Andreas Voigt on Ordinal and Cardinal Utility in 1893

Torsten Schmidt University of New Hampshire

Christian E. Weber* Seattle University

Very Preliminary. Please do not quote without permission.

ABSTRACT

The earliest argument for an ordinal treatment of utility is usually ascribed to an unpublished 1898 paper by Vilfredo Pareto. However, in 1893 Andreas Heinrich Voigt had published “Zahl und Mass in der Ökonomik” (“Number and Measurement in Economics,” Zeitschrift für die gesamte Staatswissenschaft), where he argued that utility admits only an ordinal characterization. Voigt maintained also that any attempts to associate a cardinal measure with utility came from ill-conceived efforts to mimic the physical sciences. Our paper discusses Voigt and his training in both economics and mathematics. It also discusses key developments in nineteenth century mathematics on which he drew as he wrote. Finally, it discusses the views on utility and in particular ordinal utility which Voigt set forth in his 1893 article.

J.E.L. classification #'s: B13, B21

* corresponding author Address correspondence to: Christian E. Weber Dept. of Economics and Finance Albers School of Business and Economics Seattle University Seattle, WA 98122 Ph.: (206) 296-5725 FAX: (206) 296-2486 e-mail: cweber@seattleu.edu We would like to thank Dean Peterson and participants at the 2006 History of Economics society Meetings at Grinnell College for their comments on an earlier version of this paper. Of course, any errors which may remain are entirely the responsibility of the authors.

1. Introduction

The mid and late 1930’s were a watershed era in the development of modern utility and

demand theory. Within a span of just six years, Harold Hotelling (1932, 1935) pioneered

mathematical comparative statics analysis within Anglophone economics, John Hicks and

R.G.D. Allen (1934) argued for a purely ordinal theory of utility and demand and further

developed the comparative statics analysis of consumer demand, and Paul Samuelson (1938a,

1938b) developed the complete comparative statics implications of the utility maximization

hypothesis and introduced what would later become known as the revealed preference approach

to demand theory. The next year, as if to crown these important accomplishments, Hicks (1939)

provided both a lucid verbal summary of most of these then recent results and a mathematical

treatment of demand theory which would remain definitive for decades.1

At least in the English speaking world, the theory of utility and demand had started the

1930’s as a rather loosely knit collection of ideas which, with relatively few exceptions (e.g., the

contributions of Fisher (1892) and Johnson (1913)), had hardly progressed beyond the exposition

in Alfred Marshall’s Principles (Marshall, 1890). However, by the close of the decade, the

combined efforts of a handful of the discipline’s leading theorists had established this branch of

economic theory as a rigorous, active field of research capable of yielding refutable restrictions

on observable price-quantity data, at least in principle. Furthermore, demand theory would

eventually serve as one of the main conduits through which economics would emerge by the mid

1950’s as a rigorous mathematical science.2

1 Of course, other important contributions during this period include the work done by Allen (1932, 1933, 1934a, 1934b) just prior to his collaboration with Hicks, the theoretical and empirical analyses of Henry Schultz (1933, 1935, 1938), the seminal work of Nicholas Georgescu-Roegen (1936) on the integrability problem. 2 It is no exaggeration to argue that only the evolution of general equilibrium theory between the mid 1930’s and the

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The typical “pre-history” of this important period ascribes important roles to Irving

Fisher (1892) and Vilfredo Pareto (1898, 1900) as early, if inconsistent and perhaps even

confused, advocates of an ordinal approach to utility and to Pareto (1892-1893, 1909) and Eugen

Slutsky (1915) as pioneers in the development of the comparative statics analysis of consumer

demand.3 This paper fills in an important gap in our understanding of the pre-history of 1930’s

developments in utility and demand theory. Specifically, we show that both the earliest

statement of the idea that utility should be viewed as a purely ordinal rather than a cardinal

magnitude and the earliest use of the cardinal versus ordinal terminology appear in the work of a

German mathematician and economist, Andreas Heinrich Voigt. Voigt published this

fundamental contribution to economic theory in 1893 in the German language journal Zeitschrift

für die Gesamte Staatswissenschaft (today known also as the Journal of Institutional and

Theoretical Economics).

In addition to discussing Voigt’s views on ordinalism, we show that these views grew

directly out of his knowledge of then recent developments in mathematics. As a result, the

discussion below of Voigt’s early contribution to ordinalism also furthers our understanding of

the links between developments in mathematics and subsequent innovations in economic theory.

Thus, this paper sheds further light on the question, as Roy Weintraub (2002) has recently

mid 1950’s rivals demand and utility theory in importance as a vehicle for “mathematizing” economic theory. 3 See, inter alia Joseph Schumpeter, (1954, pp 1062-1064), Jürg Niehans (1990, pp. 263-264, 272), and Ernesto Screpanti and Stefano Zamagni (1993, pp. 203-206). Christian Weber (2001) provides a detailed discussion of Pareto’s views on cardinal versus ordinal utility functions, while Peter C. Dooley (1983a) and Weber (1999) discuss the connection between Pareto’s early comparative statics analyses of consumer demand and the later and ultimately more successful contribution of Slutsky. – Niehans (1995, p. 20) credits Auspitz and Lieben (1889) for the ordinal utility function, along with Fisher. Several of the utility functions in their book are of entirely general appearance in notational terms, and they did argue that an equiproportionate change of the utility scale would be of no material consequence (p. 501). But allowing for a multiplicative change of the utility index is not the same as allowing for an arbitrary, positive monotonic transformation, and their discussions of the shape of the utility surface retained wholly cardinal elements (particularly see appendix 2, section 3).

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phrased it, of “how economics became a mathematical science,”4 Although the period under

consideration here somewhat predates that examined by Weintraub.

Finally, this paper also constitutes a partial sequel to John Chipman’s (2005) recent

encyclopedic history of utility theory on Germany during the nineteenth century. By sheer

coincidence (the authors did not learn of Chipman’s paper until they had already completed an

earlier draft of this paper), the contribution by Voigt with which this paper is primarily

concerned appeared almost exactly at the end of the roughly eighty-five year period (ca. 1805-

1890) which Chipman surveys. Thus, this paper picks up a crucial part of the history of utility

theory in Germany just at the point where Chipman’s paper ends.

The remainder of the paper is organized as follows: Since Voigt is largely forgotten

today,5 we begin in section 2 by providing a brief biography of the main protagonist or our story,

Andreas Heinrich Voigt. This section will highlight Voigt’s unique (for his era) professional

training as both an economist and a mathematician. Then, since the particular contribution of

Voigt on which we focus was firmly rooted in his formal training as a mathematician, section 3

reviews the key developments in mathematics in the late nineteenth century on which Voigt drew

as he thought about utility theory and measurement in 1892 and ’93. Thus, section 3 provides an

essential backdrop for section 4, in which we discuss Voigt’s pioneering contribution to ordinal

utility theory. Section 5 concludes our paper.

4 One might pose the question in reverse, how mathematical developments have come from dealing with economic matters, resulting in methods that would (also) become part of mathematics; this is not quite the one-way street it is often made out to be. The emergence in the twentieth century of linear and nonlinear programming methods and of game theory would be a case in point. Consider also the earlier development or discovery of the number e, the exponential function, and logarithms, attached to compound interest calculations. A closeness of economic thinking and mathematical thinking is very old: the word ‘number’ derives from the Greek νόµος, it in turn being derived from νέµειν meaning ‘to deal, distribute, hold, manage’ (Oxford English Dictionary). 5 So far as we can tell, the only modern reference to Voigt by an economist (aside from the present paper) is Peter Dooley’s (1983b) citation of a reference by Francis Edgeworth (1894) to Voigt’s (1893c) suggestion that economists treat utility as being defined ordinally rather than cardinally; however Dooley does not cite any of Voigt’s works explicitly. Dooley’s reference to Voigt inspired the further research which lead to this paper.

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2. Andreas Voigt: A Biographical Sketch

While biographical information on Andreas Heinrich Voigt is somewhat difficult to find,

we do know the following:6 Voigt was born in 1860 in Flensburg in what is now Northern

Germany.7 He began his university studies in Berlin in 1882, attending lectures in philosophy,

political economy, mathematics, and the physical sciences. While in Berlin, he studied

economics with Adolph Wagner, so we know that his interest in economics dates to his early

adult years. After two years, in 1884 he transferred to the University of Freiburg, where he

studied philosophy, the physical sciences, and mathematics. In Freiburg he was first exposed to

Ernst Schröder’s (1841-1902) Der Operationskreis des Logikkalkuls (Schröder (1877)), which

would turn out to be important for his later development. After a little more than two years in

Freiburg, Voigt moved to the University of Kiel in 1886, the university nearest to his home town,

and then to Heidelberg in 1887, concentrating on mathematics and physics. In 1889 he passed

the examinations in the state of Baden to become a teacher. His major subject for these

examinations was mathematics, and Jakob Lüroth (1844-1910) of the University of Freiburg

served as his chief examiner. After passing this examination, Voigt began teaching in Freiburg,

but also found time to begin writing a dissertation on the algebra of logic (Voigt, 1890) and to

apply for doctoral candidacy at the University of Freiburg, being admitted in November 1889.

In his postgraduate work, Voigt studied philosophy and physics but primarily

mathematical logic. Jakob Lüroth and his close friend Schröder, of Karlsruhe, read and reported

6 The biographical sketch is based largely on Hamacher-Hermes (1994) and Pulkkinen (1998). See also the brief biography of Ernst Schröder on the University of St. Andrews Mathematics and Statistics website at www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Schroder.html, (Anonymous) (1901), and Fehling (1926). 7 At that time, Flensburg was part of Denmark.

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on Voigt’s dissertation on logic. 8 At the time, Schröder was teaching at the Technische

Hochschule (technical university) in Karlsruhe, Germany (renamed Universität Karlsruhe in

1967, when all German “technical schools” were reclassified as “universities”), and was the

leading German contributor to the relatively new field of mathematical logic. Indeed, Willard

Quine (1951, p. 1) lists Schröder among only seven explicitly named late nineteenth and early

twentieth century founders of mathematical logic: “it was from Boole through Peirce, Schröder,

Frege, Peano, Whitehead, Russell, and their successors that mathematical logic underwent

continuous development and reached the estate of a reputable department of knowledge.”9 In the

1880s, Charles Sanders Peirce used Ernst Schröder’s (1877) Operationskreis des Logikkalkuls as

a textbook for his course on logic at Johns Hopkins University, and Peirce and Schröder

corresponded for a number of years (Houser 1990-91).

Voigt completed his dissertation in 1890. With his dissertation complete, several of

Voigt’s supporters suggested that he write a habilitation thesis in mathematics, but Voigt

declined, since by this time his primary academic interest had shifted to economics. The same

year, Schröder published (at his own expense) the first volume of his three-volume treatise on the

algebra of logic, Vorlesungen über die Algebra der Logik, (Schröder, 1890). Soon after the

publication of the Vorlesungen, Voigt (1892b, 1893a) defended his teacher’s algebra of logic

8 Both Lüroth and Schröder had done their doctoral work at Heidelberg with Otto Hesse and Gustav Kirchhoff; Kirchhoff himself had been a doctoral student of Hesse at Königsberg. (The Mathematics Genealogy Project at genealogy.math.ndsu.nodak.edu has been most helpful in identifying these and other connections.) Another doctoral student of Hesse was Carl Neumann, the older brother of Friedrich Julius Neumann whose 1892 paper on value theory would be cited by Voigt in “Zahl und Mass;” Friedrich Neumann both cited and credited him with most helpful conversations. Carl Neumann was also co-founder and long time co-editor of Mathematische Annalen. 9 Aside from Schröder, the references here are to the work of British logician George Boole (1815-1864), American mathematician Charles Sanders Pierce (1839-1914), German mathematician Gottlob Frege (1848-1925), Italian mathematician Guiseppe Peano (1858-1932), and of course the British logician-mathematicians Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970). Similarly, Howard Eves (1983, p. 470) rates Schröder as Boole’s near equal as a founder of symbolic logic, which he terms the Boole-Schröder algebra. As a mathematician, Schröder ranks in rarefied company indeed!

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against criticisms leveled at it by the German philosopher Edmund Husserl (1891a, 1891b,

1893). These were among Husserl’s earliest philosophical publications, after his habilitation

thesis and a revision of it on number theory and logic (Husserl 1887, 1891c); see also Pulkkinen

(1998). Of course, Husserl (1859-1938) went on to develop phenomenology as a distinct branch

of modern philosophy and to become one of the leading Continental philosophers of his age.

In 1892, while the debate with Husserl was still in progress, Schröder helped Voigt obtain

a post teaching mathematics at the Technische Hochschule in Karlsruhe. At about this time,

Voigt prepared a habilitation thesis in economics at Karlsruhe which was rejected. In 1896, with

several publications in economics to his credit (Voigt 1891, 1892a, 1893a, 1893b, 1893c, 1895),

Voigt took a position in political economy at the newly created Institut für Gemeinwohl (Institute

for Public Welfare) in Frankfurt, holding this position until 1903. Beginning around 1899, Voigt

was active in efforts to found a new, non university-affiliated school of business in Frankfurt, the

Akademie für Social- und Handelswissenschaften (see Voigt, 1899). Created jointly by the

Institut für Gemeinwohl and the Frankfurt chamber of commerce, the Akademie opened its doors

in 1901 with Voigt as its chief administrator. In 1903, when he left the Institut für Gemeinwohl,

Voigt was appointed Professor of Political Economy at the Akademie.10 Then in 1914, the

Akademie was joined together with the Institut für Gemeinwohl and several other local scientific

institutes and granted university status.11 Voigt was appointed as the new University’s first

Professor of Economics (Professor der wirtschaftlichen Staatswissenschaften). Voigt retired

from the University in 1925, but remained active as a scholar even in retirement (Voigt 1928a,

1928b), and even became an early member of the Econometric Society (anonymous, 1934). He

10 Fehling (1926) discusses the evolution of German higher education and business education in particular during the late nineteenth and early twentieth centuries. 11 In 1932 the University was named Johann Wolfgang Goethe-University on the centenary of Goethe’s death.

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died in 1940.

Voigt seems to have been a man of wide intellectual interests. Despite devoting both his

teaching and his administrative careers to economics from 1896 onward, he continued to publish

in mathematics (Voigt, 1911) and also wrote on accounting (Voigt, 1922). Within economics,

his interests were also fairly broad. In addition to his contribution to utility theory, which we

discuss in some detail below, he also studied land use and urban land prices and rents (Voigt and

Geldner, 1905 and Voigt, 1907) contributed to monetary theory (Voigt, 1920), discussed German

tariffs (Voigt, 1912), wrote both a book on “social utopias” which was went through several

German language editions and was translated into Russian (Voigt, 1906) and timely pamphlets

on economics and socialism during wartime (Voigt, 1916) and on the post-war economic order

(Voigt, 1921), and on labor arbitration (Voigt, 1928a).

Since much of Voigt’s formal education included a considerable amount of mathematics,

and especially since he made reference to then recent developments in mathematics as he

developed his ideas on ordinal utility, a clear understanding of hisviews on utility will require

some knowledgeof the mathematical milieu within which he formed his ideas. Thus, before we

discuss Voigt’s contribution to ordinalism, the next section we briefly review the particular

developments in mathematics during in the two decades prior to 1893 which seem to have

exerted a strong influence on Voigt’s views.

3. Developments in Mathematics, ca. 1870-1890

The mathematical ideas which formed the backbone of Voigt’s approach to ordinalism

were developed by a handful of German mathematicians between ca. 1870 and 1890. This of

course was the dawn of the era in mathematics between 1870 and 1940 which historian of

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mathematics I. Grattan-Guinness (2000) has described as “the search for mathematical roots,”

involving logic, the theory of sets, and certain aspects of number theory.

Working on the foundations of number theory and, among other things, exploring the

origin of the concept of number, Hermann von Helmholtz (1887), Leopold Kronecker (1887),

and Richard Dedekind (1888) all made arguments to the effect that ordinal numbers embody a

more fundamental conception of number than cardinal numbers.12 These contributions, and only

these contributions, would be cited by Andreas Voigt (1893c) as the mathematical point of origin

for his subsequent arguments favoring an ordinal conception of utility.

The three authors named did not cite much prior work. According to Dedekind (1888) in

the preface of his acclaimed Was sind und was sollen die Zahlen? (which soon after appeared in

English translation as part of Essays on the Theory of Numbers, 1901), it was in fact the 1887

appearance in the same Festschrift volume of the essays by Helmholtz and Kronecker that had

induced him to put down on paper his own thinking, much of which he said was developed

before 1887 and a continuation of his work on the nature of numbers in his earlier Stetigkeit und

irrationale Zahlen (1872). Excepting that earlier Dedekind book, the earliest source given in any

of the three papers was Ernst Schröder’s Lehrbuch der Arithmetik und Algebra für Lehrer und

Studierende (1873), which was cited by Helmholtz and Dedekind. Helmholtz mentioned it as a

source of inspiration, and Dedekind more as a general reference, although he cited it near his

citation of Helmholtz and Kronecker. As already noted, Schröder would later advise Voigt on

his dissertation.

In the first chapter of this text, Schröder offered many observations on measurement and

the denomination of units. Also in that chapter, Schröder noted the distinction between cardinal

12 Translations are available as Helmholtz (1999), Kronecker (1999), and Dedekind (1901).

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numbers (Cardinalzahlen) and ordinal numbers (Ordinalzahlen), as indicating the total number

of a group of objects vs. position of an object in a sequence, as well as the distinguishing

property of the cardinal numbers that the result of counting a collection of objects is independent

of the order of counting.13

For the ordinal numbers, Schröder remarked that in order to identify a particular unit in a

series, the name of the associated number would be quite sufficient. Roughly the conception

would later reappear in much more detail in Helmholtz (1887) and Kronecker (1887). Schröder

took up issues of measurement in the section immediately following, which was the particular

section cited by Helmholtz, but did not link the two sections.14 Yet, by Helmholtz’ own account

in his essay “Numbering and Measuring from an Epistemological Viewpoint,” his reading of

Schröder had influenced his thinking:

Among more recent arithmeticians, E. Schröder has also essentially attached

himself to [Hermann Grassmann (1878) and Robert Grassmann (1872)], but in a

few important discussions he has gone still deeper. As long as earlier

mathematicians habitually took the ultimate concept of number to be that of a

cardinal number [Anzahl] of objects, they could not wholly free themselves from

the laws of behavior of these objects, and they simply took it to be a fact that the

cardinal number of a group is ascertainable independent of the order in which

they are numbered. To my knowledge, Mr. Schröder (§12) was the first to

recognize that here a problem lies concealed: he also acknowledged – in my

opinion justly – that there lies a task here for psychology, while on the other hand

those empirical properties should be defined which the objects must have in order

13 We are showing the original terminology because the terminology, both in the original and in translation, turns out to be important; more on that below. 14 Interestingly, Schröder made repeated use of economic examples for illustration, such as counting money (p. 4), coins of specific type (p. 7), the usefulness of money for exchange of goods (p. 10), repayment of debt (p. 12), the amount paid for a good in relation to its quantity (p. 12), and the use of numbers as serving the “ultimate aim of all human action: the coming together or coinciding of needs with their satisfaction,” thus establishing the “value of numbers for the economic affairs of life” (p. 13).

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to be enumerable …. One must ask: what is the objective sense of expressing

relationships between real objects as magnitudes, by using denominate numbers;

and under what conditions can we do this? (1999, pp. 729, 730).

Helmholtz, who was primarily a physicist but whose interests had drifted into mathematics and

epistemology, apparently shared Schröder’s concern with giving meaning to numbers in the

context of practical measurement but, unlike Schröder, moved on to make a connection between

[i] measurement and [ii] the distinguishing between ordinal and cardinal numbers.

Helmholtz noted discussions by Paul du Bois-Reymond (1882) and Adolf Elsas (1886).

Du Bois-Reymond’s (1882) deliberations on the concept mathematical magnitude were all

grounded in the human ability to perceive, and his minimum standard for magnitude (Grösse)

was that of an ordered collection of notions (Vorstellungen, p. 14) which might or might not be

‘lineary’ (lineär).15 He defined a ‘lineary magnitude’ as one meeting the standard that the

difference between two magnitudes of the same kind is also a magnitude of the same kind. Thus

the distance on a line is a lineary magnitude whereas the pitch of a tone is not: one can tell which

of two tones has the higher pitch, but our ears do not perceive the difference in pitch between the

two tones itself as the pitch of a tone. Although it is therefore not a lineary magnitude pitch is

nonetheless a mathematical magnitude according to Bois-Reymond – and it has a clear ordering.

An ordinal-cardinal characterization is clearly evident, even if Bois-Reymond did not use this

terminology.16 Elsas (1886) meanwhile, referring mostly to Bois-Reymond (1882), rejected the

idea that sensations could ever be the subject of scientific investigation (esp. see p. 70),

15 The term lineär is rare and it is not included in comprehensive dictionaries, such as the Duden. As such, it seems only appropriate to use ‘lineary’ in the English as it is so similar and is listed as obsolete in the Oxford English dictionary. Hermann Grassmann (1878, p. 245) used the term before Bois-Reymond (1882). 16 Not relevant for Helmholtz’ purposes but potentially quite important for utility theorizing, du Bois-Reymond (1882) was substantially concerned with the identification of sensations following stimuli and, going further, of consequent moods or frames of mind (Stimmungen). He also brought out their role as inducement to human action (p. 37).

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describing it as purely self-delusional to use a mathematical symbol to represent strength of

sensation (p. 66).

In the description offered by Helmholtz, ‘numbering’ at its most essential consists of

affixing a series of arbitrarily chosen symbols or names to a given sequence of real objects.

Whatever these symbols or names might be, they could then be attached in the same order to

other series of objects. With repetition and always used in that same order, these symbols in

combination came to be thought of as the natural number series:

Its being termed the ‘natural’ number series was probably connected merely with

one specific application of numbering, namely the ascertaining of the cardinal

number [Anzahl] of given real things … This sequence is in fact a norm or law

given by human beings, our forefathers, who elaborated the language. I

emphasize this distinction because the alleged ‘naturalness’ of the number series

is connected with an incomplete analysis of the concept of number …. the number

series is impressed upon our memory extraordinarily much more firmly than any

other series, which doubtless rests upon its much more frequent repetition. This is

why we also prefer to use it in order to establish, through association with it, the

recollection of other sequences in our memory; that is, we use the numbers as

ordinal numbers [Ordnungszahlen]” (1999, pp. 730-731, emphasis applied as in

the original, and to the original terms).

Thus the primitive meaning of a particular ‘number’ is that of its position in the series of

symbols or names. On this basis – a pure ordering – Helmholtz took up axioms enabling basic

arithmetic operations, referencing and comparing his account with the axioms and theorems of

the Grassmann expositions. Those discussions preceded his introduction of the cardinal number

[Anzahl] of a group of objects, calling n the “cardinal number of the members of the group” if

the complete number series from 1 through n was required to match up a number with each

element (1999, p. 738). Helmholtz kept on reiterating his notes of caution on measurement of

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real objects (“Whether these conditions are obeyed for a specific class of objects can naturally

only be determined by experience,” p. 739), then moved on to denominate numbers which

represent physical magnitudes of specific units, allowing for an empirical determination. For

investigation of the magnitudes of like objects he said it would normally suffice to work with

arbitrarily chosen units to determine only the “values of proportional numbers

[Verhältnisszahlen], until those units are reduced to universally known ones (absolute units of

physics)” (p. 741).

Leopold Kronecker (1887), later in the same Festschrift volume in his essay “On the

Concept of Number,” entertained very similar reasoning, particularly the notion that ordinal

numbers were more fundamental, though Kronecker came to this from a different perspective:

unlike Schröder and Helmholtz, Kronecker was concerned solely with the concept of number in

the abstract. Even so, he corroborated the case made by Helmholtz:

The ordinal numbers [Ordnungszahlen] are the natural starting point for the

development of the concept of number. In them we possess a stock of signs,

ordered in fixed succession … We combine the totality of the signs thus applied

into the concept of the ‘cardinal number [Anzahl] of objects’ of which the

collection is composed; and we attach the expression for this concept

unambiguously to the last of the applied signs, since their succession is rigidly

determined …. the ‘Anzahl’ is expressed by the ‘cardinal number’ [Cardinalzahl]

n corresponding to the nth ordinal number, and it is these cardinal numbers which

are designated simply as ‘numbers’ [Zahlen] …. If one ‘counts’ a collection of

objects – that is, if one adjoins the ordinal numbers in succession as signs to the

individual objects – then one thereby gives the objects themselves a fixed

ordering … [But] the result of the counting is independent of the order followed

or given by the counting. The ‘Anzahl’ of the objects of a collection is therefore a

property of the collection as such, that is, of the totality of the objects, thought of

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independently of any particular ordering (1999, pp. 949-951).17

Straightforward as these observations might appear, that is not how they were received.

Dedekind (1888) offered a far more detailed and lengthier account than either Helmholtz or

Kronecker. Dedekind’s book is made up of 171 sections (44 pages in the reprint of the English

translation; see Ewald, 1999), all leading up to the identification of the concept of number – the

word ‘number’ in any form does not appear until section 73 (“… then these elements are called

natural numbers or ordinal numbers [Ordinalzahlen] or simply numbers …,” p. 809), and

cardinal numbers would not even be mentioned until section 161 (“If numbers are used to

express accurately this determinate property [of ‘how many’] of finite systems they are called

cardinal numbers [Kardinalzahlen],” p. 831). Thus Dedekind’s organization and logic also put

the ordinal numbers ahead of the cardinal numbers, consistent with both Helmholtz and

Kronecker, but more closely aligned with Kronecker as he was not at all concerned with applied

measurement.

Other developments in the concept of number had been underway in the 1870s and

1880s, primarily associated with the work of Georg Cantor which was not completed until the

second half of the 1890s (Cantor 1874, 1883a, 1883b, 1895, 1897). 18 These must be very

carefully differentiated from the former developments, for the simple reason that the various

writers employed similar or identical terminology while the substance was different.

Cantor distinguished between cardinal numbers (Cardinalzahlen) and (Ordnungszahlen).

Both of these are a conception of total number (Anzahl, always translated as ‘cardinal number’ or

17 The translated passage is modified in a minor way from Ewald (1999); for “Anzahl der Objecte” (Kronecker 1887, p. 266, double quotes in the original), given here as ‘cardinal number [Anzahl] of objects’, Ewald had ‘Anzahl of objects’ (1999, pp. 949-950). This may look like splitting hairs but, as is to be seen, the terminology in the German and the English turns out to be quite an important matter for discerning the substance. 18 As a general reference for this paragraph and those to follow, see the editorial content by P. Jourdain of Cantor (1911), Sierpinski (1965), Dauben (1979), Ewald (1999), Aczel (2000), and Grattan-Guinness (2000).

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just ‘number’). Cantor’s cardinal number is the number of elements in a set that is not

necessarily well-ordered, or not ordered at all, whereas his ordinal number is the total number of

elements in a well-ordered set. Each element of a well-ordered set except the first element has a

unique predecessor. E.g. the set of all integers is not a well-ordered set because the set of

negative integers has no least element. Also, ties are not permitted. Thus a set may or may not

possess an ordinal number. The sets considered need not be finite: and Cantor found his ordinal-

cardinal distinction to be crucially important for his analysis of transfinite numbers, literally

those beyond the finite, introducing a new and clearer account of different orders of infinity than

had been previously available.19 For finite well-ordered sets, according to Cantor, the ordinal

and cardinal numbers share all the same properties (1890, p. 14; 1897; p. 220). That alone, of

course, sets his ordinal-cardinal distinction completely apart from the more conventional

distinction entertained by Schröder, Helmholtz, Kronecker, and Dedekind.

Cantor’s conception of ‘ordinal number’ appears to be the dominant meaning of the term

in modern mathematics, not the earlier and more conventional meaning of indicating position in

a sequence. But it was the earlier meaning that Voigt (1893c) had in mind when, following

Helmholtz and Kronecker, he referred to ordinal numbers as Ordnungszahlen with the exact

same meaning as these have in modern economics, that is, as indicating the position of an object

(e.g., and indifference surface) in a sequence. (Schröder and Dedekind had used the term

Ordinalzahlen.) Clearly a potential for confusion also exists in German, not just in English

19 Cantor (1874, 1883a) proved that the order of infinity of the rational numbers (ratios of integers) is identical to that of the natural numbers, even though in any finite interval such as the unit interval there are infinitely many rational numbers. The proof is based on finding a way to arrange the rational numbers in a unique sequence that can then be matched up with the natural numbers one-to-one, without ever running out of either the naturals or the rationals. Thus the rational numbers are enumerable (countable), and similarly for the set of algebraic numbers. The set of real numbers, on the other hand, is of a higher order of infinity on any interval, as it includes the transcendental numbers (irrational numbers that are not algebraic numbers). The set of real numbers is not countably finite. For a definitive treatment, see Sierpinski (1965).

16

language, due to the common translations as ‘ordinal numbers’ of both Ordnungszahlen and

Ordinalzahlen.’20

It is therefore unsurprising that Cantor (1887, 1890) would raise this as a serious concern

with the Helmholtz and Kronecker essays, if only because this showed that they were not aware

of or even deliberately ignored his work, but also for creating confusion. In Cantor’s eyes they

had plainly misused his term Ordnungszahl for what he called Ordinalzahlwort (literally,

‘ordinal number word’). More importantly perhaps, for Cantor the Ordinalzahlwort was the

“last and least essential” aspect of the theory of numbers (p. 16) and certainly not the foundation

of anything.

Cantor raised a whole broadside of other objections. He pointed out that the Helmholtz-

Kronecker view of constructing numbers from a fixed sequence of names was not an original

idea. He offered a lengthy (and convincing) quotation from the introduction of a book by Louis

Bertrand (1778), describing a shepherd’s using a memorized sequence of words to “count” the

returning sheep in order to verify if all of the flock had returned: if, and only if, the shepherd

reached the same word each night, he would could rest assured that all had in fact returned. 21

Cantor complained specifically of Kronecker as trying to make irrational numbers altogether

20 In the present-day German the situation is actually worse than in the 19th century. Cantor’s term Ordnungszahlen now appears to be almost exclusively used for ‘atomic number,’ relating to the number of protons in an atom and Mendeleev’s periodic table, and in mathematics Cantor’s concept is now primarily called Ordinalzahl. 21 The relevant passage of Développement nouveau de la partie élémentaire des Mathématiques as quoted by Cantor (1890, p. 18): “Dans les commence-mens, les hommes furent chasseurs ou bergers. Ces derniers eurent d’abord occasion de compter: il leur importait de ne pas perdre leurs bestiaux; et pour cela il faillait s’assurer le soir si tous étaient revenus du pâturage: celui qui n’en aurait que quatre ou cinq, aurait pu voir d’un coup d’œil si tous étaient rentrés; mais un coup d’œil n’aurait pas suffi à celui qui en aurait eu vingt. Considérant donc ces bestiaux revenant les uns après les autres, il aurait imaginé une suite de mots en pareil nombre, et gardant ces mots dans sa mémoire il les aurait répétées le lendemain à mesure que ses bestiaux seraient rentrés; afin d’être sûr, s’ils eussent cessé d’entrer avant qu’il eût achevé ses mots, qu’autant qu’il lui restait de mots à prononcer, autant il lui manquait des bestiaux etc.”

17

superfluous and rejecting the concept of ‘actual infinity,’ 22 referencing a key footnote in

Kronecker (1886), a different paper. And, going further yet, he accused Kronecker of having

assumed from the outset what he had set out to prove, the irrelevance of order for enumerating a

group of objects.23

Edmund Husserl too, put some effort into expressing reservations on the Helmholtz and

Kronecker papers in Philosophy of Arithmetic (1891), and he, too, challenged the notion that

ordinal numbers could somehow be put before cardinal numbers. This was in the revision of his

1887 habilitation thesis, to which Husserl had added a ten-page appendix dealing with just the

Helmholtz and Kronecker papers. He called this effort an empty game with symbols (p. 172 in

Husserliana XII), claiming Helmholtz had simply confused cardinal and ordinal numbers (p.

174), and describing Helmholtz’ remarks on the essential character of the ‘natural’ number series

as conspicuously polemic (p. 176). Like Cantor, Husserl indicated that the idea of ‘number

names’ lacked novelty, citing an even earlier source, George Berkeley’s Principles of Human

Knowledge (1710).24 Husserl paid less attention to Kronecker, and mentioned Dedekind as well,

22 Cantor described as ‘potential infinity’ the notion of an ever increasing sequence of finite numbers, in contrast with ‘actual infinity’ which he regarded as a fixed number that is greater than every finite number, and as the proper limit point of the unlimited sequence of ever greater finite numbers. 23 Cantor’s pointed criticism has to be viewed on the background of many years of personal conflict with Kronecker, who had been one of his doctoral advisers at the University of Berlin in the 1860’s but who, a few years later, turned on Cantor (Dauben 1979, Aczel 2000). Kronecker had mounted vehement opposition to Cantor’s notions of the infinite, in particular ‘actual infinity’ as opposed to potential infinity, and ‘transfinite numbers’. Kronecker must have known of Cantor’s work in general if not in detail, but utilized the same terminology with a different meaning, but Kronecker would not even acknowledge awareness of what Cantor (1883) regarded as his path-breaking contributions to number theory, at least not in explicit form. And Cantor’s work may well have been the intended target of certain remarks, such as “… all the results of the profoundest mathematical research must in the end be expressible in the simple forms of the properties of integers” (1999, p. 955). At around the same time, Kronecker famously said “God made the integers, everything else is the work of man” (Ewald 1999, p. 942). 24 A Treatise on the Principles of Human Knowledge went through many editions, even before 1891. Husserl did not give a year of publication, and so it is unclear which edition he used, but the stated section numbers of 121 and 122 make sense. From section 121: “And, lastly, the notation of the Arabians or Indians came into use, wherein, by the repetition of a few characters or figures, and varying the signification of each figure according to the place it obtains, all numbers may be most aptly expressed; which seems to have been done in imitation of language, so that an exact analogy is observed betwixt the notation by figures and names, the nine simple figures answering the nine first numeral names and places in the former, corresponding to denominations in the latter” (emphasis added).

18

but in the end they did not fare much better; all of them were said to have mixed up the substance

of the matter with the symbol used to represent it (p. 177).25

It is clear, then, that the idea of ordinal numbers (in the older sense) as representing a

more fundamental conception of number did not enjoy the unanimous support of

mathematicians. (Neither, for that matter, did Cantor’s concept of ordinal number.)

For our purposes, it is important to emphasize in the clearest terms possible that [a]

Cantor’s ordinal-cardinal distinction does not coincide with [b] the distinction made by the those

writers who are most directly relevant for Voigt’s work on utility theory, and, at any rate, that [c]

Cantor’s body of work utilizing his ordinal and cardinal numbers was not even completed until

several years after the appearance of Voigt’s “Zahl und Mass” in 1893 (Cantor 1895, 1897).

Equally important is that the present-day understanding of economists of ordinal numbers, unlike

that of the mathematicians, reflects [b] alone. That is because Hicks and Allen (1934), who

popularized the cardinal-ordinal distinction among economists, employed the term with its older

meaning after Cantor’s terminology had already gained currency among mathematicians.26

We must emphasize also that even though Voigt referenced the writings of Dedekind,

Helmholtz, and Kronecker, and even though it makes sense that his awareness of their work had

exerted an influence on his thinking, the (non-Cantorian) ordinal-cardinal distinction must have 25 Husserl may have been influenced directly by Cantor, having written his habilitation thesis on philosophical aspects of number theory at Halle, after earning a doctorate in mathematics a few years earlier at the University of Vienna. In Vienna Husserl had worked with Leo Königsberger who had done his doctoral work in Berlin under Karl Weierstraß and Ernst Kummer. A few years later, Weierstraß and Kummer would advise Georg Cantor (see the Mathematics Genealogy Project database at genealogy.math.ndsu.nodak.edu). Lastly, Cantor may have been of influence in other ways. In the preface to his book on continuity and irrational numbers, Dedekind (1872) mentioned a paper draft by Cantor (1872) and a published paper by Eduard Heine (1872) as relevant – Heine was Cantor’s senior colleague at the University of Halle, and as Cantor (1890, pp. 20-21) would later point out, he had had a weighty influence on the Heine article, as duly acknowledged by Heine. Further, the year 1872 marked the beginning of an intensive correspondence between Cantor and Dedekind, with Dedekind acting as a sounding board for Cantor’s deliberations and numerous attempted proofs (Dauben 1979). 26 Interestingly John von Neumann (1923), in his very early work, contributed a definition of Cantorian-type ordinal numbers that seems now to be the common textbook definition.

19

existed long before the late 1880s and would have been available to Voigt without their

influence. But it also appears very likely that their reflections made Voigt acutely aware of how

his reasoning could or should be constructed. Plus, referencing “recent developments on the

concept of number” by two leading mathematicians and a physicist would lend greater authority

to utilizing the ordinal vs. cardinal distinction in economics, and in utility theory in particular.

On purely mathematical grounds, there is but one loose end that should be taken care of.

Schröder, Helmholtz, Kronecker, and Dedekind were all working with the natural number series

in their discussions of the uses of a set of numbers in an ordinal or cardinal sense. For purposes

of utility theory the natural number series are not enough. The natural numbers are discrete

objects, whereas in economics we are dealing with real-valued utility functions: the individual’s

preference ordering is continuous defined on the real continuum (in n dimensions) and thus

cannot be represented by the natural numbers. Even though the set of natural numbers is

infinitely large, there are simply not enough natural numbers to represent all conceivable ranks

of bundles. Considering any two bundles that are not valued the same, A and B, with continuity

one can always find another bundle C of intermediate value, then another valued between A and

C, and so on. But on any finite interval only a finite number of natural numbers is available.

Here Cantor and Dedekind should be mentioned once more. Cantor’s work on transfinite

numbers specified precisely the difference in terms of their degree of infinity between the natural

numbers and the real numbers, and Dedekind (1872) had introduced the so-called ‘Dedekind cut’

to conceptualize and characterize the irrational numbers which are part of the real continuum,

and which in a sense make up virtually all of it. Of course, once it is taken for granted that the

concepts of equality and inequality apply just as well to the real continuum as to the natural

numbers, there is no remaining difficulty in utilizing the real continuum in a purely ordinal

20

sense, but it is an extra step.27

4. Andreas Voigt on Ordinal Utility

With these introductions both to Voigt and to certain late-nineteenth century

developments in mathematics in mind, we can review Voigt’s prescient contribution to ordinal

utility theory.28

We begin with Voigt’s views as expressed in section II of “Zahl und Mass.” Voigt’s

discussion here builds both on his mathematical background and on several papers he had

recently published on the theory of value (Voigt 1891, 1892a, 1893a, 1893b). Nominally, he

was responding to a footnote in Friedrich Julius Neumann’s (1892) paper on physical laws and

economic laws. 29 Neumann had argued that “the increase of sensations […] eludes

measurement. There are no units for it, and thus also no measure or numeric expression. Just as

one cannot have become 1211 more courteous or amiable, the desire for a thing cannot turn out to

be 1211 as intense as it was previously. It is time for this to be accepted as fact, finally” (note 1

on pp. 442-43). Thus, according to Voigt (1893, p. 582), Neumann – along with others not

named – had challenged “the legitimacy of the most fundamental premise of mathematical

27 Cantor (1890) himself apparently did not see that as an impediment in giving examples of sets made up of elements ordered in multiple dimensions, sets that are not well-ordered. One such example concerned the set of points of a painting, with each point described by four dimensions on the continuum: the two spatial dimensions, color (as indicated by wave length), and intensity of color. Voigt (1893c), too, moved from discrete to the continuous case without a hitch. 28 In a companion to the present paper (Schmidt and Weber (2006)), we compare Voigt’s views on ordinal utility to those of four of his contemporaries, two economistys, and engineer, and a mathematician/physicist, all of whom argued in one way or another for an ordinal view of utility within eight years of Voigt’s “Zahl und Mass in der Ökonomik. 29 This paper also obtained a perfunctory citation by Marshall (1920, p. 33).

21

deduction, the measurability of basic economic phenomena.” Voigt introduced ordinalism into

economic theory at least partially to respond to Neumann’s challenge to the subjective theory of

value.

Voigt took up three separate issues in rapid succession. Given his advanced formal

training in mathematics, it is not surprising that he started by referring to recent developments in

that field:

In accordance with the fundamental conceptions of the nature of numbers which

mathematics has developed in recent times1, it is in ordinal numbers

[Ordnungszahlen] and not in cardinal numbers [Kardinalzahlen] that we see the

primary manifestation of the number concept. More particularly, measurement

relies upon an ordering of objects as a series according to size, or the magnitude

of some other characteristic. This is especially apparent for the primitive, less

refined types of measurement. (Footnote 1 reads: See Dedekind, Was sind und

was sollen die Zahlen? Braunschweig, 1888. Kronecker in the Festschrift for Ed.

Zeller’s 50th doctoral anniversary. Also Helmholtz in the same volume.) (Voigt,

1893c, pp. 582-83, all translations by T. Schmidt)30

Three features of Voigt’s argument here merit particular attention: First, to our

knowledge, this passage marks the first appearance of the words “ordinal” and “cardinal” in a

paper on economics and in particular in a paper on utility theory. This very strongly suggests

that it was Voigt who originally introduced these terms into the economics lexicon. Second,

observe that Vogt begins by asserting, with Dedekind, Kronecker, and Helmholtz cited as

authorities, that within mathematics ordinal numbers, not cardinal ones, embody the “primary

manifestation” of what it means to be a number. Voigt does not restate the arguments of any of

these authorities, and it is not precisely clear what he means when by the “primary manifestation

of the numeric concept,” but it does seem clear that the rhetorical purpose of this reference to 30 An appendix to this paper shows the translation of the entire section II in one place.

22

recent results in pure mathematics was to convince skeptical economists that if they would think

of utility as an ordinal rather than a cardinal number, they would somehow be using a deeper,

more meaningful concept of number.

Third, Voigt concludes this brief passage with an argument clearly intended to pave the

way for thinking particularly of utility in ordinal terms when he refers to the particular value of

ordinal measurement in cases where measurement is “primitive and less refined”. Voigt does not

mention specific examples such as distance or time where measurement is less primitive or more

refined, nor does he discuss specific areas where measurement is “primitive and less refined”.

However, while he does not state it explicitly, the further modifying phrase, “as, for example, in

the case of utility” is clearly implicit here, and is as good as explicit initial reference to Neumann

(1892). Obviously, Voigt is starting to lead the reader to the inevitable conclusion that utility

must be understood in purely ordinal terms.

Having appealed to mathematicians’ views on the “primary manifestation” of the concept

of number – with or without general agreement among mathematicians – Voigt next briefly

discussed the measurement of the hardness of minerals and the measurement of temperature, two

cases, both from the hard sciences, where cardinal measurement was not possible. Then he

launched into a discussion of what we can know about utility:

Elementary magnitudes in economics, such as pleasure and displeasure, utility,

and desire are obviously capable only of such a subjective ordering. All

measurement thereof consists only of the determination of ordinal numbers,

assigned to them in a series of magnitudes of like kind (Voigt, 1893c, pp. 583-84).

The argument here follows from a simple and straightforward epistemology: In Voigt’s view,

the fact that the magnitudes of pleasure, dissatisfaction, utility, and desire are entirely subjective

implies that any external observer will be incapable of assigning cardinal numbers to these

23

magnitudes; at best the observer can only assign them ordinal numbers. To put the matter even

more simply for modern economists who have long since abandoned the study of pleasure,

dissatisfaction, and desire, Voigt is arguing here that by itself, the fact the utility is subjective

suffices to imply that it must be interpreted as an ordinal quantity.

Given the important connection which Voigt made between the subjective nature of

utility and his claim that utility should be understood as an ordinal magnitude, it is not surprising

that he thought it necessary to underscore the subjective nature of utility, for immediately

following the last passage quoted, he added:

Such series [of ordinal numbers assigned to different magnitudes of utility] have

only subjective meaning for that person who constructed them, everyone else will,

according to his personal inclinations, make an ordering of the same goods that is

different, more or less, value more highly what another has put at a lesser rank,

and vice versa (Voigt, 1893b, p. 584).

Voigt summarized his view on ordinalism by asking and answering the fundamental

questions of whether it is appropriate to think of utility as a number and if so what type of

number:

Is it then legitimate to speak of the utility of a good, the desire for one etc. as

definite magnitudes? So long as one is mindful of the special nature of such

magnitudes and refers to them only in connection with a particular person making

the valuation and, furthermore, so long as one treats the ordinal numbers so

assigned only as such and does not attribute to them the meaning of proportionate

numbers [Verhältniszahlen] and speaks of a utility twice or even one and a half

times as large and, finally, so long as one does not attempt to introduce units of

utility and desire whose existence requires such proportionality, there are no

grounds on which to object to the use of the term magnitude (Voigt, 1893b, p.

584).

But he goes on to sound a warning with which modern critics of “physics envy” in

24

economics will certainly find appealing:

Any efforts in the direction of attributing the same nature to economic magnitudes

as have the extensive units of geometry and mechanics which are measurable in

units3 come from a misguided emulation of the physical sciences, based on the

erroneous premise that objectively measurable magnitudes are always the more

complete. This would be as erroneous as it would be to rank the sciences

according their scientific “degree of precision” and to declare as most complete

those that are mathematically deductive. Because mathematical deduction is the

ideal of physics, it has erroneously been elevated to being the scientific ideal as

such, as if historical investigations would not forever maintain their legitimacy

alongside physics. (Footnote 3 reads: Fisher ([1892], § 4) makes this attempt by

constructing a definition of the proportion of two utilities. He says that the utility

of a good A is twice as large as that of B if that of A is equal to that of C and that

of B under otherwise identical circumstances is equal to ½ of C. Thus he

generally assumes that the utility of C is twice that of ½ C and thereby contradicts

experience as well as his own assumptions elsewhere.) (Voigt, 1893c, p. 584).31

In summary then, Voigt’s argument for an ordinal theory of utility emerged from his

knowledge of then recent mathematical developments in the concept of number32 from his

knowledge of the economic theory of his day, especially Fisher (1892) and Neumann (1892), and

from his epistemological misgivings concerning the possibility of objectively measuring utility, a

possibility which any cardinal theory of utility must presuppose, at least implicitly.33 But his

31 We should not be amiss to point out that this was a less than entirely accurate representation of Fisher (1892). In the cited section, §4, Fisher (1892, p. 65) was expressing himself contingently for the case of perfect substitutes: “The essential quality of substitutes is that the marginal utilities or the prices of the quantities actually produced and consumed tend to maintain a constant ratio.” This certainly was not Fisher’s contention for the general case: “But few articles are absolutely perfect representatives of … the competing … group” (p. 66). 32 Even the title Voigt chose for this paper, “Zahl und Mass …” is an apparent nod to Helmholtz’ (1887) “Zählen und Messen, …” Of the three sources named by Voigt, this paper and Kronecker (1887) and Dedekind (1888), only Helmholtz showed interest in issues arising of measurement. 33 As if to lend further proof to the proposition that an ordering alone is what matters for many purposes, the Voigt article skips from page 592 to page 595: there are no pages 593-594, and page 595 simply continues the sentence from the bottom of page 592.

25

ordinalism did not take him down the same path which later ordinalists, in particular, Pareto,

Slutsky, and Hicks and Allen, followed, since he expressed strong misgivings as to whether

economists should borrow their methodology from the physical sciences. Certainly, he believed

that any such imitation should not elevate the status of economics relative to other fields of

inquiry.

5. Conclusion

This paper has discussed important late nineteenth century developments in mathematics,

and in particular in mathematicians’ understanding of the concept of number, and documented

how those changes within mathematics exerted and almost immediate impact on at least one

economic theorist. Thus, Voigt’s argument on behalf of an ordinal cview of utility ,marks one of

the earliest cases where recent changes within mathematics had an important impact on

economics. While the twentieth century would witness a number of additional such cases,

Voigt’s contribution to ordinalism (along with Pareto’s (1892-1893) contribution to comparative

statics analysis) was among the very first.

While the story of Voigt and his early argument for an ordinal approach to utility is

interesting both in its own right and for the light it sheds on the mathematization of economic

theory, it raises the further question of why such an important intellectual contribution should

have been almost completely forgotten along with its creator. We address this question in a

companion paper (Schmidt and Weber (2006)). There, we show that in fact Voigt is not merely

some long forgotten pioneer who argued for an ordinal view of the utility function five years

before Pareto (1898) and whose concept of ordinal utility was later developed independently by

26

Hicks and Allen (1934). Rather there is strong evidence that Hicks and Allen borrowed the

cardinal/ordinal terminology from Edgeworth and almost incontrovertible proof that Edgeworth

in turn had learned it from Voigt. Thus, we argue that Voigt emerges both as an early

contributor to ordinalism and as the original source within economics of the cardinal/ordinal

terminology which has been so important within utility theory ever since Hicks and Allen

popularized it.

SELECTED WORKS BY ANDREAS HEINRICH VOIGT: Voigt, A., 1890, Die Auflösung von Urtheilssystemen, das Eliminationsproblem, und die

Kriterien des Widerspruchs in der Algebra der Logik. Leipzig: A. Danz. Voigt, A., 1891, “Der Begriff der Dringlichkeit.” Zeitschrift für die gesamte Staatswissenschaft

47, issue 2, 372-377. Voigt, A., 1892a, “Der ökonomische Wert der Güter” and “Der ökonomische Wert der Güter:

Nachtrag.“ Zeitschrift für die gesamte Staatswissenschaft 48, issue 2, 193-250 and 349-358.

Voigt, A, 1892b, “Was ist Logik?” Vierteljahresschrift für wissenschaftliche Philosophie 16,

289-332. Voigt, A., 1893a, “Produktion und Erwerb,” in two parts. Zeitschrift für die gesamte

Staatswissenschaft 49, issues 1 and 2, 1-30 and 253-283. Voigt, A., 1893b, “Eine Erweiterung des Maximumbegriffes.” Zeitschrift für Mathematik und

Physik 38, 315-317. Voigt, A., 1893c, “Zahl und Mass in der Ökonomik. Eine kritische Untersuchung der

mathematischen Methode und der mathematischen Preistheorie.” Zeitschrift für die gesamte Staatswissenschaft 49, issue 4, 577-609.

Voigt, A, 1893d, “Zum Calcul der Inhaltslogik. Erwiderung auf Herrn Husserls Artikel.”

Vierteljahrsschrift für wissenschaftliche Philosophie 17, 504-507. Voigt, A., 1895, “Die Organisation des Kleingewerbes.” Zeitschrift für die gesamte

Staatswissenschaft 51, issue 2, 267-299. Voigt, A., 1899, Die Akademie für Social- und Handelwissenschaften zu Frankfurt a. M.: Eine

Denkschrift vom Geschäftsführer des Instituts für Gemeinwohl. Frankfurt: A. Detloff. Voigt, A., and P. Geldner, 1905, Kleinhaus und Mietkaserne: Eine Untersuchung der Intensität

der Bebauung vom wirtschaftlichen und hygienischen Standpunkte. Berlin: J. Springer. Voigt, A., 1906a, Die sozialen Utopien: Fünf Vorträge. Leipzig: G.J. Göschen’sche

Verlagshandlung; second printing in 1911. Russian translation: Sotsial’nyia utopii, St. Petersburg: Brokgauz-Efron, 1906.

Voigt, A., 1906b, “Die Staatliche Theorie des Geldes.” Zeitschrift für die gesamte

Staatswissenschaft 62, issue 2, 317-340.

28

Voigt, A., 1907, Zum Streit um Kleinhaus und Mietkaserne: Eine Antwort auf die Angriffe von

Dr. Rudolf Eberstadt in Berlin und Prof. D. Carl Johannes Fuchs in Freiburg i.B. Dresden: O.V. Boehmert.

Voigt, A., 1911, Theorie der Zahlenreihen und der Reihengleichungen. Leipzig: G.J. Göschen’sche Verlagshandlung.

Voigt, A., 1912a, Mathematische Theorie des Tarifwesens. Jena: G. Fischer. Voigt, A., 1912b, “Technische Ökonomik.” In L. v. Wiese (ed.), Wirtschaft und Recht der

Gegenwart, Tübingen: J.C.B. Mohr, 219-315. Voigt, A., 1916, Kriegssozialismus und Friedenssozialismus: Eine Beurteilung der

gegenwärtigen Kriegs-Wirtschaftspolitik. Leipzig: A. Deichertsche Verlagsbuchhandlung W. Scholl.

Voigt, A., 1918, “Probleme der Zinstheorie”, in two parts. Zeitschrift für

Sozialwissenschaft, N.S. 9, 61-83 and 174-206. Voigt, A., 1920, “Theorie des Geldverkehrs.” Zeitschrift für Sozialwissenschaft, N.S.,

11, 486 ff.

Voigt, A., 1921, Das wirtschaftsfriedliche Manifest: Richtlinien einer zeitgemäßen Sozial- und Wirtschaftspolitik. Stuttgart and Berlin: Cotta.

Voigt, A., 1922, Der Einfluss des veränderlichen Geldwertes auf die wirtschaftliche

Rechnungsführung. Berlin, Verlag des “Industrie-Kurier” Abt. Buchverlag. Voigt, A. 1928a, Das Schlichtungswese als volkswirtschaftliches Problem. Langensalza:

H. Beyer. Voigt, A., 1928b, “Werturteile, Wertbegriffe und Werttheorien.” Zeitschrift für die

gesamte Staatswissenschaft 84, issue 1, 22-101.

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APPENDIX

Excerpt from Andreas Voigt, 1893, “Zahl und Mass in der Ökonomik. Eine kritische Untersuchung der mathematischen Methode und der mathematischen Preistheorie”, Zeitschrift für die gesamte Staatswissenschaft 49, issue 4, 577-609 (transl. by T. Schmidt).

II.

In accordance with the fundamental conceptions of the nature of numbers which mathematics has

developed in recent times,34 it is in ordinal numbers [Ordnungszahlen] and not in cardinal numbers

[Kardinalzahlen] that we see the primary manifestation of the number concept. More particularly,

measurement relies upon an ordering of objects as a series according to their size, or the magnitude of

some other characteristics. This is especially apparent for the primitive, less refined types of

measurement. The determination of the degree of hardness of a mineral is based on a sorting of minerals

according to their hardness, by way of the principle that the softer mineral will be scratched by the harder

one. The degrees of hardness of stones assigned in this manner are merely the ordinal numbers of that

series, which has been given a certain stability only by reference to a standard series, the Mohs scale. The

numbers indicate only that a stone is harder than another, but they do not indicate the relative degrees of

hardness, in the sense that a stone of hardness 4 would be twice as hard as one of hardness 2.

The measurement of temperature by means of a thermometer is not that much more advanced.

This, too, is only an ordering of sources of heat by means of the height of a column of mercury that

increases with the temperature. The degrees on the thermometer do not indicate proportionate

temperatures. A similar ordering could be obtained more directly as well, by way of the sensation of heat.

On the one hand, one would have to make do with few distinguishable grades, perhaps with those that can

easily be described in words and without the aid of numbers. On the other hand, this ordering suffers

from an additional defect in comparison with the thermometer ordering. It is purely subjective, i.e. it

depends upon personal, temporal, and local sensitivity to heat, whereas the other one has objective

validity for all those who accept the dependence of the height of the mercury column on the temperature.

34 See Dedekind, Was sind und was sollen die Zahlen? Braunschweig, 1888. Kronecker in the Festschrift for Ed. Zeller’s 50th doctoral anniversary. Also Helmholtz in the same volume.

36

All measuring in psychophysics consists of a subjective ordering of sensations according to their

intensity, where the grades correspond to just noticeable differences.35

Elementary magnitudes in economics, such as pleasure and displeasure, utility, and desire are

obviously capable only of such a subjective ordering. All measurement thereof consists only of the

determination of ordinal numbers, assigned to them in a series of magnitudes of like kind. Such series

have merely subjective meaning for the person who constructed them; everyone else will, according to his

personal inclinations, make an ordering of the same goods that is different, more or less, value more

highly what another has put at a lesser rank, and vice versa. Is it then legitimate to speak of the utility of

a good, the desire for one etc. as definite magnitudes? So long as one is mindful of the special nature of

such magnitudes and refers to them only in connection with a particular person making the valuation and,

furthermore, so long as one treats the ordinal numbers so assigned only as such and does not attribute to

them the meaning of proportionate numbers [Verhältniszahlen] and speaks of a utility twice or even one

and a half times as large and, finally, so long as one does not attempt to introduce units of utility and

desire whose existence requires such proportionality, there are no grounds on which to object to the use of

the term magnitude. If there were, one should also not refer to temperature and hardness as magnitudes.

Any efforts in the direction of attributing the same nature to economic magnitudes as have the extensive

units of geometry and mechanics which are measurable in units36 come from a misguided emulation of

the physical sciences, based on the erroneous premise that objectively measurable magnitudes are always

the more complete. This would be as erroneous as it would be to rank the sciences according their

scientific “degree of precision” and to declare as most complete those that are mathematically deductive.

Because mathematical deduction is the ideal of physics, it has erroneously been elevated to being the

scientific ideal as such, as if historical investigations would not forever maintain their legitimacy

alongside physics.

Whereas subjectivity of measures would be a great defect in the physical sciences, it is an

essential attribute of economics, and it would make no sense at all to wish for its eradication. Physics

seeks to eliminate subjectivity to the greatest extent possible, whereas economics not only tolerates it but

has it as one its most essential foundations. If the subjective ordering of the desire for goods did not

differ from person to person, exchange of goods would not be possible. 35 See Wiener, “Die Empfindlichkeit und das Messen der Empfindugsstärke,“ Wiedemann’s Annalen, New Series Volume XLVII, p. 659. 36 Fisher (op. cit., § 4) makes this attempt by constructing a definition of the proportion of two utilities. He says that the utility of a good A is twice as large as that of B if that of A is equal to that of C and that of B under otherwise identical circumstances is equal to ½ of C. Thus he generally assumes that the utility of C is twice that of ½ C and thereby contradicts experience as well as his own assumptions elsewhere.

37

Not even the fact that the economic magnitudes are only estimated, i.e. ordered in perception, and

not measured, i.e. ordered in themselves, may be viewed as a shortcoming. This may be the source of

many practical illusions; but because the perceived and not the real utility is the motivating force of

economic activity, economics accepts estimation with its errors and leaves to the field of ethics any

criticism thereof.

We may summarize our finding as that the fundamental concepts of economics represent

subjective magnitudes of certain degrees, and we believe it to be important to emphasize that fact.

Quantitative definitions of these concepts and quantitative identification of the fundamental principles can

and must be demanded in this limited sense. If we demand that concepts be mathematically precise, this

does not necessitate that we make them the basis of mathematical deductions. Whether such deductions

are possible on the basis of mere ordinal numbers, and what kind of objective and theoretical value they

might have, will now be demonstrated using an example from the theory of barter.