first persons ever in making explicit the assumptions ofLeontief's theoretical apparatus was Georgescu-Roegen (1950),but perhaps the first writer to draw attention to the fact thatsuch apparatus is grounded upon a labor theory of value wasBurgess Cameron (1952). Cameron showed that the proposition'the [Leontiet] price of a commodity (in terms of the wage rate)will equal the number of man-hours to produce it' is derivablewithin Leontief's system.22 Morishima and Seton (1961) took forgranted that in obtaining the former result Cameron also estab­lished

that in a Leontiefmodel the price ofa commodity in terms oflabor("wage price") will equal its Marxian "value" under certain condi­tions which include (i) competitive long-run equilibrium, i.e., zeroprofits in each sector, and (ii) perfect divisibility of the economyinto "primitive sectors", i.e., sectors producing single homogeneouscommodities.23

Now, since assumption (i) in destroying the "surplus" (i.e. zeroprofit) clearly "reduces the discovered equality to a merely for­malone", Morishima and Seton's purpose was "to inquire into

56 CHAPTER TWO

the general relationship between Leontief price and Marxianvalue when both assumptions (i) and (ii) are relaxed".24 In doingso, Morishima and Seton developed a less idealized version ofthe theory and -within this still very idealized version- theywere able to obtain mathematically for the first time (their ownversions ot) two classical Marxian results, namely, (1) "The rateof exploitation will always exceed the rate of profit"; and (2) "AMarxian type of 'economic progress' (where capital accumula­tion steadily reduces the share of wages in the total costs of allsectors) will normally entail a fall in the rate of profit, unless ac­companied by a countervailing in the rate ofexploitation".25 Re­sult (1) is close to what we have called the Fundamental MarxianTheorem. On chapter 1 we formulated this theorem in the fol­lowing terms (see Theorem 6): There exists a price system atwhich the rate of profit is positive if and only if the rate of ex­ploitation is positive. Okishio (1963) provided an independentproofofa version of the theorem, which is even closer to this for­mulation, using assumptions similar to those of Morishima andSeton.26 The assumptions and structure ofthe classical model ap­pear in a rather explicit form in this paper of Okishio's, but thefirst thorough examination and description of the same modelappeared one decade later in what is now one of the classics ofthe Marxian literature, Morishima's Marx's Economics.27 In thisbook there is an exhaustive list of the assumptions of the classi­cal model and detailed proofs of all the central theorems. A re­sult which is particularly important from the point of view of thephilosophy of science is the proofof the existence of unique pos­itive solutions of the Leontieflabor-value equations and a proofof the equivalence between value understood as socially neces­sary labor-time and value understood as the sum of the amountof direct labor and "congealed" labor embodied in the meansof production which is transferred to the product,28 Using theseprecisely defined Marxian values, Morishima developed a con­siderable amount of Marx's economic theory in a scientific rigor­ous way, although ofcourse within the limitations imposed by theidealized assumptions on which the proofs concerning the exis­tence of unique positive values are based. These assumptions

THE PROBLEM OF FOUNDATIONS 57

(enlisted in the previous section) were criticized by Morishimain the same book, whose conclusion is that MTV has to be aban­doned and replaced by a new theory that combines features ofMTV and of Von Neumann's economic theory.

Morishima (1974) addressed the problem of constructing anew Marx-Von Neumann theory ofvalue. Unlike the prototype,this new theory drops assumption (51), that all capital goodshave the same period of rotation, and makes possible a bettertreatment of capital age-structure problems when the time fac­tor is introduced (which are intractable within the prototype).To achieve this, it admits both joint production and choice oftechniques, thus droping also assumptions (53) and (54). Mor­ishima was able to prove the Fundamental Marxian Theoremwithin this Marx-Von Neumann theory, using "optimum" valuesinstead of "actual" values. If actual values are obtained by calcu­lating the embodied labor contents of commodities on the ba­sis of the prevailing production coefficients, the optimum valuesare shadow prices determined by a linear programming prob­lem which is dual to another linear problem for the efficient uti­lization of labor. Although optimum values are not necessarilyunique, the rate of exploitation is well defined and -as I saidbefore- the Fundamental Marxian Theorem can be proved,provided that labor is assumed to be homogeneous.

On the same line of Morishima's, John E. Roemer (1980,1981) produced a series of more general theories and derivedthe existence of Marxian equilibria from the assumption thatthe firm's behavior consists of maximizing profits given a set ofpossible production processes and certain restrictions of capitalavailability. In Roemer's theories the value of a bundle of com­modities is defined as the minimum labor required to producethe bundle, given the technological possibilities of the economy;thus, Roemer's definition is analogous to that of Morishima's,the difference being that according to Roemer values are notnecessarily determined by a linear program. In Roemer's the­ories the exploitation rate is well defined for each productionprocess and a more general version of the Fundamental Marx­ian Theorem is proved. I n Roemer's models, like in the one built

58 CHAPTER TWO

by Morishima on Von Neumann's, the role of the exploitation ofworkers is clarified as a condition for the growth and reproduc­tion ofthe economy. Roemer's models are fairly general but theyare still based on assumptions (52), (55)-(58). Moreover -as it isalso the case for the Marx-Von Neumann theory- no satisfac­tory account of the Law of Value can be given.

In particular, assumption (52), the assumption that labor ishomogeneous, is very restrictive. According to Elster (1985),

the presence of genuinely and irreducibly heterogeneous labor is amajor stumbling block for Marxist economics. If taken seriously, itprevents the labor theory ofvalue from even getting off the ground,since the basic concepts cannot be defined. 29

It seems to me that this remark of Elster's must be taken veryseriously, because one ofthe most outstanding traits ofcapitalismis precisely a very rich division oflabor. We shall see that with thisremark Elster has really hit the core of the foundational prob­lems of MTV. It will be apparent that in order to deal with thisproblem it will be necessary to return to the very basis of MTVas given by Marx in the first chapter of C.

Before attempting to address the problem of heterogeneouslabor, I would like to close the present chapter by presenting insome mathematical detail the achievements of Morishima andRoemer.

Leaving aside the intrinsic limitations of (53) and (54), Mor­ishima found that it is impossible to provide a consistent treat­ment of aging capital goods within these assumptions. Even as­suming that (53) and (54) are satisfied by processes producingbrand new goods, serious difficulties arise in the treatment ofthe wear and tear of fixed capital goods. Morishima did showthat

There will not be universal consistency between 'the replacement ofthe wear and tear portion of the value in the form of money' and'the replacement of fixed capital in kind', unless we can get rid ofthe neoclassical method ofdepreciation and obey the Von Neumanngolden rule in the evaluation of capital costs. 30

THE PROBLEM OF FOUNDATIONS 59

According to Von Neumann's golden rule, used capital goodsmust be treated as by-products in the labor process. But this istantamount to the rejection of (53) and (54), and so the adop­tion of Von Neumann's treatment of capital goods implies thesubversion of an important portion of the very foundations onwhich the existence oflabor-values had been made to depend inthe construction of the prototype.

In his 1974 paper Morishima intended to re-establish the Fun­damental Marxian Theorem by reformulating it in terms ofopti­mum values. Thus, droping assumptions (53) and (54), but stillmaintaining the other assumptions, Morishima considered thefollowing system. Let A be the matrix of input coefficients, Lthe row vector of labor inputs, and B the matrix of output coef­ficients, i.e. the ith column of B is the vector of outputs of pro­duction process Xi. Moreover, let b be the total consumption vec­tor of the working class. According to Morishima, the necessarylabor-time, i.e. the labor-time socially necessary to produce theworker's consumption basket b, min Ly, is obtained by solvingthe following linear program:

(PI) Minimize Ly, with respect to y, subject to

By ~ Ay + b, y ~ O.

Thus, the necessary labor-time is the minimum time required toproduce b. The dual of this program is

(P2) Maximize Ab, with respect to A, subject to

AB ~ AA + L, A ~ O.

If yo is a solution to PI and Aois a solution to P2, then the DualityTheorem implies that

Aob = Lyo' (1)

and so the components of Ao can be interpreted as some kind oflabor-values, which are designated by Morishima as 'optimumvalues'. 5ince there may be infinitely different solutions to P2,

60 CHAPTER TWO

these optimum values are not uniquely determined, althoughthe value of the consumption vector per worker c = N-lb(where N is the number of workers) is the same for every op­timum value vector. For let T be the length of the working dayand suppose that the workers do not save. Then the exploitationrate is defined as

_ TN - Lyo€ - L .

YoSince Lyo = min Ly is uniquely determined (even though yo isnot), equation 1 implies that

€ = TN - Aoc

Aoc

is the same for every solution Ao. As Morishima put it:

Like actual values, optimum values may not be unique ifjoint out­puts and alternative methods of production are admitted. But un­like actual values, they give a unique evaluation of c; that is to say,Aoc takes on the same value for all optimum value systems. 31

In a similar fashion, although in a more general technology,Roemer (1980, 1981), defined the concept of labor-value as thesolution to a programming problem. Instead ofa Von Neumanntechnology, Roemer considered a closed convex32 set Y contain­ing the null vector 0 in which every bundle of commodities canbe produced and every positive output requires labor in orderto be produced. Assuming (52), that labor is homogeneous, Roe­mer defines the labor-value of a bundle y as the number

min{x : [x, X, xl E Y and x - X ~ y}.

Clearly, this number is not necessarily a solution to a linear pro­gramming problem, because the set Y is described in very gen­eral terms. Like Morishima's definition oflabor-values, however,Roemer's definition also allows a unique determination of theexploitation rate for every labor process in Y.

Roemer's definition oflabor-value, which clearly encompassesthat of Morishima as a special case, has been criticized by some

THE PROBLEM OF FOUNDATIONS 61

Marxian economists. The standard criticism is that the labor­value of a bundle of commodities should be determined by theaverage techniques being actually used in the economy, not bythose which are the most efficient in the utilization of labor. Ac­cording to Roemer, this objection

is a mainly semantic matter. By defining the labor-value of y as wehave done, we are asking for the labor-efficient way of producing y,using the aggregate prod uction set Y. If a "socially average" tech­nique is inferior to this, then we would be injecting some sort ofinefficiency into our conception of labor-values, which is not in theMarxian spirit.33

At any rate, the main problem of his definition of labor-value isnot so much that it characterizes value in terms oflabor-efficientprocesses, but rather that it characterizes value in such a way thatit presupposes that labor is homogeneous.

Independently of his definition of labor-value, Roemer hasprovided a new and important theoretical framework for MTV,within the convexity assumptions which are nowadays usual inmathematical economics, and proved outstanding results re­garding the existence of Marxian reproducible equilibria in avariety of interesting models. These results stand, even if Roe­mer's definition of labor-value is rejected, and will provide theframework for the formulation of the axiomatic foundations ofMTV in the present book. I do not claim that this frameworkis absolute: As I said, it depends on the convexity assumptionswhich contemporary mathematical economics takes for granted,but these assumptions could be eliminated as the science of eco­nomics progresses.34 As a matter of fact, we shall maintain as­sumption (55), which is a blend of convexity and linearity.

Yet, the main reason to adopt Roemer's framework is not thefact that he adopts the standard terms and assumptions of con­temporary mathematical economics, but rather the fact that Roe­mer does consider the behavior of the firm, as well (in some cases)the existence of financial capital markets. This is required to for­mulate the market conditions that produce a uniform profit rate.The task of a Marxian theoretician would be to show that even

62 CHAPTER TWO

under such equilibrium-productive behavior, prIces are regu­lated by socially necessary labor-time.

Thus, in view of what has been previously said, the first foun­dational problem that I want to formulate here is this: Is it pos­sible to generalize Roemer's theoretical framework in order to get rid of(52)? In mathematical terms, what the elimination of (52) meanswithin Roemer's theoretical framework is that the labor-powervector x of any production process in Y is not necessarily unidi­mensional, i.e. a real number, but in the general case it is an nvector forn ~ 1. Obviously, in such a case the definition oflabor­value, as the minimum amount of labor required to produce abundle ofcommodities as a net output, breaks down. How to de­fine the concept of labor-value within such a generalized struc­ture?

Before addressing this question, it will be convenient to intro­duce the methodological tools that shall be used. The next twochapters are devoted to this. In the fifth chapter we will addressthis question. The remaining assumptions (56)-(58) will be dis­cussed in subsequent chapters.

Chapter 3

STRUCTURES AND REPRESENTATIONS

The present chapter is devoted to introduce the main metathe­oretical concepts that shall be used to attack the foundationalproblems of MTY. Basically, they are two, namely, the conceptof a structure and that of a representation. Roughly, a struc­ture is an array of sets, and of relations over such sets, subject tocertain conditions and laws. Notice that, in this sense, the usualalgebraic structures are indeed structures. I will introduce thisconcept of structure in the first section of the present chapter. Insome cases, structures can be used to represent real situations.As a matter offact, the structures that we shall consider in subse­quent chapters intend to represent market capitalist economiesor some of their aspects. The structures used in the sciencesmayor may not contain numerical functions (like mass, force,temperature, and the like). Some of them include qualitative re­lations -like the abstract labor relation-, and when that is thecase sometimes it arises the problem whether it is possible to mea­sure such relations by means of numerical functions that some­how "mirror" the structure of the relation in some appropriatemathematical structure. Such a function -when it exists- iscalled a representation of the relation. I shall devote the secondsection of this chapter to discuss the concept of representation.

63

64 CHAPTER THREE

3.1 STRUCTURES

In order to motivate a general definition of structure, let us startby considering structures of first order languages. Afirst orderlogic is a family of symbols divided into categories (a first orderlanguage), together with certain rules of formation, transforma­tion and interpretation. The symbols of a first order languageare mainly divided into logical and non-logical symbols. The log­ical symbols of the language are

), ('" ---',

and the countable sequence of individual variables

) and ( are called grouping symbols or parentheses; " ---', are theconnectives. It is optional to include among the logical symbolsthe identity symbol '=', which is considered as a (constant) two­place predicate. The parameters consist of the universal quantifierV and, for each positive integer n, of a set (possibly empty) ofsymbols called n-place predicate symbols; of a set (possibly empty)ofsymbols called constant symbols; and, for each positive integern,of a set (possibly empty) of symbols called n-place function symbols(a minimal language contains at least one predicate symbol). Theremaining usual connectives

t\, V, +-+,

as well as the existential quantifier :3, can be defined in terms ofthe previous logical symbols and the universal quantifier. This isdone below.

An expression is any finite sequence of symbols. The formationrules stipulate how to build expressions out ofgiven expressions.There are two kinds of expressions: terms and formulas. Theseconcepts are defined recursively by means ofcertain operations.In order to define the concept of a term, for each n-place func­tion symbolf the term-building operation F j is needed, that takes

STRUCTURES AND REPRESENTATIONS 65

expressions Ch ••• , Cn and the n-place function symbolf to buildthe expression f C I ••• Cn' In terms of the term-build ing opera­tions, the set of terms is defined as the smallest set that containsthe individual variables and the constant symbols, and which isclosed under the term-building operations. Thus, if 'CI" 'C2' areconstant symbols 'f' is a one-place function symbol and 'g' is atwo-place function symbol, then examples of terms are 'X54" 'CI"

'C2', 'fCI' and 'gX5JCI" Analogously, in order to define the con­cept of a formula, the formula-building operations F~, F_, Vi arerequired. These operations yield the following results, for arbi­trary formulas ¢, 'l/J and positive integer i:

F~(¢) = (-,¢)

F_(¢, 'l/J) = (¢ -+ '1/')

V i(¢) = VXi¢

An atomic formula is an expression of the form

where P is an n-place predicate and t l , •••, tn are terms. In termsof the former operations, the set of well-formed formulas (wfl)can be defined as the smallest set containing the atomic formulaswhich is closed under the formula-building operations. The re­maining connectives and the existential quantifier are defined bythe following conditions. For arbitrary formulas ¢, 'l/J, and vari­able Xi:

(¢ 1\ 'l/J) abbreviates -,(¢ -+ -,'l/J)

(¢ V 'l/J) abbreviates ((-,¢) -+ 'l/J)(¢ ~ 'l/J) abbreviates ((¢ -+ 'l/J) 1\ ('l/J -+ ¢)

3Xi¢ abbreviates (-,(VXi(-'¢)))'

A variable Xi may be free in a formula. This concept is definedrecursively as follows: 1

66 CHAPTER THREE

(1) For atomic ¢, Xi occurs free in ¢ iffXi occurs in (is a symbolof) ¢

(2) Xi occurs free in -,¢ iff Xi occurs free in ¢

(3) Xi occurs free in (¢ --+ 'I/J) iff Xi occurs free in ¢ or 'I/J(4) Xi occurs free in VXj¢ iff Xi occurs free in ¢ and Xi t= Xj.

A sentence is a wffin which no variable occurs free.The transformation rules stipulate which formulas can be in­

ferred from given formulas. There are many possible sets oftransformation rules but it is possible to select a rather large setof logical axioms and only modus ponendo ponens as an inferencerule. In such a case, if A is the set of logical axioms of the system,¢ any wff, and r a set ofwff, a deduction of ¢ from r is a sequence(Qo, ••• , Q n ) of wff such that Q n = ¢ and for each i ::; neither

(1) Qi is in r U A, or

(2) for some j and k less than i, Qi is obtained by modus po­nens from Qj and Qk, where Qk = Qj --+ Qi.

A formula ¢ is said to be a theorem of r (in symbols: r f- ¢)iff there is a deduction of ¢ from r. For a particular selection ofthe set A of logical axioms, the reader is referred to Enderton(1972).~

The rules of interpretation for a first order language are clus­tered around the concept of a structure for the same language.A structure3 Q{ for a first order language is a function from the setof parameters into some family of sets that satisfies the followingconditions:

(1) Q{ assigns to the quantifier V a nonempty set lQ{j, calledthe universe of Q{

(2) Q{ assigns to each n-place predicate symbol P an n-aryrelation p'll ~ 1Q{ln; i.e. p'll is a set ofn-tuples of membersof the universe

(3) Q{ assigns to each constant symbol c a member c'll of theuniverse IQ{I

(4) Q{ assigns to each n-place function symbolf an n-ary op­erationf'll on 121/; i.e.f'll: 1Q{ln --+ IQ{I.

STRUCTURES AND REPRESENTATIONS 67

Given a sentence 4> in a language, and a structure Qt for thelanguage, it is of the utmost importance to ask whether 4> is truein Qt. To give a precise meaning to this question, the concept oftruth needs to be properly defined. In order to deal with theoccurrence of variables -which may be free- in the formulas,the set ofall functions 5 from the set ofvariables into the universeIQt/ is introduced. Each of these functions 5 assigns an element ofthe universe to each variable and can be extended to a functions, which is defined by recursion as follows:

(1) For each variable Xi: S(Xi) = 5(Xi)

(2) For each constant symbol c: s(c) = c21

(3) IftJ, ...,tn are terms andJ is an n-place symbol, then

If s is one of the former functions, s(xdd) denotes the functionwhich is identical to s except that it assigns the element d of theuniverse to Xi.

Using the functions s and their extensions S, we shall definethe notion of a formula 4> being satisfied by Qt with s (in symbols:1=21 4> [5]). This is done as follows. For arbitrary formulas 4>, 1/; andvariable Xi:

(1) 1=21 t 1 = t2 [s] iff s(t l ) = s(t2 )

(2) IfP is an n-place pred icate symbol,

(3) 1=21 -'4> [5] iff ~21 4> [s]

(4) 1=21 (4) ---7 1/;) [5] iff either ~21 4> [5] or 1=21 1/; [s] or both

(5) 1=21 'v'Xi4> [s] iff for every d E IQtI, it is the case that

Naturally, a wff 4> is said to be satisJiable iff there is a structureQt and a function s such that 1=21 4> [s]. A set of formulas r is

68 CHAPTER THREE

satisfiable iff there is a structure 2{ and a function s such that 2{

satisfies every member of r with s. The crucial concept oflogicalimplication can be defined now. Let r be a set of wffs and </> awff. r is said to logically imply </>, or </> to be a logical consequence ofr (r F </» iff, for every structure 2{ for the language and everyfunction s from the set of variables into the universe of 2{; if 2{

satisfies every formula in r with s, then 2{ also satisfies </> with s.It can be shown that if </> is a sentence and 2{ any structure,

then either </> is satisfied by 2{ with every s, or not at all. In thefirst case we say that </> is true in .2{, or that 2{ is a model of </>. Thus,in particular, if </> is a sentence and r a set of sentences, then </> isa logical consequence of r iff every model of r is a model of </>.

I have just described a first order logic in a succint way.The most important result within first order logic is Godel'sCompleteness Theorem, according to which every logical con­sequence of a set r of formulas is also deducible from r. Thisresult implies the Compactness Theorem (also known as Finite­ness Lemma). The Compactness Theorem asserts that if r is aset of formulas such that every finite subset of r is satisfiable,then r is also satisfiable. Using this theorem, Robinson (1961)proved the existence of a proper extension of the real numbersystem that contains both infinitely large and infinitely small (yetnonzero) numbers. The extreme formal rigour and precision offirst order languages, and of their semantic, is required to obtainresults like this. Indeed, many important aspects of structurescan be studied by means of first order methods.

Given a structure 2{ for a first order language, we define thetheory of 2{ (Th 2{) as the set of all the sentences of the languagewhich are true in 2{. Analogously, if 8' is a class of structures,the theory of 8' (Th 8') is the set of all sentences true in each ofthe elements of 8'. A theory is defined as a set of sentences of thelanguage which is closed under logical implication, i.e. T is atheory iff

TF</> :::} </>ET.

Let E be a consistent set of sentences and denote by Mod E theset of all models of E. The set of consequences of E (Cn E) is de-

STRUCTURES AND REPRESENTATIONS 69

fined as Th Mod E. It is easy to see that a set of sentences T is atheory iffT = Cn T.

Roughly speaking, a set E of sentences is decidable if there isan algorithm which can tell, in a finite number of steps, whethera given sentence is or is not an element of E. 4 A theory T isaxiomatizable iff there is a decidable set of sentences E such thatT = Cn E. In particular, if E is finite, then T is said to be finitelyaxiomatizable.

Some philosophers of science attempted to identify the scien­tific theories as sets ofsentences that could be translated into firstorder sentences and put together in a consistent set T; further­more, they also claimed that Twas axiomatizable, so that thescientific theories could actually be .identified with a decidableset E, a set of axioms for T. These assumptions are the basis ofthe so-called statement view of theories. The statement view oftheories is misleading because it suggests that the structures usu­ally found in the sciences are structures for first order languages.Unfortunately (because first order model theory is beautiful andcontains very deep, useful results) many structures in science arenot structures for a first order language. As examples, considerthe topological and the probability spaces. Recall that a topolog­ical space is a pair X = (X, T) such that X is a nonempty set andT is a family of subsets of X containing the empty set, the set Xitself, and closed under finite intersections and arbitrary unions.It is easy to see that the topological spaces are not models of afirst order theory (i.e. of the language in which the sentences ofsuch theory are formulated). A first attempt to provide a firstorder language for which the topological spaces would be thestructures would have to start by noticing that the universe of astructure for any such language would have to be the power setp(X) of X, since the theory needs to be able to refer to all thesubsets of X and, moreover, T needs to be thought of as a one­place predicate (i.e. a subset of p(X)), n needs to be thought ofas a binary operation over p(X), and (/) and X have to be consid­ered as constants, i.e. as elements of the universe p(X). From thispoint ofview, the first two axioms of topology can be formulatedas follows:

70 CHAPTER THREE

0Er/\XEr

VxVy«x E r /\y E r) -- X ny E r).

The problem is -as the perspicacious reader must have alreadynoticed- that there is no way of formulating the third axiom,because u would have to be treated as an infinitary operationover 8J(X). The problem remains even if we move onto secondorder languages, because the problem is not the lack ofquantifi­cation over predicate or function variables, but rather the needto treat U as an operation over the universe of the structure.

An apparent way out is to treat U (and perhaps also n) as unaryoperations over 8J(8J(X», Le. if Y ~ 8J(X), then UY E 8J(X) andnY E 8J(X). But then the new problem arises that in such a casewe would have to consider 8J(8J(X» as the universe (since the op­erations in first order logic are treated as functions having someCartesian power of the universe as domain), in which case theoperations would not be closed in the universe. Another short­coming of this approach is that the "individuals" would be fam­ilies of subsets of X; thus, in particular the topology r would bean individual, but then the objects of level "lower" that r -likethe elements of r- cannot be referred to in any way whatso­ever and so none of the fundamental axioms can be formulated.Similar problems arise when a first order formulation of the Kol­mogorov axioms for probability spaces is attempted.

The root of the former problems is that many theories con­sider simultaneously several ranks of objects, i.e. roughly speak­ing, individuals, sets of individuals, sets of sets of individuals, oreven more complex constructions, whereas first order structuresinvolve only individuals, finitary operations over individuals andrelations over individuals.

The notion of a rank of objects can be given a precise defini­tion. Consider an arbitrary nonempty set X and define

n

Xo = X; Xn + J = 8J(UX;).i=O

STRUCTURES AND REPRESENTATIONS 71

The superstructure over X is the set Ui<w Xi and is denoted by X.An object x E X is said to be of rank k > 0 iff x is an element ofX k and not an element ofXj for j < k (if x E Xk , then x E Xm form > k). We shall introduce a concept of structure that is able tohandle theoretical structures having any finite number ofobjectsof arbitrary ranks. In order to do so, notice that X can be seenas a model ofZermelo-Fraenkel theory of sets. The language ofthis theory contains the two-place predicate symbol 'E' as uniqueprimitive. The axioms of this theorl guarantee the existence ofthe sets Uy and ny for any nonempty set y EX. Thus U andn can be seen as (definable) unary operations over X. Takingadvantage of this fact the concept of a topological space can nowbe defined as as a pair (X, r) of entities in Xthat satisfies the firstorder sentences:

(T1) 0 E r /\ X E r

(T2) Vx( (x ~ r /\ finite(x» ---+ nx E r)

(T3) Vx(x ~ r ---+ Ux E r).

(Indeed, the predicate 'finite' is definable within ZF).This first order axiomatization of topology suggests that any

structure can be axiomatically characterized within a first orderlanguage, provided that the generator X of the superstructureX is suitably chosen. This can be done even when the structureunder consideration is composed by sets as dissimilar as one ofphysical things and another of numbers. For instance, supposethat we have a structure (A, R, rn), where A is a set of physicalbodies, and m a function rn: A ---+ R assigning to each element ofA a positive real number (say, its mass). The minimum super­structure required to build this structure is the one having asgenerator the set X = A U R. Sentences with restricted quan­tifications of the form 'for every x E A.. .' can be reformulatedwithin the language as sentences of the form

Vx(P(x) /\ ... ,

provided that a predicate 'P', to be interpreted as the set A, isintroduced into the language.

72 CHAPTER THREE

The former discussion suggests a general notion of structurewhich is appropriate for the purposes of axiomatizing scientifictheories. Roughly, such a structure is nothing but a finite ar­ray of sets Db ... , D k , and of set-theoretic constructions R 1, •••, Rm

over such sets, that satisfies certain sentences formulated withinan extension of the language of ZF. For instance, a topologicalspace is a structure in this sense, where Dk = X and R m = T,

that satisfies the sentences TI-T3 above. The sentences a struc­ture satisfies are indeed first order sentences, but this does notmean that the structure is a structure for the language in whichthese sentences are formulated. Rather, they are elements of astructure (X, E) for the same language. This prevents us fromexploiting certain results of first order model theory, but at leastwe can refer to all the entities involved in a very natural way and,furthermore, we can freely employ all the results of ZF for ourpurposes. Since, in particular, classical mathematics can be de­veloped within the framework of ZF, it follows that we have allof classical mathematics at our disposal. This is not the place todevelop the mathematical concepts that are being used in the re­construction of MTV; the interested reader can find a detaileddevelopment of the real number system in Suppes (1972) andLandau (1966). The concepts from linear algebra used here, likethose of vector and linear space, are developed for instance inMcLane and Birkhoff (1967).

This concept ofstructure can be given a very precise formula­tion. What it is required to do is to identify a whole class ofstruc­tures in the most general terms. For instance, topology needs toidentify the class of all the topological spaces, probability theorythat ofall probability spaces, and so on. This is done through theconcept of a structure species, which was introduced originallyby Bourbaki (1968), but better exposed by Balzer, Moulines andSneed (1987). My own exposition will follow closely this last one.The definition of structure species is the last of a series startingwith the definition of a k-type.

For each positive integer k, the concept ofa k-type a is definedinductively as follows:

STRUCTURES AND REPRESENTATIONS 73

(1) for each positive integer i ~ k: i is a k-type

(2) if U is a k-type then p(u) is a k-type

(3) if Uj and U2 are k-types then (Uj x (2) is a k-type.

From an ontological point of view, k-types are set-theoretic enti­ties build up from positive integers (which themselves are sets).But we shall identify each k-type U with an operation assign­ing to certain sets a set-theoretic construction based on thosesets. These sets have to be k in number and shall be denoted byDJ, ..., Dk ; the construction that the k-type U associates to thesesets will be called the 'echelon set of type u' over DJ, ..., D k anddenoted by u(D j , •••, D k ). The definition of this concept is alsoinductive and proceeds as follows:

(1) if u is some i (i ~ k) then u(D], ..., D k ) = D i

(2) If u] is a k-type and u has the form p(u]), then

(3) If Uj and U2 are k-types, and u has the form (uJ x (2) then

As it is usual in set-theory, we may identify the objects ofset theory with their own names and think also of the k-typesas term-building operations. Consider an extension of the lan­guage of ZF containing all the terms generated from the termsD j , ..., D k by such operations. If 'c' is a constant symbol of thelanguage denoting a set, then we shall call any formula of thelanguage of the form

a typification.I t is usual to find in scientific structures a distinction between

base sets and auxiliary sets. Base sets are those that contain theempirical objects the theory deals with, whereas auxiliary sets

74 CHAPTER THREE

are mathematical sets used to represent magnitudes of objectspertaining to base sets. That is why in the definition of a type itis convenient to talk of (k + m)-types, where k is the number ofbase sets and m is the number of auxiliary sets. Thus, a type T isan array (k, m, 0"1, •••, O"n) such that

(1) k is a positive integer and m is a nonnegative integer

(2) 0"), ••• , O"n are (k + m)-types.

If T = (k, m, 0"), •••, O"n) is a type, a structure of type T is a tuple(D 1, ••• , Dk , AJ, ..., Am' R 1, ••• , Rn) such that DJ, ..., Am are sets and,for each i :::; n, the typification

is true in any model of ZF generated from a set containing(U~~l D j ) U (U::l AI).

Let T = (k, m, 0"), ••• , O"n) be a type, and

a structure of type T. Consider an extension of the language ofZF containing constant symbols for the sets

A formula of such language containing no ocurrences of con­stant symbols other than symbols among these is said to apply to::D. 1fT is a type as before, a structure species of type T is a tuplep = (T, cPl' ..., cPs) such that, for all i :::; s, cPo applies to some struc­ture of type T. Naturally, a structure species is a structure speciesof some type 1". Among the formulas cPo (1 :::; i :::; s) that apply tosome structure of type T, some contain occurrences of symbolsfor the base sets Dl , ••• , Dk and for precisely one of the relationsRJ, ..., R n • Formulas of this type characterize these relations andso they will be called characterizations; notice that, in particular,typifications are characterizations. Scientific laws are not charac­terizations because the least that can be said about them is thatthey establish connections between different relations.

STRUCTURES AND REPRESENTATIONS 75

We have thus reached the definition of the concept we wereseeking. We shall identify a scientific structure precisely with astructure of some species p. If p = (r, <PJ, •••, <Ps) is a structurespecies of type r = (k, m, 0"1, ... , O"n), then a structure of species p isa tuple

where DJ, ...,Am are sets, that satisfies certain formulas <Ph ..., <Psthat apply to it. The relevant concept ofsatisfaction is the follow­ing. The structure

satisfies the formula <P iff <P applies to :::D and <P is satisfied inany model of ZF generated from a set that contains (U;=J D j ) U(U;::I AI)' By the theory of:::D we understand the set ofall sentencesthat apply to:::D which are true in :::D, i.e. which are satisfied by:::D.We say that the theory of:::D is axiomatizable if there is a decidableset E ofsentences that apply to :::D such that every element in thetheory of:::D is a logical consequence of E. Let r be a set of sen­tences. A model of r is any structure :::D such that the sentences inr apply to :::D and are satisfied by :::D.

From an ontological point ofview, structures are abstract enti­ties (entia rationis, as the scholastics would say) that can be objectof mathematical research. Some of these entities, in addition,represent real things, systems or processes. Representation the­ory is a philosophical discipline that aims to gain insight into theway structures are carriers of knowledge about aspects or partsof the real world. The main problem of representation theory isto give an account of the way mathematics applies to reality. Thepresent chapter is quite far from pretending to provide a com­plete account of the problems and results of that discipline: Itsaim is just to sketch how such a discipline would look like and tointroduce the concept of representation. It will be useful for thispurpose to discuss an important useful case of representation.

76 CHAPTER THREE

3.2 REPRESENTATIONS

3.2.1 THE ONTOLOGICAL FRAMEWORK

According to cosmology and natural history, the Earth, the plan­ets and millions of stars were already quite ancient when homosapiens sapiens began to marvel at the surrounding world. Whatthis means is that the human mind occupies a rather humbleplace in creation, far away from the place ofcreator ofthe naturalworld. On the contrary, man has always been trying to dominatethe natural phenomena for his own ends, a task whose success isalways limited by the vastness of the world. Since the times whenmen were afraid of thunder and ray, that task has been accom­plished through an increasing knowledge of the natural phe­nomena. Originally, this knowledge was rather empirical andunsystematic, but since the rise of astronomy in Babilon, andespecially in classical Greece with Eudoxus, that knowledge hasbecome theoretical and systematic. After the scientific revolutionin the seventeenth and eighteenth centuries, many more scien­tific disciplines have arised, including disciplines that deal withsocial phenomena as well. Among other things, these disciplinesare constituted by conceptual structures and theories by meansof which the scientists -as before- intend to obtain knowledgeof some parts or aspects of the real world.

The development ofmathematical logic in the current century-a discipline that may be divided into model theory, proof the­ory, recursion theory and set theory- has provided new tools toanalyze the conceptual structures and theories produced by thedifferent sciences, giving a new shift to the theory of science, adiscipline inaugurated by the Bohemian philosopher and math­ematician Bernhard Bolzano with his Wissenschaftslehre in 1837.From the point of view advocated here, the theory of sciencecan be defined as the discipline in charge of determining thelogical structure and foundations of scientific theories, as wellas the way these theories connect to the real phenomena withwhich they deal. This definition of the theory ofscience presup­poses that there is a way of referring to the real world which isindependent of, and in some sense "previous" to the language

STRUCTURES AND REPRESENTATIONS 77

of any given scientific theory. This presupposition is true: Thatway is provided by the conceptual system and language of thephilosophia de ente, enriched with proper and common names re­ferring to natural kinds, substances and accidents of substances.By means of such a language we certainly can refer to the entiawith which the sciences deal, and formulate propositions whichare true about such entia, in a way independent of the technicallanguage of these sciences. The philosophia de ente contains theconceptual apparatus required to discuss, for example, whetheran object considered by a theoretical model is one ens, or merelyseems to have individual unity just because of the way in whichthe mind considers it; or whether an object denoted by the lan­guage of a scientific discipline is really or merely conceptuallyexisting -ifit is an idealization.

Someone might object that if we had an independent way ofreferring to the entia with which the sciences deal, then the math­ematical structures representing them would be red undant anduseless. The reply to this is that one thing is to refer to somethingand quite another to represent it mathematically. The reason topursue a mathematical representation ofa real being, process orsystem is mainly to probe deeper into its proper operation, rela­tions or intrinsic nature, and quite often to obtain measurementsof some quantity inhering in it, that is needed for practical ortheoretical purposes.

The philosophia de ente I have in mind -also known as ontol­ogy or metaphysics- was initiated by Aristotle some five cen­turies Be and perfected by the scholastics; in a very outstandingand systematic way by Francis Suarez. The philosophia de ente isprevious to any science in the sense that its language and con­ceptual apparatus do not presuppose that of any science. This isdue to two facts: (1) Its subject matter is being, the most generaltopic ofall; and (2) being is what is first given to the understand­ing. I take this last sentence to mean that the basic structure andunsystematic comprehension of being are given in an experi­ence that does not need to be scientific. For instance, the distinc­tion substance-accident, or the distinction quality-relation, can

78 CHAPTER THREE

be and in fact were established independently of every scientifictheory. They are the result of a theoretical elaboration of expe­riences mainly obtained in the transformation of nature by manthrough labor. This does not mean that there cannot be any feed­back from science to ontology. Clearly, the concept of a system,or that of a relation non-reducible to qualities, are examples ofontological concepts, not belonging to the original corpus of thephilosophia de ente, strongly demanded by contemporary science.But ontology cannot be reduced to any science and has to becultivated with its own methods and proced ures.

Hegel referred to ontology in his lesser Logic as part of thatdiscipline that he labeled 'Metaphysic of the Past'. Yet, whatHegel had in mind here was mainly Christian Wolf's meta­physics, which included ontology as one of its branches, the oth­ers being cosmology, rational psychology and natural theology.Hegel claimed that this metaphysics -which he also charac­terized as a metaphysics of the understanding (Verstiinde)- wasdoomed to failure in its treatment of "infinite objects", i.e. God,the Soul and the World, due to the fact that it intended to treatthese objects with the finite categories of understanding. Yet,Hegel made no objection to its treatment of finite being:

In finite things it is no doubt the case that they have to be char­acterized through finite predicates: and with these things the un­derstanding finds proper scope for its special action. Itself finite, itknows only the nature of the finite. Thus, when I call some action atheft, I have characterized the action in its essential facts; and sucha knowledge is sufficient for the judge. Similarly, finite things standto each other as cause and effect, force and exercise, and when theyare apprehended in these categories, they are known in their fini­tude. But the objects of reason cannot be defined by these finitepredicates. To try to do so was the defect of the old metaphysic.6

I think that the defects that both Kant and Hegel saw (in dif­ferent ways) in Wolf's metaphysics (in his ontology, in partic­ular) cannot be attributed without any further considerationsto Suarez and the previous scholastics,7 but leaving aside theproblem whether Hegel's criticism of Wolfian ontology can be

STRUCTURES AND REPRESENTATIONS 79

extended to scholastic ontology (which I think is a very inter­esting problem), the former remarks about finite being surelyapply quite well to the Aristotelian-scholastic treatment of finiteentia, i.e. to what some philosophers would refer to as the "on­tic" aspects of reality. It is important to make these remarks herebecause the ontological framework that was behind Marx's con­struction of Capital was precisely Hegel's logic. I shall addressthe problem of characterizing Marx's dialectical method below,as well as his relation to Hegel's metaphysics, but it is importantto stress at this point that the theory of measurement I am advo­cating makes use of rather general characterizations of finite en­tia in terms ofcategories of the "understanding", as a "moment"in the whole process oftheoretization.8

The guiding line of our research shall be the problem of theapplication of mathematics to reality. The concept by means ofwhich I will attempt to explicate this application is that of fun­damental measurement. In order to introduce this concept, weshall define an ontological structure as a structure of some species,of the form (A, R I , ... , R n ), that satisfies certain ontological sen-tences </>1, , </> .. such that the elements ofA are real entia and theR j (i = I, , n) are set-theoretic relations among the elements ofA that represent real relations or operations among the same el­ements. If r is a real relation among substances, we say that Rrepresents riffR is a set of tuples of substances related by r. Moreprecisely, let 'Fx) ... Xk' express the fact that the entia Xl, ... , Xk arerelated by r. Then we say that R represents r iff

A numerical structure is a structure like the previous one, ex­cept that the underlying set A is a set of real numbers. Afun­damental measurement of the real quantities, operations or rela­tions rl, ..., rn of or among entia XI> ..., Xm is a homomorfism <.p

from the structure Q{ = (A, R I , ...R n ) into a numerical struc­ture !B = (B, SI> ... , Sn), where A = {Xl, ... , xn} and R; represents rj(1 ::; i ::; n). When such a homomorphism exists, we say that !B

80 CHAPTER THREE

represents 2t, and also that <p represents the relations R,. Thus, thefunction <p:A -+ B represents the relation R, (and so, indirectly,also the relation ri) iff for every (Xl, ..., xn ) E Ri:

A representation theorem for an ontological structure 2t is a state­ment asserting the existence ofa function representing the rela­tions of2t, and also establishing up to what point is that functionunique, that can be derived logically from the ontological sen­tences 1>1> ...,1>,. The clause of the representation theorem as­serting the existence of the representation is called the existencepart; the one asserting the degree of uniqueness is the uniquenesspart.

The role of ontology in the establishment of a rejJresentation theoremconsists of providing the conceptual apparatus required to discuss andformulate the ontological axioms 1>1, ..., 1>,. It will be profitable to il­lustrate this role with an example. This is the contents of thenext section.

3.2.2 A CASE OF REPRESENTATION

The situation is the following. There is a material substance-say a wooden beam- having the shape of a parallelepiped.The height of any of these parallelepipeds is the length of any ofthe segments orthogonal to its bases and enclosed by these bases.According to Francis Suarez, who follows the Philosopher in thisrespect,9 these segments are not imaginary, since they are realbeings in the category of quantity, inhering in the given beam.Our task is to make ontological sense of the measurement ofthelengths of these segments -which we shall call main segments­and their potential subsegments.

Indeed, not any assignment of numbers to the segmentswould count as a measurement of their lengths. The first con­dition that a measurement has to fulfill is that it must assign thesame number to segments of the same length, and a larger num­ber to the longest segment ofany pair of segments. Another req­uisite is that if a segment can be divided into two segments x, y,

STRUCTURES AND REPRESENTATIONS 81

then the numbers assigned to the segment and its parts x, y mustbe such that the sum of the numbers assigned respectively to x,yhas to be equal to the number assigned to the whole segment,i.e. the measurement has to be extensive. A question that natu­rally arises is how fine is the main segment grained, what arethe smallest segments into which a segment can be divided, orwhether the division can continue indefinitely. This raises theold metaphysical problem of the composition of the continuum,a problem which was characterized by Leibniz as one of the twolabyrinths of the human mind. lO This deep metaphysical prob­lem has a direct bearing on the choice of the axioms guarantee­ing the existence and uniqueness of the measurement. Krantzet al. (1971) introduced a regularity axiom for extensive mea­surement that can only be interpreted in two ways: The seg­ments can be infinitely divided (this is the Leibnizian view), orthere is a smallest subdivision d such that the length of any othersubdivision is a multiple of that of d (this is a version of the op­posite view). In general, the opposite view is precisely that thesegments can be divided into a finite number of smallest parts,their lengths not necessarily being multiples of the smallest part.According to contemporary physics, matter cannot be divided adinfinitum, and so it would seem to support the finitistic metaphys­ical view.

The question that arises now is whether the mereologicalstructure of any segment guarantees the existence of lengthmeasurements, assuming that the finitistic view is true. Clearly,since length measurement consists ofcomparing any length witha common length taken as unit, i.e. in determining "how manyconcatenated replicas" of this unit are equivalent to any givenlength, if there is no smallest part, or segment, such that everyother segment is a multiple of the smallest part, then no mea­surement is possible. The only way of measuring the segmentsinto which the height of the segment can be divided is then tobring a unit from outside the segment, such that all those seg­ments are multiples of the given unit. Notice that this presup­poses (i) that there is another material substance such that it has

82 CHAPTER THREE

a line segment of the type required, and (ii) that the segmentsinto which the beam heights can be divided are commensurable.

In the case of the beams, homogeneity considerations makeit plausible that the thinnest beam slice has the same width ev­erywhere, and so in this case the regularity axiom holds with theinterpretation that there is a smallest subdivision d such that thelength of any other subdivision is an integer multiple of that ofd. Notice that the regularity axiom we shall introduce is true alsounder the Leibnizian conception of the continuum. Obviously,the required representation can be constructed just by assign­ing the number 1 to any smallest subdivision d and the numberk to a segment which is equivalent to "k concatenated copies ofd". But this is a rather sloppy operationalistic way of describingthe construction. A proper philosophical treatment requires theintrod uction of more precise conceptual tools.

Consider the set X having as elements a main particular seg­ment, orthogonal to the beam bases and determined by thesebases, as well as all those potential parts of it, i.e. the segmentsinto which it can be divided, down to the smallest segments. Ifx,y are any elements of X, we write x :::: y iff the magnitude ofx is greater than or equal to the magnitude of y. As usual, wewrite x >- y if x :::: y but not y :::: x, and x rv y if both x :::: y andy :::: x. Notice that these relations are independent of the actualcomparison of the segments by any agent, and so they should notbe conceived operationalistically. We define now a direction onthe beam, for instance with respect to the hands, and designatea left and a right on the beam. We introduce now a set Y of pairs(Y, z) E X x X as follows: The pair (Y, z) is in Y iff there is a seg­ment x E X such that x can be divided into y and z, and y is thesubsegment ofx to the left ofz; in this case we writex = yEBz. No­tice that the fact that we choose to determine the direction of thesegment with respect to the hands does not make the direction asubjective entity. A direction is a real ordering among the partsof a body; in the present case, there are two orderings amongthe parts of the segment, each one determined by an extremeof the beam, which is the first element in the corresponding or­dering. In selecting a right and a left in the beam we are only

STRUCTURES AND REPRESENTATIONS 83

chosing one of these two previously existent orderings. The fol­lowing definition introduces axioms which are jointly sufficientto prove the required representation theorem. I discuss theirmeaning and metaphysical justification below.

DEFINITION 1: (X,:::, Y, EEl) is an Aristotelian extensive structure iff

(AI) (X,:::) is a weak order;

(A2) (Congruence) Ifboth x EEl y, Z EEl ware defined, and x '"'J Z,

Y '"'J w, then x EEl y '"'J Z EEl w

(A3) (Dominance) If (x, y) E Y, x ::: z and y ::: w, then there areu, v E X such that (u, v) E Y and u '"'J Z, V '"'J w. Moreover,xEEly:::uEElv

(A4) (Decomposition) If (x, y) E Y and x EEl y '"'J z, then there existu, v E X such that (u, v) E Y, u '"'J x, V '"'J Y and z = u EEl v

(AS) (Associativity) (x, y) E Y and (x EEl y, z) E Y iff (y, z) E Y and(x,y EEl z) E Y; and when both conditions hold,

(x EEl y) EEl z = x EEl (y EEl z)

(A6) (Positivity) If (x, y) E Y, then x EEl y >- x

(A7) (Regularity) Ifx >- y, then there existz,w,u E X such thatz '"'J x, W '"'J y, (w, u) E Y and z ::: w EEl u

(AS) (Archimedean Axiom) Every strictly bounded standard se­quence is finite. We say that

is a standard sequence iff there is a sequence

such thatYi '"'J Yj and (yi,yi+l) E Y (i,j = 1, ...,n, ... ) and,moreover, there is another sequence

84 CHAPTER THREE

defined by Zl =Yh Zn = Zl EEl Zn-l> with Xk rv Zk; it is strictlybounded iff there is an x E X such that x >- Xn for all Xn inthe sequence.

Axiom (AI) asserts that any two segments in X are comparedby the relation :::, i.e. that of any two x, y E X either the lengthofx is greater than or equal to the length ofy, or viceversa. Also,that the relation::: is transitive. It can be seen that this axiom ismetaphysically true.

(A2) asserts the congruence of any two segments whose partsinto which they are divided by two are congruent. The axiom isclearly true.

(A3) affirms that if there is a segment divided into two, whoseparts repectively dominate two segments, then there is anothersegment also divided into two, whose parts are respectively con­gruent to the dominated segments, and which is itselfdominatedby the original longest segment. A little spatial reflection revealsthat this is correct.

(A4) asserts that if a segment is congruent to a segment divisi­ble into two parts, then the first segment can be divided into twoparts which are congruent to the parts of the second segment. Itis easy to see that the axiom is true.

(AS) is obvious because the order in which a segment divisionis given is immaterial.

(A6) is also obvious, because the length of a segment is alwaysgreater than the length of any of its proper subsegments.

(A6) is the regularity axiom discussed above in connectionwith the problem of the composition of the continuum. Far frombeing clearly true -let alone obviously true- it expresses a par­ticular solution to the metaphysical riddle of the nature of thecontinuum. The axiom is true, however, if the line segments in­hering in the bodies are continuous in the sense accepted both byLeibniz and Aristotle (in Physics, Book VI, 23I a 20-23I b 21), or ifthey are composed ofa finite number ofcongruent subsegmentsas discussed above.

The sense of axiom (AS) is that if we have a sequence of seg­ments, the length ofany term in the sequence being greater than

STRUCTURES AND REPRESENTATIONS 85

the length of the previous term exactly by the same difference asthat of any other pair of consecutive terms, and all the terms ofthe sequence are strictly dominated by the same segment, thenthe sequence is finite. Roughly speaking, this prevents the exis­tence of infinitesimal lengths, i.e. of congruent lengths such thatno finite addition of the same -no matter how large- will eversurpass the length ofa given finite segment. The axiom appearsto be also true of real lengths.

These metaphysical assertions are sufficient to prove the fol­lowing theorem.

THEOREM: Let (X,~, Y, $) be an Aristotelian extensive structure.Then there exists a function <p: X --;. R such that, for all x, y E X,

(i)

and

x ~ Y iff <p(x);::: <p(y)

(ii) if (x,y) E Y, then <p(x $ y) = <p(x) + <p(y)

If another function <p' satisfies (i) and (ii), then there exists an a > 0such that for all nonrnaximal x E X <p'(x) = a<p(x).

The proof of this theorem is involved and requires the con­struction ofanother structure, as well as the proofofseverallem­mas concerning this other structure. In order to construct it, Ishall introduce the following definition.

DEFINITION 2: If (X,~, Y, $) is an Aristotelian extensive struc­ture, we let Z be the set of all pairs (x, y) E X x X such that thereare z, wE X with z "'"' x, w "'"' y and (z, w) E Y. If (z, w) E Y, we saythat z $ w is defined.

The required new structure will be constructed as follows. Itshall be the structure (A, ~,B, 0) such that A is the quotient setX / "'"' ofX with respect to "'"', ~ is the relation given by

86 CHAPTER TH REE

B is the set {([x], fy]) : (x,y) E Z}, and 0 is the operation assigningto each pair ([x], fy]) ofelements ofB the element [x]o fy] = [z$w],for some Z "-' x and w "-' y. Notice that "-' in these last two expres­sions is the relation over X; I shall use the same symbols ?:, ~ and"-' in both structures, since the context will prevent any possibil­ity of confusion. The strategy of the proof of the representationtheorem consists of showing that (A, ?:, B, 0) is an Archimedean,regular, positive, ordered, local semigroup. Lemma I below es­tablishes that the relation ?: and the operation 0 are well de­fined. The remaining lemmas establish that the structure sat­isfies the conditions defining local semigroups of the type justmentioned.]]

LEMMA I: Both?: and 0 are well defined.

Proof: First of all, we want to show that if [x], fy] are elements ofB such that [x]?: fy], and x "-' x',y "-' y', then [x']?: fy']. But this isclear, because the given assumptions imply that x' ?: y', by (AI).

Next, we want to establish that the result of the operation 0

does not depend on the particular selection of the equivalenceclass elements, i.e. we want to prove that if[x], fy] are classes suchthat ([x], fy]) E B, and x', y' are elements of X such that x' "-' x,y' "-' y, then ([x'], fy']) E B and [x] 0 fy] = [x'] 0 fy']. The givenassumptions imply that there are elements z, w, z', w' E X suchthat z "-' x, w "-' y, z' "-' x', w' "-' y', z $ wand z' $ w' are defined,[z $ w] = [x] 0 fy] and [z' $ w'] = [x'] 0 fy']. By (A2), we have that[z $ w] = [z' $ w'] and so the desired result follows. 0

LEMMA 2: (A,?:) is a simple order.

Proof: For any [x], fy] E A, either x ?: y or y ?: x (AI), whichimplies that?: is connected in A. If[x] ?: fy] and fy] ?: [x], thenx ?: y and y ?: x, i.e. x "-' y, from which follows that [x] = fy].Finally, if [x] ?: fy] and fy] ?: [z], then x ?: y and y ?: z, whichimplies thatx?: z (AI) and so that [x]?: [z]. 0

LEMMA 3: If([x], fy]) E B, and [x]?: [z], fy]?: [w], then ([z], [w]) EB.

STRUCTURES AND REPRESENTATIONS 87

Proof: Since ([x], [y]) E B, it follows that there are u, v with u "" x,v "" y and u EB v defined. It follows that u ?:: z and v ?:: w, and sothere are u', v' with u' "" z, v' "" wand u' EB v' defined (A3). Thisshows that (z, w) E Z and thus that ([z], [w]) E B. 0

LEMMA 4: If ([z], [x]) E B and [x] ?:: [y], then ([z], [y]) E Band[z] 0 [x] ?:: [z] 0 [y].

Proof: By Lemma 3, ([z], [y]) E B. The desired conclusion followsfrom (A3). 0

LEMMA 5: If ([x], [z]) E B and [x] ?:: [y], then ([y], [z]) E Band[x] 0 [z]?:: [y] 0 [z].

Proof: By Lemma 3, ([y], [z]) E B. The desired conclusion followsfrom (A3). 0

For some reason, the proofofone of the most apparently sim­ple properties of a binary operation, associativity, is rather in­volved.

LEMMA 6: ([x], [y]) E B and ([x] 0 [y], [z]) E B iff ([y], [z]) E Band([x], [y] 0 [z]) E B; and when both conditions hold, ([x] 0 [y]) 0 [z] =[x] 0 ([y] 0 [z]).

Proof: In order to prove the sufficiency part of the biconditionalfirst, assume that ([x], [y]) E B and ([x] 0 [y], [z]) E B. Then thereare x', y', u, z' such that x' "" x, y' "" y, [u] = [x] 0 [y], z' "" zand (x', y') E Y, (u, z') E Y. Also there are x", y" such that [u] =[x]o[y] = [x"EBy"] with x" "" x andy" "" y. It is immediate thatu ""x" EB y" and so (A4) there are' x"', y'" such that x'" "" x", y'" "" y",(x"', y"') E Y and u = x'" EB y"'. Thus, we have both (x"', y"') E Yand (x'" EB y"', z') E Y. Th us (A5), (y"', z'), (x"', y'" EB z') E Y ~ Zand therefore, since [x] = [x"'], [y] = [y"'] and [y] 0 [z] = [y'" EB z'],we have that ([y], [z]) E B and ([x], [y] 0 [z]) E B. The necessitypart of the biconditional is proven in an analogous way.

Suppose now that both conditions hold. By (A4), for anyx, y, z E X there are x', y' and z' such that x' "" x, y' "" y, z' "" zand

([xl 0 [y]) 0 [z] = [(x' EB y') EB z'].

88

Thus, by (A5),

CHAPTER THREE

([x] 0 [y]) 0 [z] = [x' EI7 (y' EI7 z')]

= [x] 0 [y' EI7 z']

= [x] 0 ([y]o[z]). 0

LEMMA 7: If ([x], [y]) E B, then [x] 0 [y] >- [x].

Proof: Assume that ([x], [y]) E B. Then (x,y) E Z and so [x] 0 [y] =[z EI7 w] for z ,....., x, w ,....., y. By (A6), z EI7 w >- z and so [x] 0 [y] =[zEl7w] >- [z] = [x]. 0

LEMMA 8: If[x] >- [y], then there exists [z] E A such that ([y], [z]) E Band [x] ~ [y] 0 [z].

Proof: Assume that [x] >- [y]. Then x >- y and so (A7) there areu, v, z E X with u ,....., x, v ,....., y (v, z) E Y and u ~ v EI7 z. Hence,(y,z) E Z and

[x] = [u] ~ [v E17z] = [y] 0 [zl 0

For any [x] E A we define the expression n[x] as follows. Ifyl EI7 ... EI7 yn is defined and x ,....., yi for any i (1 ~ i ~ n), we letn[x] = [yl EI7 ... EI7 Yn] and say that n[x] is defined. We also let NIx)

be the set {n EN: n[x] is defined}.

LEMMA 9: {n: n E NIx) and [y] >- n[x]} is a finite set.

Proof: By definition, for every n E NIx) there are Yl> ...,Yn E Xwith yi ,....., x for every i = 1, ..., n, such that

n[x] = [yl EI7 ... EI7 Yn].

Thus, to the sequence l[x], 2[x], ..., n[x], ... (which is defined be­cause m E NIx] for every m < n) there corresponds in a one toone fashion a standard sequence XI, X2, ..., xn , ... in X. The con­dition [y] >- n[x] implies that this sequence is strictly boundedand therefore (A8) it is finite, which in turn implies that the set{n : n E NIx) and [y] >- n[x]} is finite. 0

STRUCTURES AND REPRESENTATIONS 89

Proofofthe Representation Theorem: Let (X, :::, Y, EEl) be an Aristote­lian extensive structure, and define a new structure (A, :::,B, 0) asfollows. SetA = Xj"-J the quotient set ofX with respect to "-J; forany [x], [y] E A, let[x] ::: [y] iffx ::: y; let B = {([x], [y)) : (x, y) E Z};finally, if([x], [y)) E B then [x] 0 [y] = [z EEl w] for z "-J x and w "-J y.By lemmas 1-9, (A, :::, B, 0) is an Archimedean, regular, positive,ordered local semigroup. Thus, by Theorem 2.4 in Krantz et. al.(1971), p. 45, there is a function 'lj; from A to R+ such that, for all[x], [y] E A,

(i)

and

[x] ::: [y] iff 'lj;([x)) ~ 'lj;([y))

(ii) if ([x], [y)) E B, then 'lj;([x] 0 [y)) = 'lj;([x)) + 'lj;([y)).

Moreover, if'lj; and 'lj;' are any two functions from A to R+ satis­fying conditions (i) and (ii), then there exists 0' > 0 such that forany nonmaximal [x] E A,

'lj;' ([x)) = O''lj;([x)).

Let <p be the function from X to Ir defined as follows: ifx EX,set <p(x) = 'lj;([x)). Then we have

x::: y iff [x]::: [y]

iff v,([x]) ~ 'lj;([y))

iff <p(x) ~ <p(y).

and also, if (x, y) E Y, then ([x], [y)) E Band

<p(x EEl y) = 'lj;([x EEl y))

= 'lj;([x] 0 [y))

= 'lj;([x)) + 'lj;([y))

= <p(x) + <p(y).

90 CHAPTER THREE

Suppose now that <p' is another function satisfYing conditions (i)and (ii) of the theorem. Let 'ljJ' be the function from A into R+such that 'ljJ'([x]) = <p'(x). Then it is easy to show that 'IjJ' is arepresentation of (A,:::, B, o) different from 'ljJ. Notice that x ismaximal in X iff [x] is maximal in A. Hence, there is a positive 0'

such that for all non maximal x

<p/(x) = 'ljJ/([x])

= O''ljJ([x])

=O'<p(x).

The representation theorem is thus proved. 0

The former demonstrations have provided an example of theconcept of fundamental measurement, i.e. of the proof of theexistence of a numerical structure representing an ontologicalone. Sometimes -as it turns out to be the case in connectionwith abstract labor- the representation is not fundamental, be­cause the structure to be represented is already a mathematicalone. In these cases, when the structure to be represented is itselfrepresenting some real system or things, or is an idealization,the representation is mediated by a mathematical structure. Thistype of mediated representation is very common, e.g., in util­ity theory, where the existence of utility functions is often estab­lished for subsets of the Cartesian space that satisfy certain spe­cial conditions. The elements of the corresponding structuresare therefore not more or less preferable real bundles of goods,but rather vectors representing such bundles. This fact requiresthe introd uction of a concept of representation more generalthan that of a fundamental one. Obviously, the generalizationrequires only that the structure to be represented be allowed tobe any structure, not just an ontological one.

Chapter 4

THE DIALECTICAL METHOD

The aim of the present chapter is to provide a general philo­sophical framework for the reconstructive enterprise that hasbeen undertaken in the present book. Usually, those philoso­phers working in the field of the theory of science, especially inthe foundations of some discipline, have had an empiricist up­bringing and their way ofentering these fields has gone throughattempts to solve problems of empiricist philosophy in connec­tion with scientific knowledge. Yet, by no means the concern withfoundational problems needs to involve a commitment to em­piricist positions. In particular, the framework that I adopt hereis not empiricist, and that will have some impact on the formalreconstruction of MTY.

I want to discuss here Hegel's dialectical method and its crit­icism by Marx, in order to provide my own version of such amethod (if it can be called a method at all) within a more gen­eral philosophical framework. I will show that the way Marxdescribes the dialectical method in the Grundrisse suggests aprocedure to deal with certain aspects of formal scientific ax­iomatic systems. In the first section I will discuss Hegel's dialec­tic, whereas in the second I will consider the problem of Marx's"inverted" adoption of Hegel's dialectical method. My own viewof dialectic I will present in the final section of this chapter.

91

92 CHAPTER FO U R

4.1 HEGEI.:S DIALECTICAL METHOD

What is Hegel's dialectical method? How does it work? In pop­ular literature Hegel's dialectical method is described by meansof the triad: thesis-antithesis-sinthesis. As a matter of fact, Hegelnever uses this terminology, which is due to Fichte, but in theEncyclopedia Logic he asserts that

in point of form Logical doctrine has three sides: (0-) the Abstractside, or that of understanding; (fJ) the Dialectical, or that of neg­ative reason; (1) the Speculative, or that of positive reason. [... ]These three sides do not make three parts of logic, but are stagesor 'moments' in every logical entity, that is, of every notion or truthwhatever. J

This declaration of Hegel's has been taken at its face value bymany, and it is widely believed to provide a general descriptionof the dialectical method, not only as it works in the Logic, but inHegel's thought in general as well. Indeed, the famous "thesis"would be none other than the Abstract moment; the "antithe­sis" would be the Dialectical one; whereas the Speculative mo­ment would be represented by the "sinthesis". Many have beenwilling to read into the quoted paragraph a rigorous formula­tion of the dialectical method, and some have even intended toprovide a formalization of the same. Most notably, Kosok (1966)has tried to show that the Logic can be obtained through a ratherrigorous application ofa formalized version of the procedure de­scribed in the quoted paragraph. Certainly, at least the first triadin the Logic, namely Being-Nothing-Becoming, seems to fit theschema in a very precise way, but as one advances in the Encyclo­pedia (even within the Logic, which is the first part) the transitionsappear to fit less and less such triadic pattern. It is not that sucha departure from the original schema should be surprising. Af­ter all, Hegel warns in the same section (§79) that "the statementof the diving lines and the characteristic aspects oflogic is at thispoint no more than historical and anticipatory". Furthermore,in the third part of the Logic, in the Zusatz to §161, it is said thatthe transition along the three moments is the method appliedin the first part, in the doctrine of Being, while other methods,rather different, are applied in the subsequent two parts:

THE DIALECTICAL METHOD 93

Transition into something else is the dialectical process within therange of Being: reflection (bringing something else into light), inthe range of Essence. The movement of the Notion (Begriff> is de­velopment: by which that only is explicit which is already implicitlypresent.2

As a matter offaet, nevertheless, it must be distinguished whatHegel says about his method from how he actually uses it. As Findlayhas pointed out,

The devices by which the Dialectic is made to work are, in fact, inex­haustible in their subtlety and variety. Hegel admits [as we just saw]that they change systematically from one section of the Dialectic toanother, but the change is much greater and less systematic than heever admits. McTaggart, in the brilliant fourth chapter of his Studiesin Hegelian Dialectic, has gone further in systematizing them than hasany other writer on Hegel, and ifhe has failed to reduce them com­pletely to order, it would be vain for anyone else to hope to succeed.3

What this means is that the dialectical method (which perhapsshould not be called a 'method' at all) is inaccesible to any at­tempt at formalization. It is impossible to provide a codificationof the dialectical transitions (let alone a proof of their "correc­tion") in the style of a formalized logical system (as the one Ipresented in chapter 3). And the reason for this is deep, be­cause the very aim of the dialectical method is to overcome therigidity and isolation of the unilateral conceptions of the Un­derstanding. How could any method succeed in this endeavorif it were to use the typical procedures of the Understanding,of which mathematical logic is the epitome? Whereas the appli­cation of the axiomatic method requires precise concepts withsharply defined boundaries (idealizations), concepts implicitlyand exclusively defined by the axioms and kept apart from otherconcepts into which "they naturally shade, and without whichthey can have no significant application",4 the dialectical methodpretends to be precisely a method of concrete reasoning, i.e. amethod that unifies the isolated concepts by letting them mergeinto other concepts with which they are logically related formingfamilies ofconcrete notions. Indeed, ifno such families naturally

94 CHAPTER FO UR

existed, the chains of definitions we find or construct in sciencewould be entirely artificial and would not reflect, not even ap­proximately, corresponding analogous chains of concrete (non­idealized) concepts. In such a case, it would be altogether impos­sible to connect the concepts thus defined with real phenomena.

This non-formalizable character of the dialectical method isresponsible for the reluctance that many philosophers expressto accept the existence of something like a dialectical logic. Thereasons for this reluctance are compelling, because nowadaysthe term 'logic' is associated with thought proced ures that canbe somehow codified and systematized. But if the dialectical"method" is not a logic in the current sense of the term, howcan we describe it in general terms?

I shall restrict myself to consider the former question in con­nection with the dialectical method as it appears in the (Encyclo­pedia) logic, which is after all the doctrine where Marx obtainedhis own version of dialectic. First of all, it must be noted thatHegel's logic is actually a systematic and thorough presentationof ontology, the kind of discourse that the scholastics put underthe heading of ,metaphysica' or 'prima philosophia'. In fact, Hegel'slogic (ontology) is quite Aristotelian, only that it is presented ina very special way. This way is the path drawn by spirit in its ef­fort to overcome the "contradictoriness" of the concepts it findsin the journey that starts with a consideration of the concept ofpure being. This path ends up with the Idea, which includes as"moments" all the concepts of ontology. Hence, in a sense, theIdea is the logical structure of being, it is the complete system ofontology.

Two notions that appear as crucial in the account of dialecticare those of spirit and contradictory concept. Regarding the lat­ter, it is balling for all those educated in analytical philosophyany talk about contradictory concepts, since in the first place 'con­tradiction' can only be predicated of sentences or propositions. Yet,if we accept that concepts (predicates) can be defined by meansofaxiomatic conditions (as they can), then we could define a con­tradictory concept as one defined in terms of an inconsistent set

THE DIALECTICAL METHOD 95

of conditions. For instance, we can define the concept of a berk­dome in terms of the clause: 'x is a berkdome iff x is a dome andx is round and x is square'. Clearly, the concept of a berkdomewould be then a contradictory one. When Hegel characterizeshis notion of contradictory concept he seems to have in mindsomething like this. The most disturbing tenet of Hegel's philos­ophy is his claim that, moreover, such concepts have instances,that there are actual contradictions in the world. Yet, again, onething is what Hegel says about the existence of contradictions inthought and reality (which sounds like mumbo jumbo) and quiteanother the sense of'contradiction' (Widerspruch) as determinedby his actual use of the term. According to Findlay,

it is plain that he cannot be using it in the self-cancelling manner thatmight at first seem plausible. By the presence of"contradictions" inthought or reality, Hegel plainly means the presence of opposed,antithetical tendencies, tendencies which work incontrary directions,which each aim at dominating the whole field and worsting theiropponents, but which each also require these opponents in order tobe what they are, and to have something to struggle with. 5

In the sphere of thought, in particular, contradictory concepts(in the Hegelian technical sense) appear whenever Understand­ing seeks to give to each of a couple of opposed "reasonable"concrete concepts (like "what is essential" and "what is acciden­tal") "its distinct empire, or when it sharpens or exaggerates ei­ther so as to dominate the whole field and to eliminate its rival".6When a concrete concept is given "its distinct empire" it becomesisolated, cut-off from those other concepts into which it naturallymerges; when it is sharpened or exaggerated it becomes distortedand one-sided. Idealizations are distorted isolated concepts andso are contradictory in the already stipulated sense.? Indeed, itis possible to arrest concepts artificially in this way and to stickindefinitely to them, taking them as the "truth". I shall discussbelow the philosophical conditions of possibility for this to hap­pen, the ontological and epistemological views favored by thosewho stick to these concepts as if they were "ultimate truth".

96 CHAPTER FOUR

The dialectical method is at work whenever the deeds of Un­derstanding are being emended. Dialectic is the art of follow­ing the logical dynamic of concepts, "which determines them tomove forward in certain directions when pushed in unwontedways".8 This dynamic of concepts is not purely a priori, but isgrounded in experience, as these concepts also are, since theywere grasped in the first place as universal kinds exemplified byobjects of experience:

[...] in the actual working of the Dialectic there is a recourse to ex­perience which is simply a recourse to experience, and which is notbased on the demand ofabstract argument. [...JIn casting about forsomething that will serve as an opposite, a complement or a recon­ciling unity ofcertain phases, Hegel has constant recourse to natureand history: he introduces forms that would never have arrived atthrough the abstract development of concepts. 9

As should be plain by now, there cannot be a general formu­lation of dialectic as a "method" ready to be applied to whatevercontent turns up. Yet, this does not mean that there is no stan­dard whatsoever in its application. This brings us to the secondcrucial notion in the account of dialectic, the notion ofspirit. Ac­cording to Findlay,

the lower categories and forms of being really break down becausethey are felt to be inadequate approximations to the sort of self­differentiating unity which is to be found only in self-consciousspirit. This is the secret standard by which all ideas and performancesare judged, and the lubricant without whose secretely applied unc­tion the dialectical wheels and cranks would not turn at all. Any­one who does not feel impelled to think in terms ofthis sort of self­differentiating unity, will not find his inferior categories breakingdown, nor leading him to Hegelian results. 10

Hence, the philosophical pressuposition that makes dialecticwork is the operation of spirit in Hegel's sense. Before I pro­vide some criticism of this notion (in §3 of this chapter), it will benecessary to attempt first a characterization of the same.

Hegel's concept ofspirit -even ifits immediate historical an­tecedents are to be found in Kant and Fichte- is notoriously

THE DIALECTICAL METHOD 97

similar to Aristotle's concept of nous, recovered by the Latintradition as intellect agent. What is new in Hegel is, above all,his endowing of this nous of a creative power similar to that ofGod, except that spirit does not create nature in a conscious way(like Jehovah) but develops pursuing its own telos, which is self­consciousness. The "essence" of spirit is to strive from "the be­ginning" toward self-consciousness, and in its endeavor to doso, but only as a blind step necessary to provide itself with an"other", it creates nature, which is thus a lower manifestation ofspirit. It is only through man that spirit reaches the character ofself-conscious spirit. What this means is that one and the samespirit, intellect agent is present in individual men, which works inthem not just as the "universal in action", that mental power thatdiscovers the universal concept in the objects of perception, en­visaging these objects as instances of a universal substance-kind,but which also possesses the "absolute negativity of the notion".By this term Hegel understands that capability of spirit in menby which it can abstract from any objective content whatsoeverin order to concentrate in itself, to become its own object:

the essential, but formally essential, feature of mind [spirit] is Lib­erty: i.e. it is the notion's absolute negativity or self-identity. Con­sidered as this formal aspect, it may withdraw itself from everythingexternal and from its own externality, its very existence; it can thussubmit to infinite pain, the negation of its individual immediacy: inother words, it can keep itself affirmative in this negativity and pos­sess its own identity. All this is possible so long as it is considered inits abstract self-contained universality.]]

Yet, this is only in its formal aspect, for spirit must continueits journey to become absolute, precisely by following Hegel'sEncyclopedia, which in this manner constitutes the last stretch toperfect self-consciousness. What this means is that Hegel's logic,the philosophy of nature and the philosophy of spirit are effec­tive parts of this grandiosejourney ofspirit toward its telos; spiritmust go across Hegelian philosophy in order to reach its end.

In this sense, spirit is the "truth" of everything, since inthe end everything is spirit in some degree of development

98 CHAPTER FOUR

and everything is made by spirit for the sake of its own self­consciousness. Under this light dialectic appears as the work ofspirit in the last stages of its journey, as the last part of the effortof spirit toward its finality. Only that this work now producesprecisely concepts, whereas previously it had prod uced the worlditself. Dialectic is the activity of spirit in prod ucing the concep­tual material in order to reach, first, the Idea. In the philosophyof nature spirit acquires consciousness of those inferior aspectsof itself previously developed as it moved itself positing natureas a presupposition of conscious mind:

From our point of view mind has for its presupposition Nature, ofwhich it is the truth, and for that reason its absolute prius. In thisits truth Nature has vanished, and mind has resulted as the 'Idea'entered on possession of itself. 12

In the logic spirit obtains consciousness of its own develop­ment "previous" to the creation of nature. This is why Hegelsays that the logic is "the presentation of God as He was inHis eternal essence, before the creation of Nature and finiteSpirit".13 Through the philosophy of spirit, spirit acquires con­sciousness of itself as rnan, as subjective, objective and absolutemind. What this means is that even man is just a stage, albeitthe crowning one, in the way of spirit toward self-consciousness.Individual men are only provisional depositaries of spirit, andtheir creations -the State, art, religion and philosophy- are atbottom creations of self-developing spirit. It is seen, then, whyspirit is "the central notion in terms of which his system may beunderstood".14 The apparent obscurity and difficulty of Hegel'sphilosophy is removed (at least partially) if the concept of spiritas the absolutely fundamental and central notion of his systemis kept in mind:

In terms of this notion many ofHegel's most obscure transitions willbecome lucid: [their point can be seen] when we realize them to beturns on the path leading up to Spirit. 14

We shall proceed now to see how Marx pretended to makedialectic work without spirit and without the possibility of anysystematic codification of the dialectical transitions.

THE DIALECTICAL METHOD

4.2 THE MARXIAN "INVERSION" OF HEGEL'S DIALECTIC

In the Postface to the second edition of C, Marx wrote:

99

My dialectical method is, in its foundations, not only different fromthe Hegelian but exactly opposite to it. For Hegel, the process ofthinking, which he even transforms into an independent subject,under the name of 'the Idea', is the creator of the real world, andthe real world is only the external appearance of the Idea. Withme the reverse is true: the ideal is nothing but the material worldreflected in the mind of man, and translated into forms of thought.

Several remarks are in order here. In the first place, Marx isright in saying that for Hegel the process of thinking (nous) isan independent subject and creator ofthe world. Nevertheless,Marx is not careful to point out that the Idea is only a momentof spirit, which is the only true reality. Be that as it may, his claimthat the ideal is "nothing but the material world reflected in themind of man, and translated into forms of thought" never wasquite developed or clear. In the first place, what is the "ideal"?Is it the "world" of ideas in the minds of men at a certain stageof history? In the second, what is a "reflection in the mind"? Isit true that whatever 'reflection' means, every idea is a reflectionof the "material" world? Unfortunately, Marx never provided afurther development of these very briefremarks and so it is hardto say what he really meant by them.

One of the crucial notions, the notion of matter, is incredi­bly obscure, but it seems to have been devised by Marx to ex­pel from his ontology whatever entity that seemed "mystical" tohim, quite notably the Hegelian spirit. There does not seem tobe any deep philosophical consideration behind this concept ofmatter, but a rather arbitrary instinct, according to which cer­tain entities are to be allowed as existent whereas others are not.More than a metaphysically developed notion, 'matter' seems tobe just a label to attain this purpose. The history of the effortsto develop this embryonic notion of Marx's into a cogent philo­sophical concept is long and involved. Indeed, Soviet philosophysince the publication of Lenin's Materialism and Empirio-Criticism

100 CHAPTER FOUR

exhausted every possible definition and argument trying to clar­ify and defend a Marxist philosophical concept of matter. Unfor­tunately for Marxism, however, these efforts were prey from thevery beginning to unsolvable inconsistencies and dilemmas thatcondemned them to failure. This has been shown in a very de­tailed way by Lobkowicz (1978), who claims that

all such inconsistencies and dilemmas are, in the last resort, dueto the basic paradox of Marxism-Leninism, namely, that it wants tobe a materialism without leaving the heights of Occidental meta­physics which, to Soviet philosophers, is [was in 1963] exemplifiedby Hegel. l5

Marx's conviction seems to have been that there are onlyspatio-temporal entities like bodies, properties and relationsamong these entities, which somehow give rise to society andconsciousness. Marx seems to want to leave out of this pictureGod, the angels, and any property or relation that is not instan­tiated by some body or another, or at least causally connected tothe action of some physical or social object. Clearly, such a viewgets rid, in particular, of the Hegelian spirit.

The first step in the inversion of Hegelian dialectic seems tobe thus the claim that all ideas are nothing but reflections of ma­terial objects (in the given sense) in the mind of man. The term'reflection' leaves open a wide room for conceptions about theway men acquire concepts, but the point seems to be that allthese concepts must somehow refer to material objects, on painof being meaningless.

The second step in the inversion of Hegelian dialectic seemsto be the recovery of the "general forms of motion" introducedby Hegel:

The mystification which the dialectic suffers in Hegel's hands by nomeans prevents him from being the first to present its general formsof motion in a comprehensive and conscious manner. With him it isstanding on its head. It must be inverted, in order to discover therational kernel within the mystical shell. 16

THE DIALECTICAL METHOD 101

Whatever the Hegelian "forms of motion" adopted by Marx,it is clear that these forms of motion, which for Hegel were mo­ments of development of spirit toward self-consciousness, aretaken by Marx in a rather unsystematic way, deprived of anyleading thread. Engels made an effort to present the "laws ofdialectic" in the Anti-DUhring and the Dialectics of Nature but, asElster (1985) has established, Marx never took seriously these ef­forts of his partner. 17 As a matter of fact, "[a]lthough he repeat­edly intended to set out the rational core of the Hegelian dialec­tics, he never got around to doing so". 18 Elster finds in Marx (notin Engels) two rather disconnected strands of Hegelian-like rea­soning. The first is "the quasi-deductive procedure used in cen­tral parts of the Grundrisse and in the opening chapters ofCapitalI, inspired above all by Hegel's Logic". The second is "a theoryof social contradictions, derived largely from the Phenomenologyof Spirit". 18

Elster claims that this theory of contradictions "emerges as animportant tool for the theory ofsocial change", but I shall not beconcerned with it here, since I am more interested in the methodof Capital, which was declaredly inspired in Hegel's Logic. Ifin Hegel's Logic we see the self-development of spirit from ab­stract being to the Idea, in Capital we see the development of theconcept of capital from the production process to the concreteprice determined by supply and demand. The structural analo­gies between the first and the second of these processes havebeen studied by Enrique Dussel in a very careful and detailedway throughout several books, especially in Dussel (1990), whichcontains the exposition that I shall follow here.

According to Dussel, for Marx the first moment of capital iscapital as money, as financial capital ready to be invested in theproduction of commodities. Thus, notice that here capital playsthe role that the absolute played in Hegel's Logic, so that if thefirst definition of the absolute there is 'the absolute is being', inC the first definition of capital is 'capital is financial capital'. Thenegative moment is the negation of capital as financial capital,i.e. 'capital is not financial capital', that is, capital is labor-power

102 CHAPTER FO UR

and means of production, which is the determined negation offinancial capital. The "speculative moment" is the unity of fi­nancial capital, labor-power and means ofproduction: this is theproductive process, which corresponds to the Hegelian categoryof becoming. Daseyn (ens) appears here as commodity, which isthe negation of the production process. The first return-into­self is then the negation of commodity, which is now capital asaccumulation. In this rather ingenious way Marx develops theschema ofhis theory ofcapitalism, proceeding to unfold the con­tent of each of these dialectical moments, in a very detailed way,along C.

The former procedure is quite Hegelian indeed, but there issomething queer in conceiving accumulation as the identity offinancial capital with itself. After all, accumulated capital, even ifit is financial capital, has a greater magnitude than the originalfinancial capital with which the cycle had started. Yet, the purelyHegelian method presents this increase of capital as a develop­ment ofcapital itself, as something produced by capital alone (re­member that the method isjust unfolding the "moments" ofcap­ital). It is at this point that Marx's departure from Hegel's dialec­tic appears, even though (as we have seen) Marx makes use ofHegelian procedures. The departure lies, more than in an "in­version" of Hegel's, in a "breaking of the bottom" of his system(that is, of the one that results from a purely Hegelian dialecticaldevelopment of the concept of capital) by postulating a source(QueUe) of value which lies outside the categories of the system and can­not be sublated by the concej)t ofcapital in any way. This source is noneother than live labor (lebendige Arbeit), the living worker who isbefore anything else a person, a human being.

Hence, it turns out that Marx makes use of seemingly Hege­lian dialectical procedures just to be able to break the bottomof the resulting product. Strictly speaking, this is the most un­Hegelian way of proceeding, since in this form the source whichis postulated cannot be sublated by the totality of the system (the"Idea"), which is to say that somehow it remains isolated. IfDus­sel is right, Marx produced what he thought would be a Hegelian

THE DIALECTICAL METHOD 103

political economy just to introduce an element alien to it as cre­ator of value. In the cycle M - C - M' it would seem that theincrease in the returned financial capital M' is a product merelyof capital itself, of the value of the production means and thelabor-power (represented by the paid wage). Marx point is, onthe contrary, that the secret of such increase is to be found in afactor which lies outside these categories, in live labor.

The systematic expresion of the last two heuristic result is theLaw of Value. It can be said that the whole point of Capital isto prove that, essentially, all the different moments of capital arenothing but live labor, even though they do appear as the self­development of capital itself

Marx tries to show, then, that all the moments ofcapital (its determi­nations: commodity, money, means of production, product, value,surplus value, benefit, price, interest, rent, and so on) are, thanksto the "Law of Value", objectified "live labor", production of valuewhen it is reproduced or replaced; it is creation of value out of thenothingness of capital in the case of surplus value. 19

The whole point of the Law of Value is to show that every typeof benefit (in industry, commerce and, the land) is nothing butsurplus value, unpaid live labor. Now, since Marx -for ethicalreasons- does not want live labor to be sublated by capital, it iskept by him out of the system and so the methodological pro­cedure in the development of Marx's theory of capital becomesinstaurated that surplus value -and hence value- must be de­termined "independently of its form ofappearance" (exchange­value). Since live labor remains isolated from the remaining con­cepts of the system, it cannot be defined in terms of such con­cepts, and so Marx is compelled to introd uce the concept of livelabor independently of virtually every other notion of capital,merely as "labor pure and simple, the expenditure of humanlabor in general". As we saw along the first two chapters, thismethodological decision lies at the bottom of the foundationalproblems of MTV and so we can say that Marx's concept of asource of value, however plausible it might seem as the philo­sophical foundation ofa humanist critical ethic, as the ground of

104 CHAPTER FOUR

the revolutionary overcoming of capitalism, just does not workfrom a logical point of view. Marx got an ethic but not a satis­factory scientific theory of market economies. We can thereforeconclude that Marx's "inversion" or "breaking of the bottom" ofHegel's system is what lies at the basis of his failure to providean acceptable formulation of the Law of Value. I shall considerin the next section whether a new formulation of the dialecticalmethod can be given, one that avoids the shortcomings both ofHegel's and Marx's dialectics.

4.3 A NEW FORMULATION OF DIALECTIC

4.3.1 THE THEOLOGICAL FRAMEWORK

In spite of the brilliant insights obtained by Hegel in the mostdiverse areas of philosophy, his characterization of the "processof thought", nous or the intellect agent as creator of the worldand as unique intellectual principle ("the universal in action")in all men is utterly unacceptable and unsatisfactory. Before anycriticism against it is advanced, however, it must be granted thatit constitutes the most complete and thorough development ofan interpretation of Aristotle's philosophy that arises from a pe­culiar interpretation of On the Soul 111-5, where the Philosopherwrote:

Since in every class of things, as in nature as a whole, we find twofactors involved, a matter which is potentially all the particulars in­cluded in the class, a cause which is productive in the sense that itmakes them all [... ], these distinct elements must likewise be foundwithin the soul.

And in fact thought [i.e. the nous patetik6s or passive intellect], aswe have described it, is what it is by virtue of becoming all things,while there is another [i.e. just nous, what the Latins later called'intellect agent'] which is what it is by virtue ofmaking all things: thisis a sort ofpositive state like light; for in a sense light makes potentialcolours into actual colours.

Thought in this sense of it is separable, impassible, unmixed,since it is in its essential nature activity (for always the active is su­perior to the passive factor, the originating force to the matter).

THE DIALECTICAL METHOD 105

Actual knowledge is identical with its object: in the individual, po­tential knowledge is in time prior to actual knowledge, but absolutelyit is not prior even in time. It does not sometimes think and sometimesnot think. When separated it is alone just what it is, and this aboveis immortal and eternal [... ], and without this nothing thinks. 20

This passage has always been a matter of much debate but itis clear that at least under some interpretation it describes nousin terms which are similar to Hegel's characterization of spirit:nous is active, maker of all things, identical with its object, priorto potential knowledge in men, always thinks (its nature is think­ing), it is immortal and eternal, and no man can think withoutit. It is pretty clear to me that Hegel's system can be profitablyseen as the most serious attempt to rebuild the whole of Aristo­tle's metaphysics upon this conception ofnous. Yet, even thoughHegel was the first to stress the creative aspect of nous, puttingits self-development as the backbone of his system, certainly hewas not the first one in claiming that nous was one and the sameintellectual principle in all men. In point of historical fact, Aver­roes was in the past the champion of this view and, indeed, mostof the motivation of St Thomas Aquinas' effort to reconstructAristotle's philosophy sprang from the need felt by the Chris­tian philosophers to refute Averroes. Averroes claimed that boththe passive and the active intellects were respectively unique andseparated from the human soul, so that there is only one passiveintellect for all men and only one active intellect for them all:my intellects are numerically identical to yours and to those ofany other man. In De unitate intellectus contra Averroistas Aquinasmakes clear that Averroism is repugnant to Christian faith (re­pugnet veritati fidei christianae), asserting that

once the diversity of the intellect is subtracted from us, which aloneamong the parts ofthe soul appears as immortal, it follows that afterdeath nothing of the soul of men remains, except the unity of theintellect; and thus the retribution of prizes and punishments, andtheir diversity, would be supressed. 21

I n this work Aquinas attempted to prove that the Averroistposition is "at least as contrary" to the principles of (Aristotle's)

106 CHAPTER FOUR

philosophy "as it is against faith".2J Yet, his target in this opus­culum is not Averroes' claim that the intellect agent or activeis separated, as his claim that the passive one is so. His argu­ments against the separateness of the active intellect he gives inthe Summa Theologica (la, 79, 4-5) as well as in the Summa contraGentiles (2, 76). Aquinas interprets the De anima passage quotedabove as teaching the individual character of the intellect agentin individual men, and provides arguments to sustain the truthof such a teaching. His main argument, which must also be di­rected against Hegel's notion of the "universal in action" is thatif nous were the same in all men, its functioning would be in­dependent of the will and control of the individual and -sinceits very essence is thinking- it would be continuously thinking;but it is clear by experience that we can pursue or abandon ourintellectual activity at will. Another argument is that a separatedactive intellect would be more perfect that one limited by sensa­tion, and so it becomes hard to understand why such a separatedagent would need the help of the senses to perform its function(which, as the interpretation of Aristotie's passage claims, per­forms anyway previous to the thinking of individuals). More­over, if it can perform it that way, then, being numerically thesame in men or separated from them, how could it get the limi­tation ofrequiring the senses of men to grasp the universal? Thisdifficulties constitute -in my view- very strong reasons to re­ject that interpretation (even if it is what Aristotle really meantto say) and stick to Aquinas' view that the active intellect is notuniversal, but each man has his/her own individual active (andalso passive) intellect as constituting an essential aspect of his/herbeing.

Hegel's way out of the former objections (which, it seems tome, does not answer the one advanced by Aquinas) would beto grant that, indeed, that germ of spirit that self-develops intothe higher forms of consciousness cannot act as nous except throughthe senses of men, that it requires to produce such senses in or­der to become -properly speaking- the universal in action. Inother words, men are necessary instruments of the Idea in order

THE DIALECTICAL METHOD 107

to get consciousness and self-consciousness. As Westphal (1989)has shown, Hegel proves in the Phenomenology of Spirit that self­consciousness is possible iff individual human beings are con­scious of objects. 22 This means that there is no other form forthe Idea to become self-conscious than through experience ofthe world posited by itself as precondition for this to happen.

As can be seen, the prodigious coherence of Hegel's systemmakes it hard to find cracks through which such a disturbingidea of a being that produces its own determinations in orderto think itself can be criticized. As a matter of fact, the Marxistattempt to crack Hegel's system by means of that little poor andconfused notion ofmatter is pitiful. Appeal to the "intuition" thatthere cannot be such a thing as the self-developing Idea is not atall an argument against the solidity of Hegel's system, but rathera renouncement to philosophy, unless we restrict philosophy tothose petty analyses in which empiricism takes so much pride.

For those who believe that there is something valuable in thebasic view of the world afforded by Aristotle, the confrontationwith Hegel's system is unavoidable, as Hegel is one of the mostcoherent Aristotelians that have ever existed. Indeed, basicaUythe only other interpretation is the one provided by (roughly)Aquinas and the scholastics. And this is no accident. As Schellingnoticed in his time quite clearly, the only way to notch the knifeof Hegel's spirit is to oppose to it the sword of the Word of Godas revealed in the Scriptures. In his 1bward a History of ModemPhilosophy, Schelling synthesized in the following form the globalvision of Hegel's philosophy:

God, the Father, before all, is the pure logical concept, which iden­tifies itself with the pure category of being. That God must manifestitself, because his essence includes that necessary process; such rev­elation or alienation of himself in the world is God, the Son. ButGod must sublate or bring back on himself that alienation: it is thenegation of his pure logical being: negation that is accomplishedthrough humanity in art, religion and fulfilled in philosophy; thathuman spirit is equally the Holy Spirit, by which God takes for thefirst time consciousness ofhimself. 23

108 CHAPTER FOUR

Quite against Hegel's pantheism of a nous that thinks itself,Schelling claims in his Philosophie der Offenbarung that the Cre­ator is previous to being, and is beyond, being an other-worldlyreality. Perhaps to prevent the revealed concept of God fromgetting confused with the absolute defined by the motion of theHegelian categories, Schelling says that God is not being, butrather the creator of being and his Lord, the Herm des Seins. 24

This view that the creator of being is not itself (or Himself) be­ing appeared for the first time in Plotinus.25 It was also for thefirst time refuted by St Augustine, who asserted that "the creatorof being is".26 Moreover, according to the same Revelation thatSchelling is appealing to, the very name of God is 'being':

[... ] God said to Moses, "I AM WHO I AM" [... ] This is My nameforever, and this is My memorial to all generationsY

Schelling seems to have thought that that move was required(even though in fact it is not) in order to avoid -among otherthings- the unacceptable consequence that the human spirit is"equally the Holy Spirit". Most certainly, according to Revelation(and experience), the human spirit is not the Holy Spirit. More­over,. contrary to what Hegel's system seems to imply, namelythat "God" (the Absolute Spirit) is a sort of rational result whichspeculative philosophers can reach by following Hegel's systemup to its consummation, the Scriptures teach that even the leastintellectual men can be saved, because salvation has nothing todo with merits, be they intellectual or otherwise, but with faith.According to Revelation, it is only by accepting Christ Jesus asLord and Savior that men can be saved. And only those who areredemeed in this way can be regenerated in their spirit, in par­ticular in their intellect, as a necessary condition to understandheavenly matters. Hegel seems to suggest that heavenly matterscan be understood by speculative methods alone, thus makingof faith an unnecessary accessory.

Kaufmann claimed that Hegel

tried to do from a Protestant point of view what Aquinas had at­tempted six hundred years earlier: he sought to fashion a synthesisof Greek philosophy and Christianity, making full use of the laborsof his predecessors.28

THE DIALECTICAL METHOD 109

Unfortunately, if that is what Hegel was trying to do, he failed,as can easily be seen from the Scriptures. His claim that revealedreligion and speculative philosophy have the same content, ex­cept that the first uses "pictorial language" is just not correct.Speculative philosophy cannot be taken as a good elucidation ofthe Bible. Another consequence of Hegel's notion of spirit is thesuggestion that salvation is a political endeavor, since it seems toconsist merely in the emergence of spirit in the form ofa "Chris­tian" community and an ethical State, something which seemsto depend only on the will of that supraindividual entity. Thisclearly goes against the Christian teaching according to whichsalvation is a purely individual endeavor, and has as result eter­nal life for the individual that has been saved, even though theredeemed constitute a godly people in the form of visible com­munities. Very much in an Averroist fashion, Hegel's philosophydoes not make room for etemallife for the individual, but onlyfor spirit. No matter what Hegel might have said on the con­trary, what else beside this supraindividual spirit could remainafter the death of the individual men?

Avineri (1972) clearly shows how Hegel ascribed to the polit­ical situation of the decadent Roman Empire the origin of thecategory of individual as required by Christianity to prosper asa massive religion. According to Hegel, Christianity could onlythrive in a historical situation in which men were deprived oftheir political rights by the Emperor, being reduced to mereholders of private property. In contradistinction to the ancientpolis, in which the identity of the citizens was the State and sothey were ready to die for it, in decadent Rome the citizen wasable to identify himselfonly with his own property -a very finiteand transient thing- and that is how the fear of death arose.29

This is very suggestive of the role Hegel attributes to the ethicalState, which is something like a very earthly realization of theKingdom ofGod.30

The idea that the Kingdom of God has to be realized histor­ically on earth like a political institution seems to have inspiredMarxist communism. Once deprived of the concept of Geist and

110 CHAPTER FO UR

inverted, what remains of Hegelian philosophy (Marxism) cando without the "pictorial language" of religion and go all theway to claim that it is necessary to realize on earth somethinglike an analogue of the Hegelian heavenly kingdom, somethinglike the Kingdom ofGod, but inverted. If Hegel's philosophy ofmind reduces the worth of the individual to that of being a (pro­visional) bearer of spirit, in Marxist philosophy the worth oftheindividual is reduced to nothing, since every allusion to "mys­tical" entities and other-worldly realities is out of the question.This is the very essence ofTotalitarianism, because a logical con­sequence of this inverted Hegelian view is that there cannot beanything more valuable than "the totality", i.e. a purely humanState, a State that is not accountable to anything else, not evento Absolute Spirit!

Against this despicable view of man, and unless we are pre­pared to step again on the slippery slope to Gulag and Buchen­wald, it is necessary to assert the integral individuality of the hu­man soul, including the intellect, and the accountability of allmen (and the State) to a transcendent Holy God (the God, I be­lieve, ofIsrael and Christ Jesus). This is the only absolute foun­dation for the infinite worth of each individual man or woman.This is also the absolute foundation of religious freedom, be­cause the conception of the human spirit that it involves makesroom for all faiths, as well as for those who have not had any reli­gious experience by which they should at all come to believe. Inmy view, this is the theological presupposition for a fresh recon­sideration of the insights and novelties introduced by Hegeliandialectic.

4.3.2 DIALECTIC REVISITED

The central notion of Hegelian dialectic, the "lubricant with­out whose secretely applied unction the dialectical wheels andcranks would not turn at all", the concept of Geist, is not accept­able on theological and metaphysical grounds. Yet, even thoughno "interpretation" of Hegel is acceptable that does away with

THE DIALECTICAL METHOD III

Geist, there is no doubt that Hegel has been one of the greatestphilosophers ever and that he has very important results and il­luminating insights. How can we make use of these results with­out falling back into Geist? In a very strict sense, this is impossi­ble, because these results cannot be understood properly out ofthe Hegelian system, in which they belong. The most we can dois use them as suggestions to build other philosophical theories,without pretending that they are Hegelian except in the widesense in which Haydn or the young Beethoven can be said to beMozartian. Certainly, Hegel's theory of the State,3J his accountof civil society,32 his ethics,33 his critique of Kant's conception ofthe object coupled with his own view,34 his overcoming ofskepti­cism through a brilliantly argued epistemological realism,35 areall magnificent philosophical masterpieces even though Geist isquite an indigestible item.

The method that I propose to recover whatever can be recov­ered from Hegel's valuable theories is to substitute 'human ac­tive intellect' for Geist whenever the context indicates that Hegelis speaking of the human mind, and see what turns Up.36 It isobvious that many things will have to go but also many insightscan result. It will be interesting, for instance, to see what turnsup if this procedure is followed in Kenneth R. Westphal's recon­struction of Hegel's argument for epistemological realism in thePhenomenology of Spirit. Even though this argument is essential toshow the inadequacy ofempiricism as a theory of knowledge (ac­cording to Hegel, empirical knowledge or "sensuous conscious­ness" is the roughest and most elementary form of knowledge),I shall not be concerned with it here so much as with Hegel'stheory of universals as substance-kinds.

According to Kant, the objects that we find in (sense) expe­rience are unities constructed in the transcendental subject, inapplying the categories to the manifold of intuitions. Hence, itdoes not make sense to talk about objects as they are "in them­selves", since the unity and structure objects have is nothing butthe result of the synthesizing activity of the understanding. Ac­cording to Kant, the categories applied by the understanding

112 CHAPTER FOUR

are a priori and definite in number. But there is only one stepfrom here to consider all concepts as playing a role in the syn­thesizing activity of the transcendental subject. This leads to aconceptual relativism that makes dependent the nature of ob­jects on the peculiar conceptual apparatus that we have availableat a certain moment. Clearly, since these objects are somehow acreation of the concepts we happen to have, the finer the con­cepts the sharper will be our objects of experience. It is clear thatto a theory of knowledge of this kind the problem of the rigidityof the conceptions that Hegel called "of the understanding" justdoes not arise. Those who maintain this view are thus able tostick indefinitely to determinations which ,otherwise are clearlyabstract and idealized.

The demand for extreme standards of rigour in science andphilosophy, usually coupled with deep contempt against any no­tion or procedure not amenable to logico-mathematical treat­ment, and a thoroughgoing (if sometimes clandestine) attach­ment to this Kantian view of the object, or to an empiricist one,springs from a reluctance to deal with the difficulties of dialec­tical thinking, to accept the possibility of a mode of thought notcontrolable by mechanical or standard devices. Indeed, someparoxistic versions of this view even postulate as a main taskof philosophy the "clarification" of concepts, meaning by thisthat philosophy should increasingly get rid of all those "vague"and "imprecise" concepts, replacing them by their correspond­ing "elucidations", which are (and can be) nothing but abstractidealizations of concrete, albeit harder to handle notions.

Since barely only (if any) the inert or relatively sim pIe objectsusually studied by physical science can be adequately thought bymeans of precise concepts, or they can be thus conceptualizedwithout too much distortion, usually the philosophers of scienceassociated to this view tend to concentrate almost exclusively inphysics, and, since the objects and concepts of the social sciencestend to be inerradicably vague, there is a tendency to neglect thetreatment of these sciences or to dismiss them as utterly "unsci­entific".

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The so-called "problem ofincommensurability", for one, is an(undesirable) consequence of this view of the unity of the object.This problem arises for transcendental idealism because it can­not see the objects but as creations of the intellect through givenconcepts. Thus, objects constructed with different concepts mustbe different objects as well. In this form, transcendental idealistphilosophers of science become barned by the outcome that spe­cial relativity and classical mechanics have no objects in common!But everybody else knows that many of the phenomena (strictlythe same) treated by one theory are also treated by the other,even though there may be some differences in the way they aredescribed.

The problem ofincommensurability is a Kantian problem, be­cause it arises from the attempt to represent the scientific phe­nomena in terms of purely idealized concepts, pretending thatit is impossible to refer to them in a way independent of theabstract theory in question, namely in terms of the substance­universals they exemplify (if they have a real existence at all).The problem of incommensurability disappears once we rejectthe Kantian view as untenable and realize that not all conceptshave the same status. This is one of the central tenets of Hegel'sepistemological realism: both the rejection of Kant's conceptionof the object and his claim that the unity individual objects havedoes not depend upon the unity of apperception (of the sub­ject), but is due to the very nature of these objects, which existas instances of indivisible substance-universals:

[... ) according to Hegel, objects are not in fact mere 'combinations'of sensible properties, as the Kantian model suggests, and on whichhis doctrine of synthesis depends. Instead, as we shall see, Hegelargues that individual objects exist as manifestations of indivisiblesubstance universals, which cannot be red uced to a set of propertiesor attributes; he therefore holds that the object should be treatedas an ontologically primary whole. As a result Hegel adopts a meta­physical picture which enables him to argue that the object formsan intrinsically unified individual: because the individual is of suchand such a kind (a man, a dog, a canary) it cannot be reduced toplurality of more basic property-universals, while it is the universal

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that confers this substantiality upon it. In this way, Hegel replacesKant's 'bundle' model of the object with a more holistic picture,which treats the individual as a unity, in so far as it exemplifies asubstance-kind. It is this ontology ofsubstance which lies behind hisrejection of the latter's doctrine of synthesis.37

It is in this sense that substance-universals can be said to be"structures in the world", and so whenever one grasps one ofthese universals one is grasping the nature ofany of its instances.I claim that these substance-universals are the less idealized con­cepts available, even though some of their instances are "truer"than others. The status of substance-universals is quite differentfrom the status of idealized concepts. The concept of homo oeco­nomicus, for instance, is not a substance-universal and, in fact, itdoes not even have instances. It is a fictitious entity that behavesall of the time and exclusively in a form in which human beingsbarely do sometimes under very special circumstances.

Even though 1 do not believe that there is an entity that positsitself as being and then begins to engender the categories ofHegel's logic (ontology), 1 still believe that that ontology is cru­cially important because it provides the foundation for the doc­trine of the notion, in which a non-Kantian theory of essentialpred ication and the object emerges, as Stern (1990) has clearlyshown. 1 propose to see the logic -and dialectic in general- asthe development ofcategories that break down from the point ofview of the human intellect just because they are inadequate inview of unwonted experiences and/or an implicit undevelopedview of a totality. From this point of view the "I dea" is not Godbut only a conceptual construction that constitutes the systemof ontology, the "logical structure of the world in its relation toGod". What I suggest is that what really moves the dialectic isthe desire intrinsic to the human intellect to reach systematicand concrete totalities of thought, to put isolated notions in awider unifying context, to relate such notions -and the total­ity as well- to factual reality. Hegel's dialectic is appealing andimmortal as a cantata of Bach's because it constitutes one of thegreatest efforts to fully satisfy this drive. This very drive of the

THE DIALECTICAL METHOD 115

intellect toward concreteness and system is witnessed in actionin the tendency to get rid of idealizing assumptions in economictheory. We shall see and explain in some detail this dialectic inthe final subsection of this chapter.

4.3.3 THE DIALECTIC OF THE THEORY OF VALUE

As we saw in §2, Marx never got around to set out the "rationalcore" of the Hegelian dialectic. Marx was always very brief in hiswritten declarations about the nature of the dialectical method.As it was correctly pointed out by the late Professor Jean vanHeijenoort,

Marx has not bequeathed us explicit teachings on dialectic compa­rable in extent and precision to his economic doctrines. On dialecticthe great theoretician left, on the one hand, fragmentary formula­tions scattered in his works and correspondence and, on the otherhand, the product of his dialectical method, the monumental Cap­ital. We have the fruit of the method, but no systematic expositionof the method itself38

There is no doubt whatsoever that Marx thought of doing suchan exposition. For instance, in a letter addressed to Engels, datedJanuary 14, 1858, Marx wrote the following:

If there should ever be time for such work again, I should greatlylike to make accesible to the ordinary human intelligence, in twoor three printer's sheets, what is rational in the method that Hegeldiscovered but at the same time cloaked in mysticism.

"Unfortunately -adds Van Heijenoort- Marx died withouthaving written those two or three printer's sheets which doubt­less would have forestalled many subsequent discussions".39 Thisdoes not mean, however, that Marx did not write anything at allabout dialectic. The work where he explained himself most inthis respect was the Foundations of the Critique of Political Econ­omy, also known as the Grundrisse, written in 1857. Section 3 ofthe Introduction, labeled "The Method of Political Economy" isparticularly important in this connection, since it is there whereMarx characterizes the dialectical method in the most explicit

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way, namely as a method that consists in "rising from the abstractto the concrete". Regarding this paragraph, Van Heijenoort saidthe following:

[the same] represents in my opinion the most important method­ological document we possess to fill the void left by the absence ofthose "two or three printer's sheets" on dialectic which Marx neverhad the leisure to write.40

This document represents the most explicit statement of thequasi-deductive deductive procedure used in the Grundrisse itselfand the opening chapters of C but, as we saw in §2, Elster claimsthat there is yet a second strand of Hegelian dialectic in Marx'sthought, namely a theory ofsocial contradictions largely derivedfrom the Phenomenology of Spirit. Thus, at the very least it is clearthat such document is the most explicit presentation of one ofthe strands of Hegelian dialectic in Marx's thought, and hence itis possible to argue that what legitimatelly must be understood by thatstrand of Marx's dialectic is precisely the contents of such document.

As Hamminga (1990) has shown in a very detailed way, themethodology that Marx pretended to apply in Capital can benaturally seen as a dual motion that builds idealized models andthen proceeds to eliminate the restrictive assumptions definingthem, in order to yield less idealized models, maintaining at thesame time the validity of the original fundamental laws evenwithin these less idealized models. We shall also see that this dualmotion can be analyzed in terms of the concept of rising or pass­ing to the concrete.

I devote the first part of this subsection to propose a new in­terpretation of the process of "rising to the concrete", in logico­mathematical terms, trying to stick as far as possible to the text of"The Method of Political Economy"; to that effect, I use an easyexample taken from classical physics. In the second part I takeas point of departure certain results in Nowak (1980), and thehistory of Marx's MTV, to propose an elucidation of the afore­mentioned dual motion as well as to interpret the situation ofthis theory previous to the developments introduced in the sub­sequent chapters of this book. In the third part I discuss the

THE DIALECTICAL METHOD 117

dialectical method again, in connection with this particular in­terpretation of it. I close this chapter arguing that Marx in factfailed to apply correctly this method, due precisely to his ethicalreluctance to sublate the concept of living labor in the categoryof capital, which makes that concept into an abstract, isolatedone.

4.3.3.1 DIALECTIC AS MODEL CONSTRUCTION

In the already mentioned "The Method of Political Economy"4JMarx characterized the dialectical method -in a very Hegelianfashion, as we can see- as the one that, departing from ab­stract determinations, "rises to the concrete" (vom Abstrakten zumKonkreten aufzusteigen) reproducing the real concrete in the pro­cess of thought (im Weg des Denkens) as a concentration of multi­ple determinations, as unity of the diverse (Zussamenfassung vielerBestimmungen, Einheit des Mannigfaltigen). In the sections previ­ous to the one mentioned, Marx applied this method -whichhe considers as "the correct scientific method"- to elaborate aneconomic discourse in which the concepts of production, distri­bution, exchange and consumption are presented as "the artic­ulations ofa totality, differentiations within a unity".42 This dis­course is interesting but its incipient character does not make itapt to illustrate the application of the dialectical method that Iwant to present in this section. According to this application, theproced ure of passage from the abstract to the concrete consists,fundamentally, in the construction of singular models of givenscientific theories in order to represent determined real concretesituations. I will illustrate what I mean in what follows by meansof a relatively simple example taken from classical mechanics.

As a case of "reprod uetion of the concrete in the process ofthought" (Reproduktion des Konkreten im Weg des Denkens), considerthe problem of constructing a physico-mahematical representa­tion of the phenomenon of free fall of a steel sphere which isleft to fall from the roof of a building. In this case the real con­crete can be identified with the motion of the body since the

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moment in which it starts falling until it hits the sidewalk forthe first time. The "abstract determinations" (abstrakten Bestim­mungen) which serve as point of departure are the following.

(I) A set P having as a unique element the center ofthe steelsphere, which is said to be a material point or particle, i.e.it is assumed that the mass of the body is concentrated inits center.

(II) The concept of time of motion of the particle, which isidentified with a closed interval of real numbers T =[0, t'], where t' is a parameter that measures the num­ber of seconds that the motion lasts.

(III) The concept of instantaneous position, which is a func­tion r : P X T -. R3 assigning to the particle and each in­stant t ETa vector in the linear space A3

• It is usual andconvenient in this case to identify the unidimensionalsubspace oflR3

, generated by the unit vector k = (0,0, 1)(the z-axis), with the straight line determined by the pointwhere the falling starts together with the point on thesidewalk hit by the sphere. It is also convenient to assumethat this last point is the origin of the space determinedby the straight line on which the falling takes place andthe plane through the origin orthogonal to this line.

(IV) The concept of mass of the particle, which is representedby a positive real number rn.

(V) The concept of acceleration due to terrestrial gravity,which is identified with the constant number g, andwhose value depends on the point of the Earth whereit is measured, but which in the present case we can taketo be equal to 980 cm/s2

•

(VI) The concept of forces exerted on the particle, which areidentified with vectors in the linear space IR3

• Apart froma mutiplicity offorces whose resultant is small and negli­gible, it is usually assumed that there are two forces act­ing on a freely falling body, to wit, the force of gravityand the resistance of the air. The available information

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about the phenomena of free fall suggests, however, thatin this particular case it is even possible to neglect the airresistance, by which the number of forces to consider isreduced to one, which is none other than the weight ofthe sphere, and hence mg.

(VII) The definition of the concept of acceleration as the sec­ond derivative of the position function with respect totime, i.e.

d2ra = dt 2 •

(VIII) Newton's Second Law, which asserts that force is equal tomass multiplied by acceleration:

F = mao

(IX) Concepts and techniques pertaining to differential andintegral calculus, as well as to linear algebra.

Out of (I)-(lX) equation

(1)

is obtained and thereafter, by means of (IX), the function r asan explicit function of t is obtained by integrating out the righthand side member of (1) divided by m:

(2)

Expression (2) is still undetermined, since the specific values ofVo and ro, which represent respectively the initial speed and thedistance from the sidewalk to the roof, have not been establishedyet. These values are not abstract determinations, but rathermeasurements pertaining to a concrete situation. Nevertheless,they are essential to obtain the numerical determined represen­tation of the phenomenon subject to study. What this means is

120 CHAPTER FOUR

that beside the "abstract determinations" empirical data aboutthe situation subject to study are necessary in the construction ofsingular models. An examination of this situation and appropri­ate measurements reveal that the initial speed Vo is zero, whereasthe distance ro from the place where the body begins to fall to thesidewalk is (say) 10 m. Using these data the following function isobtained in perfectly explicit determined and concrete terms:

(3)

Adding one more empirical datum, the value of the mass of thebody (let us say one kilopond), we get the relational structure

llJ = (p,T,r,m,f,g),

where P is the singleton whose unique element is the sphere; Tis the interval [0, (2000jg)l/2];13 r is the vectorial function givenby (3); m = 1 kp; f (=F) is the vector (-980· 103)k, which rep­resents a force of 980 thousand dynes directed toward the side­walk; and g is the adopted value of terrestrial gravity. The struc­ture llJ is a physico-mathematical representation of the situationwe have been considering; it is a "totality of thought" (Gedanken­totalitat) which, in fact, reproduces the real situation as a "con­crete of thought" (Gedankenkonkretum). Departing from abstractdeterminations and empirical data, we have in effect reachedthe reproduction of the concrete real-the falling of the spherefrom the roof- in the process of thought, that is to say, of em­pirical and mathematical reasoning, as a synthesis of multipledeterminations, as unity of the diverse. This synthesis or con­centration of multiple determinations, is none other than therelational structure llJ, whose unity is the unity correspondingto an object, pertaining to the universe of set theory, that satis­fies certain special conditions (laws); i.e. it is a structure in thesense defined in chapter 3.

THE DIALECTICAL METHOD 121

In general, in this specific application, the dialectical methodof passage to the concrete consists in constructing singular mod­els of real situations out of abstract concepts and empirical data,by means of laws pertaining to a determined scientific theory,which is assumed as something previously given. In the exam­ple given the model represents a real situation but in some cases-as we shall see in the next section- it is useful or necessary toconstruct very idealized models not having a real counterpart,except in the sense that they are a quite remote approximationto a real situation.

Before passing to the next section, perhaps it will be conve­nient to say a few words about the sense that the terms 'abstract'and 'concrete' have received in the present context. Sometimesthe term 'abstract' is taken as a second intention term to refer toobjects such as concepts, propositions, sets, or numbers, i.e. whatsome authors call 'abstract entities'. According to this use of theterm 'abstract', the term 'concrete' is used to designate just thatwhich is not abstract, i.e. real beings. It must be clear that Marx'susage of these terms does not follow those conventions. Accord­ing to the interpretation offered here, abstract determinationsare actually for Marx abstract entities (in the current sense nowa­days), since they are objects such as propositions and concepts.Nevertheless, Marx distinguishes the "concrete real" from the"concrete thought": The concrete real is a real being (and henceconcrete in the current sense nowadays), whereas the concretethought is a conceptual structure (and thus abstract in the cur­rent sense). For Marx -as for Hegel- the distinction betweenthe abstract and the concrete, in the plane of thought, is a dis­tinction between degrees of complexity and articulation. For in­stance, the concept of mass is abstract with respect to the modelsof classical mechanics, because it is a concept that by itself doesnot provide a complete and finished comprehension ofany par­ticular reality, i.e. it is being taken in an isolated way, without itsbeing specifically articulated in any of the totalities-of-thoughtthat correspond to it (the models of the theory to which it be­longs).

122 CHAPTER FOUR

4.3.3.2 DIALECTIC AS THEORY CONSTRUCTION

According to Leszek Nowak, the Marxian method of passage tothe concrete coincides with what he calls "method of idealiza­tion":

The method which is able to reveal inner connections is called byMarx the method of proceeding from the abstract to the concrete,i.e. it is the method of idealization according to my interpretation.44

According to Nowak the method of idealization consists in thepostulation of a series of idealized (false) assumptions about theobject under study, and in showing that within these assump­tions a particular case ofa scientific law is satisfied. As an exampleof an application of this method, Nowak proposes the formula­tion by Marx of the Law of Value in Capital as an "idealizationalsentence", i.e. as a counterfactual conditional of the form:

(Tk) If G(x) /\pJ(x) = 0/\ ... /\Pk-l(X) = 0 /\Pk(X) = 0

then F(x) =Jk(HJ(x), ...,Hn(x».

"where G(x) is a realistic assumption whilepJ(x) = 0, ... ,Pk(X) = 0are idealizing assumptions (k > 0)".15 In the case of the theory ofvalue Nowak says that the propositional function G(x) defines the"universe of discourse" of the theory, whereaspl -Pk (fork = 8)are simplifying conditions that define an ideal economic system,similar to the "ideal gases, perfectly rigid bodies, and other con­structs of the type".16 According to Nowak, Marx's method inCapital consisted precisely in the construction of a sequence ofconditionals r, T 7

, ••• AT2, that departing from r eliminated

the idealizing assumptions until it obtained a conditional T 2 inwhich only two simplifying assumptions remain and whose con­sequent is a modification (in fact a generalization) of the originalversion of the Law of Value. The conditional AT2 asserts that ifthe antecedent is satisfied then the Law of Value holds with a cer­tain degree ojapproximation, so thatAT2 is a weakening ofT2

• Thisprocedure is called by Nowak 'concretization'.17

THE DIALECTICAL METHOD 123

We shall see that the proced ure of concretization is a particu­lar cases of passage to the concrete, but at this point it is impor­tant to prevent the terminology from confusing us. Irrespectiveof the terminology that is deemed as more adequate (I shall dis­cuss this point later), it seems to me that the procedure Marxattempted to follow in Capital, as well as the recent history ofMarx's theory ofvalue, can be adequately described as a processquite similar to what Nowak calls 'concretization'. In terms ofmodels and structures, what Marx did in Capital was to describea series of more or less idealized models and to presume thatthe so-called Law of Value (never formulated precisely) shouldbe satisfied in such structures. I say that Marx presumed such athing because, certainly, he never proved that the mentioned lawwas satisfied in any of those models (in fact it can be proven tohold under certain conditions, as we saw in chapter 1, in the pro­totype ofMTV). But not even in the prototype, which is the mostidealized of them, would have been Marx able to prove the Lawof Value, since the explicitation of the assumptions required forsuch endeavor required a development of the conceptual appa­ratus of mathematical economics that did not take place until themiddle of the current century, thanks to the work ofSamuelson,Leoritief, Arrow and Koopmans. 48 As we saw in chapter 2, mak­ing use of such conceptual apparatus Michio Morishima (1973)provided the first complete listing of the assumptions requiredin the derivation of the Law of Value -thus defining preciselythe linear model of the theory of value, also known as the Leon­tief model or the prototype of MTV- and effectively derivedthe Law of Value from these assumptions, in a way similar tothe one we followed in chapter 1. Taking as point of departurethis model, Morishima developed the totality of Marx's economictheory in a rigorously scientific way, although of course withinthe limitations imposed by the very idealized asssumptions uponwhich the proofs concerning the existence ofunique positive val­ues are based. As we saw in chapter 2, these assumptions (most ofwhich were taken for granted by Marx himself in Capital), were

124 CHAPTER FOUR

severely criticized by Morishima in the same book, whose con­clusion is that Marx's theory of value (i.e. what really is the pro­totype of the same theory) should be abandoned and replacedby a theory combining aspects of (the linear model of) Marx'stheory and of that of Von Neumann.

In the terms already discussed in the second chapter, Mor­ishima himself (1974) addressed the problem of constructing amodel of a "new" theory that reinterprets Marx in Von Neu­mannian terms. Unlike the classical model, this model allows abetter treatment of the problems related to the age structure ofcapital goods when the time factor is introduced (which seemto be unsolvable within the classical model), and admits jointproduction as well as choice of techniques. Morishima was ableto prove both the Law of Value and the Fundamental MarxianTheorem within this Von Neumannian model using "optimal"instead of "real" values. If real values are obtained in the linearmodel computing the contents of labor incorporated into thecommodities on the basis of the prevailing technical coefficients,optimal values are shadow-prices determined by a linear pro­gram which is dual to another linear program for the efficientutilization of labor. Even though optimal values are not neces­sarily unique, the exploitation rate is well defined and -as Ipointed out before- the Fundamental Marxian Theorem canbe proven, under the assumption that labor is homogeneous.We also saw in chapter 2 that, along the same lines, John E. Roe­mer (1980, 1981) prod uced a series of more general models andderived the existence of Marxian equilibria from the assump­tion that the behavior of the firms consists of maximizing profitsgiven a set of possible processes of production and certain re­strictions in the availability of capitals. In Roemer's models thevalue of a bundle of commodities is defined as the minimumlabor required to produce the bundle, given the technologicalpossibilities ofthe economy. It is thus seen that Roemer's defini­tion of value is analogous to that of Morishima's, the differencelying in that according to Roemer the values are not necessarilydetermined by a linear program. In Roemer's models the ex­ploitation rate is well defined for each production process and

THE DIALECTICAL METHOD 125

a more general version of the Fundamental Marxian Theoremcan be proven within them. Nonetheless, the problem of trans­lating values to prices -the so-called Transformation Problem­becomes an unsolvable problem and the Law of Value cannot beproven, which leads Roemer to conclude that Marx's theory ofvalue is not a theory ofcommodity exchange. In Roemer's mod­els, as well in the one built by Morishima on Von Neumann's, therole of the exploitation ofworkers, as a condition for the growthor reproduction of the economy, is well clarified. The models ofRoemer are fairly general but they are still based upon the ide­alizing assumption that labor is homogeneous. We shall see insubsequent chapters how can we get rid of this assumption aswell.

4.3.3.3 THE DIALECTICAL METHOD IN AXIOMATIC SYSTEMS

Once we have had the opportunity to observe the dialecticalmethod operating in the sphere of axiomatic systems throughsome examples, it will be profitable to discuss in general its pe­culiar way of functioning in these cases, trying at the same timeto eliminate some confusions in the terms in which it is oftendescribed, i.e. terms like 'rising to the concrete', 'idealization','concretization', 'general', 'particular' and other related terms.Throughout this work of elucidation I will try to reach a uni­tary and articulated vision of the dialectical method as appliedto axiomatic theories.

We saw that according to Nowak the method that proceedsfrom the abstract to the concrete is none other than what hecalls the "method of idealization", and that he characterizes thismethod as the one "which is able to reveal inner connections".Nevertheless, if by 'revealing inner connections' is understoodthe discovery of regularities capable of being expressed as scien­tific laws, then a distinction must be made between the methodof passage to the concrete and that of idealization. Restrictingourselves to the domain of scientific theories that can be formu­lated by the definition ofa set-theoretic predicate, it seems to me

126 CHAPTER FOUR

that the primary sense of the term 'rising to the concrete' mustbe identified with the procedure that consists in the construc­tion of determined models of given theories in order to repre­sent determined real concrete situations. By 'determined model'I mean a model whose relations posses certain fixed values andwhose constants, in particular, have been assigned certain values,i.e. I mean one model strictly speaking and not a class of models.This term is exemplified here with model q3 presented above,since all its parameters are perfectly determined; but, if insteadof that, we had parameters like ro, Vo and g undetermined, thenwe would rather have afamity of models and not just one. WhenI say that the model represents determined real concrete situa­tions, what I mean is that his axioms are true of such situationsand so the values of the parameters predicted by the model havea pragmatically acceptable degree ofapproximation with respectto the corresponding available empirical data about those realsituations. Notice that, in this sense, "concrete" models are mod­els that have instances in the real world.

Understanding primarily in this way the term 'rising to theconcrete', it is possible to observe other processes analogous tothe one described by this term. One of them is the constructionof a singular model which does not represent any real situation.This method is useful, since it fulfills the purpose ofbuilding eas­ily computable counterexamples (when they exist) to particulartheses that someone wants to disprove within the class of modelsto which the mentioned model belongs. Yet the resulting mod­els are not "concrete" but rather idealized, and so the processis not exactly one of "rising to the concrete", but rather one ofdeisolation and articulation of separated notions into a totality­of-thought. Another, epistemologically more important processconsists in the construction of a class of models of the theory, bymeans of a general description of the models in the same class,that leaves at least some of the parameters of the class unspeci­fied. The case can also obtain that no element ofsuch class comesto represent real situations, or that it represents them only ina very rough approximated way. The procedure of idealization

THE DIALECTICAL METHOD 127

-according to the present view- consists of the prod uction ofaclass of models none of which constitutes a pragmatically accept­able approximation to the class of real situations being theorizedabout, precisely because some of the axioms defining such classare false in such situations. Hence, we can see that against whatNowak claims the method of idealization does not coincide withthe one that moves from the abstract to the concrete, althoughidealization is indeed a case ofdeisolation and articulation of no­tions.

There is no doubt that at least in some cases the creation ofa scientific theory begins with the prod uction of idealized struc­tures which become afterward a particular class of models ofthetheory, once the theory grows and its assumptions are made ex­plicit. What Nowak calls 'concretization' is precisely the proce­dure which, taking as a point of departure a family of idealizedstructures, relaxes the assumptions defining the same structuresby way of getting rid of false assumptions, thus obtaining a fam­ily of structures in which the fundamental law(s) which charac­terizes the original structure still hold. What Nowak calls 'con­cretization' is, therefore, a generalization process as well as a con­struction process of more realistic structures. It is a generaliza­tion process because the original idealized structures become aparticular case of a wider class of structures, and it is also a con­struction process of more realistic structures because the elimi­nation of the false assumptions defining the original structuresgives rise to models in which every fundamental previous lawstill holds but which no longer depend on those false assump­tions. The construction of these more realistic structures is, ofcourse, a case of passage from the abstract to the concrete in thesense already explained, since it is not just a case of model con­struction out of abstract determinations, but one in which thereis a tendency toward building a model with instances. Hence, itis seen that at least in the case of axiomatically defined modelsthe dialectical method consists in a constructivistic procedure ofdeisolation and articulation of concepts, moved by a drive to­ward more concrete (i.e. instantiated) structures.

128 CHAPTER FOUR

Thus, contrary to what Maki (1991) suggests, it is not the sameto concretize than to deisolate determinations. Also Hegel andMarx speak sometimes as if this were so. In the case of Hegel per­haps deisolating coincides with concretization, given the kind ofconcepts Hegel is using in the logic, where he is dealing withcategories that have instantiations. In this sense, it would bemore accurate to say that such concepts are concrete throughand through. But if in the construction of a model one startswith idealized notions, the resulting model itself will never beconcrete, inheriting the idealized character of the notions withwhich it was built, no matter how tightly articulated it may be.The prototype built by Marx, even though it is a unity ofdiversedeterminations, still is too idealized to have instances. Hence,more than a mere articulation of isolated concepts into a total­ity is required in order to concretize. What is required is that theconcepts have instances in the real world. An economic discoursebuilt up exclusively with such kind of concepts I shall call "con­crete"; an economic discourse that contains idealized concepts(like 'convex technology', 'continuous preferences', and so on)I will call 'idealized', and the economy it describes a Meinongianeconomy, because it is a nonexistent object. As a matter of fact,Marx's prototype is a Meinongian economy which fails to deiso­late properly the concept oflive labor, failing to connect that con­cept with the other concepts of the theory, his prototype being infact the most idealized mathematical structure that can be builtin economic theory. On the other hand, his fear to connect suchconcept with the determinations ofcapital is rather strange, andseems to have arisen from a misunderstanding. Having live la­bor in a given economic theory as labor-power, deeply connectedwith the determinations of capital, does not commit anybody tobelieve that human beings are nothing but a moment of capital.To say this is still to be playing with the idea of an Idea that de­termines itself. Dialectic in the form I have tried to explain onlyfinds conceptual connections and only sublates concepts, not peo­ple. In the next chapter I will try to overcome the "contradic­toriness" of Marx's concept of live labor through a new conceptof abstract labor.

Chapter 5

ABSTRACT LABOR

As we saw in chapter 2, all well known mathematical formula­tions of MTV 1 are based upon Marx's assumption that labor ina capitalist economy is homogeneous, and so that the value ofthe goods it prod uces can be determined in terms of its meanor minimal temporal duration. In chapter I we saw Marx pre­tending that therefore value is an object that can be consideredin the pure sphere of prod uction, "independently of its form ofmanifestation" in the sphere ofexchange. My point ofdeparturein this chapter is the fact that Marx's methodological decision tosever value from exchange is in fact a dialectical mistake, that ineffect value cannot be defined independently of the market con­cept. The argument to show this is simple and conclusive: theattempt to define value in that way inevitably leads to the trans­formation problem, a problem that according to Theorem 7 ofchapter I is unsolvable unless we adopt the arbitrary assumptionthat all firms have the same value-composition of capital.

Marx himself provides in C, and perhaps more clearly in AContribution to a Critique of Political Economy, elements to over­come the contradictoriness of his value concept. Recall that in CMarx almost said that the market is the social process, "that goesbehind the back of the producers" that in effect reduces hetero­geneous labors to a common unit. Even though in the Contribu­tion to a Critique of Political Economy Marx seems to have already

129

130 CHAPTER FIVE

maintained his final view that value is determined in the sphereof production, from a certain point up in that work he beginsto talk as if he were attributing a role to the market in that pro­cess. He says in part 1, chapter I of that work, for instance, thefollowing:

As exchange-values of different magnitudes they [i.e. commoditiesof different types] represent larger or smaller portions, larger orsmaller amounts of simple, homogeneous, abstract general labour,which is the substance of exchange value.2

This passage sounds much the same as those passages in C whereMarx defended the market-independent view of value. Never­theless, as A Contribution progresses Marx begins to insist that ina commodity economy

the labor of different persons is equated and treated as universallabour only by bringing one use-value into relation with anotherone in the guise of exchange-value.3

Or that in the exchange process

universal social labor is [... ] not a ready-made prerequisite but anemerging result. 4

Naturally, putting together the first quotation with these last twoa perplexity cannot but arise:

[...] a new difficulty arises: on the one hand, commodities must en­ter the exchange process as materialised universal labour-time, onthe other hand, the labour-time of individuals becomes materialiseduniversal labour-time only as the result of the exchange process..5

It is pretty clear me that the contents of this last quotation is themanifestation of an ambiguity which is present both in Capitaland in A Contribution. In both works Marx starts by describingthe "substance" of value as homogeneous labor, which in addi­tion he seems to identify with abstract labor and simple or unskilledlabor. The impression that one gets reading the first sections ofsuch books is that the "substance" ofvalue can be defined almostin purely technological terms that can be applied to economies

ABSTRACT LABOR 131

which are not necessarily capitalist, since Marx seems to charac­terize this "substance" -as we saw- as "an expenditure of hu­man labour-power, in the physiological sense". Despite Marx'sallegations that

it is only the expression of equivalence between different sorts ofcommodities which brings to view the specific character of value­creating labour, by actually reducing the different kinds of labourembedded in the different kinds of commodity to their commonquality of being human labour in general,6

one always gets the distinctive feeling that the so-called "form ofvalue" is just a sociological appendage which is entirely irrele­vant for the quantitative determination ofvalue in terms of timesocially necessary for the production of a commodity. This feel­ing is reinforced by Marx when he writes, in the same section,the following:

Whether the coat is expressed as the equivalent and the linen asthe relative value, or, inversely, the linen is expressed as equivalentand the coat as relative value, the magnitude of the coat's value isdetermined, as ever, by the labour-time necessary for its production,independently of its value-form.

The fact that Marx is so ambiguous, or even inconsistent, indealing with the concept of abstract labor, does not mean thathis theory cannot be reconstructed as based on the concept ofabstract labor understood in a different way, in a way that takesthe market process into account. The first economist in attempt­ing this task was Isaak Illich Rubin in his Essays on Marx's The­ory of Value, published for the first time in the Soviet Union inthe mid twenties.? Rubin distinguishes three concepts of laborin Marx's theory, namely, (1) physiologically equal labor, (2) sociallyequalized labor, and (3) abstract or abstract-universal labor, i.e. "so­cially equalized labor in the specific form which it acquires in acommodity economy".8 Physiologically equal or homogeneouslabor is labor insofar as it is just an average productive expen­diture of human energy in any form. Socially equalized laboris labor equalized not just as homogeneous labor, but rather as

132 CHAPTER FIVE

compared by some social factor or process; for instance, Rubinpoints out that in a socialist commune the different types oflaborcan be compared by a specific organ ofthe commune for the pur­pose of accounting and distribution of labor.9 Labor comparedin this way is not abstract labor. Abstract labor is labor comparedthrough the market in a capitalist economy, where the firms areformally independent producers which are interconnected bythe market (and not, say, by a central organ in charge ofassigningproduction quotas to each firm). The category ofabstract labor ishistorical and relative to the capitalist mode ofproduction. It ex­presses the fact that in a commodity economy the different typesof labor are compared and reduced to a common "substance"precisely in the process ofcommodity exchange, where the com­parison of the different products induces a comparison of thecorresponding labors that produced them. This is the form inwhich the social comparison of the different types of labor takesplace in a capitalist economy.

The question that arises now is whether MTV, reconstructedon the basis of the concept of abstract labor as it has just beencharacterized, can be formulated in precise mathematical terms.The first effort in this direction was done by Professor UlrichKrause at the end of the seventies in one book and two papers. lO

Roughly speaking, Krause was able to show that in economicsystems that have certain special properties (that we shall see inthe next section), the exchange of commodities induces a reduc­tion of the different types of labor to a common measure, a re­duction that can be represented by means of a linear functionaldefined on the set of all labor expenditures, understood as asubset of a linear space. Representing the labor-power appliedin the prod uction processes by means ofn-d imensional vectors,as indicated in chapter 1, §3, the idea is to show that the ex­change relations among the corresponding products do in factinduce an ordering among such vectors that can be representedby a linear functional. Assuming that certain special conditionsare satisfied by the technology, this is precisely what Krause did,opening in this way the door to a mathematical treatment of themarket-dependent view of value.

ABSTRACT LABOR

5.1 KRAUSE'S TREATMENT OF ABSTRACT LABOR

133

His most general treatment of abstract labor Krause gives in his1980 paper "Abstract Labour in General Joint Systems". In thispaper Krause introduces an economy that he labels a 'generaljoint system'. A general joint system is a triple of nonnegative ma­trices (A, B, L) such that A is the m x l matrix of joint materialinputs, B is the m x l matrix ofjoint material outputs, and L is then x l matrix ofjoint labor inputs. The matrix ofjoint net outputsis defined as the difference C = B - A and a price system is apositive m vector p = [Pl'" pm]. In the notation of chapter 1, Ais a matrix of the form

:t] .-ml

The columns of this matrix can be seen as the transposes of inputvectors of processes XI, X2, ... , x/, which are all the processes inthe economy. B is just the matrix whose columns are the trans­poses of the output vectors of these same processes:

An activation or state of the joint system (A, B, L) is an l x 1vector s ~ O. This vector may be thought as indicating a givenintensity of production of the economy; for instance, if s is thevector [1 ... 1], the prod uct

[

XI I + .: . + XII ]Bs = :

Xml + ... +Xml

134 CHAPTER FIVE

indicates the entire amounts ofgoods produced in the economy,their production requiring the consumption of inputs

[

~II + .: . + ~11 ]

Bs = :x + ... +X-m) -ml

and labor inputs

[

XII + . : . + XII ]

Ls = : .Xnl + ... + Xnl

At this intensity, the economy is producing the following vectorof net outputs:

[

X)) + .: . + XII ]Cs = : .

Xml + ... +XmlHence, the set of all possible net outputs is just the cone C ={Cs Is ~ O}, whereas the set ofall possible expenditures oflaboris the cone £ = {Ls Is ~ O}.

According to Krause, "abstract labor means labor homoge­nized via the market by the exchange of products oflabor". Thishomogenization of labor can be elucidated observing that anyprice system pER'; induces a binary relation ~ over the cone£, as follows. For any x, y E £, define ~ by the condition:

x ~ y iff pi ~ py.

Now, even though a relation like this is defined for every pricesystem, not all price systems define the "right" relation betweenlabor vectors. For it would seem that if the productive process ispends more or the same amount of labor-power than processy (i.e. x ~ y), then the price of the net output of i should beno less than the price of the net output of y (pi ~ py). A pricesystem p with this property is called 'admissible' by Krause, butI will call it 'valid' (giiltig) from now on.

ABSTRACT LABOR 135

While exchange, which is the homogenization of productsthrough money, can be described by prices collected in a pricesystem p, the homogenization of labor can be described by re­duction coefficients, collected in a reduction r, which is a positiven x 1 vector. Krause defines an 'admissible' reduction as a reduc­tion r that satisfies the condition:

If x ~ y then rx ~ ry.

Krause then proves that any binary relation ~ induced by anadmissible (that is, valid) price system is represented by an ad­missible reduction and that to any admissible reduction oflaborthere corresponds a valid price system. Hence, the price systemsthat define the "right" relation among vectors of labor inputsare precisely those that are valid. Abstract labor is a structure(£, ~), where the relation ~ is induced by a valid price. Hence,in terms of the representation theory discussed in chapter 3, if~ is induced by a valid price p, an (admissible) reduction r cor­responding to price p is just a representation of ~; i.e., for everyx,y E £:

x ~ y iff rx ~ ry.

The labor-value of net output x E C, >'(x), is defined as the num­ber rx, and so labor-value is nothing but a representation of ab­stract labor in the sense of chapter 3 (notice, however, that it isnot afundamental measurement, since the structure (£,~) is notontological).

Regarding the existence of valid prices (or admissible reduc­tions), Krause proves that for given matrices A and B of inputsand ouputs there exists a semipositive price system (and a cor­responding semipositive admissible reduction) for any L iff allcommodities are separately producible, i.e. if for each bundleofcommodities b there is an activation s ofthejoint system suchthat b = Cs. Krause also proves that there is a positive price (anda positive admissible reduction) for every L iff the previous con­dition is satisfied and in addition all processes are indispensable,i.e. ifwhenever the net output Cs is semipositive, the activation s

136 CHAPTER F1 V E

is positive. Unfortunately, the conditions of separate producibil­ity and indispensability are unduly restrictive.

The supposition that any good be separately producible is re­strictive because it requires the existence of a labor process inP = {[Ls, As, Bs] Is ~ O} whose net output is a vector that haszeros everywhere except at a specified place. Since P is a cone,the condition can be formulated as follows:

(S9) Vi = 1, ... , m: there is a x E P such that the ith entry of xisone, and all the other entries are zero.

(S9) holds in Leontief technologies but fails in some joint sys­tems. For instance, it fails to hold in systems where two wageor luxury goods are jointly produced. For let the wage or luxurygoods i and) (k + 1 ::; i,) ::; m) be always jointly produced. Then,no matter in which process x they appear as outputs (x), sincethe entries i and) are always zero in ~, the same places in thenet output vector x = x- ~ are also nonzero, and so none of thegoods i and j can be separately produced. Hence, (S9) impliessome form of no joint production.

The condition that all processes be indispensable is also re­strictive because it requires that in order to produce any semi­positive net output all processes ofthe economy be activated. Butthere is no reason why this should be so in general. Consideragain the Leontief economy and let ej be the m column vectorthat has zeros everywhere except at the ith place (k + 1 ::; i ::; m),where it has a one, so that ei represents one unit of a wage orluxury good. In order to produce precisely the bundle ei, thevector Si of capital goods must be produced in such a way as tosatisfy the equation II

AS i + ~J = Sj.

Let xT = Sj + ei and think of the vector xT as an activation of thejoint system. Notice that this activation has some zero entries, inthe places corresponding to the wage or luxury goods other than

ABSTRACT LABOR 137

i, because Si has zeros at all places corresponding to non-capitalgoods. Then we have

Hence,~T -T T [ ]X = X - ~ = Si + ei - Si = ei.

This shows that there is an activation that has some zeros andyet such that the net output is semipositive. Thus, the conditionof indispensability, which can be formulated as

(510) '<Ix E P: if x ~ 0 then x > 0

is violated for instance in a Leontieftechnology.Using (59), (510) and the Farkas-Minkowski Lemma, Krause

proved the existence of a reduction, i.e. a positive vector r suchthat

rx ~ ry iff x ~ y,

for any labor input vectors x, y E L, for given matrices A and Band for every L. This result is in itself very important but un­duly strong. What we require is not a proof of the existenceof valid prices and admissible reductions for given matrices Aand B and every L, but rather for every joint system that satisfiescertain reasonable conditions. It is clear that the existence of ameasurement of abstract labor cannot depend on assumptionsas arbitrary as (59) and (510). I n other words, the basic problem inthe foundations of MTV is to prove the existence of a cardinal measure­ment of abstract labor, without assuming conditions (S9) and (S10). Ishall tackle this problem in the next section.

5.2 THE CONCEPT OF ABSTRACT LABOR

Imagine a capitalist market economy in which there are l inde­pendent producers (capitalist firms). At the beginning ofan eco­nomic cycle, each of these producers choses the production planthat he believes will yield the maximum profit for him. A produc­tion plan is a production process that has certain properties that

138 CHAPTER FIVE

will be formulated when the concept of a prod uctive structure isintroduced below. Recall that ifrn goods are being produced inthe economy, and there are n different types of concrete labors,then we can represent a prod uction process by means ofa vectorofthe form i = [x, & xl, where x is a nonnegative n vector whoseith component represents the amount of concrete labor of typei (1 ::; i ::; n) expended in the process i, ~ is a nonnegative mvector whose ith (1 ::; i ::; rn) component represents the amountof goods of type i employed as means ofproduction in i, and x isalso a nonnegative m vector whose ith component represents theamount of goods of type i (1 ::; i ::; rn) prod uced in i. The vectorx is called the vector of labor inputs, ~ is the input vector, andx is called the output vector. Notice that by definition the nullvector 0 is a production process, but at any rate it follows thatany production process xis nonnegative and has 2m +n compo­nents. It will be convenient also to have a separate notation forthe net output of process i, i.e. for the difference x-~; this shallbe denoted (as before) as x. In order to represent the produc­tion plans ofthe firms, a set ofproduction processes must possessthe following additional property, namely, that each type ofcon­crete labor be utilized at least in one of these plans and also thateach type of good be either produced or used as a means of pro­duction in some production process; this property will be called'nontriviality'. The former concepts are formally introduced inthe following definition.

DEFINITION 1: Aproductionprocess is a vector [x,~, xl in the lin­ear space R2m

+n, where m, n ;::: 1, such that the n vector x, as wellas the m vectors ~ and x, are all nonnegative. The vector x iscalled the vector of labor inputs, ~ is called the vector of means ofproduction or input vector, and x is the product vector or output vec­tor. The difference x = x-~ is called the net output ofi. A set Q ofprod uction processes is called nontrivial if for each i (i = 1, ..., n)there is a vector x E Q such that the ith component Xi of the vec­tor of labor inputs x of x is positive, and for each j (j = 1, ..., m)there is a vector i E Q such that either the jth component ~j ofthe input vector of i is positive, or the jth component Xj of theoutput vector of xis positive.

ABSTRACT LABOR 139

Assuming that there is one price for each commodity, the pricesystem is a nonnegative m vector p = fPl·· ·pm]. According togeneral equilibrium theory, to each prod ucer h (h = 1, ..., I)there corresponds a production set yh of possible productionprocesses representing his limited technological knowledge, andhis behavior consists ofchoosing a point Xh in yh that maximizeshis profit given the price system p. I am not so much concernedhere with the accuracy of this description (none of the results es­tablished in this chapter depends logically on it) as I am with thefact that as a result ofthis, 1> 0 (possibly equal) production plansare implemented in the economy and, in fact, the different typesoflabor are indirectly compared among themselves through thecomparisons between net outputs effected by the market forcesby means of the price system. What this means is that there isa relation:: among the vectors of labor inputs XI, ..., Xl of theproduction plans Xl, ..., XI chosen by the firms, such that

(h, i = 1, ..., I)

i.e. such that Xh :: Xi (to be read as: "Xh represents at least asmuch social labor as x;") whenever the price of the net outputproduced by the labor expenditures represented by Xh is greaterthan the price of the net output produced by the labor expen­ditures represented by Xi. From a logical point of view, :: is abinary relation which is connected, reflexive and transitive onthe set of labor input vectors. Moreover, defining'" as usual,namely, by the condition

iff

we could also prove that'" is an equivalence relation over thesame set. What this means is that any system of market pricesdoes in fact induce a comparison of the different types oflaboroperating in the economy. The strict dominance relation >- isdefined, as usual, by means of the condition

iff

140 CHAPTER FIVE

Not every relation induced by a price over the labor inputs,however, counts as abstract labor. It seems intuitively clear thatin order for a relation over a set oflabor inputs to be thought asabstract labor, the exchange relation among their correspondingnet outputs must be valid (giiltig). The very minimum conditionfor validity is that exchanges be ruled out, in which net productsthat "contain" more concrete labor are traded for net productsthat "contain" less. This can be expressed symbolically by the twofollowing conditions:

(C 1) Ifthe vector oflabor inputs Xh ofprocess Xh is equal to thelabor vector Xi of process Xi (i.e. Xh = Xi), then Xh rv Xi.

(C2) If some component of the vector of labor inputs Xh ofprocess Xh is strictly greater than the corresponding com­ponent of the labor vector Xi of process Xi, and no com­ponent of Xi is strictly greater than the correspondingcomponent of Xh (i.e. Xh 2': Xi)' then Xh >- Xi·

What (Cl) says is that each vector oflabor expenditures repre­sents a fixed quantity of social labor or, what is the same, that nolabor vector represents more social labor than itself. (C2) assertsthat ifvector oflabor inputs Xh contains at least as much labor ofeach type as labor vector Xi, and in fact more concrete labor ofat least one type than labor vector Xi, then Xh represents moresocial labor than Xi. It is easy to see that a necessary and suffi­cient condition for conditions (CI) and (C2) to hold is that theabstract labor relation be induced by a price which is admissiblein Krause's sense. For reasons that were apparent in chapter 1,I prefer the term 'valid', a term which is precisely defined now.

DEFINITION 2: LetP be any set ofproduction processes. Apneesystem (or simply a price) for P is a positivem vector p = [Pl" ,pn].If, in addition, p satisfies

I f X ~ Y (resp. 2':), then pi 2': py (resp. »,

then the price p is called valid for P.

ABSTRACT LABOR 141

In terms of the concept ofvalid price system, the definition ofthe concept ofabstract labor is straightforward and can be intro­duced at this point. According to Rubin (1972), the introductionofthis concept is what distinguishes Marx's labor theory ofvaluefrom that of Ricardo's. It will be obvious from the definition thatabstract labor is a relation generated by a valid price on a systemof production processes and that such relation actually repre­sents an ordering of all the labor expenditures of the system.

DEFINITION 3: Let P be a set of production processes. The setI:- = {x : [x, X. xl} of all the vectors oflabor inputs of processesin P is called the set of labors of P. If P is a valid price for P and::: is the binary relation ind uced by p over 1:-, then::: is calledabstract labor and the structure .£ = (I:-, :::) is called the abstractlabor structure corresponding to P and p.

This concept of abstract labor elucidates in precise termsMarx's suggestion that the reduction of the different types ofconcrete labors to a common measure is effected by a processthat goes behind the backs of the producers. As I suggested inchapter 1, in a capitalist economy this process cannot be but themarket. It is the market the process that red uces these laborsto a common unit. Marx spoke of reducing all kinds oflabor tosimple unskilled labor, but there is no need to take this type ofla­bor as the standard and in fact any other would do equally well.I shall discuss in the next section the conditions that guaranteethe existence of valid price systems and therefore of abstract la­bor. My approach shall be different from that of Krause's. Aswe saw, he proved that a valid price system necessarily goes to­gether with an admissible reduction; he then proceeded to showthat for joint systems satisfying (59) and (510) both determina­tions exist. My strategy shall be instead to try to find conditionsover prod uction processes, wages and profit rates that guaranteethe existence of the abstract labor relation. We know that to anyvalid price system there corresponds a nonnegative reduction,but we do not know whether this reduction is positive. In the fi­nal section ofthe present chapter I will show that every abstractlabor relation can be represented or measured (in the sense ofchapter 3) by a positive reduction.

142 CHAPTER FIVE

5.3 THE EXISTENCE OF ABSTRACT LABOR

A question that naturally arises concerns the conditions underwhich a price system is valid in an economy, i.e. the conditionsunder which the price system actually induces the abstract la­bor relation. In order to discuss this problem, I will consider ajoint system, i.e. the convex polyhedral cone of labor processesspanned by the production processes actually operated by thefirms in the economy. I will introduce the concepts of wage sys­tem and profit rate. A wage system is an n vector w whose ithentry is a positive number which represents the hourly wage ofworkers of type i (1 ~ i ~ n). Thus, if x is the vector of laborsof prod uction process X, then wx represents the monetary costsof labor power required to operate x, i.e. what Marx called the"variable" capital (in terms of money) of prod uetion process x.The profit rate of x, on the other hand, is the number 7r(x) thatsatisfies the following equation:

px = [1 + 7r(x)](p~ + wx).

That is, the profit rate multiplied by the cost of production yieldsthe net profit of process x. More formally, these concepts can beintroduced as follows.

DEFI NITION 4: Let P be a set of prod uction processes. A wagesystem for P is a positive n vector w = [WI··· wn ] of real numbers.If w is a wage system, the profit rate of a labor process x is thenumber 7r(x) given by the equation

7r(x) = px - wx,p~+wx

if x is nonnull, and zero otherwise.

Returning to the problem of the conditions under which aprice system is valid in an economy, it can be shown that if therate of profit is uniform in the economy, then the price p hap­pens to be valid, provided that a certain normality assumption

ABSTRACT LABO R 143

connecting expenditures of labor and means of production isgranted.

Consider the cone P spanned by the actually operated pro­cesses in the economy, the elements of Po = {XI> ..., XI}. Ourapproach to the problem of the existence of valid prices for Pand so for Po, which is the set that mainly interests us, is closelylinked to the existence ofa uniform profit rate. We would like toprove that if in the economy determined by processes XI> ..., Xllabor is both productive and indispensable,12 and the profit rateis uniform, then the price is valid. How can we do this?

It would seem that the clue lies in the concept of efficient pro­cess. A process X is more efficient than a process y iff X pro­duces more than y using the same or less amounts of labor, orproduces the same or more using less amounts of labor. Moreprecisely, we shall follow the use of the term introduced by thefollowing definition.

DEFINITION 5: Let X = [x, ~ x] and y = [y, y, y] be two pro­duction processes. We say that X is more eJJicientthan y iff the netoutput of X is greater than the net output of y, even though Xdoes not expend more labor of any type than y, or the net out­put ofx is equal or greater than the net output ofy even thoughX spends less labor of some type than y. In symbols:

iff either x ~ y and x 2 y,or x ~ y and x ~ y

where 'XE)" stands for 'x is more efficient than y'.Indeed, if no process is more efficient than any other, then it

follows that x ~ y implies x ~ y for arbitrary processes X and y.Hence, in such a case the inequality x ~ y implies px ~ py; i.e.it follows that p is valid. Thus, if we could show that processeswith different degrees of efficiency cannot have the same rate ofprofit, that would be enough to establish what we want, namely,that prices that equalize profit rates are valid.

Unfortunately, this proposition is false. Consider an econ­omy with two processes X = [(1, 1), (3/2, 1), (5, 3/2)] and y =

144 CHAPTER FIVE

[(2,2), (1/2, 1/4), (3,1/2)], a price system p = [1,2] and a wagesystem w = [1/4, 1/4]. We have in this economy x = [7/2, 1/2],Y= [5/2, 1/4], and so we see at once that x is more efficient thany in the sense of Definition 5. Nevertheless, quick computationsusing Definition 4 show that 1I"(x) = 1 = 1I"(y).

An examination of the counterexample just given reveals acurious situation: How is it possible that the workers in processy, working twice as much as the workers in process x, spendabout 3 times less means of production than those in processx? Clearly, this would be the case only if either the workers inprocess x were wasting means of production, or the workers inprocess y were working too slowly, but according to Marx thissituation cannot obtain in the valorization process, since in sucha process "the time spent in production counts only in so faras it is socially necessary for the prod uction of a use value" .13

According to Marx, this has various consequences, one of thembeing that

alI wasteful consumption of raw material or instruments oflabor isstrictly forbidden, because what is wasted in this way represents asuperfluous expenditure of quantities of objectified labour, labourthat does not count in the product or enter into its value. 14

Another consequence is that

the labour-power itselfmust be of normal effectiveness. I n the tradein which it is being employed, it must possess the average skilI, dex­terityand speed prevalent in that trade [...and ... ] it must be ex­pended with the average amount of exertion and the usual degreeof intensity. J5

It seems intuitively clear that the two former consequencespreclude situations like the one of the former counterexample.Indeed, consider two workers of the same trade fulfilling thejust given conditions, say two painters. Since none of them worksfaster than the other, and none ofthem wastes paint and brushesunnecessarily, it is utterly unreasonable to expect that any ofthem will spend more means of production per unit time thanthe other. Let us call 'normal' the labor-power satisfying this cou­ple of conditions. From a quantitative point of view, we say that

ABSTRACT LABOR 145

the labor power employed in two process x and y is normal ifwhenever x ~ y it follows that If ~ y. More precisely, I intro-duce -

DEFINITION 6: Let Q be a set of production processes. We saythat the labor employed in the processes in Q is normal iff noneof them is null and, for every X, y E Q, x ~ y (~) implies thatIf ~ Y(~). We shall also call the set Q itselfnormal when the laboremployed in it is normal.

The following theorem, which answers the question that wehave been considering, is an important consequence of a set ofproduction processes' being normal.

THEOREM 1: Let Q be a normal set ofproductwn processes. If therate ofprofit is the same for every process in Q, under the pme p andthe wage system w, then p is valid.

Proof" Let x, y be any two elements of Q and assume that theprofit rate is uniform for all the elements of Q, so that 1I"(x) =1I"(y). If x ~ y then we have two cases: either (i) x = y or (ii)x ~ y. If (i) is the case, since w is positive and p semipositive,wx = wy and Plf = pro Hence, wx + Plf = wy + pr and so

pi - wx = 1I"(x) . (wx + plf) = 1I"(y)(wy + pr) = py - wy.

It follows that pi = py.If (ii) is the case, then wx < wyand Plf ~ py. Hence, wx +

Plf < wy + pr and so -

pi - wx = 1I"(x) . (wx + plf) < 1I"(y)(wy + pr) = py - wy.

Adding wx at both sides of the extreme expressions, we get pi <py + (wx - wy) < py, because wx - wy is negative. 0

Theorem 1 shall be used as a lever to build a couple of in­teresting models of the theory in subsequent chapters. We shallalways suppose that the aggregate technology possibility sets arenormal, thus implying that in particular the cone spanned by

146 CHAPTER FIVE

the set Po of processes actually chosen by the firms is also nor­mal. This will follow from the fact that such cone is in fact a subsetof the aggregate technology possibility set and that any subset ofa normal set is also normal (Theorem 2). We shall also exploittheorems 3 and 4.

THEOREM 2: If the set ofproduction processes Y is normal, and X isany nonempty subset of Y, then X is normal.

Proof Assume that Y is normal and that X ~ Y. Let X, y E X,and suppose that x ~ y. Since X, y E Y also, it is immediate thatIf ~ I and so X is normal. 0

THEOREM 3: IfP+ is the positive hull sjJanned by Po = {Xl, ..., XI},and the profit rate r is the same for all the elements ofPo under the pricesystem p, then it is also the same for all the elements ofP+.

Proof We can assume, without any loss of generality, that Po islinearly independent (otherwise,just drop some of its elements).Ifx E P+ then there exist unique nonnegative numbers <Xl, ... , <XI

such that x = <XIXI + ... <XIXI. Thus, as a matter of fact, x ='\'1 '\'1 - '\'1 - d ~L...-,h~1 <XhXh, If = L...-,h~J <Xhlfh' X = L...-,h=1 <XhXh an so x = x - If =I:~~l <Xh(Xh - lfh ) = I:~~I <XhXh. Hence,

= <XI(PXI - WXI) + + <XI(PXI - WXI)

<XI (plf l + WXI) + + <X1(plfl + WXt)

Thus, setting

ABSTRACT LABOR

we have L~=l (3h = 1 and get

(_) CtI(3I(PXl - WXl) CtI(3t(PXI - WXI)

7r x = + ... + --:~~:....-_--:.

CtI(PX1 + WX1) CtI(PXt + WXt)

I

= L(3hrh=l

=r. 0

147

THEOREM 4: If P+ is the positive hull spanned by Po = {Xl, ..., xt},the profit rate r is the same for all the elements of Po under price systemp, and P+ is normal, then P is valid for P = P+ U {O}.

Proof" This follows immediately from Theorems 1 and 3. 0

Roughly speaking, a productive structure is the convex conegenerated by the prod uction processes chosen by the capitalistfirms. The only reason why this cone is introduced is to makecomputations feasible; the reader may interpret this cone as rep­resenting the technology actually chosen by the firms. Additionalaxioms defining the concept of prod uctive structure are that theset of production processes chosen by the firms is nontrivial andthat no input vector of these processes is null, i.e. that every pro­duction process requires some positive amounts of means ofpro­duction; to this condition will be added the assumption that laboris both productive and indispensable and that the cone is nor­mal. It is easy to see that the former conditions imply that in ev­ery production plan positive amounts of goods are obtained outof positive amounts of means ofproduction and concrete labors.The concept ofa productive structure is formally introduced bythe following definition.

DEFINITION 7: The structure I:J3 = (Po, P) is a productive struc­ture iff it satisfies the following axioms:

(PI) Po is a nontrivial finite set of labor processes

(P2) P is the convex cone generated by Po

148 CHAPTER FIVE

(P3) For every (x, & x) E Po, x ~ 0 and labor is productiveand indispensable, i.e. the following implications hold:

(P4) P is normal.

The properties ofthe production processes in Po expressed by(P3) also hold for all processes in the cone P. This is establishedas

THEOREM 5: For every (x, & x) E P,

~ ~ 0 => x ~ 0 ¢} x ~ o.

Proal Let x = [x, ~ x] be any production process in P. Thenthere exist nonnegative O'I, ••. , 0'/ such that

If x ~ 0, then some of the 0'], ... ,0'1, say 0'1, ••. , O'k are positive.In such a case, since (by (P3»

x· > 0 => Xi > 0 ¢} Xi > 0-, - - - -

It follows that Xi ~ 0 ¢} Xi ~ 0, which implies that

k k

X = L O'iXi ~ 0 ¢} x = L O'jXj ~ 0

I shall conclude the present section by formulating the fol­lowing methodologically useful theorem, which follows almostimmediately from theorem 4 and 5.

THEOREM 6: Let 'i3 = (Po, P) be a productive structure, I: the setof labors of P and :: the relation over I: generated by price p. If theprofit rate of the production processes in Po is nonnegative and uniformat prices p, then (I:, :=:) is an abstract labor structure.

ABSTRACT LABO R 149

Theorem 6 is methodologically important because it showsthat in order to establish the existence of abstract labor in aproductive structure it is sufficient to show the existence of aprofit equalizing equilibrium. This solves the dialectical contra­diction, pointed out in previous chapters, between the observedtendency toward a uniform profit rate in a market economy andlabor-value. In fact, according to Theorem 6 this tendency leadsstraightforwardly to abstract labor, to reduce all labors to a com­mon standard.

Roemer (1980, 1981) has established the existence of a profitequalizing equilibrium for several models of MTV. These mod­els are built upon the assumption that labor is homogeneous anddo not depend on (P4), but they can be modified by introducingheterogeneous labor in such a way that the existence of the re­quired profit equalizing equilibria is also provable within them.I will provide the details of this in chapters 7 and 8.

5.4 THE REPRESENTATION OF ABSTRACT LABOR

The present section is devoted to the main aim of the chapter,which is to prove the existence ofa mathematical representationof abstract labor. We saw on chapter 3 what a mathematical rep­resentation is and, in particular, we introduced there the conceptof a fundamental measurement. The measurement of abstractlabor as provided in the present chapter is indeed a representa­tion in that sense, but not a fundamental measurement, since thestructure to be represented -the abstract labor structure- isnot ontological but is already a mediated representation of con­crete labor; this structure is itselfa mathematical representationofan aspect of the production processes taking place in a marketeconomy.

A mathematical representation of abstract labor is a function<p: £ --l- R such that <p(x) ~ <p(y) whenever x ::: y, for everyx, y E £. As a matter of fact, there are quite a few of such rep­resentations. Just consider the function <p that assigns to eachx E £ the number pi (which is unique according to Definition

150 CHAPTER FIVE

2). If 'lj;: R --+ R is any strictly increasing function, then the com­position 'lj; 0 <P is another representation ofabstract labor. Repre­sentations of this type are called in the literature "ordinal" repre­sentations. This is not the type of representation with which thepresent chapter is concerned. This chapter is concerned with avery specific type of what is called in the literature a "cardinal"representation. More precisely, the representation we are look­ing for is a very specific type of what is called in Krantz, et at.(1971) an additive conjoint measurement. An additive conjointmeasurement of a binary relation R is a family of real-valuedfunctions {<Pi}iEN' where N = {I, ..., n}, such that

L<Pi(Xi) ~ L<Pi(yi) iff [XI" 'Xn]Rfyl' "Yn],iEN iEN

where [XI" ·xn] and fyl" 'Yn] are any two vectors in the field ofR. In the particular case of abstract labor it can be shown thatfunctions <Pi can be found that act upon the vectors that take asarguments as the tensor "taking the inner product with a fixedpositive n vector", i.e. if[xl ... xn ] is a vector in £, then there is apositive vector r = [rl" ·rn] such that <Pi(Xi) = riXi, and so

L <Pi(Xi) = rx.iEN

If we call any vector r having the former characteristics a reduc­tion, then the aim of the Representation Theorem can be de­scribed as that of establishing the existence of a reduction. Theuse of this term is justified because it effects a reduction of thedifferent heterogeneous labors to a common measure (which bydefinition is none other than labor-value).

To show that there is a reduction for a particular abstract la­bor structure is tantamount to showing that the market actuallyassigns a specific weight to each and every type of labor, eventhough this assignment may not be unique. Perhaps a physi­cal analogy will be useful here. Imagine a multiarm star-shapedbalance whose arms are of equal length and each pair of which

ABSTRACT LABOR 151

forms a constant angle, so that if the balance is hanged from thecenter of the star the arms are balanced. Supposing that thereare l production processes in the economy, we compare eachprocess with an arm of one such balance with l arms, assumingthat each arm has n numbered marks where lead spheres can behanged. The ith mark (i = 1, ..., n) in armj (j = 1, ..., l) indicatesthe place where the ith sphere must be hanged, and the distancefrom this mark to the center of the star is the amount ofconcretelabor of type i expended in thejth process. The problem then isto find l equal sets ofn spheres such that, hanging on each armthe ith sphere on the mark numbered i, the arms of the balanceare again in equilibrium, i.e. all of them still are parallel to thefloor. These spheres are the physical analogues of the weightsthat the market assigns to each type of concrete labor, and thefact that n spheres can always be found for equilibrium meansthat the value of each net output is decomposed as a contribu­tion of all the concrete labors that gave rise to it. Notice thatthe problem of the representation of abstract labor can alwaysbe posed in these terms, because given the cone spanned by anyfinite set ofproduction processes the assumption of constant re­turns to scale guarantees that multiples of the same can alwaysbe taken such that the price of the net output is the same forthem all. Actually, this proced ure will be followed in the proof ofthe Representation Theorem below.

Some reader might wonder whether the weights assigned bythe market to the different heterogeneous labors are not just thehourly wages assigned to them. Actually, we can build an exam­ple where this is so. Consider a productive structure in whichmaterial inputs are a linear function of labor inputs. More pre­cisely, for every type ofgood j (j = 1, .., rn), there are nonnegativescalars J11» ..., J1nj such that, for every x in Po with vector oflaborinputs x = [Xl" 'Xn] and vector of inputs X = [,~-1 .. '~m]'

n

~) = LJ1i)Xii=l

152 CHAPTER FIVE

where ~j is the jth component of ~ and Xi is the ith componentof x. Let p be a price system and let M be the matrix

[

j1JI

M = j1;2J

j1ml

j112

j122

j1m2

j1Jn ]j12n. .

j1mn

Then it is easy to see that the vector r given by

r = (1 + r)w + rpM

is a reduction. In fact, if x E A,

rx = (1 + r)wx + rpMxT

= (1 + r )wx + rp~

= (1 + r)(p~ + wx) - p~

= px - p~

= px.

It is easy to see that when the rate of profit is zero, r = w, i.e. thewage vector is a red uction of labor.

Nevertheless, it can also be shown that the wage vector is notalways a reduction. Consider the productive structure spannedby the set Po = {x, y}, where x = [(1,0), (2, 20), (1,45)] and y =[(0, 1), (2, 14), (1, 35)]. Then

M = [~ i~].

Set w = [1,2], P = [1, 1]. It follows that 11"(x) = 1r(Y) = 1and trivial computations show that in spite of the fact that byTheorem 4 p is admissible, and pi = 24 > 20 = pr, still wx =1 < 2 = wy.

Very much in the style of chapter 3, I shall proceed now toprovide a representation theorem for abstract labor structures.

ABSTRACT LABOR 153

I will prove that there is at least one positive reduction and thatthis reduction is unique up to affine transformations. The theo­rem does not depend logically upon Axiom (P4) but this does notmatter, since the aim of that axiom is only to prove that the com­petition ofcapitals leads to abstract labor whenever the processesin the economy are normal. At any rate we have the followingcentral result.

THEOREM 7: (Representation Theorem). Let (£,:::) be the ab­stract labor structure corresponding to the set ofproduction processes Pand the price system p, and suppose that P is a convex polyhedral conegenerated by a set Po that satisfies (P1) and (P3). Then there exists apositive vector r such that, for every x, y E £,

rx 2: ry iff X::: y.

If r' is any other such vector, then there exist 0' > 0 and a vector b suchthat r' = O'r + b.

Proof' Let {XI, ..., xd (k ::; n) be a set of nonnull, linearly inde­pendent vectors of labor inputs spanning the cone £, chosen insuch a way that PXI = ... = PXk, where Xl = [Xl> ~I' xd, ... ,Xk = [Xk, ~k' xd are production processes in P, in which XI> ••• ,

Xk appear as labor input vectors. By (P3), it follows that Xi 2: 0and so (since p is valid) p = px; > 0 for all i (i = 1, ... , k).

Let X be the matrix

[XUX2J Xnl

-PjXl2 X22 X n 2 -pX= . . .

Xlk X2k Xnk -p

If the system XT c 2: 0 has no solution, then Stiemke'sTheorem l6 implies the existence of a vector d = [15 1 " • Dn+d > 0such that XdT = 0, i.e. such that

154 CHAPTER FI VE

In such a case, setting ri = O;jOn+1 and r = [rl" ·rn ] we get

+rnXnl =p

or, in abbreviated notation,

rXi = p

+rnXnk =p

(i = 1, ...,k)

Thus, if x, yare any elements of L, then there are unique non­negative numbers aJ, ..., ak and (31, ... , (3k, such that x = L~=] aiXiand y = L~=I (3iXi. It follows that there are productive processesxand y in P such that x= [x, & x], Y= [y, y, y], i = x - ~ and~ - h ""k ""k - (l - ""k -Y = Y - y, w ere ~ = LJi-1 a~i' Y = LJi=] tJ~i' X = LJi~1 aXi,- ""k fi- ~ ""k ~ ~ ",,7< (l~ S IY = LJi~1 tJXi, X = LJi~l aXi, y = LJi~l tJXi. 0 we 1ave

k k

rx 2: ry ¢:> 2:= a irxi 2: 2:=(3i rxii=1 i=1

k k

¢:> (2:= a;)p 2: (2:= (3i)Pi=) i=1

¢:> pi 2: py¢:> x ::: y.

Thus, it will suffice to show that XT c 2: 0 has no solution.In order to derive a contradiction, assume that there is a so­lution c = bl'" IdT of XT c 2: O. Since [-p . .. - P]c 2: 0,it follows that not all the nonzero coordinates of c are posi­tive. Analogously, since the remaining rows of XT are semipos­itive, not all nonzero coordinates of c are negative. Mter col­umn permutations and renumbering, we can let II> ... , Ih be the

ABSTRACT LABO R 155

nonnegative components and Ih+h '0" Ik be the negative onesof c. It follows that there are elements u and v in I:- whichcan be written as u = E~~l/iXi and v = E~~h+I(-/i)Xi' Since[-p ... - P]c = -p [E~~l/i + E~=h+l I;] ~ 0, it follows that

(1)k h

(2:: (-/i»)P ~ (2:: li)Pi=h+l i=]

or, what is the same,

(2) pv ~ plio

Also, since the column vector that results from dropping thelast component of XT c is nonnegative, and this is none otherthan the transpose of E~=l liXi, it follows that u = E~=l ,iXi ~E~=h+l(-/i)Xi = v and so, since p is admissible,

(3)

Hence, from (2) and (3),

pli ~ pVo

~ ~

pu = pVo

On the other hand, the following chain of implications holds:

pli = pv ::} p(li - v) =°k

::} p(2:: 'iii) = °i=)

.=1

i=1

i=)

156 CHAPTER FIVE

Thus, given that XT cis semipositive, it follows that u ;::: v and sothat pii > p\'. This contradiction shows that the system XT c ;::: 0has no solution.

Any other reduction r' is a solution to

X'(r')T=[::: ::: ~:: :::](r'1'=0[1]Xlk X2k • •• Xnk

for a > O. In particular, arT is a solution to this same equation,since we have

X'(orT) = 0[1]

Therefore, (r')T differs from arT by a vector bTin the null spaceofX, i.e. r' = ar + b. This establishes that reductions are uniqueup to affine transformations. 0

As I said before, if r is a reduction and x E £, then the mag­nitude rx is called the labor-value of net output X. This conceptis so important (in fact, this is the central concept of MTV) thatit is worthy of being given a separate official definition.

DEFINITION 8: Let SlJ = (Po, P) be a productive structure, p aprice system, and £, an abstract labor structure correspondingto SlJ (i.e. to P) and p. For any net output x in the cone C ={x I[x, x, xl E P and x = x - x}, we define the labor-value of x,A(X), as the number rx, where r is a positive reduction.

Unlike the definition of labor-value provided by Marx in theprototype of MTV, that lead to a deviation of equilibrium pricesand values, the definition of labor-value in terms of reductionsleads directly to the following strong form of the Law of Value.

THEOREM 8: (Law ofValue). Let SlJ be aproductive structure and pan equilibrium price for SlJ, i.e. the profit rate of all nonnull processes inSlJ is uniform at p. Then the labor-value of any net output vector xE Cis proportional to its price.

ABSTRACT LABO R 157

Proof" It is immediate by Theorem 4 and the definition of~ thatp is valid for ~, and so there is an abstract labor structure (£,~)

corresponding to ~. Let r be any reduction representing thisabstract labor structure such that rx = pi for every x E P. Ifr'is any other reduction, for fixed nonnull x E £ let 0' be such that

,r x = O'rx.

Clearly, 0' > O. Let y be an arbitrary element of £ and f3 thenumber such that ry = f3rx = r(f3x). Then y ,...., f3x and so, sinceby hypothesis r' is also a representation of~, r'y = r'(f3x) =f3r'x = f3O'rx = O'r(f3x) = O'ry.

This chain of identities shows that the number 0' does not de­pend upon the particular vector of labor inputs x. 0

The Law ofValue can be interpreted as establishing the reflec­tion of all vectors oflabor inputs into a particular one. For givenreductions rand r', we know by the Law of Value that there is apositive scalar 0' such that r'x = O'rx for any x E £. By the con­tinuity of r (seen as a function from the cone t: into R+), there isa vector oflabor inputs Xo such that rxo = 1/0'. For every x E £,

rx = (rxjrxo)rxo = r((rxjrxo)xo).

Hence,x,...., (rxjrxo)xo = (O'rx)xo = (r'x)xo;

that is to say, the amount of abstract labor represented by x isreflected in Xo through the coefficient r'x: vector x is equivalentto r'x times vector xo. Hence, at bottom, a change of reduction isjust a change in the vector oflabor inputs which is taken as unit.This should be compared with the change of unit rod in lengthmeasurement. From now on, whenever we talk of abstract labor,reductions, and labor-values, we shall suppose that in each casea particular vector of labor inputs has been chosen as unit ofmeasurement.

Chapter 6

THE GENERAL AXIOMS OF THE THEORY

Along chapter 2 we saw the tribulations of the prototype ofMTVin the face of the hard fact that the competition of capitals leadsto a uniform profit rate, and so that values deviate from prices-contrary to what the Law of Value c1aims- unless we assumethat the organic composition of capital is the same in all indus­tries. Hence, it is easily seen that the notion of competition ofcapitals plays a central role in Marxian economics. Now, thiscompetition ofcapitals requires to be effective a financial capitalmarket, and therefore we are compelled to introduce a concep­tual apparatus to describe such a market.

Also, the very notion of a uniform profit rate is introducinga concept of equilibrium within MTY. According to Marx, theaim of the capitalists as such is to maximize their benefits. As theclassical economists would put it, the capitalists are looking aftertheir own enlightened self-interest. In real life, however, the cap­italists' looking after their own self-interest can adopt multipleforms. It can mean that the capitalists restrict their benefits forsome time, to prevent a social problem from exploding, or thatthey are prepared to give money to other enterprises in order to

prevent the collapse of the economic system. In other words, thecapitalists can and do sometimes also act as politicians pursuingthe welfare of their society, as a condition for the maintenance of

159

160 CHAPTER SIX

institutions necessary to the functioning of the market. Indeed,it would be considered as stupid nowadays to restrict the role ofthe businessman to that of the brutal horno ceconornicus, in disre­gard of all the other tasks he has to perform in his society andhis responsibilities thereto.

As Hegel pointed out, there is more to the State than the mereprotection of property. Yet, Hegel recognized that there is asphere -a "moment"- in social life where the rules ofthe mar­ket have the upper hand and completely impose themselves inall activity or transaction. This is the moment of civil or burgeoissociety (biirgerliche Gesellschaft), the moment of "universal ego­ism". Clearly, MTV intends to discover and formulate the lawsthat govern this sphere. But, within this sphere, the enlightenedself-interest of the capitalist is not to look after the general wel­fare as a citizen of the polis, but to look after the maximum benefitfor his own enterprise. This is the kind of behavior with whichMTV is concerned. Evidently, this behavior has been modeledby neoclassical economics through the notion of maximizationof certain functions over prod uction possibility sets that satisfyspecial conditions. This is of course an idealized depiction of theactual way in which entrepeneurs take decisions (usually underconditions of uncertainty), but a depiction that is necessary inorder to logically derive the central results of MTY. In this re­spect of making strong assumptions to derive important results,MTV and the theory of competitive equilibrium do not differ atall.

Furthermore, it is plain that neoclassical equilibrium andMTV are complementary and conceptually akin -despite thelong war between "bourgeois" and Marxian economists. We shallsee that the Marxian notions of equilibrium and reproducibilitynaturally lead to the neoclassical concept of a competitive equi­librium. Hence, it is not to be fashionable that these notions areintroduced here, but as a response to the very conceptual de­mands of MTY. This shall be apparent in the sequel.

In the first section I will provide a description of the kind ofeconomic objects described by the theory; these turn out to be

THE GENERAL AXIOMS 161

Meinongian economies (in the sense of chapter 4), i.e. idealizednonexistent objects that bear some resemblance to modern mar­ket economies. Due to this reason, and to the fact that they areinspired in the description of the capitalist mode of productiongiven by Marx, I will call them 'Marxian capitalist economies'.These economies are more general than the prototype of MTVas described in chapter 1, and comprise that model as a partic­ular case. The fundamental law describing these economies isthe Law of Profit Maximization, but the Law of Value is a logicalconsequence of the same. In other words, I shall prove the Lawof Value out of the postulated behavior of the capitalists. This isgiven as a theorem in the same section. In the second section Ipresent the proof of the most general version of the Fundamen­tal Marxian Theorem, according to which the exploitation of theworking class is necessary and sufficient for capitalist profit. Insubsequent chapters the consistency and relevance of the gen­eral theory presented here will be apparent, where the possibilityof the reproduction of the economy is proven, and an interest­ing specific model of the theory is developed in a quite detailedway.

6.1 A GENERAL MARKET ECONOMY

Mter the impressive collapse of the Soviet socialist monster, andthe yearning of the Soviets and Eastern Europeans after marketeconomies, the study of the essential nature of these economieshas become more interesting than ever. What are the laws andconditions that make modern capitalist economies work so ef­ficiently? In order to study in general the (reformed) Marxianresponse to this question, we shall consider a Meinongian econ­omy which can be thought ofas the capitalist economy in its idealpurity. This economy is constituted by I firms, represented bythe numbers 1, ... , I, collected in a set F. An arbitrary memberof this set shall be denoted by the index f. Every firm f E Fis governed by a decision organ -a president, a board of di­rectors, or what have you- that has to face at the beginning

162 CHAPTER SIX

of every economic cycle (say, a quarter or a year) a price sys­tem p in the standard simplex 5 = {[PI·· ·P>n] I L'::.IPi = I}(the firms are price-takers) and decide on the basis of such pricesystem which production process to operate among a set Yf ofpossible prod uction processes. The firm! possesses at the be­ginning of the cycle certain initial holdings devoted to produc­tion (it may have "in stock" other commodities) represented by avector h f E R>n. These holdings are just enough for! to operatesome process in Yf , i.e. there is a production process x E Yf suchthat h f = ~ + Bx. Firm! may operate this process, but it alsohas the option ofobtaining or providing some credit in order tooperate a different production process, including 0(the firm hasthe possibility of inaction). I shall assume that the firm might bewilling to transfer to other firms its whole initial vector of pro­ductive holdings, as well as to hire a credit with which it can buyall the initial holdings ofthe other firms. Hence, the set of creditspossible for or by! will range from -ph f to ph-ph!" Notice thatthe range of credits available for any! depends on the prevalentsystem of prices and may vary as the price system changes. Atany rate, I shall assume that the total social resources t (whichmust be greater than the aggregated initial productive holdingsof the firms) are sufficient to operate any aggregated processx E Y such that p~ + pBx ::; ph, for any price system p E 5,even though it might well be the case that ~ + Bx ~ h. In thislast case, for example, the system of credit may be used to bringbundles not belonging to the sphere ofproduction (the originalh/s), but rather to the stocks of some firm, in order to enlarge abit the original global vector of initial endowments h = LfEF h f

(this may happen when some prices in the system are zero, i.e.when some goods are free).

Given a price system p and before deciding which process tooperate, firm! must ponder which production processes can op­erate using its initial resources and shop around a little to see ifit can make some transactions allowing it to operate a differenttechnology. In other words, the directives of any firm have tomake up their minds as to which production process x E Yf are

THE GENERAL AXIOMS 163

they going to operate, so that -if they decide to do so- theywill be able to use a part or all of their initial holdings and ar­range some credit in order to acquire (or give away) means ofproduction in the amounts Cf so that h f + Cf ~ ~ + Bx for somex E Yf . I shall assume that no matter how this decision is taken(we shall see that profit maximizing drives the firms to use uptheir resources to the maximum), the firm will always choose aprocess in Yf that maximizes profits at p subject to the constraintthat p~ + pBx :s ph f + pc j' I will assume that the transferencesCf of resources are paid at the end of the cycle (when all pro­cesses end simultaneously), and so the quantity pC

fis the credit

hired by the firm.According to what has been said, the set of all possible trans­

ferences for firm! at price system p will be defined as

Cf(p) = {c Ih f + C = ~ + Bx and p~ + pBx :s ph}

for some x E Yf . Since the monetary costs of these transferencesdepend upon which price system is prevalent, the range ofcred­its possible for! at p is the set

The financial capital market is depicted in the present theoryunder the guise of credits available to or by the firms (the inter­vals If (p» and a uniform interest rate r at which credit is hired,i.e. if firm! is lent (lends) c guilders, then it has to pay back (orreceive) (1 + r)c guilders. As it can be shown, due to the conicstructure of the prod uction possibilities sets it is really indiffer­ent to the firms which credit they hire: their level of profits inproductwn is not affected by the amount of the credit taken orgiven. Hence, profit maximization does not constrain the firmsto choose this or that credit; this is left to other considerations.What matters here is that credits are available anyway, at a fixedinterest rate, and that it is possible for any financial capital toobtain the same profit anywhere, even choosing the null process

164 CHAPTER SIX

(inaction), provided that they transfer the production means tobe used up in some production process by another firm.

The production possibilities sets Yj are far more general thanthe Leontieftechnology of the simple Marxian economy studiedin chapter 1, §2. I suppose that such sets contain the possibilityof inaction, i.e. the null process O. The technologies they repre­sent yield constant returns to scale and so they are cones, but notnecessarily polyhedric and not necessarily the same for all firms(there may be imperfect entry); the assumption of constant re­turns to scale implies, in particular, that all the commodities andamounts of labor are infinitely divisible. I suppose as well thatthe limit of every convergent sequence of production processesin any production set is also in the set, i.e. the sets are closed.Furthermore, I suppose that labor is indispensable to move themeans ofproduction (~ ~ Q =} x ~ 0), that there is no free lunch(x ~ 0 =} x ~ 0), and that labor is productive (x ~ 0 ~ x ~ 0).The last assumption over the production possibilities sets is thatthe global set of possibilities of production, the aggregate setY = L~~I Yj , is normal. What this means is that labor in all theavailable possible production processes is normal in the sense ofDefinition 6 of chapter 5. This assumption means that the laborfirms can hire is immediately social, i.e. labor that has the aver­age dexterity and skill, and that in all processes is put to workwith an average speed and intensity.

As a result of the decision of the firms, a finite setPo ofproduc­tion processes is implemented. I shall consider the polyhedralcone P spanned by this set. This set P can be thought of as thetechnology actually chosen by the firms. A member ofPo (one ofthe processes chosen by the finns) thus appears just as the tech­nology P operated at a certain level. The decisions taken by thefirms shall be represented by means ofa function d: F -. Y fromthe firms to the aggregate technology set Y, such that d(h) E Y isthe decision taken by firm J E F

There is a demand function for every kind of concrete la­bor. The "reproduction" of one hour of labor of type i (i =1, ... , n) requires the consumption of the (column) vector hi by

THE GENERAL AXIOMS 165

the worker's household. All these consumption bundles are puttogether as columns of the matrix B, the consumption bundleof trade i being the ith column of B. We shall assume that theconsumption bundles b i belong to the cone C = {i I[x, ~ xl EP and i = x -~} of net outputs ofP.

These concepts are very abstract determinations. From alogico-mathematical point of view, they are abstract entities withcertain mathematical properties; from an economic pointofviewthey are notions that intend to describe historic categories cor­responding to the capitalist mode of production. Taking theseabstract concepts as a point ofdeparture, I shall follow the dialec­tical method as it was explained on chapter 4 and build a classof structures, namely the class of Marxian capitalist economies,that intend to represent a particular class of states of a capital­ist economy. Before I reach that concept, I would like to gathertogether in a unique axiomatic definition the former concepts,even though the mathematical characterization of some of themhas been already given in previous chapters. To this effect I willintroduce the concept of a basic Marxian structure.

DEFI NITION 1: A basic Marxian structure is a structure of theform

9J( = (F, Yj, 5, hj, t, d, r, B)

that satisfies the following axioms:

(1) There is a finite set of firms, represented by the set F ofthe first l positive integers.

(2) For every firm! E F, the production possibility set Yf offirm! is a subset of the nonnegative orthant n ofR2

>n+n.

Yf is a closed convex cone in which labor is both pro­ductive and indispensable, and there is no free lunch.Furthermore, labor is normal in all prod uction processavailable to the firms, i.e. the aggregate set Y is normal.

(3) The set of price systems is the standard simplex 5 ={fj}J' "pm] I "£':JPi = I}

166 CHAPTER SIX

(4) The initial endowment h f of every firmf E F is a semi­positive vector in Rm such that h f = ~ + Bx for somei E Yf ·

(5) The global social resources is a vector tERm such thath ~ t, where h = l:hEF h f > 0 is the aggregated vector ofinitial holdings of the firms. Moreover, for every pES,~ + Bx ~ t for every i E Y such that p~ + pBx ~ ph.

(6) The decision function d is a function from F into the ag­gregate prod uetion set Y.

(7) The rate of interest r is a nonnegative real number.

(8) The demand matrix B is a m x n matrix. Every columnhi of this matrix is semipositive and is in C for every i =I, ... , n and, for every kind of good i = I, ... , m: eitherthere is a process i E Y such that Xj = b, where b is apositive entry in the matrix B of consumption bundlesfor the working class, or there is a process i E Y suchthat ~i > 0, but not both.

Definition I provides the basic conceptual framework, to­gether with the mathematical characterization of the conceptsof the theory, which is required to formulate the substantial ax­ioms of MTY. Notice that Axiom (8) implies that there are twomutually exclusive kinds of goods in the economy: wage andcapital goods.

Before the substantial axioms ofMTV can be introduced, it isnecessary to develop a little bit this conceptual apparatus. Thisis the aim of the following definitions and statements

DEFINITION 2: Let p be a given price system and define

(I) The set of all possible transferences for firm f at pES is

Cf(p) = {c Ih f + c = X + Bx, i E Yj, p~ + pBx ~ ph}

(2) The set of credits possible for firmJ at pES is

THE GENERAL AXIOMS 167

(3) The set of all possible initial monetary resources for firm f atprice system pES is

I shall prove now that the just defined objects have certainproperties which are important for the development of the the­ory. Recall that the norm IIvll of a vector v = [VI' .. v.] is its dis­tance to the origin of the linear s ace to which it belongs, givenby the formula IIvll = vi + ... + v;. Say that a set X ~ Y ofproduction processes is bounded iff the set of the norms ofall pro­cesses in X is bounded, i.e. the distance of any of these processesto the null process (the origin) is never greater than a specifiedpositive real number.

LEMMA 1: IfX isa subset ofY such that X = {xl[x,~x] EX}isbounded, then X is also bounded.

Proof" IfX were unbounded, there would be an unbounded se­quence in X. Thus, it will be sufficient to show that every se­quence in X is bounded. To this effect, assume that (xd is anysequence in X. If (Xk) were unbounded, it would have a subse­quence -call it also (Xk)- such that (1lxklD would be increasingand unbounded. At any rate, the corresponding sequence of la­bor input vectors (Xk) can be assumed to converge to a limit x(not necessarily in X) because X is bounded. Let

Since Y is a cone, Yk E Y. Moreover, (Yk) is bounded becauseIIYkll :::; 1. Hence, without loss of generality we may assert that(Yk) converges to a point Y which must belong to Y, because Y isclosed. Since (IIYkID is increasing, Y 1= 0 and so, by Axiom 2 ofDefinition 1, y 2': O. On the other hand, since (Xk) ~ x,

lim yk = lim (lIxkll + 1)-1. lim Xkk-oo k-oo k-oo

= O·X

= o.

168 CHAPTER SIX

Hence, be the same axiom again, y = O. This contradictionshows that (Xk) is bounded. 0

LEMMA 2: For every f E F and p E 5, /j(p) is a nonempty compactinterval of real numbers, and so it is :=j (p).

Proof' I will show, succesively, that /j(p) is nonempty, bounded,connected and closed. Since <p(c) = pc is continuous on C j forevery p E 5, /j(p) is nothing but the range of <p, and convex­ity implies connectedness, it will be sufficient to show that Cj isnonempty, bounded, closed and convex.

First of all, since h j + 0 = h j = ~ + Bx for some x E Yj andphj S; ph, it follows that 0 E Cj(p).

Since the set X = {x IpBx S; ph} is bounded, by Lemma1 the set X = {x IpBx S; ph} is also bounded. It follows that{~ + Bx - h j Ix EX} = Cj(p) is bounded.

Let (Ck) be a sequence in Cj(p) converging to a vector c. Itfollows that there is a sequence (~k + BXk) in {y + By lyE X}such that ~k + BXk = h j + Ck and P~k + pBxk S;-ph. Let (Xk) bethe sequence of production processes in Xcorresponding to theinputs (~k + BXk)' Since X is bounded, without loss of generalitywe may suppose that (Xk) converges to a point xwhich must bein Yj , because Yj is closed. It is easy to see that

lim(~k + BXk) = ~ + Bxk-oc>

and also thatp~ + pBx S; ph.

The verification that Cj(p) is convex is simple and is left to thereader.

Since all the credits thatf can give or take have the structure

c = p~ + pBx - ph j

for x E Yj and p~ + pBx S; ph, the maximum point of /j(p) isph - phj' whereas the minimum is -ph!, In other words, /j(p)

THE GENERALAJ\lOMS 169

is the interval [-phj, ph - phj]. Hence, '=j(p) is nothing but thetranslated interval [0, ph]. 0

We have shown, incidentally, that the set of financial resources'=j(p) depends on the price system but not on the firmf. Hence,from now on I will drop the subindex 'f' from all such expres­sions, except when it is necessary to stress that it is referred tosome firm. When p is specified, the variable f. will range over theelements of '=(p). Hence, it can be substituted by an expressionof the form 'phj + c', where c is a variable ranging over Ij(p).

DEFINITION 3: Let a basic Marxian structure 9J1 be given. Foreach firmf E F we have the following concepts:

(1) For each p E 5, let .=(p) be the interval [0, ph].

(2) The financial feasibility function for firm f is the mappingB j : 5 x .= ~ Yj such that, for every (p, 0 E 5 x .=,

Sometimes I shall write B j(p, c), when I need to point outthat f. is phj plus the credit c.

(3) The profit maximizingfunction for f is the mapping II j: 5 x.= ~ III defined by

IIj(p, 0 = max{px - pBx - rc Ix E B j(p, f.)}.

where f. = phj + c for c E Ij(p).

(4) Theoptimalityfunction forf is the mappingA j :5x.= ~ Y j

such that

As before, sometimes I shall write Aj(p, c), when I needto point out that f. is ph plus the credit c.

170 CHAPTER SIX

(5) The global financial feasibility function is the mappingB: S x := -+ Y such that, for every (p,~) E S x :=,

B(p, 0 = {i E Y Ip~ + pBx ~ O·

(6) The global profit maximizingfunction is the mapping II: S x:= -+ R defined by

II(p,~) = max{pi - pBx liE B(p, ~)}.

(7) The global optimalityfunction is the mapping A: S x := -+ Ysuch that

A(p,~) = {i E B(p,~) Ipi - pBx = II(p, ~)}.

(8) Po, the set of all actually operated processes in VJl is the set ofall processes actually chosen by the firms:

{d(h) If E F}.

(9) P, the actual technology of VJl, is the convex polyhedralcone spanned by Po.

(10) L, the set of labor inputs of VJl, is the set of all vectors oflabor inputs of the processes in the actual technology ofVJl:

L = {x I[x, ~ xl E P}.

(11) The price system p is said to be feasible for firm f E F iffII/(p, c) > 0 for some c E I/. We say that a price system isfeasible if it is feasible for every firm.

(12) A good of type i (i = 1, ... , rn) is called a wage good ifthere is a process i E Y such that Xj = b, where b is apositive entry in the matrix B of consumption bundlesfor the working class. A good of type i (i = 1, ... , m) iscalled a capital good in the economy if there is a processi E Y such that~i > O.

THE GENERAL AXIOMS In

Less precise definitions of concepts (2)-(4) are due originallyto Roemer (1981).1 It is illustrative to see the motivation behindthe definition of IIf . Actually, ifcredit c is borrowed, the profit is

max{px + [c + phf - (p~ + pBx)]

-[(1 + r)c] - phf Ix E Bf(p,c)},

"where the terms are, respectively, income from production, thevalue of assets not used in prod uction but held over to the nextperiod, the value of borrowing repaid, and the value oftoday'sendowments".2 But the expressions within the brackets can besimplified to obtain clause (2) ofthe definition. Using the notionsintroduced in Definition 3 it is possible to formulate the funda­mental law that characterizes the Marxian capitalist economies.This is the Law of Profit Maximization, a law about the behaviorof the capitalist as such, according to which he always choses tooperate a productive process that maximizes his profit. The Lawof Value (which is the law that defines MTV; as I said in chap­ter 2, §1), as well as the Fundamental Marxian Theorem, can bederived from this axiom and so, in effect, MTV is a logical con­sequence of the theory determined by the following definition.

DEFINITION 4: A Marxian capitalist economy is a basic Marxianstructure in which all firms maximize their profit with respect toa feasible system of financial resources. That is to say, the follow­ing law holds for everyf E F and c Elf:

d(h) E Af(p, ~f) and L ~f = ph.fEF

Definition 4 defines the models of the Marxian economic the­ory. The Law of Value has to be proven to hold as a consequenceof the Law of Profit Maximization. I n Marxian dialectical termsthis means that the "essence" is implied by the "phenomenon",but this terminology is not well applied here, because it is as es­sential to capitalism profit maximization as it is the reduction ofconcrete to abstract labor. I n deriving the Law of Value from

172 CHAPTER SIX

profit maximization we in effect establish that the process ofre­duction ofheterogeneous labors is effected by that behavior. Thisis the import of the first theorem of the chapter, whose demon­stration requires the support of two previous lemmas.

LEMMA 3: In order to maximize its profits firm f E F activates anychosen nonnull technology to the limit ofits resources; i.e. ifx E Af(p, c)then px + pBx = phf + c.

Proof" At every point x ofAf(p, c) the foHowing equation is satis­fied:

px + pBx = ph f + c.

In effect, ify is such that

Pl + pBy < ph f + c,

then there is an a such that

a(Pl + pBy) = phf + C.

Letz be the process ay. Then p?;+pBz = phf +c and, moreover,

pz - pBz - rc = a(py - pBy) - rc

> py - pBy - rc

because a > 1. It is thus seen that at any rate it is more profitablefor f to activate the process z = ay than the less intense processy.DLEMMA 4: If process x maximizes profits at credit c then, for everycredit c' E If(p), there exists a j)ositive a such that ax maximizes profitsat credit c'.

Proof' Assume that xE Af(p, c) and suppose that, for every a >0, ax tt Af(p, c'). Then, in particular, ax tt Af(p, c') for p(ax) +pB(ax) = phf + c'. Hence, there is3 ayE Af(p, c) such thatPl + pBy = ph f + c' and

py - pBy - rc' > p(ax) - pB(ax) - rc'.

THE GENERAL AXIOMS

Thus, if j3 is such that j3(phj

+ c') = phj + c, then

j3O'(p~ + pBx) = ph j + c = p~ + pBx

and so j3 = 0'-1.

Therefore,

and yet

173

This implies that x E Bj(p, c) and also x rf. Aj(p, c), a contradic­tion. 0

THEOREM 1: The Law of Value holds true in a Ma'rxian capitalisteconomy, provided that the price system is feasible.

Proof" At point c E I j , II j adopts the value

IIj(p, c) = pi - pBx - rc

= 1I"(x) . (phj

+ c) - rc

where 1I"(x) is the rate of profit ofx as defined in Definition 4 ofChapter 5. Actually, by Lemma 4, ify maximizes II j at c', thenthere is an 0' such that O'X also maximizes IIf at c' and so

i.e. the profit rate 11" of the profit-maximizing processes is con­stant at all credits c E If. Hence, we can write

IIf(p, c) = 1I"(phf + c) - rc

= 1I"phf + 1I"C - rc

= 1I"phf + (11" - r)c.

174 CHAPTER SIX

It is obvious that Ilf is differentiable with respect to c, its deriva­tive being

ollf&=rr-r.

Since If is compact, it follows that Ilf assumes a maximum atsome point c· E If and so, at that point,

ollf& = rr-r = O.

This shows that the interest rate is actually equal to the profitrate and that Ilf is constant, its form being exactly

II f(P, c) = rph f'

i.e. the function Ilf refers only to the profit obtained over theinitial endowments, not over these plus the credits hired.

By the Law of Profit Maximization, all processes XI, ... , x/ inPo, the processes chosen by the firms, have the same profit rater, which is positive because the price system is feasible. Since thepositive hull P+ ofPo is a subset ofY, and Y is normal (by Axiom2 of Definition I), it follows by Theorem 2 of chapter 5 that P+is also normal. Hence, by Theorem 3 of the same chapter, theprofit rate of all processes in P+ is uniform and equal to r. ByTheorem 4, p is valid for P, the actual technology of VJl, andso it ind uces an abstract labor relation::: over the set I: of laborinput vectors of such technology. In other words, (1:,:::) is theabstract labor structure corresponding to the price system p andthe technology P (Definition 3, chapter 5). Since llJ = (Po, P) isa productive structure, by Axiom 2 of Definition 1, Theorem 8of chapter 5 yields the desired conclusion. 0

It is fashionable nowadays to speak of the "efficiency" of mar­ket economies. One way of making sense of this phrase is to saythat the egoist behavior of the firms guaran tees that the most ef­ficient processes (in the sense of Definition 5, chapter 5) are op­erated, consistent with the restriction in capital stocks and credit.This is certainly true in a Marxian capitalist economy in whichthere are no free goods and is established here as a theorem.

THE GENERAL AXIOMS 175

THEOREM 2: If the price system is positive and feasible, the processchosen by each firm is efficient.

Proof" Let x be the process chosen by firmf E F and w = pB. Ifxwere not efficient, there would be a process y E Yf such thaty ~ x and y ~ i, with at least one of these two inequalities beingstrict. Since p, w > 0, we have wy < wx or py > pi and so, atany rate, py - wy > pi - wx. Since Y is normal, y ~ x impliesy ~ ~ and so py + wy ~ p~ + wx ~ ph + c. This means that theprofit at y E B7(p, c) is greater than that at x, contradicting theassumption. 0

Having established how the behavior of the capitalists impliesthe Law of Value, as well as the efficiency of market economies, Ishall proceed now to prove the Fundamental Marxian Theorem,which is also a general theorem that follows from the essence ofMarxian capitalist economies.

6.2 THE FUNDAMENTAL MARXIAN THEOREM

Perhaps the most classical and single important result of MTVis the proof that the exploitation of the workers is both a neces­sary and sufficient condition for the existence of profits in thecapitalist firms. We already saw in chapter 1 this proof for thecase of the protoype. In that chapter we saw Marx's definitionof the exploitation rate, namely as the ratio of surplus-value tonecessary labor:

x-v[=--

vwhere Xi is total amount of live labor in the process operatedat the unitary level and v is the "necessary labor" of the same,i.e. the value of the subsistence basket that the workers of theprocess are able to buy with their sarary. Since we are no longerassuming that labor is homogeneous and that the technology isof Leontief type, we need to redefine these notions.

The total amount of live labor is associated now with the net­output of the process x, and is none other than the labor-value of

176 CHAPTER SIX

pi - pBx 0"----_--"---::- > .p~ + pBx

Hence, since p~+pBxcannot be negative, pi-pBx = pi-pb =>.(i) - >'(b) > O. Since b = Yfor some yEP and 7I"(y) = 71" > 0, it

the net output of this process: >'(i). The value of the subsistencebasket of the workers is equal to the labor-value >'(b) of the totalconsumption basket for the workers of the process, where thatbasket is given by b = Bx.

Hence, the exploitation rate at a non-null process x is just

(_) _ >.(i) - >'(b)

c x - >'(b) .

(By convention, we say that the exploitation rate of the null pro­cess 0 is 0.)

One of the aims of Marx in Capital was to unveil the fact thatthe exploitation of the workers by the capitalists is the "source"of the latter's profits. A precise formulation of this claim is theFundamental Marxian Theorem, that was proven for the firsttime for a Leontieftechnology with homogeneous labor by Mor­ishima and Seton (1961), and Okishio (1963) (see chapter 1 forthe details of the proof for the prototype of MTV). A proof ofthe theorem for a von Neumann economy with homogeneouslabor is provided by Morishima (1974). A proof of the theoremfor a Leontief technology with heterogeneous labor is given byKrause (1981). The following version is the most general thusfar.

THEOREM 3: (The Fundamental Marxian Theorem). For everylabor process x E P: the profit rate 7I"(x) oj x is positive if, and only if,the rate oj exploiLation c(x) oj x is positive.

Proof First of all, by the Law of Value, we may suppose thatpi = >'(x) for every x E P. Let x be any process in the actualtechnology P and suppose first that its profit rate 7I"(x) is equalto the uniform prevalent profit rate 71", which is positive. Thismeans that

THE GENERAL AXIOMS

follows that A(h) = ph > 0 and therefore the number

(_) _ A(i) - A(h)

c x - A(h)

is defined and positive.Conversely, if

A(i) - A(h) 0A(b) >,

177

since A(h) = ph = pBx cannot be negative, it must needs bepositive, and so A(i) - A(b) = pi - pBx> O. It follows that

11"(X) = pi - pBx > O. 0p~ + pBx

This theorem is absolutely general since it follows only fromthe assumptions that define the concept of a Marxian capital­ist economy. More than a mere "denunciation" of capitalism,this theorem is a very important explanatory condition for thereproduction of market economies. Besides, the concept of ex­ploitation should not be confused with the concept ofalienation:there may be exploitation without alienation and viceversa. Butnevertheless there is a tendency toward the increase of the ex­ploitation rate within capitalism in the form of a struggle of theparticular capitalist to keep the wages of his employees to themInImum.

A consequence of the FMT is that exploitation is necessary forthe feasibility of price systems. In order for a price system to befeasible, it is necessary that at that price every firm must be ableto operate a process at which the exploitation rate is positive. Ishall conclude the present chapter proving this interesting re­sult.

THEOREM 4: If the price system is feasible then every firm operates aproduction process whose exploitation rate is positive.

178 CHAPTER SIX

Proof" Suppose that p is a feasible price system. Then every firmcan operate a nonnull process yielding positive profits, and so itdoes it by the Law of Profit Maximization. Hence, the profit rateof all processes operated by the firms is positive and thereforeat price p the exploitation rate ofall the processes chosen by thefirms is positive. 0

Chapter 7

GENERAL REPRODUCIBILITY

The aim of the present chapter is to explore the conditions ofreproducibility of a general Marxian capitalist economy. I wantto consider here the conditions that make it possible for a marketeconomy to reproduce itself. Is it sufficient for reproducibility tolet the firms just pursue their enlightened self-interest, or some­thing more is required? Is it even possible to make compatiblethat behavior with reproducibility? These are the leading ques­tions of this chapter. Mter introducing a formal definition ofre­producibility, I will develop the conceptual apparatus to someextent, in order to prove the existence of what I shall call a 're­producible global decision' (RGD), i.e. to prove that the firmsneed not but can choose a technology that maximizes their prof­its and also guarantees the reproducibility of the economy_ Thisresult shall be established as the existence of a certain competi­tive equilibrium. Since the concept of (simple or extended) re­producibility is quite tipically Marxian, it will be evident that theconceptual framework ofgeneral equilibrium theory and that ofMTV are deeply related.

We have seen already (in the previous chapter) that the behav­ior of the firms guarantees the equalization ofthe profit rate, butthere is more to equilibrium than just a uniform profit rate. Theconcept ofRGD equilibrium we are interested in includes several

179

180 CHAPTER SEVEN

features: (1) it is a selection of production processes by the firmssuch that the process chosen by each firm maximizes the benefitfor that firm; (2) the processes are able to reprod uce themselves,in the sense that the aggregate process produces at least whatit consumes in terms of wage and capital goods, and perhapssome excedent; (3) the operation of the production processes isfeasible, in the sense that there are sufficient stocks accumulatedin the economy to implement the global process X. In symbolicterms, these features can be expressed as follows.

DEFI NITION 1: {XI, ... , Xl} is a reproducible global decision (RGD)iff there exist credits Cl, ... , CI and a price system p such that

(1) For every firm! E F, process Xf maximizes profits for!,i.e. Xf E Af(p, cf)

(2) The global process [x, ~ xl = LfEF Xf is nonnull andreprod ucible, i.e. x ~ X + Bx

(3) The global process is feasible, in the sense that there aresufficient stocks accumulated in the economy to imple­ment the global process, i.e. X + Bx ~ t.

(4) The capital market clears, i.e. LfEF Cf = O.

DEFINITION 2: (p', c;, ... , cn is a Ma'rxian comjJetitive equilibriumiff there exists a decision {Xl, ... , XI} that satisfies (l )-(4) of Def­inition 1 with credits c;, ... , ct and price system p'.

The aim of the following eleven, rather technical lemmas is toprepare the ground for the proof of the existence of a Marxiancompetitive equilibrium and, in this form, of a RGD.

LEMMA 1: For every (p, ~f) E S x :=j, B f (p, ~f) is nonempty, convexand compact. Analogously,for every (p, 0 E S x:=, B(p, 0 is nonempty,convex and compact.

Proof' Since ~f ~ 0 and 0 E Yf , we have

pi! + pBO = 0 :s; ~f

GENERAL REPRODUCIBILITY 181

and so 0 E Bf(P, ~f)' which establishes that Bf is nonempty.Let x,51 E Bf(P'~f) and a E [0, I]. Then p(a~) + pB(ax) =

a(p~ + pBx) ::; a~f' p[(1 - a)I] + pB[(1 - a)y] ::; (l - a)~f' andso

p[a~ + (I - a)I] + pB[ax + (I - a)y] ::; ~f

which proves that ax + (I - a)51 E Bf(p, ~f)'

Since the set X = {x Ix E Bf(p, ~f)} is bounded, becausepBx 2: 0 and pBx ::; ~f for every x E X, Lemma 1 of chap­ter 6 implies that Bf(P, ~f) is also bounded. In addition, Bf(P, ~f)

is closed, because if (Xk) is a sequence in the same set convergingto x, then (P~k + pBxk) -+ p~ + pBx ::; ~f with x E ff' whichshows that x E Bf(P, ~f)' Hence, Bf(P, ~f) is compact. The prooffor B(p, ~) is completely similar. 0

LEMMA 2: For every (p, ~f) E S X =1> the set Af(p, ~f) is nonempty,convex, and compact. Analogously, the set A(p,~) is nonempty, convex,and compactfor every (p, ~) E S x =.Proof Letf:Bf(p, ~f) -+ R be the function such that

f(x) = [-pB, _p,p]xT = pi - pBx.

Since the inner product with a fixed vector is a continuous func­tion, and Bf(P'~f) is compact by Lemma I, it follows that theset

has a maximum, by Weierstrass' Theorem, and so Af(p, ~f) isnonempty.

Let x,51 E Af(p, ~f) and a E [0, I]. Then pi - pBx =Ilf(p, ~f) = py-pBy, and so p[ai+( l-a)y]-pB[ax+(I-a)y] =a[pi - pBx] + (1- a)[py - pBy] = allf + (l - a)Ilf = Ilf . Thisshows that A f (p, ~f) is convex.

Af(p, ~f) is bounded because it is contained in Bf(P, ~f)' whichis a compact set according to Lemma 2. It is obvious that anyconvergent sequence in Af(p, ~f) converges to a maximizer inthe same set. The proof for A(p,~) is completely similar. 0

182 CHAPTER SEVEN

A correspondence <p:A - B is a functionf:A - 8J(B) from A tothe power set of B that assigns to each element x E A a subset<p(x) = f(x) ~ B. A correspondence <p is said to be lower semicon­tinuous (lsc) if, whenever (Xk) is a sequence in A converging to xand y E <p(x) , there is a sequence (yk) converging to y such thatyk E <P(Xk) for every k E w. Also, a correspondence <p is said tobe upper semicontinuous (usc) if, whenever (Xk) is a sequence in Aconverging to a point x and (yk) a sequence in B converging to ysuch thatYk E <P(Xk) for every k E w,y E <p(x).

LEMMA 3: For each ~I E 51' BI: S - YI is lsc. Analogously, B: S ­Y is lsc for every ~ E 5.

Proof' Let (Pk) be a sequence in S converging to pES, and letx E B I(P, ~/)' We have to show that there is a sequence (Xk) con­verging to x such that Xk E B I (pk, ~/) for every k. I f x = 0, we letXk be the null vector for every k and the result trivially follows.

Ifx is nonnull and px + pBx> 0 then, since (PkX + pkBx) ­px + pBx, there is a positive integer N such that

k > N implies PkX + pkBx > O.

Hence, defining

ifpkx + pkBx = 0otherwise

PkXk + pkBxk = Ak[PkX + pkBx]

= px + pBx

~ ~/'

Thus, Xk E B I (pk, ~/) for every k. Moreover, it is clear that (A k) ­1 and so (Xk) - x.

Finally, ifboth px = 0 and pBx = 0, let (Yk) be any sequenceconverging to X. Then we have (PkY ) - 0 and (PkBYk) - O.

-k

GENERAL REPRODUCIBILITY

Hence, there exists a positive integer N such that

k > N * PkY + pkBYk < (j-k

* yk E Bj(Pk,(j).

The conclusion follows if we set

183

ifYk ~ Bj(Pk,(j)otherwise.

An entirely similar argument establishes the proposition for theglobal function B. 0

LEMMA 4: (Berge's Maximum Theorem). For each (j E :=f>A j : S -+ Yj is usc. Analogously, A: S -+ Y is usc for every ( E :=.Proof" Let (Pk) be a sequence in S converging to p, and let (Xk) bea sequence converging to x such that Xk E Aj(Pk, (j) for every k.We have to show that x E Aj(p, (j).

Since Pk~k + pBXk :s; (j, it follows that

and so x E Bj(p,(j).In order to derive a contradiction, assume that x ~ Aj(p, (j).

Then there must be ayE B j(p, (j) such that

py - pBy = I > 0 = px - pBx.

It follows that the sequence (PkXk - pBXk) converges to 0 and,since Bj is Isc, there is a sequence (Yk) -+ Y such that Yk EBj(Pk, (j). Clearly, the sequence (PkYk - pBYk) converges to Iand so, for sufficiently large k, the profits that the operation ofYk would yield to firm f at prices Pk are larger than the profitsthat the operation of Xk by f E F would yield under the sameprices. This contradicts the hypothesis that Xk E Aj(Pk, (j), andso xmust be inAj(p,(j). 0

184 CHAPTER SEVEN

LEMMA 5: For every x E A(p, ph), there exists a decompositionXI, ... ,X/ ofx and credits CI, ... ,C/ such that Xj E Aj(p,cj) andLjEFCj=O.

Proof Let XI, ... , x/ be a decomposition ofx. Since

it follows that there is a Cj E C j such that

Hence,~j = phj+pcj = Plfj+pBxL

E -=j(p),andxj E Bj(p,~j).

IfYI, ... , YI are any processes with Yj E Aj(p, ~j), it follows that,for allf E F,

PY - pBYj 2: PXj - pBxrIfxj (j. Aj(p, ~j) for somef E F, we would have

PY - pBy = P(LYj) - pB(LYj)jEF jEF

> px - pBx.

Since Plf + pBx = ph, by Lemma 3 of chapter 6,

p(Lr) + P(LY) = L~jjEF jEF jEF

= LPhj + LpcjjEF jEF

= L(plfj + pBxj)jEF

= ph,

it follows that y E A(p, ph), contradicting the hypothesis thatx E A(p,ph). Finally, letcj = pCr Then LjEFphj + LjEFCj =LjEFphj implies LjEFCj = o. 0

Consider a set X and a correspondence e.p:X ---+ X. Afixedpoint of the correspondence e.p is an element x· E X such thatx' E

GENERAL REPRODUCIBILITY 185

<p(x·). The main result in mathematical economics regarding theexistence of fixed points is due to Kakutani. This shall be ournext lemma here.

LEMMA 6: (Kakutani) If X is a nonempty, compact, convex subsetof R\ and if <p is an usc correspondence from X to X such that for allx E X the set <p(x) is convex (nonempty), then <p has a fixed point.

For a proofofthis lemma, which is rather involved, the readeris referred to Kakutani (1941) or Nikaido (1968) in the Bibliog­raphy.

LEMMA 7: (Debreu). Let (: 5 -- Z be a correspondence from 5 to thecompact set Z ~ Rm. If ( is usc and, for all p E 5, ((p) is nonempty,closed, convex, and p((p) ::; 0, then there is a pin 5, and a z in ((p),such that z ~ O.

Proof In order to make use ofKakutani's theorem, a correspon­dence <p from 5 x Z into itself must be defined by means of thecondition:

<p(p, z) = J1(z) x ((p),

where J1: Z -- 5 is the correspondence that assigns to each z E Zthe set {p E 5 Ip maximizes pz on 5}. It is required to show that(i) 5 x Z is nonempty, convex and compact; (ii) <p is usc; and (iii)<p(p, z) is nonempty convex. When these conditions are fulfilledthere exists a fixed point (p', z·) E J1(z') x ((p') and we have thefollowing outcome.

p' E J1(z') implies that pz' ::; p'z' for every p E 5. z· E ((p')implies that p'z' ::; 0 and so pz' ::; 0 for every p E 5. Since forevery k E {I, ... , m} we can find a price system p E 5 whose kthcomponent,pk is 1, whilePi = 0 for every other componentj t= k(1 ::; j ::; k), it follows thatz~ ::; 0 for every k, which is tantamountto z· ~ O.

Thus, to show (i), since the Cartesian product of nonemptycompact convex sets has also these attributes, it suffices to no­tice that Z is nonempty because 5 is nonempty and the corre­spondence ( is nonempty valued. Z is by hypothesis compactand without loss of generality Z can be assumed to be convex,

186 CHAPTER SEVEN

for otherwise it can be replaced by the closure of its convex hullwithout altering the final result.

<p is usc because the Cartesian product of usc correspondencesis also usc and j1 can be seen to be usc. In effect, let (Zk) be asequence in Z converging to Z for which there is a sequence (Pk)converging to p such that Pk E j1(Zk) for every k. It is requiredto prove that p E j1(z). Let p' E j1(z) and notice that, therefore,p'z ~ pz = limk_oo PkZk. Hence, since PkZk ~ P'Zk'

lim PkZk ~ lim P'Zk = p·z.k-oo k-oo

It follows that p'z ~ pz ~ p'z and so pz = p'z, which means thatp maximizes pz on 5, i.e. p E j1(z).

<p(p, z) is nonempty convex. This follows from the fact that((p) is nonempty convex and that j1(z) can be shown to be so.Obviously, if p, p' E j1(z) then pz = p'z and so, for every 0: E[0, 1],

(o:p + (I - o:)p')z = o:pz + (I - o:)p'z

= pz

On the other hand,

m m m

"2.)o:Pi + (1 - o:)P;) = L o:Pi + L(l - o:)P;i=J i=l i=1

m m

= 0: LPi + (1 - 0:) LP;i=1

= 0: + (1 - 0:)

=1.

i=1

This shows that o:p + (I - o:)p' is a price system in 5 that maxi­mizes pz. Hence, it belongs to j1(z). This concludes the proof ofDebreu's Theorem. 0

For every p E 5, define the set

((p) = {Bx - xIx E A(p, ph)}.

GENERAL REPRODUCIBILITY 187

A correspondence is thus defined from the simplex S into the setZ = UpEs (p). We have the following lemmas concerning ( andZ.

LEMMA 8: (p) is nonernpty for every pES.

Proof' By Lemma 3, A(p,~) is nonempty for every ~ E E, in par­ticular for ph E E. Hence, for all x E A(p, ph), Bx - x E (p).o

LEMMA 9: Z is compact.

Proof' Let A be the set UPEsA(p, ph). Then Z is the image of Aunder the continuous functionf:A -+ Z, wheref(x) = Bx - X.Since the image ofa compact set in a linear space is also compact,it will suffice to show that A is compact.

Since S is bounded, {ph IpES} is also bounded and so it hasa supremum J1. It follows that if x E A, then pBx ~ ph ~ J1 andso {x Ix E A} is bounded.

If (Xk) is any sequence in A converging to x, for each k selecta price Pk E S such that Xk E A(pk, pkh). Since S is compact,without loss of generality we may suppose that (Pk) converges toa price pES. But, by Lemma 5, A: S --? Y is usc and thereforex E A(p, ph) ~ A, which establishes that A is also closed. 0

LEMMA 10: (is usc.

Proof' Let p be a point ofSand (Pk) a sequence in S converging top. Let (Zk) be a sequence in Z converging to Z such that Zk E (Pk).I will show that Z E (p).

Each Zk is of the form

where Xk E A(pk, pkh). It follows that

where J1 is as given in the proof of Lemma 9, and so the set ofvectors of labor inputs {Xk} of the terms in the sequence (Xk) is

188 CHAPTER SEVEN

bounded. Thus, by Lemma 1 of chapter 6 this sequence itselfisbounded and so we can assume that converges to a process X.Clearly, the limit Z of (Zk) must be equal to Bx - X. Since the cor­respondenceA is usc, it follows that x E A(p, ph). This establishesthat Z E ((p). 0

LEMMA 11: For every p E 5, ((p) is closed, convex, and p((p) :::; O.

Proof" Let (Zk) be a sequence in ((p) converging to z. In order toshow that Z E ((p), notice that Zk is of the form

for processes Xk E A(p, ph). Thus, pBxk - PXk :::; ph, whichimplies that (Xk) is bounded. Hence, we may infer that (Xk) con­verges to x, and likewise that (Zk) - Bx - X. Since A(p, ph) isclosed, x E A(p, ph), and so Z E ((p).

In order to show that ((p) is convex, let Z = Bx - x and u =By - Ybe any elements of ((p) and 0' E [0, 1]. Then

O'Z + (l - O')u = B(O'x + (1 - O'y)) - (ax + (l - O')y)

where X, YE A(p, ph). Since this set is convex, ax + (1 - O')y EA(p, ph) and so O'Z + (1 - O')u E ((p).

Finally, since pQ + BO :::; ph, no element x E A(p, ph) is suchthat px - pBx < O. Hence, for every Z = Bx - x E ((p), -pz =px - pBx ~ O. This proves that -p((p) ~ 0 and, therefore,p((p) :::; O. 0

THEOREM: There exists a Marxian competitive equilibrium.

Proof" From lemmas 7 and 11 follows that there is a p in 5 anda Z in ((p) such that Z ~ O. In other words, there is a processx E A(p, ph) such that Bx - x ~ 0, i.e. x ~ Bx. This establishesthat x is reproducible. Moreover, since h > 0 and p ~ 0, ph > O.On the other hand, x E A(p, ph) implies that p~+pBx = ph > 0and so x is nonnull.

GENERAL REPRODUCIBILITY 189

By Lemma 5, there exists a decomposition XI, ... , x/ ofx andcredits Ch"" C/ such that Xf E Af(p, cf) and LfEFCf = O.

Finally, since p~ + pBx ~ ph, Axiom (5) of Definition 1 inchapter 6 implies that ~ + Bx ~ t. 0

COROLlARY: There exists a global reproducible solution.

Proof' Immediate from the Theorem.

In this form we conclude the proofof the existence ofa RGD.It follows that it is possible to make compatible the global repro­ducibility of the economy with the pursuing ofself-interest by thefirms. On the other hand, it is easy to see that the firms mightchoose production processes that do not necessarily reproducethe economy. Supposedly, the law ofSupply and Demand shouldeventually lead the firms to choose precisely those processes thatreproduce the economy, but this fact (ifit is indeed a fact) is notreflected in the arguments provided here.

Chapter 8

THE PROTOTYPE REVISITED

After the long disquisition that we had to carryon in order tocope with the difficulties raised against the prototype of MTV,we are at last in an advantageous position to solve such difficul­ties. In the present chapter I will present, in a very detailed way,a reconstruction of the Leontief model of MTV with heteroge­neous labor. This model shall appear here as a model ofMTV inthe logical sense, i.e. as a Marxian capitalist economy, albeit onewith very special (and mathematically nice) properties.

In the first part of the chapter I will discuss again the "in­tended interpretation" of the mathematical theory. This dis­course is more than a mere formality: it intends to provide themathematical formulas with an economic meaning. It is just afact that the Leontief economy described thereby is a Meinon­gian economy that has no existence whatsover. At any rate, thatis the economy to which the concepts and equations of the the­ory apply, and real market economies are only roughly approx­imated by that model. I shall not discuss here what ProfessorSamuelson calls the F-twist, namely, that

It is a positive merit of a theory that (some of) its content and as­sumptions be unrealistic, since only ifit is not tailored closely to onesmall bit ofreality can it give a useful fit to a wide spread ofempiricalsituations. Unless we explain complex reality by something simplerthan itself we have accomplished little (period or by theorizing).]

191

192 CHAPTER EIGHT

Yet, it must be said that the problem of the Marx-Leontiefmodel is precisely that it fits too closely a small piece of irreality(!) and not at all the capitalist economies in their full universal­ity. It is an interesting problem in the philosophy of economicsthe usefulness of idealized models, a problem that I discussed tosome extent in chapter 4.

In the second section ofthis chapter I will present the Leontiefeconomy as a mathematical structure satisfYing the set-theoreticpredicate introduced in Definition 4 ofchapter 6 ('Marxian cap­italist economy'). Mter discussing the special properties of theRGD for this economy, I will prove that in this type ofeconomiesholds a very strong form of the Law of Value and that, eventhough the explotation rate might not be unique, yet there isan explicit function correlating these rates with the profit rate(which is unique). I conclude the chapter discussing the condi­tions under which full employment in the economy is guaran­teed.

8.1 THE LEONTIEF ECONOMY

A Leontief capitalist economy is pretty much like the simpleMarxian economy of chapter 1, except that here the firms havea whole set of production possibilities to choose and labor is notassumed to be homogeneous.

As in the simple economy ofchapter 1, at the beginning of aneconomic cycle the firms are endowed with certain initial hold­ings ofcapital and wage goods. Together with their aim to makethe greatest possible profit (this is what makes each one of thembehave like an homo ceconomu.:us), they desire to maintain the eco­nomic system indefinitely, to keep making money in the future.Hence, they also desire the reproduction (be it simple or widened)of the system (this is what makes each one of them behave like aresponsible homo politicus). A very interesting question that arisesthen is whether the capitalist can be simultaneously an homo ceco­nomicus and an homo politicus, whether the firms can have theircake and also eat it. We saw on chapter 7 that the answer is af­firmative, but we shall see here how these two ends can be coor­dinated in a Leontief economy.

THE PROTOTYPE REVISITED 193

Unlike the simple model of chapter 1, in the present one thebehavior of the firms is considered in an explicit way. We havel firms in a set F and the production possibilities set of any firmf E F is assumed to have very special properties (this is whatmakes the economy to be ofLeontieftype). The first one of thesespecial properties is that the production processes are able toproduce only one kind of good (there is no joint production),and each firm is assumed to be a specialist in the production ofone type ofcommodity. Another one is that there are no alterna­tive technologies, in the sense that all prod uction processes usethe same types of labor and means of prod uction, even thoughdifferent positive combinations of the same are allowed. Hence,"the same" technology can be operated at different levels of ef­ficiency.

There are also properties pertaining to the whole. It is as­sumed that each of the m kinds of goods can be produced bysome firm. Hence, if the price system is feasible, every good shallbe produced within the economy. It can be proven that thereexists at least one semiproductive technology, i.e. that the tech­nology has sufficient development as to be able to sustain the re­production of the economy. Quite another matter is whether the"enlightened self-interest" ofthe firms guarantees that the actualtechnology is reproducible. It is shown that there exist repro­ducible global decisions (RGDs), i.e. global decisions of the firmsthat maximize the profit of each one and which also are feasibleand reproducible. If the firms choose one of these RGDs, notonly the economy is reproducible, but also it is possible to provethat there is only one, strictly positive equilibrium price systemfor the economy. This in turn entails a very strong form of theLaw of Value, which makes it possible to assign a unique positivevalue to each commodity in the system. Finally, it is shown thatif the initial holdings of the firms are sufficiently large then fullemployment and reproduction is guaranteed.

8.2 MATHEMATICAL MODELING OF THE LEONTIEF ECONOMY

In a Marxian capitalist economy, consider a convex polyhedralcone (a technology) X contained in the aggregate set Y and let

194 CHAPTER EIGHT

Xl, ... , xm be a set of linearly independent processes spanningX. Assume that there is no joint production in these processesand also no alternative techniques. I shall call a set X with theseproperties a Leontieftechnology. More precisely:

DEFINITION 1: Let Y be the aggregate set of possibilities ofproduction of a Marxian economy. A Leontiel technology is a con­vex polyhedral cone X spanned by a finite set Xl, ... , xm of el­ements of Y having the properties that none of these processesyieldsjoint products and there are no alternative techniques, i.e.Xij > 0 iffi = j (i,j = 1, ... , m).

Following Marx's distinction into sectors, it is possible to dis­tinguish among the types of goods prod uced by X two mainkinds, namely, capital goods and wage goods. There is no loss ofgenerality in supposing that these two classes are disjoint, andso we are justified in introducing the following definition.

DEFINITION 2: A good of type i (i = 1, ... , m) is called a capitalgood in X if there is a process X E X such that ~i > O. A goodof type j (j = 1, ... , m) is called a wage good if there is a processX E X such thatxj = b, where b is a positive entry in the matrix Bof consumption bundles for the working class. Any good whichis not a wage or a capital good is called a luxury good.

In order to avoid unnecessary complications, I shall assumethat there are no luxury goods (assuming that there are just addsmore details to the proofs, as in chapter 1). Hence, by conven­tion, the types of commodities can be rearranged in such a waythat those of type 1, ... , k are capital goods and those of typek + 1, ... , m are wage goods. With this convention, the matrix ofwage and capital goods ind ustries can be written as

~ll ~lk ~lk+J ~lm

x= ~k) ~kk ~kk+l ~km

Ok+ll Ok+lk Ok+lk+l Ok+lm

Oml Omk Omk+l Omm

THE PROTOTYPE REVISITED

The matrix of wage goods can be written as

OJ 01

B= Ok Okbk+JJ bk+ 1n

bml bmn

Ifwe let

[X~' X',m]L= :Xnl X nm

195

be the matrix of labor inputs corresponding to the wage andcapital good industries, then the wage and capital inputs matrixis the sum D = X + BL. I shall call a matrix like this a globalmatrix.

I am interested in defining a concept of interconnectionamong the capital and wage good industries. This concept hasbeen already introduced in chapter 1, where I discussed the con­cept of interconnectedness of the wage and capital goods indus­tries of the prototype, represented by matrix C. I want also todefine this notion, in an analogous way, for D, but I also want todiscuss a little bit more its economic meaning and the adequacyof its mathematical representation.

Imagine an economy that can be severed into two groups ofindustries, such that the industries in any of these two groups donot need as wage or capital inputs goods of the type producedby the industries in the other group. This is an economy which isnot interconnected if, indeed, it can be called one economy at all.Let iJ, ... i." be the industries in the first group, and i.,,+J, ... , im

those in the second. The arbitrary entry di ) of matrix D is theamount ofoutput of industry i consumed by industry j either asa wage or a capital good. Clearly, the lack of dependency of in­dustries iI, ... i." with respect to ind ustries i.,,+I, ... , im means thatno good produced by these is consumed by the former, and sodij = 0 whenever j = iJ, ... i." and i = i.,,+J, ... , im • Hence, one

196 CHAPTER EIGHT

way of expressing the interrelatednes of the technology is to de­mand that the matrix D be indecomposable, which is tantamountto preclude the existence of such subsets of indices. Another im­portant property of the technology D was mentioned in connec­tion with the prototype in chapter 1; this is its semiproductivity,a purely technological property without which no economy isfeasible. I shall introduce these two properties together in thefollowing definition.

DEFINITION 3: Let X be a Leontief technology in a Marxiancapitalist economy. We say that X is interconnected iff the corre­sponding global matrix D is indecomposable. Also, we say thatX is semiproductive iffDy ~ y for some positive (column) vector y.If some Leontief technology contained in the aggregated set Yis interconnected, the set Y itself is said to be also interconnected.If some Leontief technology contained in Y is semiproductive,the set Y is also said to be semiproductive.

The prod uetion possibilities sets of a Marxian economy areclosed convex cones with certain properties. An additional prop­erty that these sets may, or may not have, is that of being tech­nologies apt to produce only one kind of good. This possibilitymotivates the concept of a specialized firm, which is introducedat this point.

DEFI NITION 4: Let F be the set of firms of a Marxian capitalisteconomy. A firm! E F is called a specialist iff its production pos­sibility set produces a unique type of good. That is to say, thereare in the economy precisely m production possibilities sets Yj ,

firm! has access only to one of these, and for this Yj there is apositive integer i (l ~ i ~ m) such that Xi > 0 for some x E Yj

and Xi 2: 0 for every xE Yj , whereas X] = 0 ifJ t i.

On top of precluding joint production, the production possi­bilities sets may also shun alternative technologies, in the sense ofallowing different proportions of inputs and outputs but avoid­ing the use ofdifferent types of inputs. In other words, these setsmay allow different degrees of efficiency to produce one kind of

THE PROTOTYPE REVISITED 197

commodity, but they do not allow techniques using essentiallydifferent types of labor and/or capital goods. This idea is moreprecisely defined as follows.

DEFINITION 5: The set Yf is said to preclude alternative technolo­gies if the following two conditions are fulfilled: (1) If there is aprocess x E Yf such that Xi = 0, then Yi = °for every y E Yf .

(2) If there is a process x E Yf such that ~i = 0, then Y =°forevery y E Yf . If in a Marxian economy all production' possibil­ities sets preclude alternative technologies, we say that there areno alternative technologies in that economy.

It can be proven that if there are no alternative technologiesin an economy and there is at least one interconnected Leontieftechnology included in Y, i.e. if Y is interconnected, then everyLeontieftechnology included in Y is interconnected. This is thefirst lemma of the chapter.

LEMMA 1: If in a Marxian economy there are no alternative tech­nologies, every good can be produced by some firm, and every firm is aspecialist, then there is one interconnected Leontief technology includedin Y iffevery Leontief technology included in Y is interconnected.

Proof" Suppose that there is one interconnected Leontief tech­nology X ~ Y. By Definition 1, this set is spanned by a finite setof processes xl, ... ,xm of elements of Y such that Xij > °iff i = j(i,j = 1, ... , m). Since every firm is a specialist, each productionpossibility set Yf produces a unique type of good. If there aremore firms than types of goods (l > m) some of these produc­tion possibilities sets Yf must be shared by two or more firms. Atany rate, there cannot be less production sets than firms, becauseevery good can be produced by some firm. Hence, it follows thatthere are exactly m production possibilities sets YI, ... Ym in theeconomy. By definition ofY, process Xi must then belong to setYi , even though two or more firms could have chosen differentprocesses belonging to this set: in such a case Xi is just the aggre­gation ofthese processes. By assumption there are no alternativetechnologies, and so all process are similar in having zeros (if at

198 CHAPTER EIGHT

all) in the same entries always. Therefore, the aggregate processXi is similar to any of the elements of the set where its compo­nents were chosen (in fact, it is a member of this set, which is acone) and so the global matrix ofX is similar to the global matrixof any other Leontief technology included in Y. It follows that ifthe global matrix ofX is indecomposable then all such matricesare indecomposable. 0

The following lemma establishes the equivalence of the nota­tion for prod uction processes in terms of 2m +n vectors with therepresentation of the same in terms of global matrices. Usually,the Leontief technology is described in terms of these matricesbut, as the lemma asserts, there is an alternative description interms of a convex polyhedral cone.

LEMMA 2: Eve-ry element X in the Leontief technology X ~ Y can berepresented as x = [Ls, Xs, s], for some state s ~ O.

Proof Let XJ, ..., xm be linearly independent processes spanningX, chosen in such a way that Xii = 1. Let L = [xi··· x;;'] be thematrix whose columns are the vectors of labor inputs of theseprocesses, let X = [~; ... ~;;,], and let x be any element of X.Then there exist unique nonnegative real numbers £xI, ... , £x msuch that x = £XIXI + ... + £xmxm. Let Xi be the state that has 1 atplace i and zeros everywhere else and notice that X = 2::::1 £XiXi.Then Xi = [LXi, XXi, Xi] and S02

m m mX = [L £X, LXi, L £XiXXi, L £xixd

i=l i=l i=l

= [Lx, Xx, x]. 0

In terms of the newly defined concepts, it is possible to in­troduce the fundamental definition of the chapter. This is thedefinition of a Leontief capitalist economy, which is a Marxianeconomy with special properties.

THE PROTOTYPE REVISITED 199

DEFINITION 6: A Leontiefcapitalist economy is a Marxian capital­ist economy .£ in which every firm is a specialist, every type ofgood can be produced by some firm, there are no alternativetechnologies, and Y is interconnected.

If the price system is feasible, as a result of the decision ofthe firms, I processes XI, ... Xl are chosen, all of these processesbeing efficient. The convex cone P spanned by these processes(the actual technology of .£) is in fact interconnected, a resultthat is stated as the first theorem of this chapter.

THEOREM 1: In a Leontief economy the actual technology is alwaysinterconnected, whenever the pme system is feasible.

Proo!' If the price system is feasible, then every firm is encour­aged to obtain positive profits by operating some nonnull pro­cess. Hence, by Lemma 1, the technology chosen by the firms isinterconnected. 0

We know from chapter 6 that if the price system is feasiblethen every firm operates a nonnull process, and these processesare chosen in such a way that the resulting price turns out to bean equilibrium price, i.e. a price at which the profit rate is thesame for all the production processes in the economy (recall thatthis is due to the existence of a financial capital market). Quiteanother question is whether the behavior of the firms, by itself,guarantees that the processes they decide to operate can be re­produced, in the sense of being able to produce the capital andwage goods they consume and perhaps obtain some excedents.A social and historical presupposition of the starting of a newproduction cycle in a market economy is the previous existenceof enough resources, as well as sufficiently efficient technologiesin all branches of the economy, as to guarantee, at the very least,the possibility of a "simple reproduction" of the economy; thatis, the possibility of replacing the worn out capital goods, and of"reproducing" that part of the working class which is employed,that is to say, of providing for the demand B of those who are

200 CHAPTER EIGHT

employed. The existence of a RGD means that there is a tech­nology which, if operated at an appropiate level, is able to re­produce the capital goods and the employed labor power oftheeconomy. Naturally, the condition required to guarantee that allthe firms operate a reproducible nonnull process is the feasibilityof the equilibrium price system.

THEOREM 2: In a Leontiefeconomy there is at least one RGD. If theequilibrium price corresponding to this RGD is feasible, then the globalmatrix corresponding to this RGD is semiproductive.

Proof" Since a Leontiefeconomy is a Marxian capitalist economy,the results of chapter 7 imply that there is a RGD {XI, ... , xm }.

Let D be the global matrix corresponding to this technology.Since all firms operate a nonnull process, because the corre­sponding equilibrium price system is feasible, the columns of Dare all nonnegative and, moreover, the output x of the globalprocess is positive. It is easy to see that the condition of repro­ducibility in Definition 1 of chapter 7, that x ~ X + Bx, is equiv­alent to Dx ~ x, i.e. to the semiproductivity ofD. 0

A very curious and special trait of Leontief economies is thefact that, for any of the actual technologies generated by a RGDwith a feasible price, there is only one uniform profit rate andonly one equilibrium price (up to multiplication by a positivescalar). This is due to the interconnectedness and semiproduc­tivity of the industries, a property that -as we saw- in theseeconomies can be formulated in terms of the indecomposabilityand semiproductivity of the global matrices. This result shall bethe next theorem of the chapter.

THEOREM 3: There is a unique (uP to multiplication by a positivescalar) uniform profit rate for the actual technology of £, whenever thedecision of the firms is a RGD with a feasible price system. Moreover,there is also a unique (uP to multiplication by a positive scalar) system ofprices at which the uniform profit rate obtains. This system of prices 'ispositive.

THE PROTOTYPE REVISITED 201

Let

Proof" Let D be the global matrix associated to the actual technol­ogy chosen by the firms (a RGD). By theorems 1 and 2, we knowthat this technology is interconnected and semiproductive, andso D is indecomposable and semiproductive.

By the indecomposability of matrix D and the Perron-Frobe­nius theorem, there is a unique positive real eigenvalue, (theFrobenius root) to which there corresponds a unique (up to mul­tiplication by a positive scalar) positive left eigenvector p:

pD = p,.

1-,71"=--.,

Since D is semiproductive, there is a positive vector y such thatDy ~ y. Hence, pDy ::; py and so , ::; I. It follows that 71" is auniquely determined nonnegative number and that 1+ 71" = ,-I.Therefore,

p = (1 + 7I")pD. 0

I shall call this price p the equilibrium price (EP), very much as inchapter 1, and the profit rate 71" shall be called the equilibrium profitrate (EPR). It can be shown that the profit rate as introduced inDefinition 4 of chapter 5 coincides with the number 71" above.That is:

THEOREM 4: For every x E P, the profit rate of x is identical to theEPR7I".

Proof" For any x E P,

px = (1 + 7I")pD

= (1 + 7I")fpXx + p(BLx)]

= (1 + 7I")fp~ + wx]

and so

O Px - wx<71"= •

p~+wx

202 CHAPTER EIGHT

Thus, 11" is in fact the profit rate of any labor process in P. D

By virtue ofTheorem 1 of chapter 6, we know that the profitrate is uniform and identical to the interest rate, and also thatthe Law of Value holds in a Leontiefeconomy. It follows also thatthe EP 11" is identical to the interest rate prevailing in the financialcapital market. The next theorem is a very strong version of theLaw of Value, since it establishes that the equilibrium prices arein fact the unique labor-values of the commodities.

THEOREM 5: In a LeontieJeconomy, suppose that the technology cho­sen by theJirms is a RGD, and let li = rXi be the amount ojabstrad livelabor expended in the production oj one unit ojgood i (i = 1, ... , m).Then the EP is inJact a system ojlabor values, in the sense that it satisJiesthe equation

P· = X·PI + ... + XPk + lI -11 -kt I

Jor every kind ojgood i (= 1, ... , m). As a matter ojJact, prices are theonly labor values in this economy, up to scale tr-ansJormations.

Proof' By Definition 8 of chapter 5, ),(Xi) = rXi, where r is apositive reduction oflabor. Hence,

m m

= LPjx) - LPJ~)j=1 j~J

m

=PiX;; - LPj~j;

m

=Pi - LPJ~Jl'J-I

Thus,

P· = X ·PI + ... + XPk + l.I -11 -kl I

Let Al and All be, as in chapter 1, the matrices

Al = [xT••. xT

]-I -k

and

THE PROTOTYPE REVISITED

AII=[XT

"'XT

]-k+l -m

203

of capital goods and wage goods industries, respectively. Let LI

and LII be the matrices given by conditions

Since AI is quasiproductive and indecomposable, Theorem 1 ofchapter 1 implies that the matrices

and

are unique and positive, where

That is to say, Ai = o:Pi for some 0: > 0 and every i = 1, ... , m.o

I shall proceed now to solve the "transformation problem" be­tween the profit and the exploitation rates for a Leontief econ­omy. Whereas in a simple Marxian economy of the type dis­cussed in chapter 1 the exploitation rate is uniform, in a Leon­tief economy need not be so, but can vary from one productionprocess to another. In fact, due to Theorem 6 the exploitationrate in a Leontief economy adopts the form introduced by thefollowing theorem.

THEOREM 6: In a Leontief economy, if the technology chosen by thefirms is a RGD, then the profit rate 1r of any process x E P+ can beobtained out of its the exploitation rate by means of the transformation

(}(~) = (C1 + ~p~ ) -I.

px-wx

204 CHAPTER EIGHT

Proof" By definition of exploitation rate,

(_) _ .\(i) - .\(Bx)

c x - .\(Bx)

= pi - pBxpBx

= px - wxwx

Therefore,

O[c(x)] = ( ~wx + ~ p~ )-1pX - WX pX - WX

= 7r(X)

= 7r. D

The firms' choosing a RGD guarantees the reproduction ofthe productive cycle but does not guarantee full employment,only the possibility of satisfying the demand of those that turnout to be employed. In order to close the present book, I willshow that full employment socially and historically presupposesthe existence of sufficiently large stocks of capital and wagegoods. To this effect I will introduce a new primitive concept,which is the set ofall possible distributions of all the available so­ciallabor power. This is determined by the technologies possiblefor the economy as a whole: the technical properties of the pos­sible technologies constrain the possible ways in which all theworkers can be allocated among the different production pro­cesses. Hence, for any Leontief technology X we may think ofthe set of all possible allocations of labor power as a subset L ofthe whole set £ of labor inputs of X (chapter 5, Definition 3).One condition that this set must satisfY is that the vector of la­bor inputs of any labor process, if properly scaled, must be inL. Clearly, if the global process x requires more labor than thatrepresented by any of the elements of S, then it cannot be oper­ated by the firms. But the firms can (in fact, they will) operate a

THE PROTOTYPE REVISITED 205

global process whose vector of labor inputs represents less laborthan any element of S, unless the accumulated vector of initialholdings is sufficiently large. If this is the case, and the global de­cision of the firms is a RGD under a feasible price system, thenthe economy can sustain itself indefinitely. The clue to full em­ployment lies, of course, in the scale of the economy. Notice alsothat there is no international trade in this model, and so all the"contradictions" that might arise are solved within this perfectlyclosed economy.

NOTES

NOTES TO CHAPTER 1

I From now on, I shall ocassionally refer to this work with the letter C. All quotationsfrom this work are taken from the Penguin edition (1976, 1978, 1981). Since thereare very different editions, instead ofreferring to page numbers in making quotationsI refer to the book, part, chapter and section, in that order. Thus, for instance, anexpression like 'Cl, pi, ch2, s3' denotes the third section of the second chapter, firstpart of book one.

2 Cl, pi, chi, s1.

3 Ibid. The italics are mine.

4 Ibid. My italics.

5 Ibid.

6 Ibid.

7 Ibid. The italics are mine.

8 Cl, pi, chi, s2.

9 Ibid. My italics.

10 Ibid. My italics.

II Ibid.

12 Cl, pi, ch 1, s3. My italics.

13 Ibid.

14 Ibid. My italics.

15 Ibid.

16 Cl, p3, ch7, s1.

17 See Corollary 2 and Theorem 7 in Kempand Kimura (1978), pp. 8-9. The Hawking­Simon condition appears as this theorem.

18 Cl, p3, ch9, s1.

19 Cl, p2, ch8, s1.

20 Morishima (1973), p. 85.

21 Morishima (1973), p. 86.

207

208 NOTES

22 Bohm-Bawerk (1896). I follow the Spanish version published in Argentina in 1974.

NOTES TO CHAPTER 2

] C3,p2,chl0.

2 C3, p2, ch9.

3 Ibid.

4 Ibid.

5 Morishima (1973), pp. 72-74.

6 C3, p2, chlO. My italics.

7 Ibid.

8 C3, p4, chl8.

9 Ibid. The italics are mine.

10 C3, p6, ch37. My italics.

I) C3, p7, ch49.

12 C3, p7, ch5!.

13 Bohm-Bawerk (1974), p. 57.

14 Ibid., p. 54.

15 Ibid.

16 Ibid., p. 58.

17 Ibid., p. 67.

18 Ibid., p. 77.

19 Nuti (1974), p. 43.

20 Ibid.

2] See pp. 8 and 9.

22 Cameron (1952), p. 193.

23 Morishima and Seton (1961), p. 204.

24 Ibid.

25 Op. cit., p. 209.

26 Cf. pp. 297 and 298.

NOTES 209

27 Morishima (1973). Many of the mathematical techniques used by Morishima in thisbook were developed mainly in the fifties by mathematical economists such as Samuel­son, Arrow and Koopmans. See for instance Koopmans (1951).

28 See Theorems 1 and 2 of chapter I.

29 See p. 131.

30 See Morishima (1973), p. 173.

31 Morishima (1974), p. 618.

32 The convexity assumptions have been used to prove the existence of equilibria inneoclassic economics (See Debreu (1956,1959». Roemer uses the convexity assumptionin order to derive the existence of what he calls Marxian reproducible equilibria (seeRoemer (1981».

33 Roemer (1981), p. 38.

34 In fact, Professor H. Scarf has developed illl portant results on the problem ofbuild­ing a theory of (neoclassical) equilibri um on non-convex finitistic assumptions. See Scarf(1981a, 1981b).

NOTES TO CHAPTER 3

I See Enderton (1972), p. 75.

2 See p. 104.

3 See Enderton (1972), p. 79.

4 For a precise definition of decidability see Boolos and Jeffrey (1980).

5 Which can be seen, for instance, in Suppes (1972).

6 Wallace and Findlay (1975), p. 50.

7 See the Susiitz to §31. Wallace and Findlay (1975), p. 51.

8 For a systematic study of the relationships between the philosophy of Hegel and thatof Aristotle, see Mure (1970).

9 See Suarez (1960), Disputation XL, Part VI, §5.

10 See Brown (1984), pp. 153, n. 12. The other "labyrinth" is the problem ofreconcilingGod's foreknowledge with human freewill.

II For a definition of Archimedean, regular, positive, ordered, local semigroup, seeKrantz et al. (1971), p. 44.

NOTES TO CHAPTER 4

1 Wallace and Findlay (1971), §79, p. 113.

210 NOTES

2 Wallace and Findlay (1971), §161Z, p. 224.

3 Findlay (1958), pp. 70-1.

4 Findlay (1958), p. 57.

5 Findlay (1958), p. 74.

6 Findlay (1958), p. 75.

7 For a view of idealization as isolation see Miiki (1991).

8 Findlay (1958), p. 77.

9 Findlay (1958), p. 71-2.

lO Findlay (1958), pp. 72-3. The first italics are mine.

II Miller and Findlay (1971), §382, p. 15.

12 Miller and Findlay (1971), §381, p. 8.

13 Wissenschafl der Logik, p. 44; quoted by Findlay (1958), p. 152.

14 Findlay (1958), p. 32.

15 See "Materialism and Matter in Marxism-Leninism", in r-fcMullin (1978).

16 In the same Postface to the Second Edition.

l7 Elster says: "I find it hard to believe that Marx would have come to accept the lawsof dialectics had he put his mind to them". See pp. 42-3.

l8 Elster (1985), p. 37.

19 Dussel (1990), p. 404. The translation is mine.

20 On 1M Soul, 430' 10-25. See Barnes (1984), volume I. The italics are mine.

21 Aquinatis (1886), p. 455. The translation is mine.

22 In the sections A. Consciousness and B. Self-Consciousness. See Westphal (1989),pp. 154 ff.

23 SCMlling Werke, v. V, p. 198. My translation. lowe to Dussel (1990) his making meaware of Schelling's criticism of Hegel.

24 SCMlling Werke, Book III, Lesson XII.

25 Plotinus, Ennead V, 4.

26 Cfr. Leclerc (1972), p. 66.

27 Holy Bible (The New King James Version). Exodus 3:14-15.

28 Kaufmann (1972), p. 21.

29 See Avineri (1972), chapter 2.

NOTES 211

30 See Dickey (1987) for a thorough study of the historical, political and theologicalcontext in eighteenth century Wurtemberg, the land where Hegel was born and wherehe grew up.

31 See Avineri (1972).

32 See Waszek (1988).

33 See Wood (1990).

34 See Stern (1990).

35 See Westphal (1989).

36 Actually, this is what Stern (1990) does in connection with the Pherwmerwlagy a/Spirit.See pp. 43-54.

37 Stern (1990), pp. 40-1.

38 This quotation is taken from an unpublished paper dated by Van Heijenoort in 1943under the pseudonym ofAlex Barbon. See Van Heijenoort (1943) in the bibliography.

39 Van Heijenoort, op. cit.

40 Ibid.

4l I follow here the German version (1974). Whenever I deem it important, I providethe original German expressions together with their translation.

42 Marx and Engels (1974), p. 630.

43 The value (2000/g) l/2 is obtained by setting - ~gt2 + 1000 = 0 (which is the positionof the particle at the end of the motion), and solving for t.

44 Nowak (1980), p. 95.

45 Nowak (1980), p. 29.

46 Nowak (1980), p. 9.

47 For a detailed, albeit a rather schematic presentation of this process in connectionwith MTV, see Hamminga (1990).

48 See Koopmans (1951) and Leontief(1941).

NOTES TO CHAPTER 5

l Morishima (1973,1974), Okishio (1963), Roemer (1980,1981).

2 Marx (1970), p. 29.

3 Marx (1970), p. 34.

4 Marx (1970), p. 45.

5 Ibid.

212 NOTES

6 In CI, pi, chi, s3.

7 See Rubin (1972).

8 Rubin (1972), p. 139.

9 Rubin (1972), pp. 139-140.

10 See Krause (1979,1980,1981) in the Bibliography.

II Cf. equation 4 of chapter 1. The proviso therein does not apply in the present case,i.e. the vectors 1fT are of dimension m.

12 More precisely, we are requiring x ;::: 0 and

for every (x, 1f. x) E Po.

13 CI, p3, ch7, s2.

14 Ibid.

15 Ibid.

16 See Kemp and Kimura (1978), p. 3.

NOTES TO CHAPTER 6

I See pp. 73-74.

2 Roemer (1981), p. 73.

3 A proofofthe nonemptyness ofA j(p, c) is provided in Lemma 2 of the next chapter.

NOTES TO CHAPTER 8

J Samuelson (1963). p. 233.

2 It would be more correct to write [L(x)T, X(i)T, x T ]. I ask the forgiveness of thereader for this little notational abuse.

BIBLIOGRAPHY

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Aquinatis, S. T., Opuscula philosophica et theologica, Tiferni Tiberini,Castello, 1886.

Arrow, K. J. and Debreu G., "Existence of an Equilibrium for a Com­petitive Economy" in Economet1ica 22, 1954.

Avineri, S., Hegel's Theory of the Modern State, Cambridge Up, Cam­bridge, 1972.

Balzel~ W., Moulines, C. U. and Sneed,j. D.,AnAnhitectonicforScience,DReidel, Dordrecht, 1987.

Barnes, j. (ed.), The Complete Works ofA11stotle, Princeton Up, Princeton,(1984).

Bohm-Bawerk, E., "The Conclusion of the Marrxian System", severaleditions.

Boolos, G., and Jeffrey, R., Computabil£ty and Logic, Cambridge Up,Cambridge, 1980.

Bourbaki, N. (pseud. ), Elements ofMathematics: Theory ofSets, Addison­Wesley, Reading, 1968.

Brown, S., Lei/miz, University of Minnesota Press, Minneapolis, 1984.

Brzezinski, j., F. Coniglione, Theo A. F. Kuipers and L. Nowak, (eds.)Idealization I: General Pmblems, Rodopi, Amsterdam, 1990.

Caldwell, B. (ed.), Appraisal and C1iticism in Economics, Allen & Unwin,Boston, 1984.

Cameron, B., "The Labour Theory of Value in Leontief Models" inThe Economic Journal 62, 1952.

Debreu, G., "Market Equilibrium" in Proceedings ofthe National Academyof Sciences 42, 1956.

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Elster,]., Making Sense of Marx, Cambridge Up, Cambridge, 1985.

Enderton, H. B., A Mathematical Introduction to Logic, Academic Press,New York, 1972.

Engels, E, Anti-Duhring, several editions.

__, Dialectics ofNature, several editions.

Findlay,]. N., Hegel: A Re-Examination, Collier Books, New York, 1958.

Gale, D., The Theory ofLinear Economic Models, McGraw-Hili, New York,1960.

Gantmacher, E R., Applications of the Theory of Matrices, IntersciencePublishers, New York, 1959.

Garcia de la Sienra, A., "Elementos para una reconstrucci6n 16gica dela teoria del valor de Marx" in C1itica 35, 1980.

__, "The Basic Core of the Marxian Economic Theory" in W.Stegmiiller, W. Balzer, W. Spohn, (ed. ), Philosophy of Economics,Springer-Verlag, Heidelberg, 1982.

__, "Axiomatic Foundations of the Marxian Theory of Value",Erkenntnis 29, 1988.

Georgescu-Roegen, N., "Leontief's System in the Light of Recent Re­sults" in The Review of Economics and Statistics 32, 1950.

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Hamminga, B., "The Structure of Six Transformations in Marx's Cap-ital" in Brzezinski et al. (1990).

Hegel, G. W. E, Wissenschajt der Logic, Surhkamp, Stuttgart, 1986.

__, Phenomenology ofSpirit, Clarendon Press, Oxford, 1977.

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Kakutani, S., "A Generalization of Brouwer's Fixed Point Theorem" inDuke MathematicalJoumal, 8 (1941).

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Kaufmann, W., "The Hegel Myth and Its Method" in McIntyre, A.(ed.), Hegel, Anchor-Doubleday, New York, 1972.

Kemp, M. C. and Y. Kimura, Introduction to Mathematical Economics,Springer-Verlag, New York, 1978.

Koopmans T. C. (ed.), Activity Analysis ofProduction and Allocation, JohnWiley & Sons, New York, 1951.

Kosok, M., "The Formalization ofHegel's Dialectical Logic" in Interna­tional Philosophical Quarterly, vol. VI, no. 4 (1966).

Krantz, D. H., Luce, R. D, Suppes, P, and Tversky, A., Foundations ofMeasurement I, Academic Press, New York, 1971.

Krause, U., Geld und abstrakte Arbeit, Campus Verlag, 1979. English edi­tion: Money and Abstract Labour, Verso, London, 1982.

___, "Abstract Labour in General Joint Systems" in Metmeconomica32,1980.

___, "Heterogeneous Labour and the Fundamental Marxian The­orem" in Review ofEconomic Studies 48, 1981.

Landau, E., Foundations ofAnalysis, Chelsea, New York, 1966.

Leclerc, I., The Nature ofPhysical Existence, George Allen & Unwin, Lon­don, 1972.

Lenin, V. I., Materialism and Empirio-Criticism, Foreign Languages Press,Peking, 1972.

Leontief, W. w., The Structure ofAmerican Economy 1919-1929, HarvardUp, Cambridge, 1941.

Lobkowicz, N., "Materialism and Matter in Marxism-Leninism" in Mc­Mullin, E. (ed.), The Concept ofMatter in Modern Philosophy, Universityof Notre Dame Press, Notre Dame, 1978.

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216 BIBLIOGRAPHY

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Morishima, M., The Theory ofEconomic Growth, Clarendon Up, Oxford,1969.

__, Marx's Economics, Cambridge Up, Cambridge, 1973.

___, "Marx in the Light of Modern Economic Theory" in Economet­1ica 42, 1974.

___ and Seton, F., "Aggregation in LeontiefMatrices and the LabourTheory of Value" in Econometrica 29, 1961.

Mure, G. R. G., An Introduction to Hegel, Oxford Up, Oxford, 1970.

Nikaido, H., Convex Structures and Economic Theory, Academic Press,New York, 1968.

Nowak, L. : The St11lcture of Idealization, DReidel, Dordrecht, 1980.

Nuti, D. M., V. K. Dmit1iev: Economic Essays, Cambridge Up, Cambridge,1974.

Okishio, N., "A Mathematical Note on Marxian Theorems" in Weltwirt­schaftliches Archiv 91,1963.

Robinson, A., Non-StandardAnalysis, North Holland, Amsterdam, 1961.

Roemer, J. E., "A General Equilibrium Approach to Marxian Eco­nomics" in Econometrica 48, 1980.

___, Analytical Foundations of Marxian Economic Theory, CambridgeUp, Cambridge, 1981.

Rubin, I. I., Essays on Marx's Theory of Value, Black & Red, Detroit, 1972.

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Scarf, H. E., "Production Sets with Indivisibilities-Part I: Generalities"in Econometrica 49, 1981a.

___,"Production Sets with Indivisibilities-Part II: The Case of TwoActivities" in Econometrica 49, 1981 b.

Schelling Werke, Schroeter, Munich, 1958.

Sneed,]. D., The Logical Structure ofMathematical Physics, D. Reidel, Dor­drecht, 1971.

Stern, R., Hegel, Kant and the Structure ofthe Object, Routledge, London,1990.

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Suarez, F., Disputaciones rnetafisicas, Credos, Madrid, 1960.

Suppes, P., Axiorruztic Set Theory, Dover, New York, 1972.

___, Introduction to Logic, D. van Nostrand, New York, 1957.

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Van Heijenoort, j., "On Marx's Method in Capital", Manuscript, 1943.

Von Bortkiewicz, L., "Zur Berichtigung der grundlegenden theoretis­chen Konstruktion von Marx im III. Band des Kapitals" inJarbilcherfilr Nationaliikonomie und Statistik 34, 1907.

Wallace, W. and Findlay, j. N., Hegel's Logic, Clarendon Press, Oxford,1971.

Waszek, N., The Scottish Enlightenment and Hegel's Account of'Civil Society',Kluwer, Dordrecht, 1988.

Westphal, K. R., Hegel's Epistemological Realism, Kluwer, Dordrecht,1989.

Wood, A. w., Hegel's Ethical Thought, Cambridge Up, Cambridge, 1990.

NAME INDEX

Alex Barbon 211Alvarez, F. 42,213Aquinas, St Th. 105, 106, 107, 108,

210,Aristotle of Stagira 77, 97, 104, 106,

107,209Arrow,K.J. 123,208,213Augustine, St 108Averroes 105, 106Avineri, S. 109,210,213

Balzer, W. 72,213Barnes,]. 210,213Broncano, E 213Brown, S., 209,213Beethoven, L. v. IIIBirkhoff, G. 72,215Bohm-Bawerk, E. v. 3, 10, 39, 42, 43,

44,49,50,51,52,53,207,208,213Bolzano, B. 76Boolos, G. 209,213Bourbaki, N. 3, 72, 213Brzezinski, J. 213

Caldwell, B. 213Cameron B. 55,208,213Christ Jesus 108, 110

Debreu, G. 128, 185,209,213Dickey, L. 210, 214Dmitriev, V. K. 54,55Dussel, E. 101,102,210,214

Elster,]. 58, 101, 116,210,214Enderton, H. B. 209,214Engels, F. 43,44,51,101,115,210,214Eudoxus ofCnido 76

Fichte,J.G.92,96

219

Findlay,J. N. 93,95,96,209,210,214,215,217

Gale, D. 214Gantmacher, F. R. 214Garcia de la Sienra, A. 214Georgescu-Roegen, N. 55, 214Godel, K. 68Gracia,]. J. E. 214

Haydn, E]. 111Hamminga, B. 116,210,214Hawking-Simon condition 207Hegel, G. W. F. 78, 92, 95, 98, 99, 100,

104, 106, 107, 108, 109, Ill, 113,121, 128, 160,210,214

Jeffre» R. 209, 213

Kakutani, S. 185, 214Kant, E. 78, 96, Ill, 113Kaufmann, W. 108,210,214Kemp, M. C. 207,212,215Kim ura, Y. 207, 212, 215Koopmans, T. C. 123,208,211,215Kosok, M. 92,215Krantz, D. H. 81,89, 150,214Krause, U. 132,133,137,176,211,215Kuipers, Theo A. E 213

Landau,E. 72,215Leclerc, l. 210,215Leibniz, G. W. E 81Lenin, V. l. 99,215LeontiefW. W. 2,54,55,123,211,215Lobkowicz, N. 100,215

Maki, U. 128,210,215

220 NAME INDEX

Marx, K. 1, 2, 3, 4, 7, 8, 9, 10, 11, 12,13, 16, 18, 19, 20, 35, 39, 42, 43, 44,45, 46, 49, 50, 52, 58, 91, 94, 98, 99,100,101,102,104,115,121,123,128129,130,131,156,210,211,215

McLane, S. 72,215McMullin, E. 210McTaggart,j. M. E. 93Miller, A. V. 210,215Morishima M. 2, 3, 29, 39, 42, 45, 47,

48,49,55,56,57,58,59,60,123,124,208,209,211,216

Moses 108Moulines C. U. 72, 213Mure, G. R. G. 209,216

Nikaido, H. 185,216Nowak, L. 116,121,122,125,126,127,

210,213,216Nuti 54,208,216

Okishio, N. 56, 176,211,216

Plotinus 108,210

Quintanilla, M. A. 213

Robinson, A. 68, 216

Roemer,]. E. 57,58,60, 124, 125, 149,171,209,211,212,216

Rubin, 1. 1. 4, 131, 141,211,216

Samuelson, P. 123, 191,208,212,216

Scarf, H. E. 209,216

Seton, F. 55, 56, 176, 208, 216

Schelling, F. W.]. 107,108,210,216

Sneed,]. D. 72,213,216

Stern, R. 114,211,216

Suarez,F. 77,78,80,209,216

Suppes, P. 72,209,216,217

Von Bortkiewicz, j. 217

Van Heijenoort,j.v. 116,211,217

Von Neumann,j. 57,58,60,123

Wallace, W. 209,217

Waszek, N., 210,217

Westphal, K. R. 107,111,210,211,217Wolf, C. 78

Wood, A. W. 211,217

SUBJECT INDEX

absolute mind 98absolute negativity of the notion 97abstract determinations 120, 121, 165activation, intensity or state of an

economy 133, 137, 198active intellect or agent 97, 104, 105,

106, IIIaffine transformation 153, 156alienation 177analytical philosophy 94approximation 121Averroism 4, 105axiomatic systems 125

behavior of the firm 57,160,179,193Bible 109,210,214Buchenwald 110

capital good 18,21,32,53,170,194age of 58industries 22, 26period ofrotation of 19,53,57

Cartesian space 90choice of techniques 1,57Christian faith 105Christianity 109circulation process 46civil or burgeois society 160classical mechanics 113, 118closed economy 19,54,205commensurability 82commodity 8, 102

separately producible 135, 136commodity economy 130commodity exchange 132communism 109competition ofcapitals 159competitive equilibrium 5, 160, 179,

209

221

Marxian 5,57, 124, 159,209competitive market economy 31, 128concentration of multiple

determinations 117, 120concrete concept 127concrete of thought 120concrete thought ll4, ll5concretization 122, 123, 125, 127cone 134, 164

closed 165convex 147, 165convex polyhedral 142, 153, 164,

193constant returns to scale 18, 53, 151,

164constant capital 35, 38consum ption basket of the workingclass 25, 31, 32, 53, 54, 176consumption bundle 165continuum 81,82contradictory concepts 94, 95correspondence 182, 184

lower semicontinuous (lsc) 182upper semicontinuous (usc) 182

cosmology 76, 78credit 162, 163, 166, 168

decidable set ofsentences 69decision function 166decision ofthe firm 164, 170, 204demand 46, 53demand function 164dialectic 128dialectic contradiction 95, 149,205dialectical forms of motion 101dialectical method 79,91, 92ff, 99, ll7,

120, 125, 127, 165formalization of 92, 94Hegelian 4,91, 92ff, 101, 104, llO,

ll5

222 SUBJECT INDEX

Marxian 79, 104, 115, 171Marx's "inversion" of 4,91,99, 100,

104, 110Marx's "breaking of the bottom"

of 102, 104mystical shell of 100rational kernel of 100, 115

division oflabor 12doctrine of the notion 114

eigenvector 33elucidation of concepts 112empirical data 119, 120empiricist philosophy 91, 107ens rationis 75equations for value 25, 36

Leontief's 56equivalence relation 139essential predication 114exchange 129exchange-value 8, 130

valid 140existence of abstract labor 149experience 96exploitation 31,58,161,175,177

factual reality 114faith 108Farkas-Minkowski Lemma 137Father, the 107financial capital 101financial capital market 159, 163,202

clearing of 180financial feasibility function 169

global 170financial resources of the firm 169firm 161, 162, 163, 165fixed point 184foundational problems 91

of MTV 8, 40, 41, 62, 103free lunch 164Frobeniusroot 33,201F-twist 191full employment 204,205fundamental measurement 3, 79, 90,

135, 149

general determination 125general equilibrium theory 139, 179general joint system 133, 142God 108,110Gulag 110

Holy Spirit 107, 108honw politicus 192honw CEconomicus 114, 160, 192honw sapiens 76homomorphism 79human intellect 114human mind 76

Idea 98, 106, 107, 128self-developing 106, 107

idealization 90, 93, 95, 125, 210method of 121, 122, 125, 126idealized concepts 114idealized models 116, 120, 122, 192idealizing assumptions 41, 114imperfect entry 164incommensurability 4

problem of 113individual men 98, 109, 110infinite divisibility 53

of goods 53, 164oflabor amounts 164of production processes 53

initial holdings or endowments 162,163, 166, 192, 205

inner prod uct of vectors 21, 150inputs 20,21isolated notions 114,210interest rate 163, 166, 202international trade 205

Jehovah 97joint production 19, 53, 57, 136, 193,

194, 196

Kingdom of God 109Kolmogorovaxioms 70

labor 12abstract 13, 15, 43, 128, 130, 131,

132, 134, 135, 140, 141, 150, 157,174,202

SUBJECT [NDEX 223

complex 12concrete 15,138,140, [47, 151homogeneous 11 13, 19,31,53,57,

58,60, 129,130, 134, 192homogenization of 134, 135heterogeneous 12, 53,58, 149, 150,

151,176,191indispensable 143, 147, [48, 164instruments of 20, 144live 34,102,103,117,128,175,202necessary 34objects of 20physiologically equal 131process 136, 142productive 143, 147, 148, 164reduction of complex to simple 12,

13, 14, 150red uction of heterogeneous to

homogeneous 14,15,17,19,129,131,135,152,156,157,172,202

simple 12, 31, 130social 42, 130, 139, 140socially equalized 131type of 139,141,164,193unskilled 20, 130, 141value-creating 131

labor-power 12, 20, 101, 131, 144in the physiological sense 131normal 144, 145, 146, 147

Lawof Profit Maximization 161,171,

174,178of Supply and Demand 189of the Cyclical Crises 1of the Tendency of the Profit Rate

to Fall 1scientific 74, 122of Value 3, 4, 5, 15, 16, 17, 39, 40,

41, 42, 43, 44, 46, 47, 48, 49, 50,51,52,53,54,58, 103, 104, 122,123,124,156,157,159,161,171,173,175,176,192,193,202

Leontief economy or technology 5,55,123, 137, 164, 176, 191, 192, 193,194, 198, 199,200,202,203,204

liberty 97linear algebra 72

linear functional 132linear model of MTV 123linear programlning problem 57, 60logic

first order 64formalized 93mathematical 76, 93second order 70

logical consequence 68luxury goods 21,32, 194

manifold of intuitions 111market 130market economy 137,161,191

efficiency of 175Marxian capitalist economy 5,161, 171,

174,175,177,179,193Marxian competitive equilibrium 160,

180Marxian theory of value (MTV) 7, 39,

41,58,115,141,160,179,210foundations of 137foundational problems of 8,40,41,

62,103Marxism 110Marxism-Leninism 100Marx-Von Neumann theory 57,58matrix 29

consumption 26, 165demand 166, 199global 195,198,200,201indecomposable 28,29,33, 196,

198, 200, 203nonpositive 29nonsingular 29of capital goods

ind ustries 23ofcapital and consum ption goods

industries 27, 195,203ofconsum ption goods industries 23ofjoint labor inputs 133, 195ofjoint material inputs 133ofjoint material outputs 133ofjoint net inputs 133, 134oflabor inputs 23quasiproductive 28,203reproducible 28semiproductive 28, 33, 200

224 SUBJECT INDEX

matter 81,99, 107concept of 100

means of production 20,35, 102, 147,193

measurement 77,80additive conjoint 150extensive 81of abstract labor 3, 137, 149

Meinongian economy 128,160,161mereology 81metaphysics 78

Aristotle's 105Hegel's 79, 105, 107, II0, II

model 3,68, 75, 125concrete 126singular 126

model construction 117model representation 126

natural history 76natural theology 78natural world 76nature 97, 98necessary labor 34neoclassic economics 160, 209Newton's second law 119nontriviality 138nous 97, 99, 104, 105, 106, 108

objects in themselves IIIobjects of experience 96, 112ontology 78, 80, 94

of substance 114scholastic 79, 107

optimality function 169organic or value composition of capital

34,44, 129, 159output 21

net 21,135,137,156,175

pantheism 108particular 125passive intellect 105, 106philosophia de enle 77philosophy ofeconomics 192philosophy of nature 97,98philosophy of spirit 97

physics 112point-in put-point-output 19positive hull 146, 147price 31,32

shadow 57price system 30,33, 133, 139, 140, 148,

152, 153, 156, 162, 163admissible or valid 9,134,135,140,

141, 142, 143, 145, 157equilibrium 34,38,46,50,156,199,

201, 202feasible 170,173,175,177,178,

193, 199,200,205private property 109producer 22, 137production plan 137, 139production price 31, 44, 45, 46, 49, 52production process 21, 102, 137, 138,

1-17, 154efficient 143, 174, 196feasible 180global 180indispensable 135, 136, 137

production set 61,139,164,165,193aggregate 164linearly independent 194normal 147, 148, 164

profit 32, 161maximization 57,124,163,179, 180net 32

profit equalizing equilibrium 149profit maximizing function 169

global 170prototype of MTV 17, 18,41,53,54,

55, 128, 156, 159, 160, 191

quantitative concept of value 7, 17, 30,40,41

rate of profit 30,33,34,36,37,43,45,46,52, 142, 177, 178,202,203equilibrium or uniform 34, 35, 44,

47,50,61,142, 143,145,146, 147,156, 159, 174, 176, 179, 193, 199,200, 201

rate of exploitation 30, 34, 35, 36, 45,56,57,60,175,176,178,203,204

rate of surplus value 35

SUBJECT INDEX 225

rate ofsurplus value 35rational psychology 78raw material 144real concrete 117, 120, 121reference 77reflection 100religious freedom 110rent 47representation 4, 63, 77

cardinal 150mathematical 149not fundamental 90ordinal 150theorem 80theory 75of abstract labor 4, 135, 151, 152,

153reproducible global decision (RGD) 5,

179,180,192,200,201,202,203,205existence of 200

reproduction of the econom y 58, 177,179,180,193,204simple 199

restrictive assum ptions 53, 116revealed religion 109Revelation 108Ricardian theory of value 7, 141rigour in science and philosophy 112rising (or passage) to the concrete 116,

117,120,122,125,126,127

salary 32salvation L08, L09satisfiable formula 67,68,75scientific method 117sentence 66set-theoretic predicate 125, 192set of labors or labor inpu ts 140, 141,

148, 170, 174simple commodity production 41, 43,

44,48,49simple Marxian economy 3, L8, 29, 30,

35,38,44,51,52,53,164,192social contradiction 116social division oflabor 46social resources, total or global 162,

166

social science 112socially necessary labor-time II, 17,20,

24,25,40,45,49,50,59, 131, 144Son, the 107soul of man 110Soviet philosophy 99special relativity 113specialist firm 193, 196, 197, 199speculative philosophy 109sphere ofdistribution 43sphere of production 16,43, 130spirit 96, 97, 99, 100, 105, 107, 109,

110, IIIabsolute 108, 110human 108, 110self-conscious 96, 97, 98, 107

standard simplex 162State 109, 160structure 3, 63, 66, 72, 75, 77

abstract labor 135, 148, 150, 153,156, 157

basic Marxian 165idealized 127Aristotelian extensive 83,85first order 70k-type of a - 72, 73numerical 79ontological 79, 135productive 147,147,148,149,151,

152, 156representation theorem of abstract

labor - 150, 151, 152, 153representation theorem of

Aristotelian extensive - 89scientific 75species 72, 74, 75super- 71type ofa - 74

substance-universals 113, 114supply 46surplus-value 7, 17, 30, 34, 35, 46, 47,

52

technology 19alternative 193,194,196,197,199actually chosen by the firms 147,

170

226 SUBJECT INDEX

interconnected 19,25,28, 195, 196,199,200,201

quasiproductive 29reproducible 27,28semiproductive 19,25,27,28, 196,

200,201tensor 150theorem 66

Berge's Maximum 183Compactness 68Coders CompletenessFundamental Marxian (FMT) 2, 30,

36, 37, 53, 56, 57, 59, 124, 161,175,176

Perron-Frobenius 33, 201Stiemke's 153

theory 68, 75axiomatizable 69finitelyaxiomatizable 69of measurement 79of models 69, 76of science 76, 91of sets 76proof 76recursion 76scientific 69, 77, 91Zermelo-Fraenkel - of sets 7 I, 72

topological space 69Totalitarianism 110totality of thought 114, 120trades 20transcendental idealism 113transcendental subject II I, 112transformation problem 4, 37, 39, 47,

124, 129,203true formula 68

understanding (Venliinde) 78,93, 95, 96synthesizing activity of III, 112

unity of apperception 113unity of the diverse 117,120universal in action 104, 106uni versal kinds 96, IIIuniversals, Hegel's theory of III

vague concepts 112

valorization process 17, 144value or labor-value 4, 9, 35, 38, 47,

49,57,59,61,62, 124, 135, 149, 150,156, 157, 175and price, proportionality of 38,39,

40as regulator of prices 40,42,43,45first definition of II, 23, 29, 30, 56form of manifestation of 9, 13, 103,

129, 131individual 48magnitude of II, 13, 131market dependent determination of

17,43, 132market independent determination

of 14, 16, 17,42, 43, 46, 130optimum 57,59,60, 124second definition of 12, 25, 29, 30,

56source of (Quell e) 102substance 10, 13, 130

variable capital 38vector 20

consumption 26,31, 60norm 167n-dimensional 132oflabor inputs 20, 138of material inputs 21,22, 138of material outputs 21, 138of net outputs 21oflabor-values 24, 25, 54of wage goods 26,32representing a prod uction

process 21wage 152

von Neumann economy 176von Neumann golden rule 58,59

wage 30,31,32hourly 151

wage good 19,21,170,194ind ustries 23

wage system 142, 145Word of Cod 107working class 161

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ISBN 90-277-0826-6; Pb 90-277-0862-2118. Not published119. J. Pelc (ed.), Semiotics in Poland, 1894-1969. Translated from Polish. 1979

ISBN 90-277-0811-8120. I. Porn, Action TheOlY and Social Science. Some Fonnal Models. 1977

ISBN 90-277-0846-0121. J. Margolis, Persons and Mind. The Prospects of Nonreductive Materialism. [Boston

Studies in the Philosophy of Science, Vol. LVII] 1977ISBN 90-277-0854-1; Pb 90-277-0863-0

122. J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical andPhilosophical Logic. 1979 ISBN 90-277-0879-7

123. T. A. F. Kuipers, Studies in Inductive Probability and Rational Expectation. 1978ISBN 90-277-0882-7

124. E. Saarinen, R. Hilpinen, I. Niiniluoto and M. P. Hintikka (eds.), Essays in HonourofJaakko Hintikka on the Occasion ofHis 50th Birthday. 1979

ISBN 90-277-0916-5125. G. Radnitzky and G. Andersson (eds.), Progress and Rationality in Science. [Boston

Studies in the Philosophy of Science, Vol. LVIII] 1978ISBN 90-277-0921-1; Pb 90-277-0922-X

126. P. Mittelstaedt, Quantum Logic. 1978 ISBN 90-277-0925-4127. K. A. Bowen, Model Theory for Modal Logic. Kripke Models for Modal Predicate

Calculi. 1979 ISBN 90-277-0929-7128. H. A. Bursen, Dismantling the Memory Machine. A Philosophical Investigation of

Machine Theories of Memory. 1978 ISBN 90-277-0933-5

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129. M. W. Wartofsky, Models. Representation and the Scientific Understanding.[Boston Studies in the Philosophy of Science, Vol. XLVIII] 1979

ISBN 90-277-0736-7; Pb 90-277-0947-5130. D. Ihde, Technics and Praxis. A Philosophy of Technology. [Boston Studies in the

Philosophy of Science, Vol. XXIV] 1979 ISBN 90-277-0953-X; Pb 90-277-0954-8131. J. J. Wiatr (ed.), Polish Essays in the Methodology of the Social Sciences. [Boston

Studies in the Philosophy of Science, Vol. XXIX] 1979ISBN 90-277-0723-5; Pb 90-277-0956-4

132. W. C. Salmon (ed.), Hans Reichenbach: Logical Empiricist. 1979ISBN 90-277-0958-0

133. P. Bieri, R.-P. Horstmann and L. KrUger (eds.), Transcendental Arguments inScience. Essays in Epistemology. 1979 ISBN 90-277-0963-7; Pb 90-277-0964-5

134. M. Markovic and G. Petrovic (eds.), Praxis. Yugoslav Essays in the Philosophy andMethodology of the Social Sciences. [Boston Studies in the Philosophy of Science,Vol. XXXVIjl979 ISBN 90-277-0727-8; Pb 90-277-0968-8

135. R. W6jcicki, Topics in the Formal Methodology of Empirical Sciences. Translatedfrom Polish. 1979 ISBN 90-277-1004-X

136. G. Radnitzky and G. Andersson (eds.), The Structure and Development of Science.[Boston Studies in the Philosophy of Science, Vol. LIXj 1979

ISBN 90-277-0994-7; Pb 90-277-0995-5137. J. C. Webb, Mechanism, Mentalism and Metamathematics. An Essay on Finitism.

1980 ISBN 90-277-1046-5138. D. F. Gustafson and B. L. Tapscott (eds.), Body, Mind and Method. Essays in Honor

of Virgil C. Aldrich. 1979 ISBN 90-277-1013-9139. L. Nowak, The Structure of Idealization. Towards a Systematic Interpretation of the

Marxian Idea of Science. 1980 ISBN 90-277-1014-7140. C. Perelman, The New Rhetoric and the Humanities. Essays on Rhetoric and Its

Applications. Translated from French and German. With an Introduction by H.Zyskind. 1979 ISBN 90-277-1018-X; Pb 90-277-1019-8

141. W. Rabinowicz, Universalizability. A Study in Morals and Metaphysics. 1979ISBN 90-277-1020-2

142. C. Perelman, Justice, Law and Argument. Essays on Moral and Legal Reasoning.Translated from French and German. With an Introduction by H.J. Berman. 1980

ISBN 90-277-1089-9; Pb 90-277-1090-2143. S. Kanger and S. Ohman (eds.), Philosophy and Grammar. Papers on the Occasion

of the Quincentennial of Uppsala University. 1981 ISBN 90-277-1091-0144. T. Pawlowski, Concept Formation in the Humanities and the Social Sciences. )980

ISBN 90-277-1096-1145. J. Hintikka, D. Gruender and E. Agazzi (eds.), Theory Change, Ancient Axiomatics

and Galileo's Methodology. Proceedings of the 1978 Pisa Conference on the Historyand Philosophy of Science, Volume I. 1981 ISBN 90-277-1126-7

146. J. Hintikka, D. Gruender and E. Agazzi (eds.), Probabilistic Thinking, Ther­modynamics, and the Interaction of the History and Philosophy of Science.Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science,Volume 11.1981 ISBN 90-277-1127-5

147. U. Monnich (ed.), Aspects ofPhilosophical Logic. Some Logical Forays into CentralNotions of Linguistics and Philosophy. 1981 ISBN 90-277-1201-8

148. D. M. Gabbay, Semanticaiinvestigations in Heyting's Intuitionistic Logic. 1981ISBN 90-277-1202-6

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149. E. Agazzi (ed.), Modern Logic - A Survey. Historical, Philosophical, and Mathemati-cal Aspects of Modem Logic and Its Applications. 1981 ISBN 90-277-1137-2

150. A. F. Parker-Rhodes, The Theory of Indistinguishables. A Search for ExplanatoryPrinciples below the Level of Physics. 1981 ISBN 90-277-1214-X

151. J. C. Pitt, Pictures, Images, and Conceptual Change. An Analysis of Wilfrid Sellars'Philosophy of Science. 1981 ISBN 90-277-1276-X; Pb 90-277-1277-8

152. R. Hilpinen (ed.), New Studies in Deontic Logic. Nonns, Actions, and the Founda-tions of Ethics. 1981 ISBN 90-277-1278-6; Pb 90-277-1346-4

153. C. Dilworth, Scientific Progress. A Study Concerning the Nature of the Relationbetween Successive Scientific Theories. 2nd, rev. and augmented ed., 1986

ISBN 90-277-2215-3; Pb 90-277-2216-1154. D. Woodruff Smith and R. McIntyre, Husser! and Intentionality. A Study of Mind,

Meaning, and Language. 1982 ISBN 90-277-1392-8; Pb 90-277-1730-3155. R. J. Nelson, The Logic of Mind. 2nd. ed., 1989

ISBN 90-277-2819-4; Pb 90-277-2822-4156. J. F. A. K. van Benthem, The Logic of Time. A Model-Theoretic Investigation into

the Varieties of Temporal Ontology, and Temporal Discourse. 1983; 2nd ed., 1991ISBN 0-7923-1081-0

157. R. Swinburne (ed.), Space, Time and Causality. 1983 ISBN 90-277-1437-1158. E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics. Ed. by R. D.

Rozenkrantz.1983 ISBN 90-277-1448-7; Pb (1989) 0-7923-0213-3159. T. Chapman, Time: A Philosophical Analysis. 1982 ISBN 90-277-1465-7160. E. N. Zalta, Abstract Objects. An Introduction to Axiomatic Metaphysics. 1983

ISBN 90-277-1474-6161. S. Harding and M. B. Hintikka (eds.), Discovering Reality. Feminist Perspectives on

Epistemology, Metaphysics, Methodology, and Philosophy of Science. 1983ISBN 90-277-1496-7; Pb 90-277-1538-6

162. M. A. Stewart (ed.), Law, Morality and Rights. 1983 ISBN 90-277-1519-X163. D. Mayr and G. Siissmann (eds.), Space, Time, and Mechanics. Basic Structures of a

Physical Theory. 1983 ISBN 90-277-1525-4164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I:

Elements of Classical Logic. 1983 ISBN 90-277-1542-4165. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. II:

Extensions of Classical Logic. 1984 ISBN 90-277-1604-8166. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. III:

Alternative to Classical Logic. 1986 ISBN 90-277-1605-6167. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. IV:

Topics in the Philosophy of Language. 1989 ISBN 90-277-1606-4168. A. J. I. Jones, Communication and Meaning. An Essay in Applied Modal Logic.

1983 ISBN 90-277-1543-2169. M. Fitting, Proof Methods for Modal and Intuitionistic Logics. 1983

ISBN 90-277-1573-4170. J. Margolis, Culture and Cultural Entities. Toward a New Unity of Science. 1984

ISBN 90-277-1574-2171. R. Tuomela, A Theory ofSocial Action. 1984 ISBN 90-277-1703-6172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical

Analysis in Latin America. 1984 ISBN 90-277-1749-4173. P. Ziff, Epistemic Analysis. A Coherence Theory of Knowledge. 1984

ISBN 90-277-1751-7

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174. P. Ziff, Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984ISBN 90-277-1773-7

175. W. Balzer, D. A. Pearce, and H.-J. Schmidt (eds.), Reduction in Science. Structure,Examples, Philosophical Problems. 1984 ISBN 90-277-1811-3

176. A. Peczenik, L. Lindahl and B. van Roermund (eds.), Theory of Legal Science.Proceedings of the Conference on Legal Theory and Philosophy of Science (Lund,Sweden, December 1983). 1984 ISBN 90-277-1834-2

177. I. Niiniluoto, Is Science Progressive? 1984 ISBN 90-277-1835-0178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative

Perspective. Exploratory Essays in Current Theories and Classical Indian Theoriesof Meaning and Reference. 1985 ISBN 90-277-1870-9

179. P. Kroes, Time: Its Structure and Role in Physical Theories. 1985ISBN 90-277-1894-6

180. J. H. Fetzer, Sociobiology and Epistemology. 1985ISBN 90-277-2005-3; Pb 90-277-2006-1

181. L. Haaparanta and J. Hintikka (eds.), Frege Synthesized. Essays on the Philosophicaland Foundational Work of Gottlob Frege. 1986 ISBN 90-277-2126-2

182. M. Detlefsen, Hilbert's Program. An Essay on Mathematical Instrumentalism. 1986ISBN 90-277-2151-3

183. J. L. Golden and J. J. Pilotta (eds.), Practical Reasoning in Human Affairs. Studiesin Honor of Chaim Perelman. 1986 ISBN 90-277-2255-2

184. H. Zandvoort, Models of Scientific Development and the Case of Nuclear MagneticResonance. 1986 ISBN 90-277-2351-6

185. I. Niiniluoto, Truthlikeness. 1987 ISBN 90-277-2354-0186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonic for Science. The

Structuralist Program. 1987 ISBN 90-277-2403-2187. D. Pearce, Roads to Commensurability. 1987 ISBN 90-277-2414-8188. L. M. Vaina (ed.), Matters ofIntelligence. Conceptual Structures in Cognitive Neuro-

science. 1987 ISBN 90-277-2460-1189. H. Siegel, Relativism Refuted. A Critique of Contemporary Epistemological

Relativism. 1987 ISBN 90-277-2469-5190. W. Callebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm

Program, with a Complete Evolutionary Epistemology Bibliograph. 1987ISBN 90-277-2582-9

191. J. Kmita, Problems in Historical Epistemology. 1988 ISBN 90-277-2199-8192. J. H. Fetzer (ed.), Probability and Causality. Essays in Honor of Wesley C. Salmon,

with an Annotated Bibliography. 1988 ISBN 90-277-2607-8; Pb 1-5560-8052-2193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science. Empirical

Studies of Scientific Change. 1988 ISBN 90-277-2608-6194. H.R. Otto and J.A. Tuedio (eds.), Perspectives on Mind. 1988

ISBN 90-277-2640-X195. D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Recent Insights

and New Perspectives on Their Relation. 1988 ISBN 90-277-2645-0196. J. Osterberg, Selfand Others. A Study of Ethical Egoism. 1988

ISBN 90-277-2648-5197. D.H. Helman (ed.), Analogical Reasoning. Perspectives of Artificial Intelligence,

Cognitive Science, and Philosophy. 1988 ISBN 90-277-2711-2198. J. Wolenski, Logic and Philosophy in the Lvov-Warsaw School. 1989

ISBN 90-277-2749-X

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199. R. W6jcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations.1988 ISBN 90-277-2785-6

200. J. Hintikka and M.B. Hintikka, The Logic of Epistemology and the Epistemology ofLogic. Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6

201. E. Agazzi (ed.), Probability in the Sciences. 1988 ISBN 90-277-2808-9202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 ISBN 90-277-2814-3203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical

Knowledge. 1989 ISBN 0-7923-0131-5204. A. Melnick, Space. Time. and Thought in Kant. 1989 ISBN 0-7923-0135-8205. D.W. Smith, The Circle of Acquaintance. Perception, Consciousness, and Empathy.

1989 ISBN 0-7923-0252-4206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism. Essays in Honor of

Arthur W. Burks. With his Responses, and with a Bibliography of Burk's Work.1990 ISBN 0-7923-0325-3

207. M. Kusch, Language as Calculus vs. Language as Universal Medium. A Study inHusserl, Heidegger, and Gadamer. 1989 ISBN 0-7923-0333-4

208. T.e. Meyering, Historical Roots of Cognitive Science. The Rise of a CognitiveTheory of Perception from Antiquity to the Nineteenth Century. 1989

ISBN 0-7923-0349-0209. P. Kosso, Observability and Observation in Physical Science. 1989

ISBN 0-7923-0389-X210. J. Kmita, Essays on the Theory ofScientific Cognition. 1990 ISBN 0-7923-0441-1211. W. Sieg (ed.), Acting and Reflecting. The Interdisciplinary Tum in Philosophy. 1990

ISBN 0-7923-0512-4212. J. Karpinski, Causality in Sociological Research. 1990 ISBN 0-7923-0546-9213. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters. 1991

ISBN 0-7923-0823-9214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein's Philosophy of

Psychology. 1990 ISBN 0-7923-0850-6215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and

Epistemological Implications of the Work of W.V.O. Quine and of N. Goodman.1990 ISBN 0-7923-0904-9

216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability.Philosophical Perspectives. 1991 ISBN 0-7923-1046-2

217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of theUniverse. 1991 ISBN 0-7923-1322-4

218. M. Kusch, Foucault's Strata and Fields. An Investigation into Archaeological andGenealogical Science Studies. 1991 ISBN 0-7923-1462-X

219. C,J. Posy, Kant's Philosophy ofMathematics. Modem Essays. 1992ISBN 0-7923-1495-6

220. G. Van de Vijver, New Perspectives on Cybernetics. Self-Organization, Autonomyand Connectionism. 1992 ISBN 0-7923-1519-7

22 I. J.c. Nyfri, Tradition and Individuality. Essays. 1992 ISBN 0-7923-1566-9222. R. Howell, Kant's Transcendental Deduction. An Analysis of Main Themes in His

Critical Philosophy. 1992 ISBN 0-7923-1571-5

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223. A. Garda de la Sienra, The Logical Foundations of the Marxian Theory of Value.1992 ISBN 0-7923-1778-5

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