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MATHEMATICS IS THE LANTERN: VITO VOLTERRA, LÉON WALRAS, AND IRVING FISHER ON THE MATHEMATIZATION OF ECONOMICS

Published online by Cambridge University Press:  18 December 2018

Sonya Scott*
Affiliation:
York University.
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Abstract

The interdisciplinary project to unite the field of mathematics with the social and biological sciences marks the work of Vito Volterra, one of Italy’s most prominent mathematicians of the twentieth century. This paper explores the connections between Volterra’s 1901 inaugural address at the Accademia dei Lincei in Rome and the work of two of his contemporaries, Léon Walras and Irving Fisher. All three thinkers were ardent advocates of the mathematical turn in economic thinking. This paper argues, however, that it is the previously unexplored relationship between Volterra and Fisher that sheds the most light on the way in which mechanical physics contributed to the project of mathematization within economics more generally. Furthermore, it explores the way in which mathematical inquiry postulated a new and coherent abstraction of the economy, at the same time that it gave epistemological authority to the economist.

Type
Articles
Copyright
Copyright © The History of Economics Society 2018 

I. INTRODUCTION

Vito Volterra was one of the first mathematicians to publicly endorse the use of mathematics in the social sciences, specifically within the discipline of economics. His endorsement in his inaugural lecture for the 1901–02 academic year at the Accademia dei Lincei was formative; he was attempting to bolster the reputation and influence of economists working towards the creation of a mathematical mode of explanation.Footnote 1 But this lecture is not significant for historical reasons alone. It also reveals an attempt to establish a new field of economic inquiry, and, as an important correlate, a new form of intellectual authority. Volterra’s lecture provides insight into the way in which very specific forms of mathematical reasoning came to facilitate a radical epistemological shift in the study of economics, including a shift in the nature of the object of study itself: the economy.

In this paper I address some of the theoretical consequences of the interdisciplinary connection between economics and mathematics. I do so by looking at the way in which the new mathematico-economic science postulates reality: that is, determines the parameters of the economic reality that is to be modeled and explained. Further, I examine two important figures that accompany this nascent form of modeling and explanation: the economic subject described in the models of the economist, and the economist himself as a thinking and authoritative subject equipped with the use of mathematical language. In addition to primary sources, I make use of archival materials in order to elaborate on the epistemological shift indicated in Volterra’s lecture. An examination of the work of Léon Walras and Irving Fisher is crucial in order to understand this shifting discipline, as both had already developed sophisticated mathematical systems of economic thought, and were impassioned advocates of the mathematical method in economics at the turn of the twentieth century.

Volterra’s lecture pushes those working on economic mathematization to go beyond the use of analogy as a form of legitimation within economic analysis. He champions the use of calculus in particular, privileging the clarity and insight that mechanical theorists had attained in physics. If one considers this endorsement in the broader epistemological history of economics, within which mathematical economics does not become generally accepted until the 1930s, then the previously unexplored Volterra–Fisher relationship offers significant insight into the changing nature of the discipline. The examination of the substantive correlation between the content of their work demonstrates the relationship among mechanics, conceptions of the economy as an object of study, and a new mathematical program. Indeed, while some have traced the roots of mathematical economics back to key moments of quantification within the discipline (Koyré 1968; Kula 1976; Porter Reference Porter, Klein and Morgan2001), the aim here is to reveal the nature of the claims regarding the economist’s knowledge, or capacity to know, and the postulates of the economic reality upon which the economist wishes to construct his models. When we consider this program of mathematization, one rooted in mechanical physics and calculus, we are actually considering a new mode of economic abstraction that gives us both a clear method and a newly delineated object of study.Footnote 2

Key to the systematic mathematization of economics is modeling as a construction of economic reason itself. And while the idea of economic modeling was not yet in play when Volterra and Fisher were pushing for a newly mathematized economics, models were nonetheless being constructed and used in order to approximate patterns of economic life. In their categorization of economic models, Mary Morgan and Tarja Knuuttila (2012) argue that there are two fundamentally different ways of viewing economic models: as idealizations of reality, or as purpose-built constructions. They argue that “idealization is typically portrayed as a process that starts with the complicated world with the aim of simplifying it and isolating a small part of it for model representation” (Morgan and Knuuttila Reference Morgan and Knuuttila2012, p. 51). This type of abstraction is derived from John Stuart Mill, where the purpose of the model is to abstract “causally relevant capacities or factors of the real world for the purpose of working out deductively what effects those few isolated capacities or factors have in practical model (i.e. controlled) environments” (ibid., pp. 51–52). However, when we approach models as constructions rather than idealizations, their purpose and use differ. Here we can see economic models as “pure constructions or fictional entities that nevertheless license different kinds of inferences,” which have “been associated with a functional account of models as autonomous objects rather than by characterizing them in relation to target systems as either theoretical models or models of data” (Morgan and Knuuttila Reference Morgan and Knuuttila2012, p. 61). If we are to understand models as concrete artefacts, as indeed the authors do, then it becomes apparent that an important epistemological function of the model is to enable the modeler to learn by constructing and manipulating the model itself (Knuuttilla 2011, p. 269). Thus, models may be understood as epistemic tools (Knuuttilla 2011) that are in and of themselves productive of knowledge even if their role is not understood as a form of abstraction attempting to distil a single causal relation that illuminates economic reality.

Much research has recently been conducted on the nature and importance of modeling in economics (e.g., Boumans Reference Boumans2005; Weisberg Reference Weisberg2007; Mäki 2009; Knuuttila Reference Knuuttila2011; Morgan Reference Morgan2012; Morgan and Knuuttila Reference Morgan and Knuuttila2012). And while the dominant position in this literature posits that models serve as mediators between our theorist and the reality that he or she seeks to describe (Weisberg Reference Weisberg2007), I propose instead that economists themselves are the mediators between the models and the economic reality they seek to describe. Economists create the models with an explicit form of mathematical reasoning, and then interpret their models in such a way that attempts to show the connection between the model and the empirical reality or universal law that they hope to illuminate. This is not to side with the perspective that models are but idealizations that seek to abstract from reality in order to distil some causal relationship (though we can certainly see some evidence of this at work in Fisher’s 1892 Investigations), but rather that the epistemological function that is attributed to economic models by Morgan and Knuuttila ought to be placed back into the head of the economist as a thinking subject. Thus, while early mathematization might justly be viewed as a process of reasoning that uses a prosthetic and that bears a semblance of objectivity and autonomy from the subjectivity of the economist as a thinking subject, it is nonetheless both subjectively and historically embodied thought. After first exploring Volterra’s and Walras’s prescriptions for correct forms of mathematical and economic reasoning, respectively, I will show how both Volterra and Fisher hold enlightenment ideals that privilege visualization in the production of knowledge. This privilege equates a rather sophisticated form of mathematical reasoning with the means to attain vision, and equates the knowing subject with the capacity to attain the truth. This knowing subject is bound to a lived context where he is attempting to construct and interpret meaning from the model he uses to explain the world, on whatever level of abstraction is possible. Yet at the same time it is precisely access to disciplinary mathematical knowledge that grants epistemological authority over the knowledge of the economic realm, and over the very interpretation of the model’s meaning and veracity in the first place. Thus, the early interdisciplinary impulse to marry mathematics with economics inspired a disciplinary epistemological privilege of the mathematician that remains entrenched within the discipline of economics today.

In order to examine these shifting grounds of epistemology and subjectivity in early twentieth-century economics, my analysis proceeds in two parts. First, I discuss the inaugural lecture in its historical context, and compare the work of Volterra and Walras in order to situate their respective approaches to the mathematical method and its empirical application within the context of the development of economics in the nineteenth century. Second, I contrast Walras’s methodology and Fisher’s mechanical approach. This differentiation leads to an examination of the hydraulic models that Fisher built in order to explain equilibrium, and clarifies the delineation of reality and subjectivity that Volterra and Fisher share. I argue that the connection between Volterra and Fisher reveals itself through Fisher’s development of a coherent and systemic mechanical model in economics, involving the use of marginal utility with a specific conception of homo oeconomicus, the expression of economics as a distinct ‘field of vision,’ and the linkage of the economic imagination with mathematical symbolism.

II. WALRAS AND VOLTERRA: PARALLEL UNIVERSES

The respective works of Volterra and Walras present distinct strategies for the application of mathematical reasoning to reality. Much has been written on Walras’s mathematical approach, and on his methodology more generally.Footnote 3 Likewise, Volterra’s influence upon economics and his role as a prominent mathematician and public figure have been well theorized.Footnote 4 Here, however, I am concerned with the parallels and distinctions in their statements on correct reasoning, and the relationship that this bears to the role of the economist, specifically the mathematical economist of the late nineteenth century.

Relevant texts to this comparative analysis include Volterra’s writings on economics: the Inaugural Lecture given at the Accademia dei Lincei in 1901; and his later review of Vilfredo Pareto’s Manuale that appeared in the Giornale Degli Economisti in 1906. And while Walras’s writings on mathematics and economics are vast, in particular I will focus on the statement of method in his Elements of Pure Economics, where we find the purest articulation of mathematical reasoning, along with his final 1909 essay, “Économique et Mécanique,” where he directly addresses the mathematician as interlocutor and authority who might grant his econo-mathematical apparatus legitimacy.

In 1900 Volterra was the chair of mathematical physics at the Accademia dei Lincei in Rome. He was one of Italy’s most prominent mathematicians, and established his reputation through his work on functionals and the development of integro-differential equations (Israel and Gasca Reference Israel and Gasca2002). Once he moved to Rome (he had previously taught mechanics in Turin from 1892 to 1900), he gained greater stature as a leading intellectual and scientific voice in Italy. His post at the University of Rome allowed him a central position from which to engage in the most important scientific developments of the day (Weintraub Reference Weintraub2002). Within this center of intellectual activity the prestige of mathematics was clear, and while debates were raging regarding the relationship between pure and applied sciences (Israel and Nurzia Reference Israel and Nurzia1989), the general impulse towards scientific development was strong. Volterra’s inaugural lecture in 1901 was thus driven by a general project to render mathematics relevant to various disciplines, and his mathematical work subsequently focused on the nature of this application to diverse objects of study.

While Volterra’s lecture was motivated by a profound respect for mathematical economists, it was also driven by a concern for the impression that would be made on those in attendance. Both in terms of his content and rhetorical style, Volterra sought to capture the interest of diverse scholars, most of whom were not mathematicians. In a letter to Giovanni VailatiFootnote 5 just prior to the composition of his lecture, he writes:

For all that it is unimportant to me, I cannot get out of the very heavy responsibility of giving the inaugural lecture for the next academic year at the University. What can I do that will not be uninteresting to scholars in other disciplines than mathematics? I have thought about doing something on attempts at applying mathematics to biological and social sciences. What would you say about a topic like that? Is it alright or should I substitute it with something else? (cited in Guerraggio and Paoloni Reference Guerraggio and Paoloni2013, p. 54)Footnote 6

While his approach to the inaugural lecture may have been tentative, the great approval and praise with which his address was later met certainlyFootnote 7 encouraged Volterra to continue his alliance with the newly budding economic science, and to develop a biological mathematics that would later produce the Lotka–Volterra (or predator–prey) equations.Footnote 8

The intellectual authority and status of the mathematician is quite apparent in Volterra’s lecture. He begins by positing that the mathematician is the most curious of all thinkers, and that with this curiosity, he wields calculus as a most powerful instrument. In fact, armed with mathematics, “he has … the key which can open the door to many of the dark mysteries of the universe, and a means to explain in symbols a synthesis which bridges and brings together the vast and disparate results of different sciences” (Volterra Reference Volterra1901, p. 437).Footnote 9 How does the secret unlocked by this key relate to economic phenomena? How does the mathematical method, as opposed to arithmetic analysis, relate to economic reality? From the outset of his lecture Volterra situates economics among the mysteries of the universe to be unlocked, and identifies mathematics as the methodology that will help us solve this mystery.

The content of Volterra’s lecture draws out the nature of the new mathematical method in economics:

Thus shape concepts so that they can introduce measure; then measure; then deduce laws; then go back to hypotheses; deduce from these by means of analysis a science of rigorously logical ideal entities, compare consequences to reality, reject or transform the already used hypotheses as soon as a contradiction appears between the results of the calculation and the real world, and in this matter succeed in guessing in new facts and new analogies, or deduce once again from the present state what the past was and what the future will be. This is, quite briefly, how one can summarize the birth and evolution of a science which has a mathematical character. (Volterra Reference Volterra1901, pp. 442–443)

The scientist, or economist in this case, is instructed to begin with the concepts that will allow for measure. The movement from measurement to laws is the movement from reality to the ideal, and then, as we move from the hypothetico-deductive to reality, the establishment of the scope of the reality about which scientific claims can be made is completed. Volterra’s method here described, which might apply to any science, is the method of determining how to constitute reality in mathematical terms, and, by so doing, constitutes its parameters and limits in economic terms as well. These are not absolute limits to reality, or claims about the objective entirety of the universe, but rather limits to the scope of the reality to which our knowledge can apply in any given model construction. Conclusions drawn within these parameters, and laws established as a result, can be subjected, of course, to revision and verification. But we can return only to an object that we have determined would require measurement and study in the first place.

Of particular interest in this respect is Michael Weisberg’s account (2007) of Volterra as a mathematician who explains real-world causal relations by modeling, as opposed to engaging in what he terms “abstract direct representation.” Weisberg posits that models mediate between the thinker and reality, and that modeling is an “indirect theoretical investigation of a real world phenomenon using a model” (Weisberg Reference Weisberg2007, p. 209). Those who engage in abstract direct representation, on the other hand, “analyze and represent the properties of a real-world phenomenon, suitably abstracted, in the first instance” (Weisberg Reference Weisberg2007, p. 222). Looking at Volterra’s biological work, particularly at his account of the post-WWI fish shortage in the Adriatic, Weisberg argues that Volterra is the quintessential modeler because

he engaged in indirect representation and analysis of a model. His equations described mathematical models of biological populations and these models were similar in certain respects to real biological systems, but the equations were not direct representations of any real system. … His characterization of the population dynamics of the Adriatic were made indirectly. (Weisberg Reference Weisberg2007, p. 216)

The important point here is that Volterra proceeded by means of mathematics in indirect investigation of a quantifiable, yet non-mathematical, phenomenon, such as population dynamics over time. And while this biological work came much later than Volterra’s inaugural lecture, it does reflect what Giorgio Israel (Reference Israel1996) describes as Volterra’s resolute attitude toward empirical content. Given that Volterra held the position that the mathematical analogy was not in and of itself beyond reproach, that it needed proof, not just vague parallels used to assert its validity, his emphasis on verification was clear (Israel Reference Israel1996, p. 73).

For a mathematician such as Volterra, this methodology implied an empirical basis to the study of universal laws. As Roy Weintraub explains, “[S]cientific models had to be based directly and specifically on underlying physical reality, a reality directly apprehended through experimentation and observation and thus interpersonally confirmable” (Weintraub Reference Weintraub2002, p. 48). The nature of the reality that Volterra aspired to illuminate may not have been the same as that of political economists who came before. If we think of the mathematician as a worldly philosopher, just as we might imagine any economist at the time, the mathematician would be dealing with some of the fundamental questions presented by the contemporary physics of the late nineteenth century in an attempt to understand the universe in new and innovative ways. This indeed is a different sort of reality from that of the economist attempting to delineate the parameters of economic equilibrium. Fundamental to the distinction between these two aims is the notion of rigor. Again Weintraub elaborates: “[T]o be rigorous in one’s modelling of a phenomenon was to base the modelling directly and unambiguously on the experimental substrate of concrete results” (Weintraub Reference Weintraub2002, p. 49). “Rigor” thus implies constraint by the concrete, rather than a model made without observation or recourse to the empirical realm. Constraint by the physical universe is substantially different, however, from constraint by elements (and only certain elements) of the social order. Although this problematic is taken up in various ways by different economists of the nineteenth century—often distinguishing between pure and applied realms, or relying on ideal states of perfectly free competition—Volterra did not address these particular difficulties in the social sciences.

At the time of his inaugural speech Volterra’s concern with the general method of inquiry in economics appears to be even more important than any given mathematical exposition. And while Volterra makes reference to the attempts of many formative mathematical economists,Footnote 10 it is the edifice of mathematical reason and the nature of rigor that are far more important than any particular theoretical insight. This is understandable because the field was relatively new to him and because he was not himself an economist. Later we see that he begins to engage more substantively and critically with economic ideas when he reviews Vilfredo Pareto’s Manuale and pays particular attention to Pareto’s replacement of “the non-measureable notion of ‘ophelimity’ with indifference curves” (Ingrao and Israel Reference Ingrao and Israel1990, p. 165). But given his limited prior engagement with the subject matter, it took a dialogical relationship for his inaugural lecture to effect a legitimating function in the world of economics. In other words, it was the support of those like Vilfredo Pareto, Maffeo Pantaleoni, and Léon Walras—all of whom were well served by his endorsement—that allowed for Volterra to gain an authoritative voice on the use of mathematics in the social sciences. In this case these economists helped specifically by ensuring publication of the inaugural lecture in the Giornale degli economisti and a later translation into French in the Revue du mois. Furthermore, Volterra’s later review (1906) of Pareto’s Manuale cannot be seen outside of this context of mutual promotion. Ultimately, however, the still inchoate and fragile intellectual support networkFootnote 11 between the disciplines cannot be read otherwise than a genuine methodological conviction that mathematical reasoning and modeling would ultimately shine more light on economic reality than literary or historical expositions had done before, even if by 1906 Volterra was indeed showing more caution about the endeavor, especially given the “overconfidence” on the part of some of the economists such as Léon Walras (Ingrao and Israel Reference Ingrao and Israel1990, p. 164).

The distinction between mathematics and mathematical reasoning is a key methodological concern raised in Volterra’s lecture. Mary Morgan has argued that “economics became mathematized at the same time as it became a modelling science” (Morgan Reference Morgan2012, p. 19). This coincidence of mathematical language and mathematical method in the new economic science stems from the mechanical approach taken by the economists of the late nineteenth century. This would have been particularly appealing to Volterra because of the influence of mechanical physics on his mathematical work. With optimism and no shortage of enthusiasm, Volterra announces the new age of a revelatory science: “This science is mechanics, and it is, together with geometry, if not the most brilliant, the most certainly solid and secure knowledge in which the human mind glories” (Volterra Reference Volterra1901, p. 443). The solidity of mechanics and geometry had at that point been established by a long and rigorous scientific tradition. To some extent empirical verification was possible, and since the seventeenth-century turn away from Aristotelian logic, experiment had indeed reinforced the capacities of mechanics by the end of the nineteenth century. With economics, and its rather amorphous object—that of economic activity in general—the connection between mathematical reasoning and empirical verification becomes more challenging. Yet Volterra quite enthusiastically remarks that “our mechanical scientist sees in the logical process for obtaining conditions for economic equilibrium the same reasoning he himself uses to establish the principle of virtual work, and when he comes across the economic differential equations he feels the urge to apply to them the integration methods which he knows so well” (cited in Weintraub Reference Weintraub2002, p. 50).

Given the nature and enthusiasm of Volterra’s endorsement of the application of mathematics to economics in order to render it scientific, it would be tempting to say that the work of Léon Walras—specifically his Éléments d’Économie Pure (1874)—most perfectly epitomizes what Volterra espouses. As Michael Turk (2002) points out, of the economists that Volterra cites as forerunners in the mathematical endeavor, Walras was the only one alive at the time, suggesting a sort of mutual admiration (2002, p. 153).

Turk also refers to a letter that Walras wrote to the Nobel Peace Prize Committee in January of 1906, which, in an attempt to promote himself for the prize, references the legitimation provided by Volterra’s recently translated lecture in the Revue du mois.Footnote 12 Yet this is hardly firm ground upon which to posit a causality of ideas. It should be noted that in his lecture Volterra also mentions the mathematical work of Fisher and Pareto, both of whom were still alive at the time and among Volterra’s more regular correspondents. In fact, Volterra published only three papers relating to economics, none of which deal with Walras’s work in any substance.Footnote 13 The only correspondence between Walras and Volterra appears to be of courteous symbolism. Their thanks and calling cards can be found in the respective archival collections of their work in Lausanne and Rome, but there is no significant content to their correspondence, no interchange of ideas or queries regarding their respective areas of research.

Nonetheless, if we compare Walras’s theory of equilibrium to Volterra’s mathematical project, an interesting connection can be seen. In Volterra’s archived work from 1890 until 1906,Footnote 14 his notes deal almost exclusively with the development of mathematical theories of mechanical equilibrium. This is relevant because equilibrium is the conceptual bridge that links forms of mathematical exposition and explanation to the economic world. Pareto later explained Walras’s role in developing mathematical economics precisely through the foundation of free competition, where economic forces tend towards equilibrium with a universality as reliable as gravity itself.Footnote 15 While Volterra and Walras may differ in both their motivations for exploring equilibrium and the nature of their studies, their connection is particularly salient if contrasted with other dominant thinkers of the time who were also working towards an increasingly rigorous and mathematically grounded science, such as Alfred Marshall.

In the vast literature on Walrasian thought and method, we see many treatments of the mathematical innovations of Walras’s work. But our purpose here is to seek a connection to the mathematical project of Volterra, and as such to examine Walras’s attempt to tie his economic science to the mathematical mode of reasoning more generally. The mathematical realm of economic thought is contained within pure economics, as opposed to his social or applied theories. For him, the logic of his Elements is purely mathematical:

This whole theory is mathematical. Although it may be described in ordinary language, the proof of the theory must be given mathematically. The proof rests wholly on the theory of exchange; and the theory of exchange can be summed up in its entirety in the following double-condition of market equilibrium: first, that each party to the exchange attain maximum utility and secondly, that, for each and every commodity the aggregate quantity demanded equal the aggregate quantity offered by all parties. (Walras [Reference Walras and Jaffé1874] 1952, p. 43)

The relationship between individual desire and collective equality is determined throughout the Elements with a priori laws. These must later be subjected to empirical verification with examination of trading patterns on markets. As a result we see Walras’s claim that

the physico-mathematical sciences, like the mathematical sciences, in the narrow sense, go beyond experience as soon as they have drawn their type concept from it. From real-type concepts, these sciences abstract ideal-type concepts which they define, and then on the basis of these definitions they construct a priori the whole framework of their theorems and proofs. After that they go back to experience not to confirm but to apply their conclusions.… Reality confirms these definitions and demonstrations only approximately, and yet reality admits of a very wide and fruitful application of these propositions. (Walras [Reference Walras and Jaffé1874] 1952, p. 71)

The need for empirical confirmation is strong in Walras’s work, and he reasserts this concern throughout the Elements after every mathematical exposition. As I have argued elsewhere, “verification serves to bridge the gap between mathematical models and empirical reality, a gap that exists because individuals do not calculate their trading schedules using complex calculus and mathematical proofs” (Scott Reference Scott2013, p. 100). But unlike Volterra, who posited that one ought to compare with reality the consequences of the deductions one has made from hypotheses and then after proceed to modify those hypotheses, Walras believes that his a priori laws will conform sufficiently to reality for us to then apply our propositions and continue without the same constraints of empirical verification. In other words verification is not a methodological constraint in the same way as it is in Volterra’s prescription. Volterra was more concerned with the rigor of the model and Walras more concerned with the purity of the mathematical exposition in such as way as to discourage premature empiricity or application.

The weight of Walras’s theoretical goals in the Elements—to systematically establish equilibrium in increasingly complex markets with the use of linear equations—pushes him to exclude questions of crisis, labor, and the state from his system, saving the latter two for his applied economics. Roberto Baranzini (Reference Baranzini2005) argues, for example, that Walras’s pure economics represents a sort of ‘positivist finalism’ whereby economic and social ideals, and the necessity of realizing them, are endogenous to social and political economy and to their epistemology. Further,

the concept of crisis is antithetical to the rationality of pure science. When Walras realizes that the crises that he is describing are not fleeting disturbances, he is forced to expel them from pure economics to relegate them to applied economics. In pure economics there remains the quantitative theory of money which is compatible with the perfection of the science. (Baranzini Reference Baranzini2005, pp. 107–108; my translation)

The perfection of the science in Walras’s case involves the construction of a system that includes the individual component of marginal utility on greater and greater scales. This involves greater acceptance of abstraction in dealing with aggregates in the course of attempting to come closer to the complexity of larger and larger markets, and finally by attempting to deal with time and the dynamics of equilibrium. Regardless of Walras’s delayed ‘return to reality’ (relative to Volterra’s methodological recommendation), an empirical reality nonetheless informs the structure of his system. For, beginning with individual exchange, it goes on to incorporate larger and larger markets in order to reproduce the complexity of the social world.

It is important to note that Walras’s competence in physics did not correspond to his vigorous application of the analogies to his Elements (Mirowski Reference Mirowski1989, p. 258). The metaphor of the ‘economic universe,’ which functioned by laws as stable as gravity, gives us the clearest link between the work of Walras and the methodology set out in Volterra’s inaugural speech. At the same time, however, this correlation is somewhat misleading. Perhaps the fact that “in their correspondence and in their published work, the early neoclassical economists recognized each other as mathematical theorists first and foremost, and when they proselytized for their works, it took the format of defending the mathematical model in the context of economic theory” (Mirowski Reference Mirowski1989, p. 195) was part of what later drove Volterra, in the footsteps of Henri Poincaré, to become wary of the premature application of mathematics to the question of desire through the concept of utility.

It will be useful now to turn to Walras’s final essay, “Économique et Mécanique” (1909), in which “he decided to address himself to physicists and mathematicians as the only scientific circles capable of truly appreciating his work” (Ingrao and Israel Reference Ingrao and Israel1990, p. 90). Indeed, Walras comments, “With [economists] it would be pointless to argue: they do not speak the same language as us. But with mathematicians it is otherwise: we can explain ourselves and maybe even understand one another” (Walras Reference Walras and Dockès1909, p. 131; my translation). For the purpose of a comparison of the respective methods of Walras and Volterra, a few points from “Économique et Mécanique” are worth exploring because Walras takes from Fisher’s Reference Fisher1892 Mathematical Investigation to explain the connection between mechanics and economics, but does not alter the content of his earlier theory of equilibrium to adapt to some of these newer claims of analogy.Footnote 16 Fisher’s account of the parallels between mechanics and economics will become relevant later on as we explore the connection between the work of Volterra and Fisher.

In “Économique et Mécanique” Walras is concerned with showing that utility is indeed measurable and can be rendered objective, rather than remaining a subjective and isolated statement of taste. This is a response to Poincaré, who had previously expressed doubts about the systemic mathematization of desire.Footnote 17 In order to do this Walras distinguishes between two categories of mathematical facts. The first category of physico-mathematical facts is constituted by external facts, those that occur outside of us, in the “theatre of nature” (p. 330). These facts, as they are external, are universally appreciable and can be objectively measured. Mechanics and astronomy use these physico-mathematical facts in their mathematical reasoning. However, there is a second category of mathematical fact, psychico-mathematical facts, which are intimate and take place within us. Each human subject hosts these facts as part of their consciousness, and for this reason they cannot be appreciated by others in the same way that we appreciate them ourselves. Thus, these psychic facts are both individual and subjective. Economics, Walras argues, uses these psychico-mathematical facts in its mathematical reasoning.

Let us turn to the reasoning that Walras uses to justify his approach to the mathematicians. Exchange value in economics is the key to quantification, but quantification alone is not enough. Walras establishes the first steps of the quantification of need through the capacity of the individual to rank his or her desire for commodities. The act of comparison allows for the appreciable determination of preference, and this, he argues, is not a moral issue or a question of the philosophy of happiness, but rather a fact of economic measurement. Walras sought to show that the psychico-mathematical method of economics was in fact developed in a manner “as rigorously identical to that of two most advanced and most undisputed physico-mathematical sciences: rational mechanics and celestial mechanics” (Walras Reference Walras and Dockès1909, p. 332; my translation).

The analogy between the mechanical and the economic method that Walras describes in his final essay involves four theories, formulated as two parallel sets. The first is the theory of the maximum satisfaction of the subject engaged in exchange, equivalent to the theory of the maximum energy of the roman scale. In the first instance the maximum satisfaction is determined by calculating the proportionality of the rareté of each good with the exchange value. In mechanics the analogous theory takes a roman scale and determines the inverse proportionality of force on the two arms of the lever. Walras cites Fisher’s Mathematical Investigations and the set of equations that Fisher makes between mechanical and economic variables, in order to justify his analogy.Footnote 18 In this way force and rareté are considered as vectors, on the one hand, and energy and utility are considered scalar quantities, on the other (Walras Reference Walras and Dockès1909, p. 334).

The second is the theory of the general equilibrium of the market, equivalent to the equilibrium of celestial bodies. This reiterates the statement of method from his Elements. Ultimately Walras argues that the only difference between these parallel theories is that the two mechanical theories are characterized by exteriority, and the two economic theories are characterized by interiority. Because we can calculate relative values with psychico-economic facts, we are able to call it a mathematical science (Walras Reference Walras and Dockès1909, p. 339).

Walras and Volterra shared an intellectual goal of rendering mathematical science a widespread form of social inquiry. Yet if Volterra’s preliminary notes for his inaugural lecture, or the correspondence between Maffeo Pantaleoni and Volterra are to be given any weight, Walras appears to have been secondary to Volterra’s interest in the work of Fisher.Footnote 19 It is further the case that Vailati pointed Volterra in the direction of many of these authors to begin with, yet this was nonetheless after Volterra had already determined the topic of his address (DeZan and Pizzarelli Reference De Zan and Pizzarelli2013). A look at Volterra’s preparatory notes for his lecture reveals that even William Stanley Jevons captured his imagination more firmly than Walras. Specifically, the fundamental importance of Jevons’s now famous claim that “it seems perfectly clear that Economy, if it is to be a science at all, must be a mathematical science” is obvious in Volterra’s notes.Footnote 20 The passage from Jevons appears to be axiomatic, an obvious imperative requiring no justification (at least not from the point of view of the mathematician). What was still required, however, was a mathematical mode of explanation sufficient to the project of economic modeling. For this we can turn to the late nineteenth-century work of Irving Fisher.

III. VOLTERRA AND FISHER: POSTULATING ECONOMIC REALITY

Irving Fisher played an important role in the promotion of economics as a mathematical science as well. While he didn’t enjoy the same position of influence as Volterra did in 1901, he was certainly widely read and became an early American advocate for change in the discipline along with thinkers such as John Bates Clark and Wesley Clair Mitchell (Tobin Reference Tobin1985; Breslau Reference Breslau2003). The trail connecting Volterra to Fisher at the point when Volterra was writing the inaugural lecture (for we will see evidence of their correspondence afterwards) can be found in Volterra’s first footnote where he references Fisher’s Mathematical Investigations in the Theory of Value and Prices (1892). He explains that an exposition of Fisher’s mechanical model can be found in Colonello Enrico Barone’s article in the Giornale Degli Economisti (Reference Barone1894). But most importantly, he references Fisher’s thorough bibliography of mathematico-economic writers in the appendix of his Investigations. The appendix would have been a valued resource for Volterra, given that he was already seeking counsel on good sources of mathematical economics. Further, Fisher’s bibliography would have been regarded as authoritative because it was based on Jevons’s famous list of mathematical economists, though it added on to this list by filling in gaps and extending past Jevons’s death, culminating with Fisher’s own 1892 publication as the final entry.

Fisher’s Reference Fisher1892 dissertation had been influenced by the suggestion of one of his mentors, William Graham Sumner, that he combine mathematics and economics (Barber Reference Barber2005, p. 46). He was also influenced by Josiah Willard Gibbs in his early years of study at Yale. The branch of physics that explored thermodynamics was particularly important for the very structure of Fisher’s thought: “In his formal mathematical model building too, Fisher was greatly impressed by the analogies between the thermodynamics of his mentor Gibbs and economic systems, and he was able to apply Gibbs’s innovations in vector calculus” (Tobin Reference Tobin2005, pp. 25–26). Marcel Boumans (Reference Boumans2001) draws an interesting parallel between Gibbs’s use of geometrical illustration and Fisher’s appreciation of visualization. This particular connection will become relevant when we consider the very nature of Fisher’s models later on: “While Gibbs saw geometrical illustrations mainly as aids to the imagination, Fisher stressed the role of visualizations because they helped one to understand a system or phenomenon. It connected the unknown to something familiar, something with which one had experience” (Boumans Reference Boumans2001, p. 322). Fisher’s published dissertation garnered him significant respect from several proponents of mathematical economics in Europe, and set him apart from his peers at Yale who were primarily influenced by the German Historical School. Of his dissertation, once published, he “observed at the time, his ‘little book’ amounted to ‘a letter of introduction to a great many people.’ It opened the way for meetings with Pantaleoni in Rome, with Pareto and Walras in Lausanne, with Böhm-Bawerk and Menger in Vienna, and with Edgeworth in Oxford” (Barber Reference Barber2005, p. 47). Thus, in many ways Fisher was a conduit from Europe to North America of budding mathematical approaches to economic thought. And while the development of the mathematical economic science did not begin in earnest until the 1930s, Fisher continued, over the course of thirty years, to develop economic ideas that diverged considerably from those of his colleagues at Yale.

While it is clear that Volterra believed Fisher to be expert in the tradition of mathematical economics, we also know Fisher likewise to have held Volterra in very high regard. Indeed, it appears it was not Walras but Fisher who turned to Volterra for encouragement, collaboration, and dialogue throughout their correspondence. Fisher first wrote Volterra in February of 1902, thanking him for his “interesting and valuable” essay—a reference to a version of his inaugural lecture that Volterra had sent to Fisher.Footnote 21 But it was not until the 1930s that the correspondence between Volterra and Fisher commenced in earnest. Viewed in historical context, this might be explained by many of the difficulties that had plagued Fisher in the late ’20s and early ’30s. He had suffered a serious financial and reputational loss due to the 1929 crash and subsequent depression. It may be possible that the latter pushed him to reach out to old supporters who had bolstered him during his early career, and that the former pushed him toward the foundation of the Econometric Society with Ragnar Frisch and Charles Roos in 1930, along with its journal, Econometrica (Tobin Reference Tobin2005, p. 19).

Thus, in 1931 Fisher reignited their correspondence by soliciting Volterra’s membership in the Econometric Society, explaining that “the purpose of this international society is the advancement of economic theory in its relation to statistics and mathematics, and we hope the society may be a significant influence in transforming economics into a real science.” Later, in January of 1932 Fisher asks Volterra’s advice regarding potential new members for the Econometric Society, and requests a bibliography of his work on econometrics. More interesting, however, is a letter dated November 1933 to which Fisher attached a copy of his “Debt-Deflation Theory of Great Depressions” from Econometrica of the same year. In his accompanying letter Fisher expresses anxiety about having first expounded his debt-deflation theory in non-mathematical language in Booms and Depressions, fearing that while this was necessary to bring about attention from and reception by non-mathematical thinkers, the originality of his ideas may be overlooked, and what he quite proudly felt to be his discovery might not appear as such, given the popularization of his ideas before the exposition of the mathematical instantiation of his cycle theory in an academic publication. Specifically, he writes:

I did not try, in “Booms and Depressions” itself, to make any specific claims but merely made a very general statement in the preface. This was partly because of my unfamiliarity with the literature (although thus far no one has found the theory definitely anticipated). Moreover, the book was written for immediate usefulness and I feared that, if I emphasized the newness of the theory, I might excite distrust in the lay reader.

But now that the theory is being widely accepted, I fear it may not be traced to the proper source and that the book may be regarded merely as a popularization of previously accepted conclusions. I would be grateful if you can supply me with any additional names of ‘cycle’ students and still more if you can indicate any anticipation of this theory other than the few partial anticipations noted in this article.

While Volterra’s errant response would surely prove to be fascinating,Footnote 22 the important point here is that if one is to make claims on the basis of interpersonal relations, the case to be made for the Volterra–Fisher connection is by far the stronger. Also, this too, like Walras’s 1906 letter to the Nobel Peace Prize Committee, is a question of relationship established after the fact (and due to the fact) of the inaugural lecture. But even, or especially, when we turn to the content of both Walras’s and Fisher’s claims regarding the mathematical methods appropriate to economic science, the correlation between Fisher and Volterra remains the more convincing case of connection.

The claim that the Volterra–Fisher connection is more relevant than the Volterra–Walras connection demands comparison of the work of Fisher and Walras. A comparison of the two reveals that Fisher’s mode of reasoning corresponds more concretely with the mechanical theory of equilibrium, which was one of Volterra’s foremost preoccupations in his mathematical work at the turn of the twentieth century.

Despite the fact that Fisher credits Jevons’s Theory of Political Economy (Reference Jevons1871) and Rudolf Auspitz and Richard Lieben’s Untersuchungen über die Theorie des Preises (1889) as the two most formative textual sources for his Mathematical Investigations (providing him with conceptions of marginal utility and symmetry, respectively), Walras was a genuine mathematical ally insofar as he developed similar formulae for the determination of marginal utility in multi-commodity markets. In the preface to his Investigations Fisher states that the only differences between their approach to markets with m commodities and n consumers “are that I use marginal utility throughout and treat it is a function of the quantities of commodity, whereas Professor Walras makes the quantity of each commodity a function of the prices” (Fisher Reference Fisher1892, p. 4). Fisher understands this coincidence in result to be an endorsement of the mathematical method in economics, offering a certain degree of objective verification, especially as he claims he was not aware of Walras’s work while he wrote his dissertation (Tobin Reference Tobin1985).

Yet these separate paths are nonetheless important. Consider, as Mary Morgan shows, the distinct mathematical tendencies of the late nineteenth century, with “the method of postulation and proof,” on the one hand, and “the method of hypothetical modelling using mathematical models,” on the other (Morgan Reference Morgan2012, p. 18). She asserts that Fisher and Walras demonstrate the distinction between these two approaches perfectly. Walras’s Elements introduces a new method of mathematical postulation and proof while other texts, such as Marshall’s Principles and Fisher’s Mathematical Investigations, were developing models with which to employ the mathematical reasoning itself: “The fact that Fisher built his hydraulic model to represent Walras’ ideas, and to figure out by exploring with that physical model the process by which the latter’s mathematically postulated and proved general equilibrium might be arrived at, shows us the difference between them” (Morgan Reference Morgan2012, p. 18). The conceptual distinction between the two approaches implies a distinction in goals (though of course they both aimed to render economics mathematical), whereby Walras, working his way up from simple exchange, desired to explain the laws of an entire economy of free exchange operating in a movement toward equilibrium, while Fisher was constructing a model of the mechanical operation of a materially constrained economy, not purporting to discern a priori laws of what Walras termed the “economic universe.”Footnote 23

The geometrical dimension of equilibrium is part of Fisher’s rather unusual perspective at the time. What specifically merits attention with respect to his connection with Volterra, given the promotional content of the latter’s lecture, is the commentary on the epistemological necessity of the mathematical method, specifically on modeling based in mechanics. Fisher’s broader understanding of the relationship between economics and mechanics exists in tandem with that of Volterra in four key areas: 1) the development of a coherent and systemic mechanical analogy in economics; 2) the use of marginal utility theory with a specific conception of homo oeconomicus; 3) the expression of economics as a distinct ‘field of vision’; and 4) the linkage of the economic imagination with mathematical symbolism.

First, for Fisher as for Volterra, “the introduction of mathematical method marks a stage of growth—perhaps it is not too extravagant to say, the entrance of political economy on a scientific era” (Fisher Reference Fisher1892, p. 109). He too points to Jevons as the first step along this evolutionary path. It is interesting to note that both Fisher and Volterra think in terms of evolutionary language in their perception of the development of economic thought, but do not employ evolutionary philosophy like Marshall, or employ natural or developmental metaphors to describe the economic system itself. More importantly, however, Fisher makes the mechanical analogy both explicit and complete.Footnote 24 As we saw later echoed in Walras’s “Économique et Mécanique,” Fisher equates the variables of mechanics—particles, space, force, work, and energy—with a parallel group in economics—individuals, commodity, marginal utility or disutility, disutility, and utility, respectively (Fisher Reference Fisher1892, p. 85). Thus, what Walras, in his Elements, was ultimately attempting to describe mathematically, regarding the equilibrium of greater and greater markets through the marker of the numéraire,Footnote 25 Fisher demonstrates as a different kind of law: “equilibrium will be where gain is maximum; or equilibrium will be where the marginal utility and the marginal disutility along each axis will be equal” (Fisher Reference Fisher1892, p. 86). Note here the parallel in mechanics: “equilibrium will be where net energy is maximum; or equilibrium will be where the impelling and resisting forces along each axis will be equal” (Fisher Reference Fisher1892, p. 86). In order to explain equilibrium Walras gives us proportional tendencies in a two-dimensional account of reality, while Fisher employs a series of relational forces in a three-dimensional mechanical model.

In the second substantive area of connection between Volterra and Fisher, the latter argues that there is no idea more fruitful than marginal utility in the history of economics as a science. While utility forms the bedrock of Fisher’s system and is perhaps the sole indicator of human agency therein, he prefers to keep the nature of subjectivity itself rather simple. With reference to the history of the term, he clarifies that “perhaps utility is an unfortunate word to express the magnitude intended. Desirability would be less misleading, and its opposite, undesirability, is certainly preferable to dis-utility. ‘Utility’ is the heritage of Bentham and his theory of pleasures and pains. For now his word is the more acceptable the less it is entangled with his theory” (Fisher Reference Fisher1892, p. 28). Ultimately subjectivity is constrained within a clear postulate: “each individual acts as he desires” (Fisher Reference Fisher1892, p. 28).

It is thus consistent that Fisher, like Volterra, eschews excessive psychologisms or non-scientific—that is, overly literal—understandings of homo oeconomicus. For Volterra, homo oeconomicus is interesting as a problematic in the resistance to the use of mathematics in economics more generally, and to a lack of mechanical knowledge more specifically. In his inaugural lecture he brings this archetypal figure to the foreground:

[T]he notion of homo oeconomicus which has given rise to much debate and has created so many difficulties, and which some people are still loathe to accept, appears so easily to our mechanical scientist that he is taken aback at other people’s surprise at this ideal, schematic being. He sees the concept of homo oeconomicus as analogous to those which are so familiar to him as the result of long habitual use. He is accustomed to idealizing surfaces, considering them to be frictionless, accepting lines to be nonextendable and solid bodies to be non-deformable, and he is used to replacing natural fluids with perfect liquids and gasses. (cited in Weintraub Reference Weintraub2002, p. 49)

A more subtle appreciation of the nature of analogy, and of the purpose of idealization, is necessary for the mechanical study of the universe. It is also necessary, implies Volterra, for the study of economic matter. But the object of the mechanical science has a well-delineated contour, and such obvious limits to study are not so clearly discerned in economics proper. What would we make of Poincaré’s reservation towards Walras’s smooth economic agents: those who are “infinitely self-interested and infinitely clairvoyant” (cited in Walras Reference Walras and Dockès1909, p. 341; my translation)? Fisher’s more basic postulation that “each individual acts as he desires” (Fisher Reference Fisher1892, p. 28) is likely more acceptable as a ‘smooth surface,’ as we will see in his theorization of consumers as cisterns. For Fisher, the postulation of economic reality is itself an idealization of economic surfaces and the limits of its scope. It is a postulation by means of omission, and what is included (commodities, individuals, and utilities) also represents an ideal world.

In his detailed hydraulic modeling of an exchange economy, Fisher represents homo oeconomicus—‘the economic man’—with a cistern, which has maximum and minimum levels of fluxion. ‘Man as cistern’ is based on the synecdoche, originating in the work of Adam Smith, that uses the stomach to represent the individual by means of his desire and capacity to consume. This particular idealization bears the clear advantage of omitting the subjective complexities of desire by limiting the question of appetite to the filling of a vessel. And since there are various appetites to be satisfied, “the economic man is to be regarded as a number of cisterns or stomachs, each relative to a particular commodity” (Fisher Reference Fisher1892, p. 30).

Figure 1. Consumers as Cisterns (Fisher Reference Fisher1892, p.32).

The idealization and abstraction of homo oeconomicus is particularly poignant because Fisher extends the model to encompass individual levels in the context of more and more commodities, until he finally reaches the ‘economic world,’ a late nineteenth-century precursor to what we might term “the Economy.” With m independent variations (instead of three), the explanatory capacity of utility extends outward. It is really only mathematics, he believes, that can allow us this more complete picture:

There is a curious glamor over ‘the fourth dimension.’ The popular interest is to prove that it ‘exists.’ Its origin historically and its present usefulness is in the interpretation of a fourth independent variation, i.e. in representing just such relations as now concern us. It seems unfortunate that only mathematicians should be acquainted with this fact. (Fisher Reference Fisher1892, pp. 79–80)

Because Fisher represents economic man with cisterns, and the economic world with a fourth dimension where commodity variation and our relation thereto can extend indefinitely, his conception of equilibrium tends to be more compact than even Walras imagined in his Elements. A narrow conception of “the economic world” emerges,Footnote 26 limited by its ideal constituents, or cisterns (or stomachs, or individuals consuming in the present). Indeed, this limitation reveals itself plainly through his treatment of comparative utilities.Footnote 27

The mathematical method itself becomes a utility in Fisher’s exposition, and the subjectivist grounding of marginal utility theory is evident, given that the utility of the method depends upon the intelligence of the economist. “The utility of mathematical method is purely relative, as is all utility. It helps greatly some persons, slightly others, is even a hindrance to some” (Fisher Reference Fisher1892, p. 107). The more basic the capacities of the thinking subject at stake—that is, the less well versed in advanced mathematics—the less useful the mathematical method will be for that subject. The truth value of the ideas attained, however, is a different matter, and it is clear that the utility function of the mathematical method in and of itself is great. It should be noted that this ‘in and of itself’ actually implies an economist who is an ideal thinking subject, a mathematician in his own right. The correlation between the universe and the economic world is glossed by the mathematical method itself. The method is capable of defining its object and thus constituting the limits of the reality it seeks to describe.

The expression of economics as a distinct field of vision implies that the economist is a particular type of thinking subject. This is the third clear point of connection between Fisher’s and Volterra’s perspectives. In many ways Fisher’s articulation of this ideal field of vision echoes the delineation of the thinking subject common to the Enlightenment. In an expression of rather exuberant optimism, Fisher declares that

the effort of the economist is to see, to picture the interplay of economic elements. The more clearly cut these elements appear in his vision, the better. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looks up in firm, bold outlines. The old phantasmagoria disappears. We see better. We also see further. (Fisher Reference Fisher1892, p. 119)

For Fisher, therefore, the economist too might fall victim to the fog and illusions of faulty and vague reasoning, but for the use of mathematics, which gives shape to the truth of economic reality.

The parallel in Volterra’s work is evident when he explains the relevance of calculus to the opening of the imagination of the economist: “Nobody will be able to foreshadow to the mathematician the wide horizons which will be opened by the narrow and thorny path to which he is constrained by his calculations” (Volterra Reference Volterra1901, p. 439). Both thinkers make their fundamental epistemological claims by distinguishing pre-mathematical reasoning as a time where vision is hazy, unclear, and imprecise. The Enlightenment concern for scientific reasoning is mirrored by the ocularcentric association between knowledge and vision. Fisher is clear in establishing the connection between good mathematical reasoning and mechanics. Demonstratively, he argues:

The truth is, most persons, not excepting professed economists, are satisfied with very hazy notions. How few scholars of the literary and historical type retain from their study of mechanics an adequate notion of force! Muscular experience supplies a concrete and practical conception but gives no inkling of the complicated dependence on space, time and mass. Only patient mathematical analysis can do that. This natural aversion to elaborate and intricate analysis exists in Economics and especially in the theory of value. (Fisher Reference Fisher1892, p. 3)

The terrain of inquiry thus must be simplified in order to be properly explained to a popular audience. And, as we have seen, in order to carry out an intricate mathematical analysis the economist must also simplify by relying on the idealizations that allow for the models to be constructed in the first place. This is, one expects, a trade-off that Fisher was aware of as he constructed his models.

A final point of connection between the substantive content of Volterra’s and Fisher’s work is the association of mathematical symbolism with the power of the economist’s imagination. Mathematical symbolism becomes, in short, the means through which the economist might better imagine the economic world. Fisher distinguishes between mathematics and mathematical method. Of course others, including Walras, had done this before him. But Fisher goes further:

We must distinguish carefully between what may be designated as mathematics and mathematical method. The former belongs ... to every science. In this sense economics has always been mathematical. The latter has reference to the use of symbols and their operations. A symbol may be a letter or a diagram, or a model. All three are used in geometry or physics. ... To avoid mathematical method is to do without the rule. Symbols and their operations are aids to the human memory and imagination. (Fisher Reference Fisher1892, p. 107)

Indeed, years later, Volterra explains that the use of mathematical symbols lends mathematics an air of mystery that separates it from the other sciences. So while it is useful, as Fisher puts it, in order to give order and aid human memory and stimulate the imagination, it also has historically served to prevent interdisciplinary crossover. As Volterra explains: “Unfortunately professional mathematicians are separated from the rest of the world by a barrier of symbols, that give a certain air of mystery to their reasoning and their work” (Volterra Reference Volterra1901, p. 440). At the same time this symbolic language is the essence of the work and reasoning itself, and thus poses a challenge to the interdisciplinary creation of the science of economics. In an ironic turn it was only ultimately through literary and non-mathematical expression that Fisher and Volterra could hope to persuade a new generation of thinkers to take up economics as a mathematical science.

V. CONCLUSION

A comparison among the respective programs of mathematization drawn out by Walras, Fisher, and Volterra has shown the nature of each thinker’s claims regarding the economist’s knowledge, his capacity to know, and the nature of the reality he wishes to describe. Further, we have discerned how a new mode of abstraction emerged within economic thought at this historical juncture. While all three were certainly engaged in the program of mathematization more generally, it has been shown that the appeal of modeling stemmed from the interdisciplinary confluence between mechanical science and economics.

The Walrasian method uses linear equations of price fluctuations in limited markets as the mathematical ground of the economic system in the Elements. Yet Walras’s use of the mechanical analogy that connects economics to the mechanical operation of the physical universe is literary and descriptive, rather than being wholly integrated into the nature of the exposition itself. For Volterra, the nature of reality corresponds with the physical universe because the applicability of mechanical science to reality is axiomatic. The study of economic reality, however, requires a particular form of methodological rigor in order to assure the coherence of the mechanical science to the object of study in the first place. To enable the application of mechanical science to the study of economics, economic reality requires clear parameters, which can be discerned through the process of abstraction. Fisher’s models achieve this postulation of reality by means of omission, limiting the variables in the economic world to individuals, commodities, and utilities, and, within these, limiting individuals to cisterns, commodities to a homogeneous fluid, and utilities to quantity as determined by level. Thus the principles of mechanics can be applied to the economics by the creation of hydraulic models, models that represent the parameters of the economic realm.

The implication of postulating reality—that is, postulating the elements that will constitute the only relational variables within an economic world—is vast. First, a clearly delineated ‘economy’ becomes our object of study. On the one hand, this represents a necessary trade-off in the quest to describe reality, and gives the economist a chance to explore the nature of certain relationships through the mathematical method. On the other hand, it renders the economy a coherent entity, an object that can be constituted by, and understood through, advanced mathematics only. The new terrain of economic knowledge, the new and extended field of vision, are available only for those who have the deep and rather disciplinary knowledge of mathematics itself. Moreover, the only subjects capable of understanding and reasoning through the economy are economists themselves (with, of course, the possible exception of a few select and predisposed mathematicians). The role of the economist as mediator becomes solidified with mathematical economics as the capacity for epistemological entry and the utility of the theory itself dwindle among the inexpert many.

Because economics is a discipline that today appears to be insular, deeply steeped in its own symbolic language, and consumed by mathematical modeling, it is rather counterintuitive to pursue an interdisciplinary history of its emergence. Yet interdisciplinary conjunctions, such as those that occurred between the early mathematizers like Jevons, Walras, Pareto, and Fisher, and mathematicians such as Volterra, in fact gave the discipline its coherence. This coherence, in the form of a near total systemic containment, in fact, engenders a general resistance towards interdisciplinary inquiry once mathematical economics takes hold. This is especially salient once the program of mathematizing economics gains mainstream acceptance from the mid-twentieth century onwards (Collander et al. Reference Colander, Holt and Rosser2004). Ironically, the interdisciplinary impulse to join the social and mathematical sciences, and to shed light on a universal truth governing the economic realm, renders the economic science more disciplinary than ever.

The distinction between the popular and specialized functions of economics helps to delineate some of the problems that have arisen from the conjunction between economics and mathematics. Fisher, with reference to the sophistication of Cournot’s early work, articulates the role of the economist in relation to the masses.

Mathematical economics is useless in a political mass meeting. But in every science there must be a differentiation of technical investigation from popular text writing or teaching. Is physics less practical because the X-rays are studied in the seclusion of the laboratory by highly mathematical methods, of which the work-a-day world has no conception? If the experience of other sciences is a guide, the best way to make economic theories practical is to make them perfect. ‘Our speculations can scarce ever be too fine, provided they be just.’ The profound technical treatise is a prerequisite of the good popular manual. What is not as clear as crystal to the writer of the first will never be clearer than Newfoundland fog to the reader of the second. (Fisher Reference Fisher1898, p. 138)Footnote 28

While the utility of the mathematical method is of little value to those not well versed in mathematical reasoning, the mathematical development of economics can nonetheless serve the public good, so long as the economist is sure to perfect his science. Yet, because mathematical symbols appear mysterious to non-experts, as Volterra and Fisher have both pointed out, the authority to develop the theories that will serve a public and political function resides with the mathematical economist alone, who must first develop these ideas free from the constraints of normative concerns.

I have tried to show how the authority necessary for the development of these theories is inextricably bound to the mathematical postulation of reality. The authority of the economist comes from first determining the proper object of economic inquiry and then determining the mathematical method he will apply thereto. The relationship between the exclusionary function of mathematical language and the authority the economist holds is revealed in the confluence between disciplines that first emerged as a genuine attempt to shed light on the world through the lens of a universal and true science.

Footnotes

I would like to thank Roberto Baranzini, Director of the Centre Walras-Pareto at the University of Lausanne, for access to archival materials. I also extend my gratitude to the Bibliothèque cantonale et universitaire Lausanne, and to the Archives of the Accademia dei Lincei Rome. I am indebted to both Daniele Besomi and Mark Peacock for their generous feedback and insight. Finally I would like to thank the anonymous peer reviewers for their careful reading and commentary.

1 This included such eminent thinkers as Stanley Jevons, Léon Walras, Vilfredo Pareto, and Irving Fisher.

2 Of course, classic mechanical physics was later unseated by developments in quantum mechanics, with consequences for the nature of mathematical modeling in economics that were not minor. In the course of this paper I aim to present the context of the mechanical science still in its prime, along with its relationship to economic reasoning. As a stylistic preference, I have chosen to write in the present tense when discussing the authors’ intentions and ideas, but do not intend to indicate the universality or contemporary validity of these positions.

3 Of course the literature on this matter is too vast to give an account of here, but particularly relevant sources in the history of mathematization include Walsh and Gram (Reference Walsh and Gram1980), Jaffé (Reference Jaffé and Walker1983), Mirowski (Reference Mirowski1989), Ingrao and Israel (Reference Ingrao and Israel1990), and Walker (Reference Walker2006).

4 Extensive and important work has been carried out by Giorgio Israel, the foremost expert on Volterra and his mathematical legacy, along with Ingrao and Israel (Reference Ingrao and Israel1990), Israel and Gasca (Reference Israel and Gasca2002), and Weintraub (Reference Weintraub2002).

5 Giovanni Vailati (1863–1909) was a mathematician and historian who served as an assistant to Volterra in Torino, and whom Volterra helped receive a post in the history of mathematics. Prior to Volterra’s inaugural lecture Vailati taught a course on the history of mechanics. Guerraggio and Paoloni (Reference Guerraggio and Paoloni2013) show the influence that Vailati had on Volterra’s conception of the history of mathematical economics, and the full correspondence between Volterra and Vailati has recently been published in its original Italian by De Zan and Pizzarelli (Reference De Zan and Pizzarelli2013).

6 The original letter, written on the first of July of 1901, reads: “Per quanto me ne sia schermito non mi è riescito di dispensarmi dal pessantissimo incarico di fare il discorso inaugurale nel prossimo anno scolastico all’Università. Che fare per non riescire del tutto privo di interesse fra i cultori // delle altre discipline fuori delle matematiche? Avrei pensato di far qualche cosa sui tentativi di applicazione delle matematiche alle scienze biologiche e sociologiche. Che ne dice di un argomento simile? Le va o lo sostituirebbe con qualche altro? Nel caso quali libri sono a sua conoscenza che mi potrebbero riescire utili?” (DeZan and Pizzarelli Reference De Zan and Pizzarelli2013, p. 97). Vailati’s response, two days later, contains bibliographical suggestions from the biological sciences (Francis Galton’s Hereditary Genius, Karl Pearson’s Grammar of Science, and John Venn’s Logic of Chance) and recommends Vilfredo Pareto and Philip Wicksteed as the best representatives of mathematical economics. He also goes on to assess Pareto’s treatment of marginal utility and ophelimity, arguing that the latter presents a false analogy and that his work was much improved when he abandoned this notion and settled in to pure quantitative concepts by means of indifference curves. Vailati’s concerns are later echoed in Volterra’s lecture. The rest of Volterra’s presentation of important mathematical economists, I believe, stems from Irving Fisher’s Mathematical Investigations in the Theory of Value and Prices (1892). This connection will become important later on in this article.

7 With the help of Maffeo Pantaleoni, the lecture was quickly published in the Giornale degli economisti in November 1901 and then translated into French and published in La revue du mois in 1906.

8 The first predator–prey theory, predating the Lotka–Volterra formulation by almost 130 years, also arose from the combination of economic and mathematical reasoning (Berryman Reference Berryman1992). Robert Malthus’s famous theory of population (whereby the population can grow exponentially but resources increase only arithmetically) gives precedent to the connection between the biological, economic, and mathematical sciences. Later Lotka and Volterra developed a more mathematically sophisticated theory. Lotka’s 1925 Elements of Physical Biology gives us insight into the core of this theory. See Berryman (1992, p. 1531).

9 All translations of the published version of Volterra’s Reference Volterra1901 lecture—“Sui Tentativi di Applicazione Delle Matematiche Alle Scienze Biologiche e Socialeare my own unless otherwise referenced.

10 Specifically, Volterra mentions each of the following economists: Stanley Jevons, Vilfredo Pareto, William Whewell, Antoine Augustine Cournot, Hermann H. Gossen, Léon Walras, and Irving Fisher.

11 See Ingrao and Israel (1990, pp. 139–171) for a discussion of the tenuous connections between economists and mathematicians.

12 Making reference to Volterra’s lecture, Walras writes to the Nobel Peace Prize Committee: “The author, as he himself states, is more a mathematician than an economist; but, just the same, no one has better recognized in our ‘analytical economics’ the proceedings by the same means as which Maxwell and Helmholtz have transformed physics and physiology, and makes comprehensible how economic questions such as free exchange, for example, which are and remained so tangled and confused in everyday language, can be brought to a solution so clear and sure through the use of the mathematical method” (cited in Turk 2002, p. 153). On Walras’s attempts at the Nobel Peace Prize, see Sandmo (Reference Sandmo2007).

13 In this particular piece—“L’Economia Matematica: Ed il nuove manuale del prof. Pareto”—Volterra argues that “Pareto has been one of the scientists, as Croce has rightly observed, who has most contributed to rendering economics a pure science giving it the character of the natural and physical mathematics. The study of economics, says Croce, has in recent times been characterized by a double-movement. On the one hand it has managed to free itself from the burden of all the political and practical questions and restricted itself to the simple consideration of effective reality; on the other hand it is becoming yet more impartial in the debate around the nature of economic facts and on the relationship between this and other aspects of reality enclosing itself in the realm of phenomena” (Volterra Reference Volterra1906, p. 296). Here the delineation of reality is quite clear, and reveals the nature of the reality that Volterra thought corresponded to the proper terrain of economic study—free from, as Joseph Schumpter would later say, the vicissitudes of the ideological realm—such as politics and other normative constraints.

14 Accademia Nazionale dei Lincei, Fondo Volterra, boxes 5–13.

15 Pareto writes: “Walras was the first to consider the connection of economic phenomena and to give them a system of equations which represents and determines such connections with the supposition of free competition. To this system of equations falls the name ‘the equations of Walras’” (Pareto Reference Pareto1906, p. 426; my translation). It was these equations that Fisher later replicated, albeit independently, in his Mathematical Investigations (1892) to construct his hydraulic models of equilibrium proper.

16 “Économique et Mécanique” has been examined by several scholars (Mirowski Reference Mirowski1984, Reference Mirowski1989; Le Gall Reference Le Gall2002), but in particular Ingrao and Israel (Reference Ingrao and Israel1990) give a detailed account of the essay and Walras’s quest (and relative failure) to gain legitimacy with the mathematicians of his time.

17 Walras in 1909 felt he had to satisfy Poincaré’s issue with the transferability of the individual and subjective fact of desire into the objective material of mathematical analysis. Poincaré had several objections to the analogy that Walras established between physics and economics. For Poincaré, the reasonable limit of analogy was crossed when the hypotheses underlying the analysis were ignored. In a letter written to Walras in 1901, he comments on the Elements d’économie politique pure (fourth edition) and elaborates: “For example, in mechanics, we often neglect friction and regard bodies as infinitely smooth. You regard men as infinitely self-interested and infinitely clairvoyant. The first hypothesis can be admitted in a first approximation, but the second would perhaps necessitate some reservations” (cited in Walras Reference Walras and Dockès1909, p. 341; my translation). Walras believed that his explanation of the psychico-mathematical fact was sufficient to alleviate these reservations, though this theory was clearly developed after he had written the Elements, and with the help of some thinking from Fisher and Antoine Cournot (Walras Reference Walras and Dockès1909, pp. 333–334).

18 This set of parallels will be of particular importance in the next section of this essay. For the moment, however, it suffices to note that Walras borrows from Fisher well after having written his Elements.

19 Pantaleoni comments, on November 4, 1901, that he remembered Volterra’s interest in the economic mechanics of Irving Fisher, and points him to Colonello Barone. He also asked Volterra to indicate Schiapparelli’s publication and to mention his geometric argument in his lecture. Three days later Pantaleoni writes again and relates to Volterra that Barone found Fisher’s manuscript fascinating (Accademia Nazionale dei Lincei, Fondo Volterra).

20 Volterra singles this passage out by separating it from the rest of his citations and bibliographic references as well as by using a larger script (Accademia Nazionale dei Lincei, Fondo Volterra, Box 10, Folder 168, cc.233).

21 All of the letters cited from Fisher to Walras are housed in the Accademia Nazionale dei Lincei’s Archivio Vito Volterra 1860–1940. The extensive Fondo Vito Volterra includes seven letters from Fisher to Volterra on the following dates: 1) 25 February 1902; 2) 21 December 1931; 3) 29 January 1932; 4) (n.d.) November 1933; 5) 20 March 1934; 6) 27 July 1934; 7) 1 August 1934.

22 My research in both the Volterra Archive at the Accademia di Lincei in Rome and in the Fisher Archive at Yale has not revealed Volterra’s response to Fisher. The letters from Fisher to Volterra that I have presented in this paper remain unpublished to date.

23 Daniel Breslau pushes the implication of this distinction even further. He argues that Fisher’s contribution was in fact deeply embedded in the nature of his hydraulic models themselves, which required both abstraction and homogenization in order to function: “[I]n all of these areas—price formation, the monetary system, interest—Fisher proceeded by translating problems that had been understood in terms of differential social actors and goods, into terms of a mechanical system equilibrating a homogeneous substance” (Breslau Reference Breslau2003, p. 399).

24 Fisher is aware of his innovation: “The student of economics thinks in terms of mechanics far more than geometry, and mechanical illustration corresponds more fully to his antecedent notions than a graphical one. Yet so far as I know, no one has undertaken a systemic representation in terms of mechanical interaction of that beautiful and intricate equilibrium which manifests itself on the ‘exchanges’ of a great city but of which the causes and effects lie far outside” (Fisher Reference Fisher1892, p. 24).

25 For example, in his solution to the multi-commodity problem, Walras posits the law of the establishment of equilibrium price as such: “Given several commodities, which are exchanged for one another through the medium of a numéraire, for the market to be in a state of equilibrium or for the price of each and every commodity in terms of the numéraire to be stationary, it is necessary and sufficient that at these prices the effective demand for each commodity equal its effective offer. When this equality is absent, the attainment of equilibrium prices requires a rise in the prices of those commodities the effective demand for which is greater than the effective offer, and a fall in the prices of those commodities the effective offer of which is greater than the effective demand” (Walras [Reference Walras and Jaffé1874] 1952, p. 172).

26 Some authors have argued that what we can call “the Economy”—as a complete and coherent entity—emerged via Fisher in the late 1920s or early 1930s in the US. See Breslau (Reference Breslau2003) and Tribe (2015).

27 Fisher’s comments on utility as a quantity are fascinating—as much for cultural and historical reasons as for economic ones—when he turns to reflect upon the capacity to compare the utilities of different people. Though this is outside the purview of his model, he sees both the ethical and economic value to such comparison: “If persons alike in most respects show to each other their satisfaction by similar gestures, languages, facial expression, and general conduct we speak of their satisfaction as very much the same. What however this may mean in the ‘noumenal’ world is a mystery. If on the other hand differences of age, sex, temperament, etc., enter, comparison becomes relatively difficult and inappropriate. Very little could be meant by comparing the desire of a Fuegian for a shellfish with that of a college conchologist for the same object and surely nothing is meant by comparing the desires of the shellfish itself with that of its tormentors” (Fisher Reference Fisher1892, pp. 86–87).

28 The passage—“Our speculations can scarce ever be too fine, provided they be just”—is from Hume’s essay “Of Commerce” from Essays and Treatises on Several Subjects, etc. (Reference Hume1758).

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Figure 0

Figure 1. Consumers as Cisterns (Fisher 1892, p.32).