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 Please do not quote without permission.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The earliest argument for an ordinal treatment of utility is usually ascribed to an unpublished 1898 paper by Vilfredo Pareto.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: [email protected] 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.
29
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30
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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.