Andreas Voigt and Ordinal Utility, 1886-1934

Torsten Schmidt University of New Hampshire

Christian E. Weber* Seattle University

Very Preliminary. Please do not quote without permission.

ABSTRACT In 1893 Andreas Heinrich Voigt published “Zahl und Mass in der Ökonomik” (“Number and Measurement in Economics,” Zeitschrift für die gesamte Staatswissenschaft), in which he argued that utility admits only an ordinal characterization. This paper compares Voigt’s views on ordinalism with those of his contemporaries, and also discusses Voigt’s impact on later economic thought. F.Y. Edgeworth soon learned of Voigt’s contribution and cited Voigt’s ordinal-cardinal distinction several times in the Economic Journal between 1894 and 1915, bringing that distinction, both the concept and the terminology, into economics outside of Germany. Since John Hicks learned of the cardinal-ordinal terminology from Edgeworth, and since Pareto was inconsistent in his treatment of utility and apparently never used the word “ordinal,” even when discussing ordinal utility functions, it appears that Voigt’s 1893 paper – via Edgeworth as early as 1894 – is the original source of the ordinal utility concept as we know it today. 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.

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1. Introduction

In a recent paper (Torsten Schmidt and Christian Weber (2006)), we have shown that a

heretofore almost forgotten late nineteenth century German mathematician and economist,

Andreas Heinrich Voigt, argued explicitly for an ordinal approach to utility in a paper published

in 1893 in the German language journal Zeitschrift für die Gesamte Staatswissenschaft. Since

Voigt’s paper appeared a full five years before Vilfredo Pareto (1898) argued for an ordinal view

of utility in a presentation to the Société Stella of Paris, this recent rediscovery of his work marks

Voigt’s paper as the earliest statement of the idea that utility should be viewed as a purely

ordinal rather than a cardinal magnitude. Furthermore Schmidt and Weber (2006) also show that

Voigt’s paper also contains the earliest use of the cardinal versus ordinal terminology into

economics.

The present paper builds on this recent contribution to the history of utility theory in three

ways: First, we compare Voigt’s approach to ordinal utility with those of four of his

contemporaries. In so doing, we provide what appears to be a complete history of ordinal utility

theory through roughly the turn of the twentieth century. Second, we trace the influence of

Voigt’s contribution on subsequent writers. Importantly, we show that 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 an ordinal utility function was later developed

independently by John Hicks and R.G.D. Allen and other writers in the 1930’s. Rather, there is

strong evidence that Hicks and Allen (1934) borrowed the cardinal/ordinal terminology from

Francis Edgeworth, and that Edgeworth in turn had learned it from Voigt.

Finally, since we show here that Edgeworth passed Voigt’s ordinalist views, as well as

his terminology, along to later generations of economists, without an unambiguous endorsement

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but also without any hint of criticism or disagreement, we will also demonstrate that

Edgeworth’s views on utility appear to have been somewhat more complex than the

uncompromising cardinalism generally attributed to him.1 This paper thus contributes to recent

efforts to develop a more nuanced view of Edgeworth’s beliefs concerning utility.2

The remainder of the paper is organized as follows: We begin in section 2 by providing

brief introductions to Andreas Heinrich Voigt, to the important developments in nineteenth

century mathematics on which he drew as he thought about utility theory and measurement in the

early 1890’s, and to his pioneering contribution to ordinal utility theory. Section 3 then contrasts

his views on ordinal utility with those of other late nineteenth and early twentieth century writers

who in one way or another glimpsed the ordinal approach to utility which Hicks and Allen and

others would advocate during the 1930’s, and which would finally come to dominate within the

economics profession by the mid 1950’s.3 Section 4 discusses Voigt’s indirect influence on later

generations of economists, including and perhaps most importantly Hicks and Allen. One of the

main goals of section 4 will be to show how Voigt influenced Hicks and Allen via Edgeworth’s

repeated references to Voigt’s distinction between ordinal and cardinal utility and his arguments

in favor of the cardinal approach. Section 5 concludes our paper.

1 See e.g., Joseph Schumpeter (1954, p. 1065), Jürg Niehans (1990, p. 282), and Ernesto Screpanti and Stefano Zamagni (1993, p. 204). John Creedy’s more complete discussion of Edgeworth’s views on utility represent something of an exception here. See, e.g., Creedy (1986, pp. 23-24). 2 For example, Weber (2005) argues that Edgeworth’s views on complementarity and substitutability were considerably more complex than most modern writers have acknowledged, and that in fact they foreshadowed important twentieth century developments in the theory of related goods. 3 Although Hicks and Allen won some early converts to their views on ordinalism, perhaps most important among them the young Paul Samuelson, true believers in the “old time religion” proved much more difficult to persuade. Oskar Lange (1934), Harro Bernardelli (1938), W.E. Armstrong (1939), Frank Knight (1944), Abba Lerner (1944), and somewhat later Dennis Robertson (1952, 1954) all argued against replacing cardinal with ordinal utility. It seems reasonable to date the final and complete triumph of ordinalism to Robertson’s apparent failure in the mid ‘50’s to win any converts to his cause.

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2. Andreas Voigt, Late Nineteenth Century Mathematics, and Ordinal Utility4

Andreas Voigt (1860-1940) spent his student years (1882-1890) at the universities of

Berlin, Freiburg, Kiel, and Heidelburg.5 His studies included philosophy, political economy,

mathematics, and the physical sciences. His teachers and thesis advisors included Adolph

Wagner in economics and Ernst Schröder in mathematics. In 1892, Schröder helped Voigt

obtain a teaching post for mathematics at the Technische Hochschule in Karlsruhe. At about this

time, Voigt prepared an habilitation thesis in economics at Karlsruhe which was rejected. In

1896, Voigt took a position in political economy at the recently opened Institut für Gemeinwohl

(Institute for Public Welfare) in Frankfurt, a position he held until 1903. While at the Institut,

Voigt worked to help create a non university-affiliated business school in Frankfurt, the

Akademie für Social- und Handelswissenschaften (see Voigt, 1899). The Akademie opened in

1901 with Voigt as its chief administrator, and when Voigt left the Institut für Gemeinwohl in

1903, he was appointed Professor of Political Economy at the Akademie.6 In 1914, when the

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

institutes and granted university status, Voigt became the new University’s first Professor of

Economics (Professor der wirtschaftlichen Staatswissenschaften). Voigt retired from the

University in 1925.

Since Voigt’s development of ordinal utility theory drew heavily on then recent 4 This section condenses material found in greater detail in sections 2, 3, and 4 of Schmidt and Weber (2006). The interested reader is referred to that paper for more detailed accounts of Voigt’s life and work, developments in the concept of number ca. 1870-1890, and Voigt’s contribution to ordinalism. 5 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). 6 Fehling (1926) discusses the evolution of German higher education and business education in particular during the late nineteenth and early twentieth centuries.

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developments in mathematics, we need to review those developments at least briefly if we are to

understand Voigt’s contribution to ordinalism.7 In his paper on ordinalism Voigt (1893c) cited

only three mathematicians, Hermann von Helmholtz (1887), Leopold Kronecker (1887), and

Richard Dedekind (1888),8 each of whom had recently argued in effect that ordinal numbers

embody a more fundamental conception of number than cardinal numbers.

In their papers, Helmholtz and Dedekind both cited Schröder’s Lehrbuch der Arithmetik

und Algebra für Lehrer und Studierende (1873),9 In the first chapter of this text, Schröder noted

the distinction between cardinal 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.10 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.

Interestingly, one of the other authors cited by Helmholtz, Adolf Elsas (1886), had

implicitly rejected the marginalist paradigm in the economics of his day when he argued that

sensations could never be the subject of scientific investigation (esp. see p. 70), describing it as

7 For the sake of brevity, this discussion covers only the authors on which Voigt drew explicitly, along with two of the works they cited. For a more complete discussion of these and other developments in the concept of number during this time period, and in particular the views of Georg Cantor and Edmund Husserl on the subject, see Schmidt and Weber (2006). 8 Translations are available as Helmholtz (1999), Kronecker (1999), and Dedekind (1901). 9 Recall that Schröder would later advise Voigt on his dissertation. 10 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.

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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 in the same order be attached 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. 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 stated and discussed axioms which could serve as the foundations of

basic arithmetic operations. These 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).

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. Nevertheless, he essentially agreed with the case made by Helmholtz (1887).

The next year, Dedekind (1888) offered a far more detailed and lengthier account of

cardinal and ordinal numbers than either Helmholtz or Kronecker, but he tended to agree with

both authors in treating the ordinal numbers as more fundamental than the cardinal numbers,

although he was more closely aligned with Kronecker in that he was not interested in applied

measurement.

Finally, we turn to Voigt’s contribution to ordinal utility theory. Part of Voigt’s purpose

in his “Zahl und Mass in der Ökonomik”, (Voigt, 1893c) was to respond to a footnote in

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Friedrich Julius Neumann’s (1892) paper on physical laws and economic laws.11 Neumann had

argued that “the increase of sensations […] eludes measurement. There are no units for it, and

thus also no measure or numeric expression.” (note 1 on pp. 442-43). Voigt (1893c, p. 582)

argued that Neumann – along with others not named – had challenged “the legitimacy of the

most fundamental premise of mathematical deduction, the measurability of basic economic

phenomena”, and took up the task of defending the subjective theory of value against Neumann’s

criticism.

Voigt considered three separate issues in rapid succession.12 He started by referring to

recent developments in mathematics. To our knowledge, this passage contains the first

appearance of the words “ordinal” and “cardinal” in any paper primarily concerned with

economics, so that Voigt appears to have been the first to introduce these terms into the

economics lexicon. Second, Voigt asserted, citing Dedekind, Kronecker, and Helmholtz as

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

manifestation” of what it means to be a number. Although he did not restate the arguments of

any of these authorities, it seems clear that the rhetorical purpose of referring to recent results in

pure mathematics was to convince skeptical economists that by thinking of utility as ordinal,

they would somehow be using a deeper, more meaningful concept of number. Voigt then argued

for the particular value of ordinal measurement in those instances where measurement is

“primitive and less refined”.

Next, Voigt briefly discussed measuring the hardness of minerals and temperature, two

cases from the hard sciences where cardinal measurement was not possible, after which he

considered what we can and cannot know about utility: 11 This paper also obtained a perfunctory citation by Marshall (1920, p. 33). 12 An appendix to this paper shows T. Schmidt’s the translation of section II in its entirety.

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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 epistemology here is as straightforward as its implications: The fact that pleasure,

dissatisfaction, utility, and desire are entirely subjective implies that no external observer can

assign cardinal numbers to them; at best the observer can only assign them ordinal numbers.

This then is the heart of Voigt’s argument for ordinalism: by itself, the fact the utility is

subjective implies that it must be interpreted as an ordinal quantity.

Finally, as if to drive home his point, Voigt explicitly underscored the subjective nature

of utility:

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, 1893c, p. 584).

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

knowledge of then recent mathematical developments in the concept of number 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.

3. Andreas Voigt’s Contemporaries on Ordinal Utility

Before we discuss Voigt’s influence on subsequent economic thought, it will be helpful

to review briefly the ordinalist views of four of Voigt’s contemporaries and to contrast those

with Voigt’s own statements on the subject. Thus, this sections discusses how Voigt’s views on

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ordinalism compare with those of Giovanni Battista Antonelli, Irving Fisher, Vilfredo Pareto,

and Henri Poincaré, each of whom at least touched on the subject of ordinal utility between 1886

and 1901.

G.B. Antonelli

So far as we can tell, the Italian civil engineer Giovanni Battista Antonelli made the first,

somewhat indirect, argument for an ordinal approach to utility in his Sulla Teoria Matematica

della Economia Politica (Antonelli, 1886). Among its other important contributions, this

monograph contains perhaps the first discussion of the integrability problem in economics. In

the process of deriving the conditions for integrability of demand functions, Antonelli discovered

that that if price-quantity data can be reconciled with the utility function, U = U(x), where x is an

n vector of goods consumed, then these data are also consistent with any monotone increasing

transformation of U, g(x) = F(U(x)). Although Antonelli did not use the ordinal/cardinal

terminology, and although he did not specifically interpret his mathematical finding in this way,

he had recognized in essence that while empirical price-quantity data might be used to recover a

utility function (if a utility function exists and if the observed values of x result from maximizing

that utility function subject to a budget constraint), any utility function thus recovered, say U(x),

would constitute only one of infinitely many utility functions, F(U(x)), all of which would be

consistent with the data.

Unlike Voigt, and despite the advanced training in mathematics which his professional

training as an engineer would have entailed, Antonelli did not cite developments in the concept

of number as an argument for ordinalism. Indeed the contributions of Helmholtz (1887),

Kronecker (1887), and Dedekind (1888) all appeared after the publication of Antonelli’s

monograph.

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Furthermore, while Antonelli’s somewhat inadvertent (and implicit) argument for an

ordinal interpretation of utility is fundamentally epistemological, since it derives from the

question “What can we know about utility from price-quantity data?,” the epistemological

foundation of Antonelli’s argument for ordinalism is both somewhat narrower and more

contingent than Voigt’s. Whereas Voigt very broadly ruled out the general possibility of

measuring utility using cardinal numbers, Antonelli had simply argued that price-quantity data

alone could not determine a unique (cardinal) utility function. Since he did not discuss the point

further, Antonelli seems to have left open, at least implicitly, the possibility that there might be

other ways to measure utility, perhaps even methods which would yield cardinal numbers for

utility. Had a psychologist or economist ever invented a “utilimeter” or “hedonimeter” for

measuring utility without using price-quantity data, this would have invalidated Voigt’s strong

and very general argument for ordinalism, but not Antonelli’s more circumspect argument.13

Irving Fisher

Part I Chapter I and all of Part II of Irving Fisher’s 1892 Ph.D. dissertation, Mathematical

Investigations in the Theory of Value and Prices, deal with utility theory. Compared to

Antonelli’s confident if off-hand treatment of the non-uniqueness of the utility function, Fisher’s

views on the ordinal nature of utility are considerably more complex.

We concentrate in particular on Fisher’s views in Part II Chapter IV, entitled “Utility as a

Quantity.” Here, Fisher developed two important cornerstones of ordinal utility theory, while

explicitly denying a third. First, he argued that

[i]t would doubtless be of service in ethical investigations and possibly in certain

13 To be as clear as possible, we should also emphasize that the different epistemological foundations of Voigt’s and Antonelli’s arguments for ordinalism have nothing to do with the implicit and somewhat indirect nature of Antonelli’s argument versus the explicit and direct approach of Voigt.

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economic problems to determine how to compare the utilities of two individuals.

It is not incumbent on us to do this. When it is done the comparison will

doubtless be done by objective standards (Fisher, 1892, p. 86).

Fisher apparently believed that interpersonal comparisons of utility are simultaneously possible,

helpful, and unnecessary. In this chapter, he also argued (Fisher, 1892, p. 88) that the

consumer’s “lines of force” or “maximum directions” in the commodity space (along an

indifference surface, the maximum direction is the vector which is locally orthogonal to the

indifference surface) are all one needs to describe consumer behavior; the observer does not need

to know how much utility increases along a maximum direction near an indifference curve.

Fisher then concluded Mathematical Investigations with the following strikingly modern

observation:

Thus if we seek only the causation of the objective facts of prices and commodity

distribution four attributes of utility as a quantity are entirely unessential, (1) that

one man's utility can be compared to another’s, (2) that for the same individual

the marginal utilities at one consumption-combination can be compared to

another, (3) even if they could, total utility and gain might not be integrable, (4)

even if they were, there would be no need of determining the constants of

integration. (Fisher 1892, p. 89, emphasis in the original.)

In summary, Fisher seems to have been the first to state explicitly that positive

economics, the description, explanation, and prediction of what is, does not require cardinally

measurable utility, even in principle. He also understood that while the nature of the indifference

map is crucial to understanding demand behavior, the size of marginal utility at any point in the

consumption space is irrelevant. However, he apparently believed that normative economics

either cannot do without cardinal measurability or at least benefits from assuming cardinal

measurability.

As with Antonelli, Fisher’s views on ordinalism clearly differ from Voigt’s. Like

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Antonelli, Fisher made no reference to recent developments in mathematicians’ conception of

numbers to buttress his views on the nature of utility. This in itself is interesting both because

Fisher’s formal education included considerable training in mathematics and because he had at

least a reading knowledge of German.14 As a result, it is at least possible that the polyhistor

Fisher would have known of the then recent innovations in the concept of number of Helmholtz

(1887), Kronecker (1887), and Dedekind (1888). However, he did not reference any recent

developments in mathematics as he discussed the nature of utility.

Beyond that, however, Fisher clearly believed in 1892 that although it was not yet

possible to measure utility objectively, it would be possible one day.15 He also believed that this

would be of value for normative economics. In contrast, Voigt left normative questions out of

his discussion of utility altogether.16 Furthermore, there are clear differences between Fisher and

Voigt concerning the epistemological issues which interested them as they wrote about the

nature of utility. Fisher asked, in effect, “Will an ordinal view of utility suffice to explain price-

quantity data?” – without using the word ‘ordinal’ – and answered this question “Yes.” In

contrast, Voigt asked, “Is it possible to measure utility objectively?,” and answered “No.”

Vilfredo Pareto

Next we come to Pareto. Pareto’s views on the nature of utility have received substantial

attention from historians of economic thought. Recent discussions include those of Roberto 14 As we discuss below, Fisher knew of at least one of Voigt’s early papers, Voigt (1892a). 15 And of course, years later Fisher (1927) contributed to the effort to measure marginal utility from price-quantity data. 16 In a note published simultaneously and cited in “Zahl und Mass,” Voigt (1893b) argued on purely normative grounds for the necessary condition for what is now called Pareto efficiency, for two individuals and two goods, as an ‘extension of the concept of a maximum:’ that each person’s utility be maximal conditional on the value of the utility of the other. The same condition was quoted later on in “Zahl und Mass,” as part of a discussion of bartering, and as a purely descriptive criterion.

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Marchionatti and Enrico Gambino (1997), Martin Gross and Vincent Tarascio (1998), Luigino

Bruni and Fransesco Guala (2001), and Weber (2001). Paul Samuelson (1974) both praises and

criticizes Pareto's utility theory, and John Chipman (1976) provides a rather generous apologia

for Pareto. George Stigler's (1950) earlier discussion of Pareto's utility theory is rather more

dispassionate. The brief treatment here aims merely to contrast Pareto’s mature views on

ordinalism with those of Voigt.

In late 1898 Pareto presented a privately published paper, “Comment se pose le problème

de l'économie pure”, to the Société Stella (Pareto, 1898). Two years later, he published the two-

part “Sunto di alcuni capitoli di un nuovo trattato di economia pura” in the Giornale degli

economisti (Pareto, 1900). In these papers, Pareto proposed a purely ordinal approach to utility.

In the “Sunto ...,” Pareto repeatedly advocated discarding utility in favor of a direct study of

choices and the constraints which impede them. He noted that if one could measure utility, the

result would give only one of infinitely many utility indices, and then categorically denied that it

is possible to measure utility. Pareto also considered arbitrary transformations of a utility

function, much like the transformations which Antonelli had considered earlier. He provided the

first explicit proof that applying a transformation to the utility function does not affect the shapes

of the indifference curves;17 it merely changes the numbers assigned to them. This is one

common way of stating that utility is ordinal. Aside from assuming that marginal utility is

positive, Pareto placed no restrictions on the utility function.

Several years later, in Chapter III of his Manual of Political Economy (Pareto, 1909),

Pareto inverted Edgeworth's use of the indifference map. Edgeworth had

assumed the existence of utility (ophelimity) and deduced the indifference curves

from it. On the other hand, I consider the indifference curves as given, and

17 This fact is really only implicit in Antonelli’s treatment of the integrability problem.

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deduce from them all that is necessary for the theory of equilibrium, without

resorting to ophelimity (Pareto, 1909, p. 119n).

This chapter also contains Pareto's famous statement that the indifference map

gives us a complete representation of the tastes of the individual ..., and that is

enough to determine economic equilibrium. The individual can disappear,

provided he leaves us this photograph of his tastes (Pareto, 1909, p. 120).18

In the Appendix to the Manual, Pareto restated and extended the ordinal utility theory of

the “Sunto ...” He proved again that arbitrary transformations of the utility function do not alter

the shapes of indifference curves and emphasized that measurability of the utility function “is not

at all necessary in order to establish the theory of economic equilibrium” (Pareto, 1909, pp. 394-

395).

Clearly, Pareto’s mature views on ordinal utility19 come closer to those of Voigt than do

those of either Antonelli or Fisher. However, unlike Voigt, but like Antonelli and Fisher, Pareto

neither referenced recent developments in mathematics nor used the words “Cardinal” and

“ordinal”. However, like Voigt he argued for an ordinal view of utility largely on

epistemological grounds: He agreed with Voigt that we cannot measure utility objectively, and

that as a result, we cannot impart cardinal characteristics to the numbers assigned to the utility

function.

Henri Poincaré

About a year after Pareto’s two-part “Sunto …” appeared in the Giornale degli

economisti, Léon Walras found himself having to defend his version of the subjective theory of

18 As Weber (2001) notes, Pareto had already made a similar sounding observation in 1900 in the “Sunto …”. 19 Pareto’s earlier discussions of utility, especially those in the “Considerazioni sui principi ..” (Pareto (1892-1893)) are much more strongly cardinal than those of “Comment se pose ..” or the “Sunto …”.

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value, which derives equilibrium prices from utility maximization, against an assault on it

mounted by the distinguished French mathematician Hermann Laurent.20 Acutely aware of his

own limitations as a mathematician, in September 1901, Walras appealed to Henri Poincaré for

support in his dispute with Laurent. In response to a second letter from Walras on the subject of

utility measurement, Poincaré wrote back to Walras:

Can satisfaction be measured? I can say that one satisfaction is greater than

another, since I prefer one to the other, but I cannot say that the first satisfaction is

two or three times greater than the other. That makes no sense by itself and only

some arbitrary convention can give it meaning. Satisfaction is therefore a

magnitude but not a measurable magnitude. Now, is a non-measurable magnitude

ipso facto excluded from all mathematical speculation? By no means. … you …

can define satisfaction by any arbitrary function providing the function always

increases with an increase in the satisfaction it represents (Quoted in Jaffe, 1977,

p. 304).

This defense of using ordinal magnitudes to represent satisfaction or utility clearly suggests that

Poincaré’s views on the nature of utility coincided most closely with those of Voigt and Pareto.

Note that he bases his argument for what amounts to an ordinal view of utility (note the explicit

reference here to monotone increasing, but otherwise arbitrary transformations of utility) on the

explicit assumption, which he shared with Voigt and Pareto that it is not possible to measure

utility. But it is worth noting even Poincaré, easily a more accomplished within mathematics

than Voigt, Antonelli, Fisher, or Pareto did not refer to late nineteenth century developments in

mathematicians’ conception of what a number is. Perhaps he knew his audience well enough to

avoid such a reference.

In summary, none of Voigt’s contemporaries made precisely the same pair of arguments

20 The following discussion of the Walras-Poincaré correspondence draws heavily on Jaffé (1977).

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for ordinalism that he had, both referring to recent changes in the concept of number within the

mathematics community, and arguing that the fact that we cannot measure utility objectively

implies that utility must be defined cardinally. However, two of them, Pareto and Poincaré,

clearly echoed Voigt’s argument that an external observer’s inability to measure utility

necessarily implies that any numbers assigned to a utility function must be understood as having

only ordinal, and not cardinal properties.

4. Andreas Voigt’s Eventual Influence on Economics

Aside from earning Edgeworth’s respect, on which more below, Voigt seems to have

attracted relatively little attention from Anglophone economists, either during his lifetime or

later. Irving Fisher (1892) included Voigt (1892a) in his update of Jevons’ bibliography of

mathematical contributions to economics without any discussion. Arthur Marget (1932) briefly

mentioned Voigt’s (1920) contribution to monetary theory, and P.N. Rosenstein-Rodan (1934)

mentioned Voigt (1892a) (along with seven other economists) favorably for his relaxation of

certain restrictive assumptions in his treatment of time in economics. In Germany, Kurt Sting

(1931) noted Voigt’s (1928) last and retrospective contribution to theory of value. In more

recent times, Peter Dooley (1983) has cited Edgeworth’s (1894) reference to Voigt’s (1893c)

suggestion that economists treat utility as being defined ordinally rather than cardinally; however

Dooley does not identify any of Voigt’s works explicitly.21

Among historians of economics, Voigt seems to be almost completely forgotten, with the

honorable exception of Dooley’s reference. For example, Joseph Schumpeter (1954) did not

mention Voigt at all. Even Karl Pribram, who should have known Voigt personally since he held 21 Dooley’s reference to Voigt inspired the further research which led to this paper.

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an appointment as Professor of Economics at the University of Frankfurt from 1928-1933,

mentioned only two of Voigt’s papers in his massive treatise on the history of economics

(Pribram, 1983). These two papers, published in 1912 and 1913, concerned epistemological

aspects of Max Weber’s call for a wertfrei (value-free) economics, and Pribram cited them

without providing titles, much less any discussion or analysis.

Rather than belabor the fact the Voigt exerted relatively little direct influence either on

his contemporaries or on subsequent economic theory, the remainder of this section discusses

both Voigt’s obvious and direct influence on Edgeworth and his apparent but indirect influence

on Hicks and Allen.

F.Y. Edgeworth

Perhaps surprisingly, the only economist on whom Voigt seems to have made something

close to an important direct impression during his lifetime was Edgeworth. Edgeworth’s

apparent respect for Voigt and especially for Voigt’s argument for ordinal utility comes as a

particular surprise since Edgeworth has a reputation as one of the most, perhaps the most,

adamantly cardinal and utilitarian of all of the major late nineteenth and early twentieth century

pioneers of utility theory in economics.22 Thus, this section contributes to recent attempts (e.g.,

Weber, (2005)) to gain a deeper understanding of Edgeworth’s views on utility.

22 Just to review Edgeworth’s defenses of and contributions to cardinalism briefly: He devoted his first book (Edgeworth, 1877) to the Benthamite allocation problem of maximizing the sum of the utilities of the members of a society. He based the utility theory of his Mathematical Psychics on the axioms that “(p)leasure is measurable, and all pleasures are commensurable” (Edgeworth, 1881, p. 59) and that “(t)he rate of increase of pleasure decreases as its means increase ... the second differential of pleasure with regards to means is continually negative” (Edgeworth, 1881, p. 61). In the same book, he discussed measuring utility, a science he termed “hedonimetry”, at some length (Edgeworth, 1881, pp. 7-9, 98-102). The only faint foreshadowing of ordinalism is Edgeworth’s (1881, p. 20) suggestion that a utility function defined over two goods x and y should be written in the general non-additive form F(x, y). Finally, he wrote a little noticed book (even the lengthy entry on Edgeworth in The New Palgrave Dictionary of Economics (Newman, 1987) mentions, but does not discuss it) on the problem of measuring probability and utility (Edgeworth, 1887).

17

Edgeworth appears to have learned of Voigt and his work in 1893 or 1894, probably as a

result of his duties as editor of the Economic Journal. In those days, the Economic Journal

published brief summaries of papers on economics which had recently appeared in other

journals, including the Zeitschrift für die gesamte Staatswissenschaft, the journal in which “Zahl

und Mass” appeared. In his role as editor of the Economic Journal, Edgeworth would naturally

have known of these summaries before they appeared in print, even if he did not prepare each

summary himself. In its March 1894 issue, the Economic Journal (Anonymous, 1894, pp. 202-

203) published a one-paragraph summary of “Zahl und Mass,” which had recently appeared in

the Zeitschrift für die gesamte Staatswissenschaft. That summary explicitly mentions Voigt’s

then novel argument for an ordinal view of utility, which was summarized in section 2 above.

In the same issue of the Economic Journal, Edgeworth (1894) entered into a debate

between Alfred Marshall and J. Shield Nicholson on the subject of consumer’s surplus and the

possible constancy of the marginal utility of income. Dooley (1983) recounts this debate, as well

as the broader contretemps which arose in response to the publication of the first edition of

Marshall’s Principles in 1890 in some detail. Within the broader set of debates over the theories

set out in the Principles, the specific issue at stake between Edgeworth (on behalf of Marshall)

and Nicholson concerned whether or not money can serve as a measuring rod for measuring

utility, with both Marshall and Edgeworth arguing in favor of using money to measure utility,

and Nicholson arguing against.

As Dooley (1983) notes, Edgeworth’s attempted defense of Marshall’s views against

Nicholson’s criticism resulted in something less than a complete victory for Marshallian

consumer’s surplus analysis.23 However, Edgeworth did include in a footnote on p. 155 of his

23 The final resolution of these questions had to wait almost fifty years for the definitive treatments of Allen (1933) and Samuelson (1942).

18

article the following observation:

On the measurement of sensation consider Dr. Voigt’s proposal to use only

ordinal - not cardinal - numbers, referred to on p. 202 of the present number of the

Journal.

Edgeworth left his acknowledgement of Voigt at that. He neither defended, nor attacked Voigt’s

proposal, and he certainly did not expand upon it. One gets the distinct impression that

Edgeworth saw Voigt’s view on the necessarily ordinal nature of utility as an interesting

theoretical curiosum which perhaps the economics profession should have called to its attention,

but little more. In fact, the tenor of Edgeworth’s entire œuvre, as well as what we know of his

personal character would suggest that Edgeworth quite likely disagreed with Voigt’s view, but

felt compelled to mention anyway it for the sake of intellectual honesty. However, we shall see

presently that understanding Edgeworth’s views on Voigt’s suggestion is rather more difficult

than this interpretation suggests.

Edgeworth would mention Voigt in print five more times (Edgeworth 1900, 1906, 1907,

1915, 1917). Four of these references (Edgeworth 1900, 1907, 1915, 1917) are to Voigt’s “Zahl

und Mass,” and of these four, three (Edgeworth, 1900, 1907, 1915) refer specifically to Voigt’s

argument for an ordinal approach to utility.24

The first of these three references to Voigt’s argument for ordinalism appeared in a

footnote to a discussion of the application of Utilitarianism to the design of optimal taxes and

railway rates (Edgeworth, 1900, p. 178) and is nearly as brief and every bit non-committal as the

1894 reference:

Attention may be called to Dr. A. Voigt’s reflections on the measurement of

24 Edgeworth’s citation of Voigt in a paper on urban land prices (Edgeworth, 1906) referred to a study by Voigt on the degree of monopolization of the real estate market in Berlin. His citation of Voigt’s 1893 paper (Edgeworth, 1917) mentions that paper not for Voigt’s views on ordinalism but apparently (Edgeworth was very vague here) for his development of the first order conditions for optimal exchange of two goods between two persons.

19

economic advantage by ordinal numbers, degrees of utility being distinguished as

first, second, &c., in the order of magnitude, but not as multiples of a unit

(emphasis in the original).

As in his earlier discussion of Voigt’s argument for an ordinal approach to utility, Edgeworth

simply cites Voigt’s “reflections” without any sort of further comment.

In contrast, Edgeworth’s last two references to Voigt’s argument for ordinalism come

surprisingly close to endorsing Voigt’s views on the subject, at least implicitly. In 1907 and

again in 1915, Edgeworth published survey articles on then recent applications of mathematics to

economic theory, both of which contain favorable references to the views Voigt had expressed in

his 1893 paper. In “Appreciations of Mathematical Theories”, Edgeworth (1907) broached the

question of the appropriate units of measurement of utility, cited the answers given by Fisher and

Pigou to this question, and then opined (in the body of the text rather than a footnote this time)

that

Perhaps it is better to say, with Professor A. Voigt, that no unit is required:

quantities like utility are to be measured only by ordinal numbers. In

confirmation of this conception, Professor Voigt refers to the view, now prevalent

among mathematicians, “which sees in ordinal number rather than in cardinal the

primary conception of number” (Edgeworth, 1907, pp. 222-223, emphasis in the

original).

It is important to emphasize that in this passage, Edgeworth did not merely suggest that the

reader “consider” Voigt’s view, nor did he simply “call attention” to Voigt’s view. Rather he

argues that “perhaps it is better” to adopt Voigt’s view, which amounts to an English language

endorsement of ordinalism 27 years ahead of Hicks and Allen (1934).

Then in a footnote to this passage, Edgeworth referred specifically to Voigt’s 1893 paper.

To back up Voigt’s (and now his) claim as to the new “primary conception of number,” among

mathematicians, in a second footnote he quoted from an article, “Functions of Real Variables”,

20

by “the eminent mathematician Professor A.E.H. Love”25 in vol. 28 of the 1902 Encyclopœdia

Britannica: “The capacity of numbers to answer questions of how many and how much – in

other words to express the results of counting and measuring – may be regarded as a secondary

property derived from the more fundamental one of expressing order.” It is worth noting that

Love had included in his bibliography Richard Dedekind’s (1888) Was sind und was sollen die

Zahlen?

As he had in 1894 and 1900, Edgeworth again failed to develop or extend Voigt’s

ordinalist view of utility, and in fact in the very next paragraph, he launched into a discussion of

the possibility of measuring utility by money, an idea he apparently just could not let go. The

key difference between this reference to Voigt’s argument for ordinal utility and his earlier

references is that in the case, Edgeworth explicitly, albeit hesitantly and inconsistently, expressed

his agreement with Voigt’s views on the nature of utility.

In 1915 Edgeworth published another, longer survey of some then recent contributions to

mathematical economics. He began the section of the paper on utility by quoting from the

exchange between Walras and Poincaré discussed above. In particular, he provided a slightly

different translation than Jaffe’s of most of passage from Poincaré quoted above.26 He then

added the observation that

Poincaré’s ruling is in accordance with the view now generally prevalent among

mathematicians, that the capacity of numbers to express he results of counting and

measuring ‘may be regarded as a secondary property derived from the more

fundamental one of expressing order. Natural numbers from a series with a

25 Augustus Edward Hough Love (1863-1940) held the Sedleian Chair in Natural Philosophy at Oxford from 1898 until his death. Among mathematicians and geophysicists, he is best remembered for his work on the elasticity of solids (also the subject of Pareto’s Ph.D. dissertation), which had important implications for understanding seismic activity. In 1911, he predicted the existence of Love waves, one of four distinct types of seismic waves. 26 The differences between the two translations involve word choice only; in substance, the two are identical.

21

definite order, and ‘greater than’ and ‘less than’ mean ‘more advanced’ and ‘less

advanced’ in this order.’ These are the words of another eminent mathematician,

Professor Love (Edgeworth, 1915, p. 58).

In a footnote to this passage, Edgeworth mentioned both Love27 and Voigt, inaccurately

referring the reader interested in further details to page 222 of volume IX of the Economic

Journal.28 As had been the case with his discussion of Voigt’s argument for ordinal utility eight

years earlier, in this passage Edgeworth appears to endorse, at least implicitly, the ordinal view

of utility which he had first learned from Voigt in 1893 or 1894. Certainly, there is nothing here

to criticize either Poincaré’s or Voigt’s statements on the nature of utility, and Edgeworth’s

instinct would almost certainly have kept him from criticizing Poincaré at any rate.

In summary, Edgeworth’s repeated references to Voigt’s views on the ordinal nature of

utility appeared over a span of more than two decades, were universally neutral or positive in

tone, and appeared alongside references to mathematicians of the stature of Love and Poincaré.

While these facts raise questions about the precise nature of Edgeworth’s views on utility, which

are almost universally acknowledged to be very cardinal and utilitarian, such questions are

beyond the scope of this paper.29 The main point demonstrated here is that Edgeworth knew of

Voigt’s argument for an ordinal approach to utility and reported it repeatedly to English speaking

economists without ever arguing against it.

27 The reference was again to the same article, Love (1902). But in the meantime the next edition of the Encyclopædia Britannica had become available, and for this new edition, Love (1910) had revised his article to omit the statement that ordinal numbers were more fundamental than cardinal numbers. 28 The correct reference would have been to page 222 of volume XVII, discussed above, not to volume IX. 29 The questions raised here also go beyond Weber’s (2005) recent demonstration that Edgeworth developed remarkably modern sounding definitions of complements and substitutes which represent a distinct advance beyond the Auspitz-Lieben-Edgeworth-Pareto definition based on the signs of the second-order cross partial derivatives of the utility function.

22

John Hicks and R.G.D. Allen

While Voigt’s influence on Edgeworth is easy to assess from Edgeworth’s published

papers, his influence on Hicks and Allen and in particular on their thinking as they wrote their

1934 masterpiece, “A Reconsideration of the Theory of Value”, was more indirect and certainly

more difficult to assess with any sense of certainty. However, we do know the following:

First, perhaps due to the 1925 publication of his Papers Relating to Political Economy30

and the 1932 reprinting of his Mathematical Psychics, Edgeworth was very much “in the air” in

the early 1930’s. Among the more than ninety papers published in English language economics

journals between 1929 and 1934 which cited Edgeworth at least once, we find the following

seminal contributions: Gottfried Haberler’s (1929) paper on comparative cost, Hotelling’s (1929,

1931) papers on market stability and exhaustible resources, as well as his pioneering paper on the

comparative statics of supply and demand functions (Hotelling, 1932), Frank Graham’s (1932)

paper on international values, Wassily Leontief’s (1933) use of indifference curves to study the

impact of international differences in demand on world prices and trade volumes, and Schultz’s

(1933) theoretical and empirical analysis of consumer demand.

Second, in several of their own single-authored papers published prior to their famous

collaboration, both Hicks and Allen had referred to Edgeworth’s work, (Hicks 1930, 1932; Allen

1932a, 1933, 1934), and Allen had in fact written a review of the 1932 reprint of Mathematical

Psychics (Allen, 1932b).

Third, in their joint 1934 paper, Hicks and Allen cited Edgeworth, including the 1925

reprints of his “Pure Theory of Monopoly” (Edgeworth, 1897) and his 1915 review of

mathematical contributions to economic theory, discussed above. In particular, they cited the

30 Edgeworth’s Papers are now available online at: http://cepa.newschool.edu/het/texts/edgeworth/edgepapers.htm.

23

section of Edgeworth’s 1915 paper in which he had quoted from Poincaré’s letter to Walras.31

Finally and perhaps most importantly, Hicks and Allen used the words “cardinal” and

“ordinal” in their paper. This is interesting because these two words had appeared together in

only two English language articles on economics prior to 1934, both of them authored by

Edgeworth (1894, 1907). The word “ordinal” had appeared (with or without the word “utility”

also appearing in the same piece) in six articles and in two lists of recent works in economics.32

Of the six articles, the word appears in connection with utility in only three, all of them authored

by Edgeworth (1894, 1900, 1907). In the other three articles, the word ordinal refers to the

ordering of something other than satisfaction. Aside from these three papers by Edgeworth, the

only other pre-1934 use of the word ordinal to describe utility which we have managed to locate

is in a footnote in Knight’s classic Risk, Uncertainty, and Profit (Knight, 1921, pp. 69-70), which

comes at the end of rather lengthy section in which Knight argues strongly for an ordinal view of

utility,33 and Hicks and Allen did not cite Knight. In particular, we have not found the words

“cardinal” or “ordinal” in any of the English translations of Pareto’s works available to us.

Although they do not provide definitive proof, these facts which surround the writing of

“A Reconsideration of the Theory of Value” do strongly suggest that Voigt did exercise an

indirect influence on Hicks and Allen, and in particular on their choice of the words “cardinal”

and “ordinal” as they wrote. We do know that, like so many of their important contemporaries,

Hicks and Allen had both read at least some of Edgeworth’s works, and it seems at least likely

31 Of course, Hicks and Allen also drew on a number of other previous authors for inspiration as they wrote their seminal 1934 paper, including among others, Fisher (1892), Pareto (1909), Johnson (1913), Marshall (1920), Ragnar Frisch (1932), and Schultz (1933), and Hicks has specifically mentioned the influence of Pareto’s Manual, especially the Mathematical Appendix, on his thinking during his formative years (Klamer, 1989). 32 One of these lists of recent works was the brief discussion of Voigt (1893c) in Anonymous (1894). 33 The reader will note the inconsistency between Knight’s view of utility in his youth (Risk, Uncertainty, and Profit began its life as Knight’s doctoral dissertation at Columbia) and in his later years (Knight, 1944).

24

that they would have read other articles by Edgeworth which they simply felt no need to cite. As

a result, it seems quite likely that they would have learned the cardinal/ordinal language from

reading Edgeworth, since, so far as we have been able to tell, Pareto had not used these terms in

connection with utility, and they had been used to describe utility in only one work not authored

by Edgeworth.

In summary, while Hicks and Allen no doubt got the idea that, at least in some

applications, it is appropriate or even best to view the utility function in purely ordinal terms

from Pareto (1909), it seems highly likely that they also got such an idea, along with the

cardinal/ordinal language from Voigt through Edgeworth. If this is indeed the case, then the

usage of this language in so much of modern microeconomics traces back ultimately to a now

virtually forgotten German mathematician and economist whose influence worked, ironically

enough, through one of the most cardinal and utilitarian of all of the second generation

marginalists!

5. Conclusion

F.Y. Edgeworth learned of Voigt’s 1893 “Zahl und Mass” shortly after it was published

and cited Andreas Voigt’s ordinal-cardinal distinction several times in the Economic Journal

between 1894 and 1915. These repeated citations of Voigt brought this important distinction,

both the concept and the terminology, into Anglophone economics. Even if Edgeworth had

never learned of Voigt’s path-breaking contribution in making the case the case for ordinal utility

in the most explicit terms, it has to be regarded as the original manifestation of the ordinal utility

concept as we know it today, not merely as a precursor or some earlier variant. Further, Voigt’s

contribution was not exactly hidden from view: it appeared in a major economics journal, and we

25

know this from Edgeworth’s representation.

But our paper has gone beyond simply giving long overdue recognition to a largely

forgotten pioneer of modern utility theory. While giving Voigt his due would certainly have

been worthwhile on its own, we have also shed further light on the development of ordinal utility

theory prior to the fundamental contribution of Hicks and Allen (1934). In doing so, we have

documented Voigt’s argument for an ordinal approach to utility as an important early case,

indeed one of the earliest cases, where recent developments at the frontiers of mathematics

exerted a major influence on the course of economic thought: Voigt’s argument grew directly

out of his exposure to recent extensions or reconsiderations of the concept of number within

mathematics. We have also discovered the apparent original source of the cardinal/ordinal

terminology in economics and discussed the likely connection between this source and other

later uses of this terminology. Our comparison of Voigt’s views on the ordinal nature of utility

to those of four of his contemporaries has demonstrated the diversity of views on ordinalism

among the doctrine’s earliest pioneers. Finally, we have shed further light on Edgeworth’s

complicated, multi-faceted views on utility beyond the recent discussion of Edgeworth’s

contribution to the theory of related goods in Weber (2005). As a result, it is now clear that

Edgeworth’s views on utility went considerably beyond the almost knee jerk cardinalist-

utilitarian doctrine usually imputed to him.

However, further work clearly remains to increase our understanding both of Voigt’s

other contributions to economic theory and of the pre Hicks-Allen history of ordinalism in

economics.

For example, after presenting his argument for ordinal utility in section II of “Zahl und

Mass,” Voigt went on in sections III and IV to discuss the two person-two good exchange

problem and to develop the criterion for an optimum allocation of the two goods across the two

26

traders, augmented by presentation of the same criterion in a separate paper in the Zeitschrift für

Mathematik und Physik (Voigt, 1893b). This of course raises the obvious question, which is

beyond the scope of the present paper, of the connection between this facet of Voigt’s work and

the similar contributions of Edgeworth and Pareto. Further, at roughly the same time Voigt dealt

with questions of value in several other papers spanning many published pages in total (Voigt

1891, 1892b, 1893a). Clearly, a full appreciation of “Zahl und Mass” both in its own right and in

the context of these other papers would be highly desirable. For the moment, however, we leave

detailed examinations of these papers and of Voigt’s other contributions to economics, and of the

connections between his work and that of his contemporaries and later economists, as potentially

important extensions of the present effort.

In addition, we have suggested an apparent indirect route through which Voigt’s

contribution to ordinalism may have influenced later economic thought, or at least the language

in which that later thought found expression. However, our discussion of Voigt’s influence on

economic thought and language also raises further questions. Specifically, it is important to

recall that the intellectual world of the 1890’s, including the world of economics, was a very

different place than it is today in at least two important ways: First, there were far fewer journals

than there are today and second, many, perhaps most economists had at least a “reading

knowledge” (and often more) of several different languages. Recall that Irving Fisher (1892, p.

124) cited Voigt’s (1892a) “Der Ökonomische Wert der Güter” during the year in which it

appeared in print, that Edgeworth published in both English and Italian, and that Pareto

published in Italian, French, English, and German, just to cite three particularly relevant

examples.

The fact that there were few economics journals in existence in the closing years of the

nineteenth century and that Pareto could read as many languages as he did clearly raises the

27

question of whether Pareto might have known directly of Voigt’s work, including “Zahl und

Mass,” or whether he might at least have learned of Voigt’s argument for ordinalism by reading

the anonymous (but probably Edgeworth’s) reference to it in the March 1894 issue of the

Economic Journal (anonymous, 1894) or Edgeworth’s (1894) own reference to it in the same

issue. Pareto certainly knew of the Economic Journal, since he had published one short paper in

it (Pareto, 1892). If he did not learn of Voigt’s ordinalist views there, might he have learned of

them from some other source some time before he first presented his own public argument for

ordinal utility in 1898?34 These questions raise the interesting possibility that Voigt may have

had some part in influencing Pareto’s 1898 conversion to ordinalism. For the moment, we leave

this fascinating possibility as a topic for further research.

34 If so, then this would beg the further question why he would have failed to acknowledge Voigt’s contribution.

28

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.

29

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.

30

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31

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