Via Po, 53 – 10124 Torino (Italy) Tel. (+39) 011 6704917 - Fax (+39) 011 6703895

URL: http//www.de.unito.it

WORKING PAPER SERIES

Were Jevons, Menger and Walras really cardinalists? On the notion of measurement in utility theory, psychology, mathematics and

other disciplines, ca. 1870–1910

Ivan Moscati

Dipartimento di Economia “S. Cognetti de Martiis”

Centro di Studi sulla Storia e i Metodi dell’Economia Politica "Claudio Napoleoni"

(CESMEP)

Working paper No. 03/2010

Università di Torino

Ivan Moscati*

Were Jevons, Menger and Walras really cardinalists?

On the notion of measurement in utility theory, psychology,

mathematics and other disciplines, ca. 1870–1910

December 2010 Abstract The paper investigates how Jevons, Menger and Walras conceived measurement in general, and the measurement of utility in particular. It argues that these three marginalists understood measurement in the so-called classical way, which is different from the representational understanding of measurement underlying current cardinal utility theory. In accord with the classical view of measurement, Jevons, Menger and Walras associated utility measurement with the possibility of identifying a unit of utility rather than with the ranking of utility differences. Thus, and contrary to the canonical narrative of the history of utility theory, the three marginalists were not cardinalists in the current sense of the term. In order to better understand the actual approach of Jevons, Menger and Walras to utility measurement, the paper situates their thought in the context of ideas of measurement in philosophy, physics, psychology, mathematics and areas of economics not related to utility theory. Focusing on the years 1870-1914, the period in which the three were active, the paper arrives at two conclusions: on the one hand, the classical understanding of measurement dominated not only economics but also other disciplines; but on the other hand, the direct relationships between the reflections on utility measurement by Jevons, Menger and Walras and contemporary non-economic debates on the topic were negligible. Keywords: Utility theory; Cardinal utility; Jevons; Menger; Walras; Measurement theory; Classical measurement; Representational measurement JEL Codes: A12, B13, B31, B40 * Bocconi University, Department of Economics. Via Röntgen 1, 20136 Milano, Italy. E-mail: ivan.moscati@unibocconi.it. I am grateful to Nicola Giocoli, Francesco Guala, Harro Maas, Massimo Marinacci, Philippe Mongin, Aldo Montesano, Mary Morgan, Fiorenzo Mornati, Margaret Schabas, Allan Hynes for useful discussions and suggestions. I thank the participants at seminars at Bocconi and Collegio Carlo Alberto, and at the 2010 meeting of the STOREP for helpful comments. I also thank the Italian Ministry for Education, University, and Research, for a PRIN grant for research on “Complexity and economic theory”, and the Centre for Philosophy of Natural and Social Science at the London School of Economics for its hospitality during part of the work on this paper. Any errors are mine.

1

1. Introduction

The canonical narrative of the history of utility theory is largely concerned with the contrast between ordinal and cardinal views of utility. According to this narrative, in a first phase, lasting roughly from 1870 to 1910, William Stanley Jevons, Carl Menger, Léon Walras and the other early marginalists treated individual utility as cardinally measurable. In a second phase, inaugurated by Vilfredo Pareto’s Manuel d’économie politique (1909) and concluded by John Hicks’ Value and Capital (1939), utility theorists moved away from cardinalism and embraced an ordinal approach to utility. The publication of Theory of Games and Economic Behavior (1944) by John von Neumann and Oskar Morgenstern imparted new vigor to the cardinal view, for the “Expected Utility Theory” presented in their book features a utility function that is cardinally measurable. This feature of the von Neumann-Morgenstern utility function generated a debate between cardinalists and ordinalists that lasted for much of the 1950s.1

This standard story overlooks the fact that discussions about cardinal and ordinal utility throughout this period were deeply interrelated with the ways in which utility theorists conceived of measurement, that is, with their understandings of what it means to measure a thing, and more specifically, what it means to measure a tricky thing such as the utility a commodity gives to an individual. The present paper belongs to a larger research project in which I bring the measurement-theoretic dimension of the history of utility theory into focus by investigating the interplay between utility analysis, the way utility theorists understood measurement, and the concept of measurement in other disciplines in the period 1870-1960. The main insight driving my research is the idea that when we keep in mind the measurement-theoretic dimension of utility analysis we gain new insights into the history of utility theory and, consequently, can correct some shortcomings in the standard narrative of it.

In particular, this paper argues that because the conception of measurement held by Jevons, Menger, and Walras was different from that underlying current cardinal utility, these three marginalists were therefore not cardinalists in the current sense of the term. As the seminal works of Jevons and Menger that initiated the so-called “marginal revolution” appeared in 1871, and as Walras published his last economic writing in 1909 (Walras [1909] 1987), the time span covered by their respective writings, approximately 1870-1910, seems an appropriate period on which to focus. In addition to an examination of these three key marginalists, the present paper also investigates contemporary understandings of measurement in other disciplines, especially psychology and mathematics, seeking for connections between Jevons’s, Menger’s and Walras’s approach to utility measurement and discussions about measurement taking place beyond economics. The paper concludes that these connections were negligible.

In Moscati 2010, which is another part of my wider research project on the measurement-theoretic dimension of the history of utility theory, I argue that other main marginalists of the period 1870-1910, such as Francis Ysidro Edgeworth, Alfred Marshall, and Irving Fisher, were also not cardinalists in the current sense of the term. A further paper (Moscati 2008) shows that the expression “cardinal utility” acquired its current economic meaning in the course of a debate on “The Determinateness of the Utility Function,” which took place in the Review of Economic

1 Versions of this narrative are offered by Schumpeter 1954, Jaffé 1977, Niehans 1990, Ingrao and Israel 1990,

Mandler 1999, Giocoli 2003, Mirowski and Hands 2006, Moscati 2007.

2

Studies between 1934 and 1938. Moreover, it is in the contributions to this debate that the first clear signs of a new understanding of measurement appear in economics. In a future paper, the final part of the research project, I will argue that utility theorists completed their shift from the traditional to the current understanding of measurement only in the 1950s, by way of discussion of the von Neumann-Morgenstern utility function. In this final paper I will also point out that, in both economic and measurement-theoretic terms, the cardinalist approach to utility which has thrived in many fields of economic theory since the 1950s diverges from, and should not be confused with the approach to utility of Jevons, Menger, Walras and the other early marginalists.

1.1. Different notions of measurement and utility To illustrate better the meaning of the claim that Jevons, Menger and Walras were not cardinalists, an outline of both traditional and modern concepts of measurement, as well as a brief review of the meaning of cardinal utility in contemporary microeconomic theory are necessary. As illustrated in more detail in Section 2, the traditional concept of measurement dates back to Aristotle and Euclid and from the time of Galileo onward dominated both the natural and human sciences until the early decades of the 20th century. According to this concept, measuring the property of an object (e.g. the length of a table) consists of comparing it with some other object that displays the same property and is taken as a unit (e.g. a meter rule), and of assessing the numerical ratio between the unit and the object to be measured (if the ratio is two to one, the table is two meters long). In the traditional view, only the assessment of this numerical ratio stands as a true measurement. In a number of works, Joel Michell (1993, 1999, 2006, 2007) has labeled this traditional understanding of measurement the “classical” view of measurement, and this paper adopts his terminology. The currently dominant concept of measurement, which Michell labels the “representational” view, was first formulated, albeit in a cursory manner, in 1903 by the English philosopher and mathematician Bertrand Russell. This representational view, however, was only fully worked out some years later (and in a period well beyond the period considered in the present paper) by, among others, the British physicist and philosopher of science Norman Robert Campbell (1920), the American philosopher Ernest Nagel (1931), and the American psychologist Stanley S. Stevens (1946). In 1971, David Krantz, Duncan Luce, Patrick Suppes and Amos Tversky presented a systematic account of the representational theory of measurement in the first volume of their monumental Foundations of Measurement. According to the representational view, measurement consists of assigning numbers to objects in such a way that the relations among the assigned numbers represent the relevant relations among the objects. Moreover, since objects may entertain different types of relations, the representational approach admits different forms of measurement, as opposed to the unique form allowed by the classical approach. The representational forms of measurement are also called “scales”, and those that are of interest for the present paper are the ordinal, the interval, and the ratio scales.2

The ordinal scale applies when only the rank-ordering of objects is possible. In this case, the numbers associated with the objects mirror only their ranking: if A is ranked over B, the number associated with A needs only to be larger than the number associated with B. The interval scale applies also when the ranking of differences between objects is possible, that is, when we can

2 The classification of measurement scales adopted here follows the one used by Stevens 1946. Krantz, Luce,

Suppes and Tversky 1971 employ a more sophisticated and partially different classification.

3

state not only that A is ranked over B, and B is ranked over C, but also that the difference between A and B is, for instance, larger than that between B and C. Accordingly, the difference between the numbers associated with A and B must be larger than the difference between the numbers associated with B and C. The ratio scale brings us back to classical measurement. It requires, in addition to the ranking of objects and their differences, also the existence of an object which is univocally associated with the number zero (the zero-point of measurement) and the assessment of the ratios between objects. There exists a straightforward hierarchy among these three forms of measurement: if something is measurable in the ratio-scale sense, it is also measurable in the interval-scale sense; in turn, if something is measurable in the interval-scale sense, it is also measurable in the ordinal-scale sense.3

Contemporary microeconomic theory adopts the representational view of measurement and understands ordinal and cardinal utility functions as two different forms of utility measurement.4 Both functions assign numbers to objects (typically bundles of goods) in such a way that the numbers represent the individual’s preferences among the objects. Ordinal utility analysis assumes that the individual is just able to rank his preferences for objects. Accordingly, the numbers associated with objects represent only this preference ranking and thus provide an ordinal measurement of utility. Cardinal utility assumes that the individual is also able to rank the difference of preference between objects and so state that his preference for A over B is e.g. larger than his preference for B over C. Correspondingly, the difference between the numbers associated with A and B must be larger than the difference between the numbers associated with B and C. This amounts to saying that cardinal utility provides an interval-scale measurement of utility.

Since at least the 1930s, mainstream economic theory has abandoned the idea that utility can be measured in the ratio-scale sense. For this reason, there is no standard expression to designate this type of utility in current microeconomics. If we label ratio-scale measurable utility as “proportional utility”, however, the relationships between scales of measurement and the corresponding types of utility can be summarized as in Table 1 below.

Table 1

SCALE OF MEASUREMENT Ordinal scale Interval scale Ratio scale

TYPE OF UTILITY Ordinal utility Cardinal utility Proportional utility

1.2. The thesis, again This preliminary aside concerning measurement theory and utility analysis allows us to state the main tenets of the present paper more precisely. The paper, then, shows that Jevons, Menger and Walras conceived of measurement in the classical, and not the representational manner, and that they applied their classical conception of measurement to the measurement of utility. This means 3 Each scale is also associated with a class of functional transformations that do not modify its capacity to mirror

the relevant relations among objects. The ordinal scale is associated with increasing transformations (i.e., with any f(x) such that f΄>0), the interval scale with increasing linear transformations (i.e., with transformations of the form αx + β, where α>0.), while the ratio scale is associated with proportional transformations not that do not modify the zero point (i.e. transformations of the form αx, where α>0).

4 A classic treatment of ordinal and cardinal utility from the viewpoint of contemporary economic analysis can be found in Fishburn 1970.

4

that in proposing different methods to measure utility, or in discussing the extent to which their theories actually relied on the measurability of utility, or even in arguing that utility is immeasurable, they referred to the classical understanding of measurement. Consequently, they focused on the possibility or impossibility of ascertaining a unit of utility and assessing utility ratios and did not, as is the case in current cardinal utility analysis, associate utility measurement with the ranking of utility differences. Furthermore, they did not even use the word “cardinal” or the cardinal-utility terminology. For all these reasons Jevons, Menger, and Walras cannot be considered cardinalists in the current sense of the term, and the canonical narrative of the history of utility theory should be amended accordingly. In particular, the ordinal revolution initiated by Pareto should be viewed, not as a shift from cardinal to ordinal utility, but as a jump from proportional to ordinal utility, with cardinal utility entering the scene only at a later stage. A related paper (Moscati 2008) shows that, in fact, cardinal utility came to be associated with the ranking of utility differences only in the course of the debate on “The Determinateness of the Utility Function,” which took place in the Review of Economic Studies between 1934 and 1938.

1.3. Objections and replies It is useful to address here some possible objections to this thesis. In the first place, one may point out that classical measurement is equivalent to ratio-scale measurement and that the latter implies interval-measurement, which is in turn associated with cardinal utility. Thus, even if Jevons, Menger and Walras looked at utility measurement from the classical viewpoint, this would be perfectly consistent with their being cardinalists. In effect, they would have been “more-than-cardinalists” or “hyper-cardinalists”. Thus, so the objection might go, the standard narrative of the history of utility theory is fundamentally correct. To this we reply that if Jevons, Menger and Walras looked for ratio-scale measurement of utility instead of interval-scale measurement, then a standard narrative that depicted them as mere cardinalists would miss an important aspect of the history of utility theory. Thus, even the “hyper-cardinalist” reading of the three founders of marginal utility theory calls for a revision of the canonical narrative, exactly along the lines put forward in the present paper. In particular, and as already mentioned, the ordinal revolution should be considered, not as a shift from cardinal to ordinal utility, but as a more radical jump from proportional to ordinal utility.

Secondly, one may observe that, from the standpoint of current economic analysis, it is easy to demonstrate that the theories of Jevons, Menger and Walras did not depend on ratio-scale measurability of utility. Some of the assumptions they made – for example, that the marginal utility of each commodity is decreasing and independent of the quantities consumed of other commodities – do indeed imply interval-scale measurability of utility and hence a cardinal approach, but not proportional utility. Thus, in labeling the early marginalists cardinalists the canonical narrative would be, from the standpoint of current economic analysis, correct. However, from the historical, not the theoretical viewpoint, depicting Jevons, Menger and Walras as cardinalists is misleading. Historically, they did not think of their discussions about the measurability of utility as related to interval-scale measurement, and for them measurement univocally meant identifying a unit and assessing ratios, not ranking differences. In other words, the categories through which Jevons, Menger and Walras addressed the problem of the measurability of utility were not those of current measurement theory and contemporary utility analysis, and the anachronistic act of reading the present categories back into this episode from

5

the past would be historically misguided.

A third possible objection is related to the assumption, common to our three founders of marginal utility theory, that the marginal utility of each commodity is decreasing. To modern eyes, this assumption entails the supposition that individuals are able to rank variations of their total utility and state that the increment of total utility obtained from the consumption of the n-th unit of the good is larger than the increment obtained from the (n+1)-th unit. This ranking, however, is no other than that which delivers an interval-scale measurement of total utility and hence cardinal utility. Since Jevons, Menger and Walras dealt with the ranking of utility differences, so this third objection goes, they were cardinalists. The problem with this argument is that the three marginalists did not associate decreasing marginal utility with the interval-scale measurement of total utility. In discussing the measurability of utility, they all referred to the classical, unit-based measurement, and they did so with respect to both marginal and total utility. Furthermore and contrary to contemporary microeconomic analysis, for them the key notion of utility theory was not total, but marginal utility. Accordingly, they conceived of the challenge of utility measurement as related to the search for a unit to measure marginal utility, which is a search going well beyond the cardinal utility analysis of total utility. Thus, even if they assumed decreasing marginal utility, Jevons, Menger and Walras were not cardinalists in the current sense of the term.

Finally, one may claim that in the canonical narrative the expression “cardinal utility” is not used in its current meaning connected with rankings of utility-differences, but in a much vaguer sense that encompasses any form of utility measurement stronger than the ordinal one and hence including even classical measurement. If this were true, standard narratives about early utility theory would not be mistaken after all. In reply to this, it can be observed that in most cases the standard story is told by scholars who are well aware of the current technical meaning of cardinal utility, so that it appears debatable to claim that they used the term in a loose and inappropriate sense. More importantly, if in the canonical story the term “cardinal utility” is used in a vague and possibly misleading sense, this is in itself sufficient reason to attempt to avoid further confusion by constructing, for the first time, a rigorously defined historical analysis.

1.4. Structure of the paper The structure of the remaining part of the paper is as follows. Sections 2-6 provide an extensive review of the notion of measurement in philosophy, physics, psychology, mathematics, and the parts of economics unrelated to utility theory, with a focus on the period 1870-1910. This review offers a picture of the broad intellectual context within which Jevons’s, Menger’s and Walras’s discussions on utility measurement took place. Among other things, the review shows that in the period under consideration the classical understanding of measurement dominated not only economics but also other disciplines.

Sections 7-9 are devoted to Jevons, Menger and Walras, respectively, and argue that the three founders of marginalism conceived measurement in the classical fashion, associated utility measurement with the possibility of identifying a unit of utility and assessing utility ratios, and thus were not cardinalists in the current, ranking-of-utility-difference sense of the term. It is also shown that the reflections on utility measurement of Jevons, Menger and Walrus were, in fact,

6

fundamentally independent of contemporary discussions in other disciplines. Section 10 summarizes the contributions of the paper.

2. Measurement in philosophy As noted above, the classical understanding of measurement dates back to Aristotle and Euclid. In Aristotle’s philosophy the notions of quality and quantity are strictly separated, and the notion of measurement associated with the notion of quantity in a strict interrelation with those of unit and number. In his Metaphysics Aristotle argued that a number is either one or a plurality of “ones”, i.e. one unit or a plurality of units (Book X, chapter 1), and defined quantity (as opposed to quality) as that which “is divisible into two or more constituent parts of which each is by nature a ‘one’” (V, 13).5 Measuring, in this philosophy, means assessing the number of units constituting a quantity; measurement is in fact the specific way in which a quantity qua quantity is known: “For measure is that by which quantity is known; and quantity qua quantity is known either by a ‘one’ or by a number, and all number is known by a ‘one’. Therefore all quantity qua quantity is known by the one.” (X, 1) Aristotle added that the quantity and the unit through which it is measured are always of the same kind: “The measure is always homogeneous with the thing measured; the measure […] of length is a length, that of breadth a breadth” (X, 1).

Similar ideas can be found in Euclid’s Elements. He defined a ratio as “a sort of relationship in respect of size between two magnitudes of the same kind” (Book V, definition 3).6 When a greater and a lesser magnitude are placed in such a relationship the lesser measures the greater, while the greater is said to be a multiple of the lesser. However, there are also incommensurable magnitudes such as the side and the diagonal of a square, for which it is not possible to establish a ratio relationship (X, def. 1). Euclid considered numbers to be special types of magnitudes “composed of units” (VII, def. 2). He proved that the ratio of commensurable magnitudes can be expressed as a ratio of numbers (X, proposition 5).

Ancient and medieval scholars recognized that some qualities, although not measurable in the classical sense, admit of predication by the qualifiers more and less. In the Categories Aristotle himself observed that a thing may be whiter, more beautiful, or warmer than another, and that the same thing may exhibit a quality in different degrees in different moments: “A body, being white, is said to be whiter at one time than it was before, or, being warm, is said to be warmer or less warm than at some other time” (Categories, chapter 5). Aristotle did not investigate measurement issues with respect to these kinds of qualities. In the Middle Ages these issues were taken up by scholars such as Peter Lombard, Thomas Aquinas, Duns Scotus and Nicole Oresme, often in the context of theological discussions such as the question of whether and how the virtue of charity can increase or decrease.7 However, these medieval debates did not modify the classical understanding of measurement. Thus, in his Rules for the Direction of the Mind, René Descartes ([1628] 1999: 64) still contrasted ordering with measuring: “All the relations which may possibly obtain between entities of the same kind should be placed under one or other of two categories,

5 This and the following translations of Aristotle are from Ross 1910-52. 6 This and the following translations of Euclid are from Heath 1908. 7 On these medieval discussions see Sylla 1972 and Solere 2001.

7

viz. order or measure.”8

The distinction between properly measurable quantities and entities only admitting predication by the qualifiers more and less returned also in the distinction between extensive and intensive magnitudes introduced by Kant in his Critique of pure reason ([1787] 1997: 288-292). For Kant, extensive magnitudes refer to the spatial and temporal dimensions, are made of homogeneous parts and thus can be measured in units. Intensive magnitudes, by contrast, are related to perception, cannot be conceived as composed of parts but have only degrees and a zero point. As instances of intensive magnitudes, Kant mentioned the same entities Aristotle referred to as qualities admitting predication by the qualifiers more and less, namely color and warmth (292). Kant’s distinction between extensive and intensive magnitudes lived on in philosophy well into the 20th century, whereby extensive magnitude generally meant a magnitude measurable in the classical sense, while intensive magnitudes were thought of as something that can be ranked but not measured.9

3. Measurement in physics Although the quantitative natural philosophy that arose in the Scientific Revolution was in many ways opposed to Aristotle’s qualitative physics, Galileo, Newton and the other main natural philosophers of the period endorsed the classical view of measurement as set down by Aristotle and Euclid. For example, in the First Article of his Arithmetica Universalis (1707), Isaac Newton defined number as “the abstracted Ratio of any Quantity to another Quantity of the same kind, which we take for Unity” (quoted in Michell 2007: 20). More than 150 years later, the eminent Scottish physicist J.C. Maxwell expressed the same view of measurement. At the very beginning of his Treatise on Electricity and Magnetism (1873: 1), Maxwell affirmed that:

Every expression of a Quantity consists of two factors or components. One of these is the name of a certain known quantity of the same kind as the quantity to be expressed, which is taken as a standard of reference. The other component is the number of times the standard is to be taken in order to make up the required quantity. The standard quantity is technically called the Unit, and the number is called the Numerical Value of the quantity.

Between 1870 and 1910 the classical understanding of measurement remained fundamentally undisputed in physics, probably because most of the entities physicists dealt with in the period were measurable in the classical sense, and thus they were not motivated to commence any fundamental reconsideration of the notion of measurement. Rather than questioning what measurement means, physicists focused on more practical issues, such as how to attain precision and reproducibility in classical measurement.10

An important physical magnitude that for a long time had eluded classical measurement was heat. We saw that Aristotle considered the heat of a body as a quality admitting predication by the 8 Here and in the following quotes from French and German texts, the reference is to their English translations,

when available. However, the original texts have always been checked and the available English translations have been modified when this seemed appropriate. When no English translation was available, the translation is mine.

9 For more on the distinction between extensive and intensive magnitudes after Kant, see Michell 2006. 10 For a concise review of measurability issues in physics between the late 19th century and the early 20th century,

see Darrigol 2003: 541-545.

8

qualifiers more and less, and this view remained the standard one for centuries. From the the 17th century onward a gradual move toward the quantification of heat commenced: beginning with Galileo, several types of thermometers were introduced and continuously improved; different temperature scales were proposed, such as the Fahrenheit (1724) or the Celsius (1742) scales; various thermal units were put forward, such as Clemént’s Calorie (1824), which is the quantity of heat required to raise the temperature of a kilogram of water by one degree. Yet this process of quantification notwithstanding, we have seen how, at the end of the 18th century, Kant still considered heat an intensive magnitude that could not be measured in the classical sense. Around 1850, however, William Thomson (Lord Kelvin) and James Prescott Joule showed how the thermal properties of gases could be used to measure temperature in an absolute, i.e. classical, way. From that point on, heat was conceived of as a classically measurable magnitude.11

Another physical magnitude that eluded classical measurement was the hardness of minerals. In 1812 the German geologist Friedrich Mohs had proposed the ordering of minerals according to their ability to scratch each other, i.e. according to their ‘hardness’: if one mineral scratches another, the former is harder than the latter. The scratching test allowed the ordering of minerals and the construction of what became known as the Mohs Scale of mineral hardness, which is a purely ordinal scale. However, the fact that the hardness of minerals was not measurable in the classical sense was not considered sufficiently important for it to stimulate an in-depth discussion of measurement in physics.

4. Measurement in psychology In contrast to the situation in physics, during the period 1870-1910 a more fundamental discussion about measurement took place in other domains, specifically psychology, mathematics and utility theory. The psychological and mathematical discussions about measurement were very much interrelated, with developments within psychology influencing mathematics rather than vice versa. The discussion within utility theory, however, remained basically autonomous, and was hardly related to either mathematical or psychological discussions. In psychology the discussion about what it means to measure a thing and under what conditions measurement is possible originated with the emergence of psychophysics, which is generally considered the immediate ancestor of 20th century quantitative and experimental psychology.

4.1. Fechner’s psychophysics Gustav Fechner was a German philosopher and scientist whose research was motivated by the philosophical ambition of overcoming the dualism between mental and physical phenomena. He attempted to achieve this goal by way of the construction of “an exact science of the relations between body and mind”, which he called “psychophysics” and expounded in his Elements of Psychophysics (Fechner [1860] 1966: xxvii). One main goal of psychophysics was achieve the measurement of sensations, by which Fechner understood classical measurement: “Generally the measurement of a magnitude consists of assessing how many times another magnitude of the same kind taken as a unit is contained in the former.” (38) As the unit of sensation, he took the just perceivable difference of sensation generated by a change in physical stimulus, and assumed 11 On the history of thermometry see Chang 2004.

9

that all just perceivable (henceforth abbreviated as j.p.) differences are equal, i.e., that they are independent of the magnitude of the stimulus.

Fechner acknowledged that sensations are not directly measurable, but argued that they can be indirectly measured by exploiting their correlation with stimuli. His idea is that a sensation can be seen as a multiple of equal j.p. increments and is measured by the number of increments in physical stimulus generating those j.p. increments of sensation:

Our measurement of sensations comes to this: split up every sensation into equal parts, i.e. equal [just perceivable] increments, which it grows from the zero-state, and consider the number of these equal parts as determined (as if by the bits of a yardstick) by the number of the corresponding, variable stimulus increments that produce equal sensation increments. […] We determine the magnitude of the sensations, which we cannot determine directly, as a multiple of the equal parts which we can determine directly; but we read off the number of parts not from the sensation, but from the stimulus which brings the sensations with it, and which can be more easily read. (Fechner [1860] 1966: 50)

The German physiologist Ernst Heinrich Weber ([1834/1846] 1996) had shown that the minimal increment of physical stimulus necessary to produce a difference of sensation varies with the magnitude of the stimulus. Thus, for example, if a subject carries an object weighting 40 grams, an increment of one gram may be sufficient to produce the perception of a heavier object, while if the object weights 160 grams it might be necessary to add four grams to produce the sensation of a heavier object. By building on Weber’s results and assuming that all j.p. differences of sensation are equal, Fechner arrived at the following logarithmic formula connecting physical stimulus to the corresponding sensation: γ = κ log (β/b), whereby γ is the magnitude of the sensation, κ is a constant depending on the specific sensation under consideration, β is the magnitude of the physical stimulus, and b is the so-called absolute threshold, i.e. the level of the physical stimulus below which no sensation is perceived.

4.2. The debate on psychophysical measurement Because it crossed so many scientific and philosophical dimensions, psychophysics stirred considerable interest and discussion among Fechner’s contemporaries. His psychophysical investigations were subsequently developed by a number of followers, among whom we may single out Wilhelm Wund (1873-74) in Germany and Joseph Delboeuf (1873) in Belgium. Nevertheless, in the 1870s and 1880s Fechner’s basic claim that sensations can be measured was attacked from many quarters. Here we shall consider only a select few of the contributions to the debate over the measurability of sensations that followed the publication of his Elements.12

The French mathematician Jules Tannery (1875: 134) argued that measurable magnitudes are characterized by the possibility of defining the equality and the act of addition between two of them, and by their homogeneity:

The only magnitudes to which the idea of measurement is directly applicable are those for which we can clearly conceive equality and addition. In fact, wherever we dealt with the measurement of

12 Titchener 1905 offers a thorough review of this debate. On it, see also Michell 1999, Darrigol 2003, and

Heidelberger 2004.

10

quantities, may they be lengths, angles, surfaces, times, or forces, we begin by focusing on what is meant by the equality and the sum of two of these quantities. [...] The essential character of directly measurable magnitudes is homogeneity: what is added to them, when they increase, is exactly of the same kind as what already exists.13

The only magnitudes to which the idea of measurement is directly applicable are those for which we can clearly conceive equality and addition. In fact, wherever we dealt with the measurement of quantities, may they be lengths, angles, surfaces, times, or forces, we begin by focusing on what is meant by the equality and the sum of two of these quantities. [...] The essential character of directly measurable magnitudes is homogeneity: what is added to them, when they increase, is exactly of the same kind as what already exists.14

For Tannery, sensations do not possess the characteristics that make a magnitude measurable: “It does not seem to me that sensations possess the homogeneity character [...]. I can conceive neither the sum of two sensations nor their difference: when a sensation increases, it becomes another [...], and I do not conceive it as the sum of the first sensation with a certain differential sensation.” (136-137) Since sensations are not measurable, while mathematical tools presuppose the measurability of the objects they apply to, the mathematical treatment of sensations made no sense. In particular, Tannery attacked the supposed differentiation and integration of sensations that Fechner had employed in order to arrive at his logarithmic formula:

Since we do not know at all what the difference between two sensations means, how can we speak of the differential of a sensation? [...] Here there is neither quantity nor continuity. [...] Since we do not know what the sum and difference of two sensations is, what does integration, i.e. the sum of differentials of sensation, mean? (113)

In Section 7 and 9 we will see that Tannery’s case against the mathematical treatment of sensations is very similar to the arguments already made between 1871 and 1873 by those who criticized the mathematical treatment of utility of Jevons and Walras. However, I have found no direct link between Tannery’s critique and those made by the critics of Jevons and Walras critics.

The German physiologist Johannes von Kries ([1882] 1995) criticized Fecher’s assumption that all j.p. differences of sensation are equal. For Kries, the increments of sensations occurring at different levels of stimuli are not immediately comparable and thus the claim that they are equal, rather than being either true or false, simply makes no sense:

If a location on the skin is subjected first to a two-pound and then to a three-pound load, and subsequently to a ten-pound and then a fifteen-pound load, the latter two sensations of pressure occupy quite a different place in the whole series of sensations than the first two. Thus the one increment is something quite different from another; at first they admit of no comparison. The claim that they may be equal makes absolutely no sense. (291)

13 As we will see in Section 5, the idea that measurability is a feature of certain magnitudes that can be defined in an

explicit way will be put forward, independently of Tannery, in the 1880s by German mathematicians discussing the notion of number and measurement.

14 As we will see in Section 5, the idea that measurability is a feature of certain magnitudes that can be defined in an explicit way will be put forward, independently of Tannery, in the 1880s by German mathematicians discussing the notion of number and measurement.

11

Following on from this, Fecher’s claim that a sensation can be divided into equal j.p. differences of sensation also makes no sense: “What it means to say that one pain may be exactly ten times as strong as another, is simply unfathomable. […] A loud tone does not conceal within itself this or that many faint tones, in the same sense that a foot contains twelve inches or a minute sixty seconds.” (292) A possible escape from the difficulty would be to stipulate by convention that all j.p. differences of sensation are equal. However, Kries (292) pointed out that this would leave Fechner’s logarithmic law devoid of any empirical content:

We can fix a convention such that all just-noticeable increments in sensation […] are to be considered equal. Having done so, we can represent a number of observed facts in that we attribute to sensation an increment proportional to the logarithm of the stimuli. But this law means nothing at all without the convention, as we have seen. And with the convention this law means nothing but the observed facts. (289)

Kries concluded that sensations are intensive magnitudes in the Kantian sense of the term, which cannot be measured in the classical, unit-based fashion.

Echoing some of Tannery’s arguments, the French philosopher Henry Bergson ([1889] 1960) argued that sensations are not even intensive magnitudes admitting predication by the qualifiers of more and less, but are rather qualities. Different sensations are qualitatively different, and the interpretation of the transition from sensation S to sensation S΄ as an increase or a decrease is arbitrary:

The mistake which Fechner made […] was that he believed in an interval between two successive sensations S and S', when there is simply a passing from one to another and not a difference in the arithmetical sense of the word. […] [Calling ΔS the transition from S to S'] is a symbolical interpretation of quality as quantity. (67-69)

For Bergson, as sensations are qualities they are immeasurable.

On the other side of the Atlantic, the American psychologist and philosopher William James (1890: 546) followed Tannery and Kries in disputing Fechner’s claim that sensations consist of the sum of j.p. increments taken as sensation-units: “We have already found ourselves […] quite unable to read any clear meaning into the notion that they [sensations] are masses of units combined. To introspection, our feeling of pink is surely not a portion of our feeling of scarlet; nor does the light of an electric arc seem to contain that of a tallow-candle in itself.”

Fechner (1882, 1887) replied to some of these criticisms, but basically did not modify his stance on sensation measurement. By contrast, in their effort to defend Fechner and psychophysics from these and other attacks, Delboeuf (1875, 1878, 1883), Wund (1880) and other psychologists developed a partially different view as to the nature of the actual object of psychophysical measurement. According to this modified view, psychophysics does not measure the absolute magnitude of sensations as in Fechner’s theory, but rather the magnitude of difference between sensations. Significantly, the measurement of sensation differences was still of the classical, unit-based kind, as the units by which these differences were to be measured were still the equal j.p. increments of sensation. This modified solution left open many problems, such as that concerning the supposed equality of j.p. differences, and it failed to satisfy critics of psychophysics such as

12

Tannery (1884, 1888). We do not need to dwell here on subsequent discussion as to possibility of the psychophysical measurement of sensation differences.15 Rather, we conclude this review of the debate on psychophysical measurement by emphasizing that the entire debate remained completely within the classical understanding of measurement based on the availability of a unit: for Fechner, sensations are divisible into equal units and thus are measurable; for Fechner’s critics, sensations are not measurable because they cannot be split into equal and homogeneous units; finally, for supporters of measurement of sensation differences the latter are measurable since they can be constructed as the sum of equal j.p. increments of sensation.

4.3. Mental measurement in the 1900s From the end of the 19th century on, quantitative psychologists began to extend their attempted mental measurements from sensations to intellectual abilities. Some pioneers in this field were the Americans James Cattell (1890, 1893) and his student Edward Thorndike (1904), the Frenchman Alfred Binet (1903) and the Englishman Charles Spearman (1904).16 For our present purposes, the important point is that, while the object of attempted measurement may have developed, the understanding of measurement remained the classical one. For instance, in his entry on “Measurement” for the Dictionary of Philosophy and Psychology, Cattell (1902: 57) asserted: “[Measurement] is the determination of a magnitude in terms of a standard unit. […] A ratio is the basis of all measurement.”

Psychologists who continued to devote their attentions to the measurement of sensations also maintained a classical understanding of measurement. One such psychologist was Edward Titchener, an English student of Wundt who after graduation moved to Cornell University where he made a significant contribution to the establishment of experimental psychology in the United States. In his Text-Book of Psychology, Titchener (1909: 207) claimed that “whenever we measure, in any department of science, we compare a given magnitude with some conventional unit of the same kind, and determine how many times the unit is contained in the magnitude.”

Only from the 1910s on did quantitative psychologists begin to realize that their measurement practices of mental variables were hardly compatible with the classical notion of measurement. This realization eventually led to Stevens (1946) representational approach to mental measurement.17

4.4. Measurement in psychology: summing up In the period 1870-1910, attempts made by Fechner and other psychologists to measure sensations and other mental variables stimulated a discussion on measurement in psychology. This discussion, however, remained within the boundaries of the classical understanding of measurement. In fact, underlying the arguments of both advocates and critics of psychological measurement was the assumption that sensations and other mental variables could only be measured by expressing them as multiples of a unit.

15 For more on this topic see Titchener 1905, section 6. 16 For more on the early history of measurement of intellectual abilities see Boring 1957. 17 On the modifications in the understanding of measurement in psychology after 1910, see Michell 1999, chs. 4

and 6.

13

The debate on psychological measurement seems to have had no influence on Jevons, Menger and Walras. As we will see in Sections 7-9, neither Fechner nor any of the other scholars involved in the debate on psychological measurement were mentioned in the main works or the correspondence of the three founders of marginal utility theory.

5. Measurement in mathematics The years 1870 to 1910 witnessed major developments in mathematics. Among other innovations, the period was marked by the rise of set theory, mathematical logic, and the axiomatic approach to geometry and other fields. The period also witnessed major advances in the theory of number and measurement, which were partly stimulated by the contemporary psychological debates on the measurability of sensations. Among other advances in this field, in these years the distinction between cardinal and ordinal numbers was introduced and investigated, the question of what conditions make a magnitude measurable in the classical sense was explicitly addressed, and (as noted above) a first, cursory formulation of the representational notion of measurement was put forward.18 Some hints of the mathematical discussions concerning number and measurement were transmitted to utility theory through the “Voigt-Edgeworth channel” and the “Poincaré-Walras channel” discussed, respectively, in Sections 5.8 and 9.4.

5.1. Schröder The distinction between cardinal numbers (Cardinalzahlen) and ordinal numbers (Ordinalzahlen) appears to have been introduced in a handbook of arithmetic and algebra published by the German mathematician Ernst Schröder in 1873. Schröder’s cardinal numbers match the classical understanding of number and measurement as stated by Aristotle and Euclid: cardinal numbers express the total number of units constituting a given quantity, e.g. five, and thus are the relevant ones for the unit-based measurement of it. Ordinal numbers, by contrast, come to the fore in the process of counting the units belonging to a quantity and express the position of a specific unit of the quantity, e.g. the fifth unit (Schröder 1873: 13-14).

5.2. Du Bois-Reymond Although he does not employ the words Cardinalzahlen and Ordinalzahlen, a cardinal-ordinal distinction is nevertheless suggested also by Paul Du Bois-Reymond, another German mathematician, in the discussion of discrete magnitudes contained in his Die allgemeine Functionentheorie (The general theory of functions) of 1882. For Du Bois, a discrete magnitude is expressed by the total number of objects that constitute it, i.e. by what Schröder had called cardinal number. Now, for small magnitudes up to about seven objects this number can be perceived directly by simply looking at the magnitude. With larger magnitudes, however, the associated number is apprehended indirectly through a process of counting in which the next number is assigned to a magnitude containing one more object than the preceding magnitude (Du

18 Grattan-Guinness 2000 and Ferreorós 2007 provide thorough reconstructions of this period of the history of

mathematics. On the history of number and measurement theory, see in particular Darrigol 2003 and Michell 1993 and 1999.

14

Bois-Reymond 1882: 16, 19). As this counting process is ordinal in nature, Du Bois’s theory suggests that cardinal numbers beyond about seven derive from ordinal numbers.

Apparently unaware of the similarities of his approach to that previously taken by Tannery (1875), Du Bois attempted to define the properties that make a magnitude measurable, where by measurement he meant classical measurement, i.e. the assessment of the ratio between the magnitude and a unit (Du Bois-Reymond 1882: 30-31, 47-48). He focused in particular on length, which he took as the archetypal form of a continuous, measurable magnitude, labeled “linear magnitudes” the magnitudes analogous to length, and proceeded to indicate six formal properties characterizing them (23-47). Among other characteristics, Du Bois argued that any linear magnitude can be obtained from the sum of smaller linear magnitudes, and can also be divided arbitrarily with the result always a linear magnitude. The technical question of whether Du Bois’s properties are necessary or sufficient to characterize measurable magnitudes in the classical sense of measurement is not relevant here. For us what is more important to notice is that Du Bois put forward – and did so more explicitly than Tannery – the idea that measurability is a feature of magnitudes that can be defined in an explicit and systematic way.

Du Bois praised Fechner for his pioneering attempt to measure sensations, and argued that warmness and other tactile sensations are linear magnitudes and hence measurable. Although Du Bois was more doubtful about the status of other sensations and psychological dispositions, he was convinced that “it is legitimate to talk, in the proper sense of the word, of a double happiness or a double disgruntlement.” (37)

5.3. Helmholtz In 1887 the German physicist, physiologist and philosopher Hermann von Helmholtz developed some of the ideas of Schröder and Du Bois. His essay, “Numbering and Measuring from an Epistemological Viewpoint,” appeared in a Festschrift in honor of the German philosopher Eduard Zeller (Helmholtz [1887] 1999).19 As a philosopher, Helmholtz had previously offered an empiricist interpretation of the axioms of geometry, treating them as propositions to be tested against experience rather than the a priori laws of Kantian philosophy. In “Numbering and Measuring” Helmholtz attempted to extend his empiricist approach to the notions of number and measurement.

Helmholtz (729) referred approvingly to Schröder’s discussion of cardinal and ordinal numbers, although he himself used a slightly different terminology: Anzahl instead of Cardinalzahl, and Ordnungszahl instead of Ordinalzahl. While Schröder treated cardinal and ordinal numbers as associated with two different but equally original functions of number, Helmholtz suggested that ordinals are more fundamental than cardinals. For Helmholtz, in fact, numbers derive from the psychological capacity of “retaining, in our memory, the sequence in which acts of consciousness successively occurred in time” (730), and thus are originally ordinal in nature. Cardinal numbers come into play only when we apply the numbers generated by the internal intuition of time to external experience, typically in the attempt to determine the amount of objects belonging to a given set. 19 Interestingly, in the 1860s Wundt had been an assistant of Helmholtz at the Heidelberg Physiological Institute,

while in 1876-77 Kries studied under him at Berlin University.

15

In accord with the classical understanding of measurement, Helmholtz defined a magnitude as measurable if it can be expressed as the sum of some kind of units (740-741). Like Du Bois, who is cited a number of times in the essay, Helmholtz put forward a set of features identifying measurable magnitudes. However, Helmholtz intended these features as empirical conditions to be verified practically or experimentally, rather than formal properties mirroring those of length as in Du Bois’s analysis. In particular, for Helmholtz there must be some practical procedure to compare two magnitudes and ascertain whether they are alike, and a further procedure to connect them in a sense analogous to mathematical addition. For instance, physical magnitudes such as the weights of two bodies are compared by placing the bodies on the two pans of a balance, and are connected by putting the two bodies in the same pan.

5.4. Kronecker In the same Festschrift that contained Helmholtz’s paper, a further contribution to fundamental number theory was published: “On the Concept of Number”, by Leopold Kronecker. Another eminent German mathematician, and like Schröder and Du Bois but without reference to them, Kronecker ([1887] 1999: 951) distinguished between ordinal and cardinal numbers, which he labeled as Cardinalzahlen and Ordnungszahlen. Importantly, he argued that ordinals are more fundamental than cardinals, and he did so in a manner more explicit than had Helmholtz: “The ordinal numbers are the natural point of departure for the development of the concept of number” (949). For Kronecker, the cardinal number n associated with a collection of objects is just the n-th ordinal number reached by counting the objects (951).

5.5. Dedekind The appearance of these essays by Helmholtz and Kroneker induced Richard Dedekind, another prominent German mathematician, to publish his own theory of numbers. In his 1888 book, What Are the Numbers and What are They For?, Dedekind ([1888] 1999: 790) acknowledged that his number theory was in many respect similar to those of Schröder, Helmholtz and Kroneker. In particular, he endorsed their distinction between ordinal numbers (which he called Ordinalzahlen) and cardinal numbers (Cardinalzahlen), and repeated the view of Helmholtz and Kroneker that cardinals are more primitive than ordinals (809, 831). However, contrary to Helmholtz and Kroneker, who grounded ordinals on the internal intuition of time or the counting process, Dedekind provided a logical foundation of ordinals based on set theory.

5.6. Cantor Not everyone agreed with the idea that ordinals are more fundamental than cardinals. Already in 1887, Georg Cantor, another major German mathematician and creator of the theory of transfinite sets, had attacked Helmholtz and Kronecker on this point: “While from my standpoint the words for ordinal numbers [Ordinalzahlwörter] appear as the last and most unessential things in the scientific theory of numbers, they have been taken in two recently published works [i.e. by Helmholtz [1887] 1999 and Kronecker [1887] 1999] as the starting point for the development of the concept of number” (Cantor [1887] 1932: 382). Cantor’s critique was mainly directed against the “extreme empiristic and psychological standpoint” on number theory maintained by Helmholtz and Kronecker (382), but his position was also related to the fact that, since the mid-

16

1880s, he had begun to use the words Kardinalzahl and Ordnungszahl within his transfinite set theory in a sense different from that intended by Schröder, Du Bois, Helmholtz or Kronecker. In Cantor’s theory, a cardinal number characterizes a family of finite or infinite sets whose elements can be put into a one-to-one correspondence, while ordinal numbers characterize well-ordered sets whose elements can be put into a one-to-one, order-preserving correspondence.20 In economics Cantor’s terminology has had no significant influence, but in mathematics the terms cardinal and ordinal have generally been used according to the sense established by Cantor.

5.7. Poincaré Outside Germany, the notion of measurement was discussed by, among others, Henry Poincaré, a prominent French mathematician, physicist, and philosopher of science. In an article on the mathematical continuum published in 1893, Poincaré (1893: 26-27) identified order as the basic property of the elements belonging to the continuum, but pointed out that order alone is insufficient to make continuum magnitudes measurable. A first condition of attaining measurability is the transitivity of the equality relation between magnitudes, which means that if magnitude A equals magnitude B, and magnitude B equals magnitude C, then magnitude A must equal magnitude C. Poincaré entered into the debate on the measurability of sensations and argued that they do not satisfy this transitivity condition: “It has been observed, for instance, that a weight A of 10 grams and a weight B of 11 grams produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grams, but that the weight A was readily distinguished from the weight C. Thus […] A=B, B=C, A<C.” (29) The intransitivity of the equality relations for sensations prevents their measurement.

As a second condition for the measurability of magnitudes, Poincaré mentioned their divisibility into n equal parts. He did not go into the subject, however, simply referring approvingly to Helmholtz’s 1887 article for a thorough treatment of the conditions required for the measurability of magnitudes. Poincaré repeated almost literally this analysis of measurable magnitude in the second chapter of his book La science et l’hypothèse (1902).

His reference to Helmholtz indicates that Poincaré, like the German scientist, understood measurement in the classical sense. Poincaré’s classical understanding of measurement is confirmed by an exchange of letters with Walras in September 1901 concerning the measurability of utility. The correspondence between Poincaré and Walras, which is analyzed in Section 9.4, is one of the few channels through which some hints of the mathematical discussions of measurement reached utility theory. However, and anticipating the discussion below, it can be stated here that Poincaré’s views did not influence Walras’s approach to utility measurement.

5.8. Voigt In the same year in which Poincaré’s essay on the mathematical continuum appeared, the German mathematician and economist Andreas Voigt (1893) published an article in the political-economy journal Zeitschrift für die Gesamte Staatswissenschaft entitled “Number and Measure in

20 Well-ordered sets are sets endowed with a complete and transitive order relation that also have a least element

under that relation.

17

Economics”.21 Since Voigt referred to some of the ideas on number and measurement that had emerged in the mathematical discussions of the 1880s, it seems appropriate to discuss his contribution here. Torsten Schmidt and Christian Weber (2008, 2009, 2010) have provided an extensive account of the background and content of Voigt’s article, and this allows us to deal with it succinctly and focus only on its measurement-theoretic aspects.

Voigt mentioned the distinction between ordinal and cardinal numbers as presented by Helmholtz, Kroneker and Dedekind, and endorsed their view that ordinals come first: “In accordance with the fundamental conceptions of the nature of numbers which mathematics has developed in recent times, we see the primary form of the number concept in the ordinal number [Ordnungszahl] and not in the cardinal number [Kardinalzahl]” (Voigt [1893] 2008: 502). As Schmidt and Weber (2008: 498) have noted, this passage would seem to mark the first appearance of the terms “ordinal” and “cardinal” in an economics paper. As we will see shortly, however, Voigt’s notion of “cardinal” was not the current one associated with interval-scale measurement and, as was the mathematical literature he referred to, in fact related to classical measurement.

Voigt suggested that ordering may be considered as a rudimentary form of measurement, and offered two examples taken from the physical sciences, namely the Mohs scale of mineral hardness and the measurement of temperature. In both cases, the numbers assigned to the objects express only order relationships and not ratio relationships among the objects themselves: “The numbers [in the Mohs] scale indicate only that a stone is harder than another, but they do not indicate the ratio [Verhältnis] of hardness, in the sense that a stone of hardness 4 would be twice as hard as one of hardness 2. […] The degrees on the thermometer do not indicate the ratio of temperatures” (Voigt [1893] 2008: 503).

Other examples of ordering as a primitive form of measurement can be found in what Voigt considered as non-physical sciences such as psychophysics and economics. Voigt apparently held some skepticism as to Fechner’s measurement of sensations, for he argued that psychophysical measurement consists basically of “a subjective ordering of sensations according to their intensity.” (503) For Voigt, the measurement of utility in economics is analogous to the measurement of sensations in psychophysics, in the sense that subjects are capable of ranking pleasures as they rank sensations. Voigt concluded that, if we keep in mind that ordering is just a rudimentary and imperfect form of measurement that cannot rely on the availability of units and does not allow for ratio assessments then it is legitimate to speak of utility as a magnitude:

So long as one treats the ordinal numbers assigned [to utilities] only as such and does not attribute to them the meaning of ratio numbers [Verhältniszahlen] and does not speak of a utility twice […] times as large, and […] so long as one does not attempt to introduce units of utility […] whose existence presumes such ratios, there are no grounds on which to object to the use of the term magnitude [Grösse]. (503-504)

The above quote shows that, his reference to ordinal numbers notwithstanding, Voigt still thought of measurement in the classical sense. For him true measurement remains an act based on units and allowing for ratio assessments. Ordering provides merely an extremely imperfect form of

21 In 1986 the journal changed its name to the Journal of Institutional and Theoretical Economics.

18

evaluation, which may be labeled as measurement only under a number of qualifications. In the rest of his article Voigt neither developed further his ideas on measurement, nor addressed the difficulties involved in considering utility as a magnitude not measurable in the classical sense. In effect, in the analysis of the equilibrium of bilateral exchange presented in Sections 3 and 4 of his essay, Voigt introduced assumptions implying the classical measurability of utility.

Neither Menger nor Walras referred to Voigt’s contribution (Jevons had died in 1882), and among the other important early marginalists only Francis Y. Edgeworth seems to have noticed it.22 Between 1894 and 1910, Edgeworth (1894, 1900, 1907) referred to Voigt’s distinction between cardinal and ordinal numbers three times, though always in a cursory way. Moreover Edgeworth associated cardinal numbers with the availability of a unit and hence with classical measurement, and did so without mentioning utility differences. In fact, the present meanings of the words “cardinal” and “ordinal” settled in economics only later, around the mid 1930s, and the expression “cardinal utility” came to be coupled with the ranking of utility differences only in the late 1930s.23

5.9. Hölder In 1901, Otto Hölder, professor of mathematics at Leipzig University, published an article on “The Axioms of Quantity and the Theory of Measurement”. In his essay, Hölder made use of the axiomatic approach to geometry launched two years previously by David Hilbert (1899) in order to refine Helmholtz’s theory of measurable magnitudes: “The theory of measurable magnitudes […] has been recently treated in depth from a number of different points of views. Nevertheless, it seems that the theory has not been treated exhaustively […]. As with geometry […] the theory of measurable magnitudes can be based upon a set of […] ‘axioms of magnitude’” (Hölder ([1901] 1996: 237-238). Hölder laid down seven axioms (238) and rigorously proved that, for magnitudes satisfying them, a unique number is associated with the ratio between any two magnitudes a and b: this number is the measure of a in terms of b and “b is called the unit” (242).24 This very result makes clear that Hölder also understood measurement in the classical sense and that the measurability warranted by his axioms was the classical one. The main problem with Hölder’s axioms is that they make it extremely difficult, if not impossible, to check whether a given kind of magnitude satisfies them.25 This limitation may explain at least in part why Hölder’s article went unnoticed until the 1930s.

5.10. Russell As we have emphasized more than once, all contributions to the theory of measurement considered so far remained within the classical understanding of measurement. Bertrand Russell’s 1903 Principles of Mathematics contains our first formulation of the representational view of measurement.

22 For the time span 1893-1910, the JSTOR database contains no references to Voigt’s article apart from those made

by Edgeworth. In the literature in German, Voigt’s article was apparently referred to only by Čuhel 1907. As Schmidt and Weber 2009 show, only after 1920 do citations became somewhat more numerous.

23 On Edgeworth’s understanding of measurement, see Moscati 2010; on the coupling of cardinal utility with the ranking of utility differences in the late 1930s, see Moscati 2008.

24 For a discussion of Hölder’s axioms, see Michell 1999: 47-59. 25 For a discussion of the difficulties in testing Hölder’s axioms, see Nagel 1931.

19

Since at least 1897, Russell had been arguing that quantities and numbers, rather than being – as supposed by the classical view – strictly interconnected, constitute two disconnected and independent structures (Russell 1897). Based on this understanding of numbers as an autonomous system of entities, Russell (1903: 158) in Part Three of the Principles of Mathematics defined measurement in the broadest sense as “the correlation with numbers of entities which are not numbers.” The entities typically correlated with numbers are magnitudes, which Russell defined as “anything which is greater or less than something else” (159). Notably, he referred to pleasure as a typical example of a magnitude. Russell added that, as greater and less establish an order between magnitudes, so the numbers correlated to the magnitudes should mirror this order:

Measurement of magnitudes is, in its most general sense, any method by which a unique and reciprocal correspondence is established between all or some of the magnitudes of a kind and all or some of the numbers […]. It will be desirable that the order of the magnitudes measured should correspond to that of the numbers. (176)

This sentence seems to express a representational understanding of measurement, in which ordering is considered as a specific but true form of measurement. However, Russell made very little of it. He immediately passed on from measurement in the broadest sense as something “vague” (177) and focused instead on a “more natural and restricted sense” (180) of measurement that he called “numerical measurement”.

Numerical measurement requires that “we may say that one magnitude is double of another” (177), and thus in arriving at it Russell had returned to classical measurement. Following Du Bois and Helmholtz, but without referring to Hölder, whom he did not to know, Russell put forward some conditions that he supposed would warrant the numerical measurement of magnitudes (181). Unlike Hölder, however, Russell did not provide any rigorous proof that his conditions would suffice to generate numerical, i.e. classical, measurement.

In Russell’s next important book on mathematics, namely the Principia Mathematica written in collaboration with Alfred North Whitehead and published between 1910 and 1913, measurement “in its most general sense” disappeared and measuring was understood classically as the assessment of ratios (Whitehead and Russell 1913: 407-457).

5.11. Measurement in mathematics: summing up This section has shown how the cardinal-ordinal terminology was introduced into mathematics in the 1870s, and how during the debates of the 1880s the distinction between cardinal and ordinal numbers became standard in mathematical discourse. Through the Voigt-Edgeworth channel the cardinal-ordinal terminology made its first albeit inconsequential appearance in economics as early as 1893-1894. However, for mathematicians as well as for Voigt and Edgeworth the word “cardinal” had nothing to do with interval-scale measurement in the current sense of the term. Rather, it was related to the decomposition of a number in units and with the possibility of measuring magnitude in the classical way. More generally, with the ephemeral exception of Russell, between 1870 and 1910 all mathematicians defined measurement in accordance with the classical view. Thus, even if Jevons, Menger and Walras had looked to mathematics to address the problem of utility measurement (which basically they did not) they would have found classical, unit-based measurement rather than interval scales.

20

6. Measurement in economics, before and beyond marginal utility theory Undoubtedly, within economics the issue of measurement is by no means confined only to utility theory. At least since Sir William Petty’s Political Arithmetick (1690), economists have attempted to measure many things aside from utility, e.g. the national income, the price level, the quantity and velocity of money or features of the business cycle.26 However, economic measurements unrelated to utility tend to fit the classical understanding of measurement because some one or other straightforward unit of measure always appears to be available for them. Consequently, these “non-related-to-utility” economic measurements never motivated economists to question the very notion of measurement: aside from utility theory, the issue was always “how to measure?” rather than “what is measurement?” In the case of utility, by contrast, the fact that there exists no obvious unit of measure, and also no straightforward way to assess utility ratios, created a tension between utility measurement on the one hand, and the classical understanding of measurement on the other, which eventually lead to the latter being superseded.

Apart from utility theory, the area of economics in which measurement issues were discussed more thoroughly was probably the theory of exchange value. Classical economists such as David Ricardo, as well as Menger, Walras and other marginaliststs addressed the problem of measuring the exchange value of commodities in terms of money or some other commodity taken as a unit. Generally speaking, the problem these authors faced was that the unit available to measure exchange value was not fixed and invariable as are, generally, the units used to measure physical magnitudes.27 Here we only stress that in discussing the measurement of exchange value, both classical and marginalist economists remained completely within the classical understanding of measurement: for them, measuring exchange value consisted of assessing the ratio, which in this case was an exchange ratio, between an object and another object taken as the unit.

Even scholars who addressed the issue of measurability of pleasure or utility prior to the emergence of marginal utility theory understood the measurement of these feelings in the classical sense. In the late 18th century Jeremy Bentham, the founder of Utilitarianism, claimed that pleasures can be measured either directly, by taking as a unit “that pleasure which is the faintest of any that can be distinguished to be pleasure” (quoted in Halevy 1901: 398), or indirectly through money, whereby a pleasure is measured by the quantity of money paid to obtain it. Both of Bentham’s measures of pleasure, regardless of any possible flaws, in principle allow for the measurement of utility in the classical sense: in both cases there is a unit, i.e. the “faintest pleasure” or the monetary unit, that makes the assessment of utility ratios possible.28

Jules Dupuit (1844) took, as the measure of the utility of an object, the maximum price a consumer is willing to pay for it. In spite of all the defects it may have, Dupuit’s measure of utility also allows in principle for the classical measurement of utility: there is a definite unit, i.e. the monetary unit, and utility ratios between commodities can be easily assessed by comparing the maximum prices the consumer is willing to pay for them.

Heinrich Gossen (1854) also conceived measurement in the classical sense and deemed that

26 On economic measurement beyond utility theory, see Klein and Morgan 2001, Boumans 2005 and 2007. 27 On Ricardo’s search for an invariable measure of value, see Sraffa 1951. On Menger’s and Walras’s discussion

about the measurement of exchange value, see Sections 8 and 9 below. 28 For more on Bentham’s measures of pleasure, see Mitchell 1918.

21

pleasures are measurable. For him, any pleasure could be taken as a unit to measure any other pleasures:

We can conceive of the magnitudes of various pleasures only by comparing them with one another, as, indeed, we must also do in measuring other objects. We can measure the magnitudes of various areas only by taking a particular area as the unit of measurement […]. Similarly, we must fix on one pleasure as our unit […]. It is a matter of indifference which pleasure we choose as the unit. (quoted in Stigler 1950: 315)

This extensive review of the notion of measurement in various disciplines – philosophy, physics, psychology, mathematics, (“non-related-to-utility”) economics – has shown that the classical understanding of measurement was unrivalled and dominated in all of them since ancient times, or at least since their earliest origins. Given this background, the fact that Jevons, Menger and Walras conceived of measurement in the classical fashion and applied this classical view to the measurement of utility should hardly appear surprising.

7. Jevons William Stanley Jevons (1835-1882) was, as Harro Maas (2005, 26) has called him, a “Victorian polymath,” who contributed to disciplines as diverse as formal logic, meteorology, political economy, statistics, and the philosophy of scientific method. As a philosopher of science, Jevons (1874, vol. I: 313-14) assigned great importance to the exact measurement of phenomena, and indeed much of his work as a scientist was related to measurement. In particular, in the 1860s Jevons attempted to measure the wear-rate of gold coins in the United Kingdom, the variation in the value of domestic gold as a consequence of the influx of gold from Australia and the United States, and the annual growth rate of coal consumption in the country. In the 1870s, he turned to the measurement of business cycles and their supposed relationships to the length of sunspots. In these enterprises in empirical measurement a straightforward unit of measure was available – be it a weight unit, a monetary unit, or a temporal unit, and thus Jevons’s efforts fitted naturally into the classical approach to measurement. In dealing with utility in The Theory of Political Economy (first edition 1871; second edition 1879), Jevons faced the problem of the apparent lack of an appropriate unit of measurement, but this difficulty did not lead him to modify his classical understanding of measurement.29

7.1. Utility measurement and economic calculus Inspired by Bentham, Jevons (1871: vii) in the Preface of his Theory famously defined economics “as a Calculus of Pleasure and Pain”. Then, in Chapter One of the book, he discussed a number of possible criticisms of this definition, in particular the objection that such a calculus is unfeasible because pleasure and pain cannot be measured (8-9, 18-19). Jevons countered this objection with three arguments, each of which reveals his classical understanding of measurement.

In the first argument, Jevons acknowledged that, at the moment at which he wrote, measuring pleasure and pain was impossible because no unit had been found: “We cannot weigh, or gauge, 29 For more on Jevons’s enterprises in empirical measurements, see Peart 1996, chs. 2-3 and 10, and Peart 2001;

Peart 1995 focuses rather on Jevons and utility measurement.

22

or test the feelings of the mind; there is no unit of […] suffering, or enjoyment” (9). He even doubted that such a unit could be conceived:

We have no means of defining and measuring quantities of feeling, like we can measure a mile, or a right angle, or any other physical quantity. […] We can hardly form the conception of a unit of pleasure or pain, so that the numerical expression of quantities of feeling seems to be out of the question. (19)

Jevons went on to note, however, that several other entities, such as probability, electricity or heat, had for a long time appeared impervious to measurement but had recently become amenable to exact measurement (9-11). Jevons speculated that the same could happen in the future to pleasure and pain: “In matters of this kind, those who despair are almost invariably those who have never tried to succeed” (9). Jevons’s reference to distance, angles, probability, electricity and heat (all of which were measurable in a classical sense at the time he wrote) as instances of successful measurement, as well as his insistence on the inextricable connection between the possibility of measurement and the availability of a unit, show that he conceived measurement in a classical fashion.

As a second argument, Jevons claimed that his theory was largely independent of the measurability of utility because it mainly relied upon the comparison of different pleasures and “seldom or never affirm[s] that one pleasure is a multiple of another in quantity.” (20) Insofar as the mind of an individual is able to compare directly different pleasures, units to measure them are unnecessary: “We only employ units of measurement […] to facilitate the comparison of quantities; and if we can compare the quantities directly, we do not need the units. Now the mind of an individual is the balance which makes its own comparisons.” (19) In arguing that the economic calculus is independent of the measurability of pleasure, Jevons reiterates, this time by negation, his classical understanding of measurement: utility measurement, if it were possible and could be employed in economic calculus, would consist of assessing how many times a pleasure is a multiple of another taken as a unit.

In his third argument, Jevons argued that, although not directly measurable, pleasure and pain can be measured through their indirect but observable effects. In this respect, he claimed, feelings are analogous to gravity, for both are unobservable but can be measured through their effects on, respectively, market prices and physical bodies:

We can no more […] measure gravity in its own nature than we can measure a feeling, but just as we measure gravity by its effects in the motion of a pendulum, so we may estimate […] feelings by the varying decisions of the human mind. The will is our pendulum, and its oscillations are minutely registered in all the price lists of the market. (14; see also p. 140)

As with most physical entities, gravity is measurable in the classical way, and the fact that in his third argument Jevons equated the measurement of feelings to the measurement of gravity confirms that for him measurement meant classical measurement.

To summarize, all three of the arguments Jevons put forward in the Theory – against the objection that the economic calculus is unfeasible because pleasure is immeasurable – illustrate that he conceived measurement as fundamentally associated with the availability of a unit and the

23

assessment of ratios; in short, his conception of measurement was the classical one.

7.2. Utility measurement and decreasing marginal utility As with the other early marginalists, Jevons (1871: 61-68) assumed that the marginal utility of each commodity, or, to be more precise and using his terminology, “the final degree of utility”, is eventually decreasing. Jevons’s economic theory relies significantly on this assumption, but as already noticed in Section 1, this does not imply that he was reasoning in terms of interval-scale measurement of total utility or cardinal utility analysis. In fact, for him, marginal utility, and not total utility, was the key notion “upon which the whole Theory of Economy will be found to turn.” (61-62) Marginal utility determines how commodities are distributed among different uses, the laws of exchange between them, and the offer of labor. Total utility is of secondary economic importance and, generally, it is more difficult to assess than marginal utility (61).

Accordingly, Jevons’s discussion of the measurability of pleasure summarized in Section 7.1 refers primarily to marginal utility, not total utility. It is marginal utility that is, at present, indirectly measured by prices (third argument), that in the future might even be measured in a direct way (first argument), and whose measurement, in the end, is not essential to the theory of political economy (second argument). Thus Jevons thought in terms of a classical, unit-based measurement of marginal utility, not a modern, interval-scale measurement of total utility.

7.3. The theory of measurement in The Principles of Science Jevons’s views can be confirmed if we turn to the way he dealt with measurement issues in his main work on scientific method, The Principles of Science (1874).30 Chapters thirteen and fourteen were devoted to measurement theory, significantly titled, respectively, “The exact measurement of phenomena,” and “Units and standards of measurements”. In these chapters Jevons reaffirmed the classical view according to which measurement consists of assessing the numerical ratio between a magnitude and another magnitude taken as a unit. Among the many passages in which Jevons presented this idea, the following one is particularly illustrative: “The result of every measurement is to make known the purely numerical ratio existing between the magnitude to be measured, and a certain other magnitude, which should, when possible, be a fixed unit or standard magnitude.” (Jevons 1874, vol. I: 331) Only four pages of the Principles (vol. II: 457-460) were devoted to political economy and the social sciences. What Jevons wrote in these pages about the measurement of pleasure and pain and their measurement adds nothing to the discussion contained in the Theory.

7.4. Utility measurement for Jevons’s critics The Theory of Political Economy received at least ten reviews, published not only in scientific journals but also in newspapers.31 Anticipating many of the arguments which were later and independently put forward by Fechner’s critics, a number of reviewers argued that Jevons’s analysis relied on the assumption that utility is measurable but that he was unable to substantiate

30 On the relationship between the Theory of Political Economy and The Principles of Science, see Schabas 1990,

chs. 3-5. 31 On the reviews of Jevons’s Theory, see Howey 1960, ch. 10.

24

this assumption. These critical claims, it should be noted, show that also Jevons’s critics adhered to the classical understanding of measurement.

The anonymous author of the review that appeared in the Saturday Review of 11 November 1871 argued that Jevons’s mathematical treatment of utility presupposed the measurability of utility in terms of a unit and the capacity of assessing utility ratios. Jevons, however, had not established the presuppositions of his analysis:

When a mathematician wishes to calculate the variations of a force, he begins by telling us distinctly what is the measure of force. […] But what is the measure of utility? To this we can discover no answer in Mr. Jevons’s book. […] We must suppose that a unit of pleasure is somehow discoverable. […] We must be able to say, for example, that the pleasure of eating a beefsteak is to the pleasure of drinking a glass of beer as 5 to 4. The words convey no particular meaning to us; and Mr. Jevons, instead of helping us, seem to shirk the question. (Anonymous [1871] 1981: 153-54)

Another anonymous reviewer, whose comments appeared in the Galsgow Daily Herald on 16 December 1871, made a similar point. The reviewer held that Jevons’s analysis required the possession of “some unit of pleasure […] by which [pleasures] might be measured and reckoned”, but, insisted the reviewer, such a unit was simply not available (quoted in Howey 1960: 62).

In his review of January 1872, published in the Fortnightly Review, the classical economist John Elliott Cairnes maintained that Jevons’s argument, that utility although not directly measurable may nevertheless be measured indirectly by prices, is tautological:

How are we to measure pleasure? […] We may take exchange-value as the criterion of utility; and this is the test that Mr. Jevons ultimately adopts. […] So that what we come to is this – exchange-value depends upon utility, and utility is measured and can only be known by exchange value. […] I own it seems to me to come so close to an identical proposition, that I am unable to distinguish it from that species of announcement. (Cairnes [1872] 1981: 150)

In a letter to Cairnes dated 14 January 1872, Jevons replied to this criticism by repeating a point already made in the Theory and noted above, namely that the method of measuring utility is analogous to that used to measure gravity, and that insofar as the latter is appropriate, so too is the former: “There is no means of measuring pleasure & pain directly, but as those feelings govern sales and purchases, the prices of the market are those facts from which one may argue back to the intensity of the pleasure concerned. Even gravity is known only in its effects” (Jevons 1977: 246). Apparently, Cairnes was not persuaded for he repeated his accusation that Jevons’s argument was tautological in his Some Leading Principles of Political Economy (Cairnes 1874: 21).32

The accusation that Jevons’s theory depends on a fallacious assumption that utility is measurable was hurled also from historicist quarters. In the lengthy entry on “Political Economy” that appeared in the ninth edition of the Encyclopaedia Britannica, John Kells Ingram argued that Jevons’s conception of marginal utility is “ingenious, but we cannot regard it as either “positive”

32 On Cairnes’s critique of Jevons, see Shabas 1985.

25

or fruitful. […] What is the unit of utility? If we cannot look for something more tangible […] than this, there is not much encouragement to pursue such researches.” (Ingram 1885: 399)

In sum, these critical reviewers and commentators associated the measurability of utility with the availability of a unit and criticized Jevons for failing to provide such a unit. Thus their criticisms drew upon their classical understanding of measurement, which they in fact shared with Jevons. Furthermore, in the criticisms there is no reference to rankings of utility differences.

Judging from the second edition of The Theory of Political Economy (1879), the critical remarks of Cairnes and the other reviewers had no effect on Jevons’s ideas concerning the measurability of utility. The opinions he expressed on this subject in the two editions of the book are the same, and all passages from the first edition quoted in the previous pages were repeated with only very slight adjustments in the second. We may conclude that, in passing from the first to the second edition of the Theory, Jevons’s understanding of utility measurement remained the classical one.

7.5. Jevons, psychophysics and mathematics Although there are similarities between Jevons’s approach to the measurement of utility and Fechner’s approach to the measurement of sensations, as well as between the criticisms Jevons and Fechner received, the research of the English economist seems unrelated to that of the German psychologist. Neither the Theory of Political Economy, nor the Principles of Science, nor Jevons’s papers and correspondence collected by R.D. Collison Black (1972-1981), contain any reference to Fechner. Jevons was a friend of the psychologist James Sully, who had studied psychology in Germany and played an important role in introducing psychophysics into England in the second half of the 1870s.33 This circumstance suggests that Jevons was probably aware of Fechner’s research. Even if this was the case, however, Jevons does not appear to have been influenced by Fechner’s work nor by the early discussions of it.

Later psychological discussion concerning the measurement of sensation or mathematical discussion of number and measurement could not influence Jevons, for he died in 1882.

8. Menger The appraisal of Menger’s views on utility measurement is trickier than those of Jevons because the Austrian economist avoided taking an explicit stance on the issue. For Menger we thus need to take a roundabout path and start from his discussion of money as a possible measure of the exchange value of goods. This discussion is useful for our purposes since it shows that Menger’s general understanding of measurement was definitely the classical one. From here we pass to Menger’s Principles of Economics and argue that the way that he used numbers to express the marginal utility of goods indicates that he applied the classical understanding also to utility measurement.34

33 On Sully see Boring 1957, ch. 20. Sully was also a friend of Edgeworth and played an important role in

generating Edgeworth’s interest in psychophysics. 34 Menger’s elusiveness on utility measurement has been noticed also by Stigler (1950: 317) and Howey (1960: 46-

47). In the following, references will be made to the first edition of the Principles and not to the second, which

26

8.1. The measurement of exchange value Menger (1840-1921) discussed the issue of whether money measures the exchange value of goods in chapter 8 of his Principles (Menger [1871] 1981: 272-280), in his article “Money as Measure of Value” (Menger [1892] 2005), and in the three subsequent versions of the entry on “Money” published in the Handwörterbuch der Staatswissenschaften (Menger 1892, 1900, [1909] 2002). The most thorough discussion of the issue can be found in the final, 80-pages long version of “Money”, but the ideas expressed here are substantially identical to those that Menger had presented in the Principles almost forty years earlier.35

For Menger, the precise question at issue was whether “the valuation of goods in money [should] be regarded as measurement [Messung] of their exchange value by the monetary unit.” (Menger 2002 [1909]: 60) In order to answer this question, Menger first of all made clear that by measurement he meant “a procedure by which we determine the as yet unknown magnitude of an object by comparison with a known magnitude of the same kind taken as a unit.” (Menger 1900: 84-85, and [1909] 2002: 60) This is no other than the classical notion of measurement and Menger claimed that, by its lights, money does not measure the exchange value of goods. In fact the exchange ratios between money and goods, i.e. prices, vary from time to time and from place to place, and it is not only absolute, but also relative prices that vary. Moreover, the variation in exchange ratios may be due, not only to modifications in the exchange value of goods, but also to modifications in the exchange value of money: unlike the units used in physical measurement that are fixed and invariable, the monetary unit changes. Nevertheless, and although for Menger money does not measure exchange value, it does provide “estimates” (Schätzungen) or “valuations” (Bewertungs) of exchange value that are useful for many practical purposes (Menger [1871] 1981: 276, and [1909] 2002: 61, respectively).

Menger’s definition of measurement, his rejection of the idea that money measures the exchange value of goods, and also the fact that he contrasted measurement against estimates and valuations, show that his general understanding of measurement was the classical one. We turn now to the question of whether Menger applied this understanding to utility measurement.

8.2. From the measurement of value to the measurement of marginal utility Menger devoted chapter 3 of his Principles to the theory of value. By “value” he meant the importance of a commodity for an individual, and he argued that this value could be measured. This claim was made explicitly, albeit cursorily, in the first footnote of the chapter in which he affirmed that “value is a magnitude that can be measured” (Menger [1871] 1981: 293). The same opinion is implicit in the fact that the second section of chapter 3 is entitled “The original measure [Mass] of commodity value.” (121)36 In particular, for Menger the value to an individual of a given quantity of a good is measured by the importance of the need-satisfaction (Bedürfnissbefriedigung) assured by the last unit of the good, that is, by what today we would

was posthumously published by Menger’s son (Menger 1923). On the limitations of the second edition of the book see Wicksell [1924] 1968.

35 On money as measure of exchange value in Menger’s thought see also Roll 1936, Streissler 2002, and Campagnolo 2005.

36 Menger’s stance on value measurement in the Principles therefore appears different from the one he expressed in a notebook from the years 1867-68, which contains material preparatory to his main work. In this notebook Menger wrote, “value can be merely estimated, not measured” (quoted in Yagi 1993: 706).

27

call the marginal utility of that unit (132). This assertion shifts the issue of measurement from the measurement of value to the measurement of marginal utility.

8.3. Menger’s utility numbers In investigating the value of consumption goods and productive factors, as well as in analyzing how bilateral exchange works, Menger expressed the marginal utility of goods by numbers and assumed that marginal utility is decreasing. For instance, he imagined that for a particular individual the first unit of food has a marginal utility of 10, the second unit has a marginal utility of 9, the units following have a marginal utility of 8, 7, 6, etc., respectively, while the eleventh unit has zero marginal utility (125-127).

During the Ordinal Revolution of the 1930s, Friedrich Hayek (1934) and George Stigler (1937) rushed to enroll Menger in the ordinal camp, claiming that his utility figures represent only the ranking of the importance of need-satisfactions. However, the ordinalist interpretation of Menger clashes with the fact that his utility numbers express decreasing marginal utility which, as was mentioned in Section 1 above, entails cardinal utility.37 But, as a matter of fact, Menger’s utility figures entail even more than cardinal utility. For he assumed that there is a zero point of marginal utility (Menger [1871] 1981: 126-7, 135, 183-6), and on two occasions also claimed that his utility figures express the ratio of marginal utilities. In a footnote to chapter 4 of the Principles, which is devoted to the theory of exchange, Menger in fact specified: “When I designate the importance of two need-satisfactions with 40 and 20 for example, I am merely saying that the first of the two satisfactions has twice the importance of the second to the economizing individual concerned.” (183) Shortly thereafter, Menger remarked that if the marginal utility of a cow to an individual is 10, while the marginal utility of an additional horse is 30, then the horse has “three times the value of a cow” (184). But all of this entails that Menger’s utility numbers measure marginal utility in the classical sense in terms of some (unspecified) unit of satisfaction.38 In the handwritten notes to his copy of the Principles as transcribed by Emil Kauder (1961: 200-203), Menger made no comment on the two above-quoted passages. Furthermore, in the second, posthumous edition of the Principles edited by Menger’s son, the passages remained unchanged (Menger 1923: 176-177).

Thus, Menger’s elusiveness notwithstanding, we can conclude that he conceived of utility measurement in the classical fashion.

8.4. Menger, psychology and mathematics In Hayek’s edition of Menger’s collected works (Menger 1934-36), and in the second edition of the Principles (which is not included in the collected works), there is no trace, allusion to, or hint of the discussions on measurement taking place in psychology and mathematics in the period 1870-1910. This fact strongly suggests that, even if Menger was aware of those discussions, they did not influence him.

37 In later writings, Stigler (1941, ch. 6; 1950) ceased to consider Menger as an ordinalist. 38 This is also the interpretation of the J. Dingwall and B.F. Hoselitz, the English translators of Menger’s Principles;

see Menger [1871] 1981: 126-27, footnote 8.

28

9. Walras Walras addressed the problem of measurability of utility on various occasions throughout his scientific career, both in published works and in correspondence. His stance on the issue as well as his understanding of utility measurement did not change over the years. In particular, Walras consistently maintained a classical understanding of measurement and never associated utility measurement with the ranking of utility differences.

9.1. As-if measurement of utility at the Académie In two meetings of the Académie des Sciences Morales et Politiques held in Paris on 16 and 23 August 1873, Walras read a paper entitled “Principe d’une théorie mathématique de l’échange” (“Principle of a Mathematical Theory of Exchange”, henceforth “Principle”) in which he outlined the theory of utility and exchange later expounded in the Elements of Pure Economics. The paper was subsequently published in the Journal des Économistes (Walras [1874] 1987).

In Section 5 of this paper, Walras (273-277) discussed how to derive the demand curves of commodities from their “intensive utility”, i.e. from their marginal utility.39 In this context, he addressed the issue of measurability of utility and assumed the stance that he was to maintain for the next forty years on the subject: although utility is not directly measurable, we can nevertheless derive from it demand curves and thus carry out a suitable analysis of demand by treating marginal utility as if it were measurable. It is to be emphasized that Walras was concerned here with the measurement of intensive or marginal utility, not of total utility. Moreover, he associated the measurability of marginal utility, however fictitious, with the availability of a unit, not to the ranking of utility differences:

Let us suppose, for a moment, that utility is directly measurable, and we shall find ourselves in a position to give an exact, mathematical account of the influence it exerts […] on demand curves and hence on prices. I shall, therefore, assume the existence of a standard of measure [étalon de mesure] of intensive utility. (274)

Walras briefly justified this as-if procedure by claiming that it is the same as that employed in physics and mechanics, where masses enter into scientific calculations despite the fact that they are not directly measurable (274).

Walras’s paper was not well-received by his audience at the Académie, and one of the objections raised against his theory of exchange bore upon the measurability of utility. Emile Levasseur, a professor of economic history, geography and statistics at Collège de France, argued that Walras’s analysis was misleading because it relied on the false assumption that utility is measurable: “It would be very good if desire and need were measurable in an exact way […] but it is not like that, quite the opposite. […] [Walras’s] data are, so to say, incommensurable; it follows that his [intensive utility] curves are without foundation; […] they are false.” (Levasseur [1874] 1987: 530)40 Levasseur denied that intensive utility is measurable, but he did not object to Walras’s classical association of measurement with the availability of a unit. Thus, and as was 39 Walras’s “rareté” is the intensive or marginal utility of the last, infinitesimal part of a given quantity of a

commodity, and coincides with Jevons’s “final degree of utility”. 40 For more on the objections to Walras’s communication raised by Levasseur and other members of the Académie,

see Howey 1960, ch. 23, and Ingrao and Israel 1990, chs. 4 and 6.

29

also the case with Jevons’s critics, Levasseur shared with his opponent the same, classical understanding of measurement.

9.2. As-if measurement of utility in the Elements Levasseur’s objection did not modify Walras’s stance on the measurability of utility. In Lesson 14 of the Elements he basically repeated the argument already made in the “Principle” the previous year: marginal utility is not measurable, but by assuming that it is, i.e., by assuming the existence of a unit, we can analyze individual demand:

It seems impossible, at first glance, to pursue it [the analysis of demand] further, because intensive utility […] has no […] measurable relationship to space or time […]. Still, this difficulty is not insurmountable. Let us assume that such a […] relationship does exist, and we shall find ourselves in a position to give an exact, mathematical account of the [influences] on prices of […] intensive utility. I shall, therefore, assume the existence of a standard of measure of intensive utility. (Walras [1874] 1954: 117)

The main difference between the argument put forward in the “Principle” and that of the Elements is that in the latter Walras did not attempt to justify his as-if approach to utility measurement by claiming that it paralleled the one employed in physics and mechanics. Possibly, Levasseur’s objections suggested to Walras that this claim left him open to further attacks from his critics and that it would be safer to remove it.

The as-if-utility-were-measurable case presented in the above-quoted passage remained in identical form in the following three editions of the Elements published, respectively, in 1889, 1896, and 1900 (Walras 1988: 105-107). The argument was also repeated in an article on the application of mathematics to pure economics published in Italian in the Giornale degli Economisti (Walras [1876] 1987), in Appendix I added to the third edition of the Elements (Walras 1988: 694), in his book on the Théorie de la monnaie (Walras 1886: 30), and in a number of letters Walras sent to colleagues between 1873 and 1900 (Jaffe 1965, letters 232, 268, 789).

The question of whether or not Walras’s as-if argument for utility measurement is sound does not fall within the scope of the present paper. What is important to stress, given our present aims, is that Walras’s association of the measurement of marginal utility with the availability of a unit shows that he conceived of measurement in the classical fashion. This interpretation is strengthened by considering Walras’s discussion of the measurement of exchange value in Lesson 25 of the Elements.

9.3. The measurement of exchange value in the Elements Lesson 25 of the Elements is entitled “Choosing an instrument of measure and a medium of exchange.” Here Walras addresses a question analogous to that discussed by Menger in chapter 8 of his Principles of Economics and later in his entry on “Money,” namely the question of how to measure the exchange value of commodities.

30

In Lesson 25 Walras examines and rebuts the popular opinion according to which exchange value can be measured by the monetary unit, e.g. the franc, just as the meter can measure length. Walras ([1874] 1954: 186-187) pointed out, as a first objection against the analogy between the franc and the meter as measuring rods, that there is an essential difference between the two: “since the meter is a fixed and invariable unit of length, whereas the franc is a unit of value which, far from being fixed and invariable, differs from place to place and varies from one moment to the next.” For Walras, this objection is correct but not decisive, since one can still reply to it that, at least at a given place and at a given moment, the value of the franc is fixed.

The second and decisive objection against the analogy between exchange value and length, Walras continued, is that, contrary to the measurement of length, the measurement of exchange value does not consist in the assessment of the numerical ratio between a given unit and a given object. “When I measure any given length, for example the length of a façade,” observed Walras, “I have to take three things into account: the length of the façade, the length of a ten-millionth part of a quarter of the earth’s meridian [i.e. the length of the meter] and the ratio of the first length to the second, which is the measure of the façade.” (187) Walras argued that in the case of value measurement, even at a given place and at a given moment, two of the three components of length measurement do not exist: there is neither an intrinsic value of the commodity to be measured, nor an intrinsic value of the franc that can be used as a unit for measurement: “The word franc [as a unit of value] is the name of a thing that does not exist.” (188) For Walras, exchange value is an essentially relative phenomenon, and what exists in assessing exchange value is only the third component of proper measurement, namely the ratio between the franc and the commodity at issue.

According to Walras, this conclusion does not entail that we cannot measure exchange value in a proper sense. All that follows from it is that to measure exchange value we must choose as a unit “a certain quantity of a given commodity [labeled as numéraire], and not the value of this quantity of the given commodity.” (188) The exchange value of a commodity is then measured by the number of units of the numéraire exchanged for one unit of the commodity at issue. Walras’s discussion about how to measure exchange value did not change in the following three editions of the Elements (Walras 1988: 223-226).

Just as with his as-if argument for utility measurement, it is not the purpose of the present paper to examine the potential problems and limitations of Walras’s stance on the measurement of exchange value.41 Again, then, we remain content to simply point out that his discussion of this second topic makes clear that he understood measurement in the classical sense: for Walras the archetypal form of measurement is the measurement of length and, in general, measurement consists of assessing the numerical ratio between the chosen unit and the object to be measured. If we take into account Walras’s general view of measurement, it is not surprising that he associated the measurement of marginal utility with the availability of a unit.

41 For instance, from a Mengerian perspective the switching from money to some other commodity does not solve

the problem of measuring exchange value. For a comparison of Menger’s and Walras’s monetary theories, see Arena and Palermo 2008.

31

9.4. The exchanges with Laurent and Poincaré on utility measurement In 1900, Walras was brought back to the issue of the measurability of utility by the criticisms leveled at him by Hermann Laurent, a distinguished French mathematician. In an exchange of letters with Walras that took place in May 1900 (Jaffe 1965, letters 1452-1456), and in a communication at a meeting of the Institut des Actuaires français held in Paris on 21 June 1900, Laurent objected that Walras’s analysis of demand and exchange was based on the assumption that utility is measurable, which basic assumption Laurent denied: “How can one accept that satisfaction is capable of being measured? Never will a mathematician agree to that.” (quoted in Jaffe 1965, vol. 3: 113) In reply to Laurent’s criticism, Walras stood by his earlier stance that his analysis was in fact independent of any actual measurement of utility: “I skip completely the standard of utility and rareté.” (119)

The issue of measurability of utility cropped up again the following year in another and more famous exchange of letters, this time between Walras and Poincaré.42 As we saw in Section 5.7, Poincaré (1893, 1902) stressed order as the basic property of continuum magnitudes, but also argued that for magnitudes to be measurable further conditions such as those indicated by Helmholtz ([1887] 1999) are necessary.

On 10 September 1901 Walras sent to Poincaré, whom he did not know, a copy of the fourth edition of the Elements, published in 1900. A brief correspondence between the two followed, and in a letter of 26 September 1901 Walras (Jaffé 1965, vol. 3: 161) solicited the opinion of the famous mathematician concerning the way in which marginal utility was dealt with in the Elements, i.e., as something that cannot be measured but may nonetheless be treated as if it were measurable: “The rareté (or intensity of the last want satisfied) […] is not an appreciable magnitude, but it suffices to think of it as such in order to derive [...] the principal laws of political economy.” To support this approach, Walras brought back into play the alleged analogy with physical sciences put forward in the “Principle” and later discharged in Elements, and claimed that his as-if-measurable procedure was analogous to procedures employed in statics: “I open Poinsot’s Statics […]; I see that he defines the mass of a body as ‘the number of molecules composing it’ […]. I notice that, by so doing, he too regards as appreciable a magnitude which is not, given that no one has ever counted the molecules in a body.” (161)43

In his reply of 30 September 1901, Poincaré observed that satisfactions can be ordered but cannot measured, and agreed with Walras that the immeasurability of satisfactions does not preclude their mathematical treatment:

Can satisfaction be measured? I can say that one satisfaction is greater than another […], but I cannot say that the first satisfaction is two or three times greater than the other. […] 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. (162-163)

42 The Walras-Poincaré correspondence can be found in Jaffé 1965, letters 1492, 1494-1498, and has been

analyzed, among others, by Jaffé 1977, Ingrao and Israel 1990, ch. 6, and Mirowski and Cook 1990. The English translations of Walras’s and Poincaré’s letters are from Jaffè 1977 and Ingrao and Israel 1990 (although I have made some minor modifications in order to improve the translations).

43 Louis Poinsot (1777–1859) was a French mathematician and physicist; Walras referred here to Poinsot 1842. On Poinsot’s influence on Walras’s thought, see Jaffé 1965, vol. 3: 148-150.

32

Poincaré pointed out that, as satisfactions can only be ordered, the mathematical function expressing them is not unique, and in particular any increasing transformation of the function represents the same sensations equally well: “You can define satisfaction by any arbitrary function providing the function always increases with an increase in the satisfaction it represents.” (163) Finally, Poincaré warned that the lack of uniqueness of the function representing satisfactions limits the significance of the results obtained by the mathematical treatment of them.

Before examining Walras’s interpretation of Poincaré’s reply, two brief comments on the latter are in order. Poincaré’s remark as to the invariance of the utility function to any increasing transformation recalls the ordinal approach Pareto had already put forward in the “Summary of Some Chapters of a New Treatise of Pure Economics” published in the Giornale degli Economisti in March and June 1900 (Pareto [1900] 2008).44 There is no evidence, however, that in September 1901 Poincaré knew Pareto’s “Summary”. Rather, Poincaré’s remark appears to be related to the German debates about the notion of number and to his own work on the mathematical continuum. More importantly, in order to argue that satisfactions are not susceptible to measurement, Poincaré argued that it is impossible to identify their ratios (“I cannot say that the first satisfaction is two or three times greater than the other”). This argument confirms that, as already argued in Section 5.7, Poincaré maintained a classical understanding of measurement and thus was, at least in this respect, completely in accord with Walras.

In the final letter of the correspondence, dated 3 October 1901, Walras self-servingly interpreted Poincaré’s comments as an unreserved statement of support of his own as-if-measurable treatment of marginal utility: “You gave me great pleasure in explaining to me, with the authority you possess, that I was warranted in representing the individuals’ satisfactions by means of functions which, although arbitrary, constantly increase with the represented satisfactions.” (Jaffé 1965, vol. 3: 167) During the following years, in several letters to colleagues as well as in an autobiographical note of 1909, Walras continued to present Poincaré as a keen supporter of his as-if-measurable treatment of marginal utility and more generally of his mathematical approach to economics.45

In conclusion, the exchanges with Laurent and Poincaré did not modify Walras’s understanding of measurement, nor his stance on the measurability of marginal utility as already expressed in the “Principle” of 1873: marginal utility is not measurable in the classical sense, but by assuming the existence of a unit of utility and thus treating it as if it were classically measurable, an exact analysis of demand can be developed.

9.5. Utility measurement in “Economics and Mechanics” In 1909, in his last economic article, “Economics and Mechanics,” Walras returned to the issue of utility measurement. As its title suggests, in this essay Walras compared economics with mechanics and claimed a kinship between the two based upon the way that both investigate quantitative facts using mathematics and arrive at equilibrium conditions that are formally

44 Walras also associated Poincaré and Pareto in this respect; see Jaffé 1965, letter 1502, footnote 13. 45 See Jaffé 1965, vol. 1: 1-15 (autobiographical note), and letters 1505, 1511, 1514, 1586, 1638, 1642, 1646, 1668,

1669, 1677, 1681.

33

identical. The nature of the facts investigated by economics and mechanics is, however, different: while mechanics studies exterior or physical facts, economics deals also with interior or psychological facts, such as need. The main dissimilarity between physical and psychical facts is, in turn, that there are units and instruments which may be used to measure the former, but this is not the case with the latter:

For each [physical] fact, there is a collective and objective unit, that is, a magnitude, which is the same for everybody and which serves to measure it. […] There are meters and centimeters to establish the length of the arms of the steelyard balance, grams and kilograms to ascertain the weight supported by these arms […]. There are no [instruments] to measure the intensities of need of traders. (Walras [1909] 1990: 206-207, 212-213)

Just as in his previous writings, Walras claimed that the lack of instruments to measure psychological facts entails no difficulty for economics. In “Economics and Mechanics” he developed this argument along three different lines.

In the first place, and as suggested by Poincaré in his letter of 30 September 1901 (which Walras published as an appendix to “Economics and Mechanics”), at the subjective level each trader compares the utility of different things and determines that which is greater for him. In this sense, although not measurable, utility is at least appreciable: “The need which we have for things, or their utility for us, is an internal quantitative fact, appreciation of which remains subjective and individual. So be it! Nonetheless it is a magnitude and, I would say, an appreciable [appréciable] magnitude.” (207) For our purposes, the distinction between appreciable and measurable magnitudes is significant because it demonstrates that Walras did not consider comparison as measurement. This is in accord with our claim that he always understood measurement in the classical fashion.

Secondly, Walras argued that the point of maximum satisfaction for the individual, as well as the point of general equilibrium for the market, is characterized by the equality of marginal utility ratios with price ratios. The circumstance that individuals maximize their utility and that markets are in equilibrium would prove, according to Walras, that traders are capable not only of comparing utilities, but also of assessing their ratios and therefore of measuring utility:

Each trader […], consciously or unconsciously, decides by himself in his internal theatre if his last needs satisfied are proportional to the values of the commodities. The circumstance that the measure is […] interior […] does not prevent it from being a measure, that is, a comparison of quantities and quantitative ratios [rapports quantitatifs]. (213)

Scholars skeptical of the measurability of utility, such as Levasseur or Laurent, would most probably have objected that this argument begs the question, as the derivation of the conditions for utility maximization and market equilibrium seems to depend on the measurability of utility. This potential criticism is irrelevant for the matter at hand in the present paper. For us the important point is that Walras’s train of reasoning confirms that he associated measurement with the assessment of quantitative ratios and understood measurement in the classical way.

The third of Walras’s arguments suggests the possibility that even the measurement of physical magnitudes may be problematic. In making this point, Walras quoted from Poinsot’s definition of

34

the mass of a body as “the number of molecules contained in it,” which he had already cited in one of his letters to Poincaré. In meeting this apparent difficulty for the physical sciences, Walras appealed again to the authority of Poincaré, and mentioned approvingly the instrumentalist definition of mass that the latter had put forward in his La science et l’hypothèse. According to this definition, “masses are coefficients which it is found convenient to introduce into calculations.” (Poincaré 1902, quoted in Walras [1909] 1990: 213) Walras suggested that utility and rareté could be considered in an analogous way, i.e., as hypothetical causes to be introduced into the calculations of economics in order to derive the empirical laws of demand, supply and exchange:

[I wonder whether] all these concepts, those of mass and force as well those of utility and rareté, might not simply be names given to hypothetical causes which should be necessarily and justifiably introduced into the calculations with a view of linking them to their effects. […] Thus […] the utilities and the rareté would be the causes of demand and supply, from which would result the value of exchange. (213)

If utility and rareté are regarded as hypothetical causes of observable effects, then the burden of measurability shifts to their effects and the lack of instruments to measure utility and rareté ceases to appear problematic. Although now presented in more sophisticated epistemological fashion, the argument was very much the same that Walras had made in the “Principle” of 1873: even if utility is not measurable, in order to construct an exact analysis of demand it is necessary and thus justified to hypothesize that it is.

9.6. Conclusions on utility measurement by Walras Consideration of the “Principle”, the Elements, “Economics and Mechanics”, and Walras’s correspondence illustrates that his stance on the question of the measurability of marginal utility as well as his understanding of measurement remained fundamentally the same throughout his scientific career: measurement is possible when a unit is available and consists of assessing the numerical ratio between the unit and the object to be measured, while other forms of evaluation such as the comparison of magnitudes do not constitute proper measurement. According to this idea of measurement, marginal utility is not measurable, but by assuming that it is we can derive the laws of demand, supply and exchange. There is no trace in Walras’s writings of a notion of measurement as related to the ranking of utility differences. Therefore we can safely conclude that Walras was not a cardinalist in the current sense of the term.

With respect to any putative relationship between Walras’s reflection on utility measurement and discussions on measurement taking place in other disciplines in the period 1870-1910, the link is negligible. The fact that neither Walras’s complete economic works (Walras 1987-2005), nor his correspondence (Jaffé 1965) contains any reference to Fechner strongly suggests that Walras was not influenced by the psychological debates over measurement. With regard to Walras’s possible connections with the mathematical discussions about number and measurement, these seems to be limited to his correspondence with Poincaré, which, as we have seen, did not lead him to modify his earlier views.

35

10. Summary and conclusions The present paper has addressed three questions: (1) How did Jevons, Menger and Walras conceive measurement in general, and utility measurement in particular?; (2) Were Jevons, Menger and Walras cardinalists in the current, ranking-of-utility-difference sense of the term?; (3) What was the relationship between their view of utility measurement and the view of measurement held in other disciplines in the period 1870-1910?

With respect to the first issue, the paper has shown that the three founders of marginal utility theory conceived measurement in the classical fashion, not the representational one. Accordingly, they associated utility measurement with the possibility of identifying a unit of utility and assessing utility ratios, rather than with the ranking of utility differences. Furthermore, they did not even use the word “cardinal” or the cardinal-utility terminology, which came into prominence in economics only in the 1930s. Therefore – and this answers the second question – Jevons, Menger, and Walras were not cardinalists in the current sense of the term.

This conclusion calls for a correction of the canonical narrative of the history of utility theory, which portrays Jevons, Menger, Walras as cardinalists. In fact, the standard narrative projects the representational understanding of measurement dominating current economics, as well as the notion and vocabulary of cardinal utility that emerged in the 1930s, back into the period 1870-1910, in which years such understanding, notion and vocabulary did not yet exist. The present paper corrects this anachronism and suggests that the canonical dichotomy between cardinal utility and ordinal utility is conceptually too threadbare and barren to tell an accurate story of utility theory. Most importantly, proportional, i.e. classically measurable, utility should be added to the picture, and the ordinal revolution should be viewed, not as a shift from cardinal to ordinal utility, but as a jump from proportional to ordinal utility, with cardinal utility entering the scene only at a later stage.

In order to address the third question – that of the relationship between the view of utility measurement of our three marginalists and discussions in other disciplines – this paper has investigated the understanding of measurement in philosophy, physics, psychology, mathematics, and also in the areas of economics not related to utility theory, in the period between1870 and 1910. On the one hand, the paper has found that in this period the classical understanding of measurement originating in the works of Aristotle and Euclid dominated not only economics but also other disciplines. This circumstance renders the classical approach to utility measurement of Jevons, Menger and Walras less surprising. On the other hand, however, the paper has found that direct relationships between the reflection on utility measurement of Jevons, Menger and Walras and contemporary non-economic debates on the topic were negligible.

This last conclusion suggests in turn that speculation on measurement in utility theory constitutes a strand of thought largely independent from other, contemporary, scientific discussions on measurement. This inference is relevant for the broad history of measurement theory because historians of science working on this topic, such as Michell (1993, 1999), Díez (1997), Darrigol (2003), and Martin (2003), have overlooked the economic discussions on measurement. By examining the reflection on utility measurement of Jevons, Menger and Walras, as well as the criticisms that their treatment of utility received by other economists and scholars, the present paper has integrated wider histories of scientific measurement theory with a hitherto missing

36

economic part of the story. Thus, the paper has contributed, not only to the history of utility theory, but also the broader history of scientific measurement.

References Anonymous [1871] 1981. Jevons on the Theory of Political Economy. In Black 1972–1981, vol. 7, 152–

157. Arena, R., and S. Gloria-Palermo. 2008. Menger and Walras on Money: A Comparative View. History of

Political Economy, 40: 317–343. Bergson, H. [1889] 1960. Time and Free Will. New York: Harper. Binet, A. 1903. L’Étude éxperimentale de l’intelligence. Paris: Schleicher. Black, R.D.C., ed. 1972–1981. Papers and Correspondence of William Stanley Jevons, 7 vols. London:

Macmillan. Boring, E.G. 1957. A History of Experimental Psychology. New York: Appleton-Century-Crofts. Boumans, M. 2005. How Economists Model the World Into Numbers. London–New York: Routledge. —, ed. 2007. Measurement in Economics: a Handbook. London: Academic Press. Cairnes, J.E. [1872] 1981. New Theories in Political Economy. In Black 1972–1981, vol. 7: 146–152. —. 1874. Some Leading Principles of Political Economy. New York: Harper. Campagnolo, G. 2005. Carl Menger’s “Money as Measure of Value”: An Introduction. History of

Political Economy, 37: 233–243. Campbell, N.R. 1920. Physics: the Elements. Cambridge: Cambridge University Press. Cantor, G. [1887] 1932. Mitteilungen zur Lehre vom Transfiniten. In G. Cantor, Gesammelte

Abhandlungen, edited by E. Zermelo, 378–439. Berlin: Springer. Cattell, J. McK. 1890. Mental Tests and Measurement. Mind, 15: 373–380. —. 1893. Mental Measurement. Philosophical Review, 2: 316–332. —. 1902. Measurement. In Dictionary of Philosophy and Psychology, edited by J.M. Baldwin, vol. 2: 57.

London: Macmillan. Chang, H. 2004. Inventing Temperature. Oxford: Oxford University Press. Čuhel, F. 1907. Zur Lehre von den Bedürfnissen. Innsbruck: Wagnerschen Universität Buchhandlung. Darrigol, O. 2003. Number and Measure: Hermann von Helmholtz at the Crossroads of Mathematics,

Physics, and Psychology. Studies in the History and Philosophy of Science, 34: 515–573. Dedekind, R. [1888] 1999. What Are the Numbers and What are They For? [Was sind und was sollen die

Zahlen?]. In Ewald 1999, 787–833. Delboeuf, J. 1873. Etude psychologique. Bruxelles. Hayez. —. 1875. La mesure des sensations. La Revue Scientifique, 46: 1014–17. —. 1878. La loi psychophysique et le nouveau livre de Fechner. Revue philosophique, 5: 34–63 and 127–

157. —. 1883. Examen critique de la loi psychophysique. Paris: Baillere. Descartes, R. [1628] 1999. Rules for the Direction of the Mind. In The Philosophical Writings of

Descartes, vol. 1, 7–77. Cambridge: Cambridge University Press. Diez, J.A. 1997. A Hundred Years of Numbers. An Historical Introduction to Measurement Theory 1887–

1990. Studies in History and Philosophy of Science, 28: 167–185 and 237–265. Du Bois-Reymond, P. 1882. Die allgemeine Functionentheorie. Tübingen: Laupp.

37

Dupuit, J. 1844. De la mesure de l’utilité des travaux publics. Annales des Ponts et Chaussées, 2nd series, 8: 332–375.

Edgeworth, F.Y. 1894. Professor J.S. Nicholson on ‘Consumers’ Rent’, Economic Journal, 4: 151–158. —. 1900. The Incidence of Urban Rates, I. Economic Journal, 10: 172–193. —. 1907. Appreciations of Mathematical Theories. Economic Journal, 17: 221–231. Ewald, W. 1999. From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2.

Oxford: Clarendon Press. Fechner, G.T. [1860] 1966. Elements of Psychophysic. Vol. 1. New York: Holt, Rinehart and Winston. —. 1882. Revision der Hauptpuncte der Psychophysik. Leipzig: Breitkopf and Hartel. —. 1887. Über die psychischen Massprincipien und das Weber’sche Gesetz. Philosophischen Studien, 4:

161–230. Ferreirós, J. 2007. Labyrinth of Thought. Berlin: Birkhäuser. Fishburn, P.C. 1970. Utility Theory for Decision Making. New York: Wiley. Giocoli, N. 2003. Modeling Rational Agents. Cheltenham: Elgar. Gossen, H.H. 1854. Entwicklung der Gesetze des menschlichen Verkehrs. Braunschweig: Vieweg. Grattan-Guinness, I. 2000. The Search for Mathematical Roots, 1870–1940. Princeton: Princeton

University Press. Halevy, E. 1904. La formation du radicalisme philosophique. Tome I: La jeunesse de Bentham 1776–

1789. Paris: Alcan. Hayek, F.A. von. 1934. Carl Menger. Economica, 1: 393–420. Heath, T.L. 1908. The Thirteen Books of Euclid’s Elements. Cambridge: Cambridge University Press. Heidelberger, M. 2004. Nature from Within. Pittsburgh: Pittsburgh University Press. Helmholtz, H. von. [1887] 1999. Numbering and Measuring from an Epistemological Viewpoint. In

Ewald 1999, 727–52. Hicks, J.R. 1939. Value and Capital. Oxford: Clarendon Press. Hilbert, D. 1899. Grundlagen der Geometrie. Leipzig: Teubner. Hölder, O. [1901] 1996. The Axioms of Quantity and the Theory of Measurement, Part I. Journal of

Mathematical Psychology, 40: 235-252. Howey, R.S. 1960. The Rise of the Marginal Utility School, 1870–1889. Lawrence, KS: University of

Kansas Press. Ingram, J.K. 1885. Political Economy. In Encyclopaedia Britannica, 9th ed., vol. 19: 346–401. Ingrao, B. and G. Israel. 1990. The Invisible Hand. Cambridge, MA: MIT Press. Jaffé, W., ed. 1965. Correspondence of Léon Walras and Related Papers, 3 vols. Amsterdam: North-

Holland. —. 1977. The Walras–Poincaré Correspondence on the Cardinal Measurability of Utility. Canadian

Journal of Economics / Revue canadienne d'Economique, 10: 300–307. James, W. 1890. The Principles of Psychology. New York: Holt & C. Jevons, W.S. 1871. The Theory of Political Economy, 1st edition. London: Macmillan. —. 1874. The Principles of Science. London: Macmillan. —. 1879. The Theory of Political Economy, 2nd edition. London: Macmillan. —. 1977. Letter to J.E. Cairnes of 14 January 1872. In Black 1972–1981, vol. 3: 245–247. Kant, I. [1787] 1997. Critique of Pure Reason. Cambridge: Cambridge University Press. Kauder, E. 1961. Carl Mengers Zusätze zu “Grundsätze der Volkswirtschaftslehre”. Tokio: Bibliothek der

Hitotsubashi Universität. Krantz, D.H., P. Suppes, R.D. Luce, and A. Tversky. 1971. Foundations of Measurement. Vol. 1. New

York–London: Academic Press. Kries, J. von. [1882] 1995. On the Measurement of Intensive Magnitudes and the so-called

Psychophysical Law. In Niall 1995, 282–302. Kronecker, L. [1887] 1999. On the Concept of Number. In Ewald 1999, 947–55. Levasseur, P.E. [1874] 1987. Compte–rendus des séances et travaux de l’Académie des sciences morales

et politiques. In Walras 1987, 529-532. Maas, H. 2005. William Stanley Jevons and the Making of Modern Economics. Cambridge: Cambridge

University Press. Mandler, M. 1999. Dilemmas in Economic Theory. New York: Oxford University Press. Martin O. 2003. Les origines philosophiques et scientifiques de la théorie représentationnelle de la mesure

(1930–50). Social Science Information, 42: 485–513. Maxwell, J.C. 1873. Treatise on Electricity and Magnetism. London: Constable and Co.

38

Menger, C. [1871] 1981. Principles of Economics. New York: New York University Press. —. [1892] 2005. Money as Measure of Value. History of Political Economy, 37: 245–261. —. [1909] 2002. Money. In Carl Menger and the Evolution of Payments Systems, edited by M. Latzer and

S. Schmitz, 25–108. Cheltenham: Elgar. —. 1892. Geld. In Handwörterbuch der Staatswissenschaften, 1st edition, vol. 3, 730–757. Jena: Fischer. —. 1900. Geld. In Handwörterbuch der Staatswissenschaften, 2nd edition, vol. 4, 60–106. Jena: Fischer. —. 1923. Grundsätze der Volkswirtschaftslehre, 2nd edition, edited by. K. Menger. Wien: Hölder, Pichler

and Tempsky. —. 1934–36. Collected Works, edited by F. Hayek, 4 vols. London: London School of Economics. Michell, J. 1993. The Origins of the Representational Theory of Measurement: Helmholtz, Hölder, and

Russell. Studies in the History and Philosophy of Science, 24: 185–206. —. 1999. Measurement in Psychology. Cambridge University Press. —. 2006. Psychophysics, Intensive Magnitudes, and the Psychometricians’ Fallacy. Studies in the History

and Philosophy of Biological and Biomedical Sciences, 17: 414–432. —. 2007. Representational Theory of Measurement. In Boumans 2007, 19–39. Mirowski, P. and D.W. Hands. 2006. Introduction. In Agreement on Demand: Consumer Theory in the

Twentieth Century, edited by P. Mirowski and W.D. Hands, HOPE 38 (supplement): 1–6. Mirowski, P. and P. Cook, 1990. Walras’ “Economics and Mechanics”: Translation, Commentary,

Context. In Economics as Discourse, edited by W.J. Samuels, 189–224. Dordrecht: Kluwer. Mitchell, W.C. 1918. Bentham’s Felicific Calculus. Political Science Quarterly, 33: 161–183. Morgan, M.S., and J.L. Klein, eds. 2001. The Age of Economic Measurement. HOPE 33 (supplement). Moscati, I. 2007. History of Consumer Demand Theory, 1871–1971: A Neo–Kantian Rational

Reconstruction. European Journal of the History of Economic Thought, 14: 119–56. —. 2008. Changing the Yardstick. From the Classical to the Representational Notion of Measurement in

Utility Theory (1934–1954). Paper presented at the HES conference, Toronto, June 2008. —. 2010. Were Early Marginalists Really Cardinalists? Paper presented at the ESHET conference,

Amsterdam, March 2010. Nagel, E. 1931. Measurement. Erkenntnis, 2: 313–333. Niall, K.K. 1995. Conventions of Measurement in Psychophysics: von Kries on the so-called

Psychophysical Law. Spatial Vision, 9: 275–305. Neumann, J. von, and O. Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton:

Princeton University Press. Niehans, J. 1990. A History of Economic Theory. Baltimore, MD.: Johns Hopkins University Press. Pareto, V. [1900] 2008. Summary of Some Chapters of a New Treatise on Pure Economics by Professor

Pareto, Giornale degli Economisti, 67: 453–504. —. 1909. Manuel d’économie politique. Paris: Giard et Brière.. Peart, S. 1995. Measurement in Utility Calculations. The Utilitarian Perspective. In Measurement,

Quantification and Economic Analysis, edited by I.H. Rima, 63–86. London: Routledge. —. 1996. The Economics of W.S. Jevons. London: Routledge. —. 2001. “Facts Carefully Marshalled” in the Empirical Studies ofWilliam Stanley Jevons. In Morgan and

Klein 2001, 252–276. Petty, W. 1690. Political Arithmetick. London. Poincaré, H. 1893. Le continu mathématique. Revue de Métaphisique et de Morale, 1: 26–34. —. 1902. La science et l'hypothèse. Paris: Flammarion. Poinsot, L. 1842. Eléments de statique. Paris: Bachelier. Roll, E. 1936. Menger on Money. Economica, 3: 455–460. Ross, W.D., ed. 1910–52. The Works of Aristotle. Oxford: Oxford University Press. Russell, B. 1897. On the Relations of Number and Quantity. Mind, 6: 326–41. —. 1903. The Principles of Mathematics. Cambridge: Cambridge University Press. Schmidt, T., and C.E. Weber. 2008. On the Origins of Ordinal Utility: Andreas Heinrich Voigt and the

Mathematicians. History of Political Economy, 40: 481–510. —. 2009. German Language Ordinalism, 1892–1932. Paper presented at the HES Conference, Denver,

June 2009. —. 2010. Forgotten Origin and Unlikely Paths: From Andreas Heinrich Voigt to the Hicks–Allen

Revolution in Consumer Theory. Economic Inquiry, forthcoming. Schröder, E.. 1873. Lehrbuch der Arithmetik und Algebra. Leipzig: Teubner. Schumpeter, J.A. 1954. History of Economic Analysis. New York: Oxford University Press.

39

Shabas, M. 1985. Some Reactions to Jevons’ Mathematical Program: The Case of Cairnes and Mill. History of Political Economy, 17: 337–353.

—. 1990. A World Ruled by Number. Princeton: Princeton University Press. Solere, J.-L. 2001. The Question of Intensive Magnitudes According to Some Jesuits in the Sixteenth and

Seventeenth Centuries. The Monist, 84: 582–616. Spearman, C. 1904. General Intelligence, Objectively Determined and Measured. American Journal of

Psychology, 15: 201–293. Sraffa, P. 1951. Introduction to The Works and Correspondence of David Ricardo, vol. 1, xiii–lxii.

Cambridge: Cambridge University Press. Stevens, S.S. 1946. On the Theory of Scales of Measurement. Science, 103: 677–680. Stigler, G.J. 1937. The Economics of Carl Menger. Journal of Political Economy, 45: 229–250. —. Production and Distribution Theories. London: Macmillan. —. 1950. The Development of Utility Theory. Journal of Political Economy, 58: 307–27 and 373–96. Streissler, E. 2002. Carl Menger’s Article “Money” in the History of Economic Thought. In Carl Menger

and the Evolution of Payments Systems, edited by M. Latzer and S. Schmitz, 11–24. Cheltenham: Elgar.

Sylla, E. 1972. Medieval Quantifications of Qualities: the ‘Merton School’. Archive for the History of Exact Science, 8: 9–39.

Tannery, P. 1875. A propos du logarithme des sensations. Revue Scientifique, 8: 876–877 and 1018–1020. —. 1884. Critique de la loi de Weber. Revue Scientifique, 17: 15–35. —. 1888. Psychologie mathématique et psychophysique. Revue Scientifique, 25: 189–197. Thorndike, E.L. 1904. Theory of Mental and Social Measurements. New York: Science Press. Titchener, E.B. 1905. Experimental Psychology: A Manual of Laboratory Practice. Vol. II: Quantitative

Experiments, part 2: Instructor’s Manual. New York–London: Macmillan. —. 1909. A Text–Book of Psychology. Vol. 1. New York: Macmillan. Voigt, A. [1893] 2008. Number and Measure in Economics. In Schmidt and Weber 2008, 502-504. Walras, A., and L. Walras. 1987–2005. Œuvres économiques complètes, Paris: Economica. Walras, L. [1874] 1954. Elements of Pure Economics, London: Allen and Unwin. —. [1874] 1987. Principe d’une théorie mathématique de l’échange. In Walras 1987, 261–281. —. [1876] 1987. Une branche nouvelle de la mathématique. In Walras 1987, 290–329. —. [1909] 1990. Economics and Mechanics. In Mirowski and Cook, 206-214. —. 1886. Théorie de la monnaie. Lausanne: Corbaz & C. —. 1987. Mélanges d’économie politique et sociale. In Walras and Walras, vol. 7. —. 1988. Éléments d'économie politique pure. In Walras and Walras, vol. 8. Weber, E.H. [1834/1846] 1996. On the Tactile Senses. Hove: Taylor & Francis. Whitehead, A.N., and B. Russell. 1913. Principia Mathematica. Vol. 3. Cambridge University Press,

Cambridge. Wicksell, K. [1924] 1958. The New Edition of Menger’s Grundsätze. In K. Wicksell, Selected Papers on

Economic Theory, 193–203. London: Allen & Unwin. Wundt, W. 1873–74. Grundzüge der physiologischen Psychologie, 1st edition. Leipzig: Engelmann. —. 1880. Grundzüge der physiologischen Psychologie, 2nd edition. Leipzig: Engelmann. Yagi, K. 1993. Carl Menger’s Grundsätze in the Making. History of Political Economy, 25: 697–724.