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 Please do not quote without permission.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.

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
1

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
2
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).
For example, Weber (2005) argues that Edgeworth’s views on complementarity and substitutability were
3 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.
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.
2

2. Andreas Voigt, Late Nineteenth Century Mathematics, and Ordinal Utility 4
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
5 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.
The biographical sketch is based largely on Hamacher-Hermes (1994) and Pulkkinen (1998). See also the brief
6 biography of Ernst Schröder on the University of St. Andrews Mathematics and Statistics website at wwwgroups.dcs.st-andrews.ac.uk/~history/Mathematicians/Schroder.html, (Anonymous) (1901), and Fehling (1926).
Fehling (1926) discusses the evolution of German higher education and business education in particular during the late nineteenth and early twentieth centuries.
3

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:
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.
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).
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.
8

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 pricequantity 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.
11

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
T his 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 utility 19 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 …”.
13

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).
14

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.
15

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).
16

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).
17

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.
18

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”,
19

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.
20

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 Love 27 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.
21

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 Economy 30 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
.
22

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).
23

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
24

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 cardinalistutilitarian 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
25

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
26

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.
27

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

OTHER WORKS CITED:
(Anonymous), 1894, “Recent Periodicals and New Books.” Economic Journal 4 (March):
196-208.
(Anonymous), 1901, “Notes.” Journal of Political Economy 10 (December): 104-105.
Allen, R.G.D., 1932a, “The Foundations of a Mathematical Theory of Exchange.” Economica
12 (May): 197-226.
Allen, R.G.D., 1932b, “Review of Mathematical Psychics .” Economic Journal 42 (June): 307-
309.
Allen, R.G.D., 1933, “On the Marginal Utility of Money and its Applications.” Economica 13
(May): 186-209.
Allen, R.G.D., 1934, “The Nature of Indifference Curves.” Review of Economic Studies 1
(February): 110-121.
Antonelli, G.B., 1886, “Sulla Teoria Matematica della Economia Politica.” (“On the
Mathematical Theory of Political Economy.”) Reprinted in J.S. Chipman, L. Hurwicz,
M.K. Richter, and H. Sonnenschein, eds. 1971, Preferences, Utility, and Demand . New
York: Harcourt Brace Jovanovich.
Armstrong, W.E., 1939, “The Determinateness of the Utility Function.” Economic Journal 49
(September): 453-467.
Bernardelli, H., 1938, “The End of the Marginal Utility Theory?” Economica 5 (May): 192-212.
Bruni, L. and F. Guala. 2001. “Vilfredo Pareto and the Epistemological Foundations of Choice
Theory.” History of Political Economy 33 (Winter): 21-49.
Chipman, J.S. 1976, “The Paretian Heritage.” Revue Européane des Sciences Sociales et
Cahiers Vilfredo Pareto 14: 65-171.
Creedy, J., 1986, Edgeworth and the Development of Neoclassical Economics . Oxford: Basil
Blackwell.
Dedekind, R., 1888. Was sind und was sollen die Zahlen?
Braunschweig: F. Vieweg. Translation available as part of Dedekind (1901).
Dedekind, R., 1901. Essays on the Theory of Numbers.
Translation by W.W. Berman of
Stetigkeit und irrationale Zahlen and Was sind und was sollen die Zahlen?
, Chicago and
London: Open Court. 3 rd printing 1924. Also reprinted in Ewald (1999): 790-833.
30

Dooley, P.C., 1983, “Consumer's Surplus: Marshall and His Critics.” Canadian Journal of Economics 16 (February): 26-38.
Edgeworth, F.Y., 1877, Old and New Methods of Ethics. Oxford: Parker.
Edgeworth, F.Y., 1881, Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences . London: Kegan Paul.
Edgeworth, F.Y., 1887, Metretike, or the Method of Measuring Probability and Utility . London:
Temple.
Edgeworth, F.Y., 1894, “Professor J. S. Nicholson on ‘Consumers' Rent’.” Economic Journal 4
(March): 151-158.
Edgeworth, F.Y., 1897, “La teorio pura del monopolio.” Giornale degli economisti 2 (July,
October, November), 13-31, 307-320, 405-414. Partial translation: “The Pure Theory of
Monopoly.” In F.Y. Edgeworth (ed.) 1925. Papers Relating to Political Economy (3 vols.) London: MacMillan. Reprinted in Pascal Bridel (ed.) 2001. The Foundations of
Price Theory , (6 vols.) London: Pickering and Chatto.
Edgeworth, F.Y., 1900, “The Incidence of Urban Rates.” Economic Journal 10 (June): 172-193.
Edgeworth, F.Y., 1906, “Recent Schemes for Rating Urban Land Values.” Economic Journal 16
(March): 66-77.
Edgeworth, F.Y., 1907, “Appreciations of Mathematical Theories.” Economic Journal 17 (June):
221-231.
Edgeworth, F.Y., 1915, “Recent Contributions to Mathematical Economics, Part I.” Economic
Journal 25 (March): 36-63.
Edgeworth, F.Y., 1917, “Some German Economic Writings about the War.” Economic Journal
27 (June): 238-250.
Elsas, A., 1886. Ueber die Psychophysik. Physikalische und erkenntnistheoretische
Untersuchungen . Marburg: N.G. Elwert’sche Verlags-Buchhandlung.
Fehling, A.W., 1926, “Collegiate Education for Business in Germany.” Journal of Political
Economy 34 (October): 545-596.
Fisher, I., 1892, Mathematical Investigations in the Theory of Value and Prices . New Haven:
Yale University Press.
Fisher, I. 1927. “A Statistical Method for Measuring ‘Marginal Utility’ and Testing the Justice of a Progressive Income Tax.” In J. H. Hollander, ed. Economic Essays Contributed in
31

Honor of John Bate s Clark.
MacMillan: New York.
Frisch, R., 1932, New Methods of Measuring Marginal Utility . Tübingen: J.C.B. Mohr.
Graham, F.D., 1932, “The Theory of International Values.” Quarterly Journal of Economics 46
(August): 581-616.
Gross, M. and V.J. Tarascio. 1998, “Pareto’s Theory of Choice.” History of Political Economy
30 (Summer): 171-187.
Haberler, G., 1929, “The Theory of Comparative Cost Once More.” Quarterly Journal of
Economics 43 (February): 376-381.
Hamacher-Hermes, A., 1994, Inhalts- oder Umfangslogik? Die Kontroverse zwischen E. Husserl und A. Voigt . Verlag Karl Alber, Freiburg and München.
von Helmholtz, H., 1887, “Zählen und Messen erkenntnistheoretisch betrachtet.” In
Philosophische Aufsätze. Eduard Zeller zu seinem fünfzigjährigen Jubiläum gewidmet ,
15-52, Leipzig, Fues. Reprinted in 1962, Leipzig: Zentral-Antiquariat der Deutschen
Demokratischen Republik. Translation by M.F. Lowe as Helmholtz (1999). von Helmholtz, H.., 1999. “Numbering and Measuring from an Epistemological Viewpoint.”,
Translation of Helmholtz (1887) by M.F. Lowe in W. Ewald (ed.), From Kant to Hilbert:
A Source Book in the Foundation of Mathematics , volume II, Oxford: Clarendon, 727-
752.
Hicks, J.R., 1930, “Edgeworth, Marshall, and the Indeterminateness of Wages.” Economic
Journal 40 (June): 215-231.
Hicks, J.R., 1932, The Theory of Wages . New York: MacMillan.
Hicks, J.R. and R.G.D. Allen, 1934, "A Reconsideration of the Theory of Value, Parts I and II."
Economica 1 (February, May): 52-76 and 196-219.
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