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 Vailati^{Footnote 5} just prior to the composition of his lecture, he writes:

While his approach to the inaugural lecture may have been tentative, the great approval and praise with which his address was later met certainly^{Footnote 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:

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

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 network^{Footnote 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:

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 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 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:

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:

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:

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

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 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:

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.