Edwin Bidwell Wilson

Wilson was born in 1879, in Hartford, Connecticut. He was trained as mathematician at Harvard University, Yale University, and at the École Normale Supérieure in Paris around 1900. Wilson subsequently became one of the “most active” members among the American research community of mathematicians during the first decade of the 1900s (Fenster and Parshall Reference Fenster, Parshall, Knobloch and Rowe1994). He, however, gradually marginalized himself from that community, disavowing the influence that David Hilbert’s structuralist mathematics was then exerting on his American colleagues and concomitantly committing to the traditional applied American mathematics that Josiah Willard Gibbs, his mentor at Yale, practiced.^{Footnote 7} Wilson’s career illustrates this process of marginalization and corollary process of incursion into other fields. First, in 1907, he became associate professor of mathematical physics at Massachusetts Institute of Technology (MIT). Second, in 1922, he accepted the chairmanship of the department of vital statistics at the newly founded Harvard School of Public Health (HSPH), opening the door to his incursion into social science and economics. In parallel spheres, since 1914, when the Proceedings of the National Academy of Science (PNAS) was launched, Wilson served as managing editor of this journal until the end of his life in 1964.^{Footnote 8}

The task that Wilson gave himself consisted of interconnecting mathematics and different subject matters. At Harvard’s department of economics, between 1932 and 1943, Wilson gave a course on Mathematical Statistics and another on Mathematical Economics (since 1935), alternately every two years since 1935. Wilson aimed his instruction at protecting students from what he disdainfully regarded as the beauty of certain pure theoretical and/or mathematical contemporary works in economics.^{Footnote 9} He thought that students of economics, by learning his American Gibbsian constrained mathematics, would learn how to behave in a scientific way.^{Footnote 10}

Paul Samuelson

Samuelson was born in 1915, in Gary, Indiana. In 1932, he entered college at the University of Chicago, where he majored in economics. He performed exceptionally well and was awarded in 1935 a selective pre-doctoral scholarship given by the Social Science Research Council (SSRC). With it, he went to Harvard, with all expenses covered. During the 1935–36 academic year, Samuelson took, in particular, Wassily Leontief’s Price Analysis course, and Wilson’s course on Mathematical Statistics. Samuelson was then only twenty-one years old. The following spring, he attended Wilson’s course on Mathematical Economics. For most students, it was difficult, but Samuelson was mathematically well trained (Samuelson Reference Samuelson, Dietzenbacher and Lahr2004). In college, he had taken a significant number of mathematical courses. Also, during the summers of 1935 and 1936, he had taken extra curriculum courses on differential equations and on the theory of equations, where linear matrix equations were treated (Backhouse Reference Backhouse2015).

Eventually, Samuelson impressed Wilson. As he wrote to Lawrence Henderson, chairman of the Harvard Society of Fellows (HSF) and Wilson’s close friend, when recommending Samuelson as a Junior Fellow of the Society, Wilson believed that “one of the most brilliant young men in political economy whom I have ever met is Samuelson.… I had him in my course in mathematical statistics and he was the most original and inquisitive of all the students.”^{Footnote 11}

In 1937, Samuelson was elected Junior Fellow of the HSF. The membership came with a scholarship, and also with the restriction that the work undertaken under the Fellowship program should be somehow different from the work that would be submitted to obtain a doctoral degree. Presumably following this rule, Samuelson did not work to complete a comprehensive and well-constructed thesis. Between 1937 and 1940, instead, he conducted research and wrote an important number of papers, not all published, on consumer theory, cost and production theory, capital and investment theory, business cycles, population dynamics, international trade and welfare economics, as well as comparative statics and dynamics. In order to fulfill the requirements of the department of economics and to graduate, however, in 1940, Samuelson took some of his fellowship projects, put them together, added three introductory chapters and a mathematical appendix, and submitted a thesis, defended in November 1940.

“You did a fine job at your doctor’s examination,”^{Footnote 12} Wilson wrote Samuelson after the defense. Concerned about career opportunities for Samuelson, Wilson was then actively supporting Samuelson’s thesis to be considered for the David A. Wells Prize, which was awarded to Samuelson in 1942.^{Footnote 13}

The Opening Page

The first instantiation of Samuelson’s commitment to Wilson in matters of mathematics, statistics, and science appeared in the opening page of the thesis, where he wrote: “Mathematics is a Language.” Samuelson attributed, rightly or wrongly, this motto to Gibbs, the legacy of whom was transmitted to him by Wilson, who defined mathematics as a sort of language.^{Footnote 16} For Wilson, mathematics as a language implied two main ideas, which Samuelson probably wanted to evoke, and which set the spirit of the thesis since its opening page.

First, mathematics as a language implied, for Wilson, defining mathematics as connected with science and meaning.^{Footnote 17} For Wilson, mathematics consisted of establishing correspondences, as translations, between purely mathematical entities that represented certain mathematical structures, which he called “postulates,” and conventional working hypotheses found in subject matters, which he called “axioms.” In these translations, postulates and axioms must simultaneously restrict each other: while postulates imposed logical structure on the subject matter, acting thus as a sort of grammar, axioms constrained freedom and abstraction of postulates and gave them meaning connectedly to the subject matter, acting thus as a sort of semantics. Without their corresponding meaning in science, mathematical structures were as beautiful and as useless as pure theoretical treatises of subject matters, Wilson felt. At the same time, without their corresponding mathematical structures, subject matters could not achieve scientific status. For him, mathematical structures were indispensable in science to mediate between theory and data, as they were necessary to determine meaning. Wilson, however, insisted that emphasis should be placed on meaning and intuition rather than on pure consistency of close mathematical systems (Wilson Reference Wilson1904). He even believed that contributions in which the mathematics was not necessarily fully consistent but in which meaning and intuition of the subject matter filled the mathematical gaps could be regarded as truly scientific and well mathematically founded. In his words: “[W]hether the [work] is mathematically complete or not does not interest me; this is unimportant. Science advances not so much by the completeness or elegance of its mathematics as by the significance of its facts” (Wilson Reference Wilson1928a, p. 244).

At the same time, for Wilson, mathematics implied immediate usefulness, which could be achieved only if correspondences between postulates and axioms were established.^{Footnote 18} In such translations, mathematical operators and operations should be used. Sometimes, new operators and operations should even be developed, in accordance with the immediate problems at hand. This “operational … side,” Wilson believed, required applying “a series of rules of operation often both dull and unintelligible,”^{Footnote 19} generally found in algebra or advanced calculus, but which could be regarded as simply as the arithmetic operations of division and multiplication. These operations, he thought, “are not in themselves of practical or intellectual interest.”^{Footnote 20} Operational thinking, Wilson believed, was hence distinct from postulational and axiomatic thinking.

Wilson’s interest in axioms, as conventional working hypotheses, reflected his belief that they corresponded to the (ontological) invariances necessary for the use of mathematics in science, as they supposedly represented things that “change so slowly that we may regard them for practical purposes as non-changing or at any rate can assign limits to their change in amount and not [in] time.”^{Footnote 21} Also, Wilson thought, scientific knowledge resulted from a plurality of working hypotheses. Scientific knowledge was therefore never to be held as universally true, but merely as partial, probable, and approximate. Because the reason for prevalence of a certain working hypothesis over another was not self-evident (Wilson Reference Wilson1920), scientific knowledge, for him, was also conventional. In this way, as a result of the possibility to “assign limits to their change in amount and not [in] time,” working hypotheses conveyed truth and meaning, relative to the problem at hand, only in a certain proportion at given moments in time. Statistics, he thought, offered an operational way of determining the most likely working hypothesis, as it could be used to quantify that range that carried truth and meaning while connecting theory and data.

All this implied that in defining mathematics as a language, Wilson believed that the mathematician/scientist needed to be familiar with certain mathematical structures (postulates), to master the conventional working hypotheses (axioms) of the subject matter of interest as well as to know how to play with his skills in (vector) algebra, advanced calculus, and (mathematical) statistics in order to develop correspondences between postulates and axioms. When establishing these translations, Wilson insisted, the mathematician/scientist should endeavor to produce much science with little mathematics, by following the idea that meaning and intuition prevailed over mathematical consistency, even when the mathematics was highly sophisticated.^{Footnote 22}

Also, for Wilson, defining mathematics as a language implied regarding mathematics and its operational (algebraic and statistical) techniques as a vernacular, which all individuals could learn (Wilson Reference Wilson1940); and as Wilson stated: “[T]here [seemed] to be no present conclusive evidence that learning a particular technique [was] impossible to any person … and, therefore, each could presumably learn any technique and use it in much the same sense as he could learn any language and write in it” (Wilson Reference Wilson1940, p. 664).

For Wilson, these operational techniques were the language that economists should learn if they wanted economics to become truly scientific. This was the language that Samuelson learnt and used in his thesis.

Introductory Chapters

The Individual Level

In 1937, Samuelson published his two first papers. He elaborated on the consumer’s (1937a) and the entrepreneur’s (1937b) behavior, by assuming that they optimized intertemporally. These papers on mathematical economics appeared in February and in May, respectively. Samuelson must have finished the first paper before taking Wilson’s course on Mathematical Economics; in the May paper, Samuelson briefly referred to Wilson’s Advanced Calculus (1911) and to Edmund Whittaker’s and George Robinson’s The Calculus of Observations (1924), both covered by Wilson in his 1936 course on Mathematical Statistics. In these papers, Wilson’s influence on the way Samuelson approached mathematical economics was not yet evident. Wilson’s presentation of Gibbs’s thermodynamical systems that “have profoundly altered [Samuelson’s] views in corresponding fields of economics”^{Footnote 27} took place almost at the same time that these two papers were published; it is unlikely that Samuelson had had the time to engage with the difficult content of Wilson’s presentation of Gibbs’s systems. It can be conjectured that once Samuelson explored more in detail Wilson’s course material on mathematical economics and thermodynamics, he started then neglecting the old Fisherian working hypothesis of intertemporal optimization, as Wilson presented the consumer maximization problem as being independent of time.^{Footnote 28}

In the thesis, with the first working hypothesis, which consisted of assuming an extremum position, Samuelson presented the consumer and the firm problem analogically. With it, he defined “individual’s equilibrium” with respect to specifically demanded and/or supplied quantities that corresponded to the optimal individual’s position. At this individual level, such quantities therefore implied simultaneously concepts of stability of equilibrium and optimality, at discrete moments in time, as his idealized consumer and firm did not optimize over time, but at all moments in time.^{Footnote 29}

Consumer Theory

After having attended Wilson’s lectures in Mathematical Economics during the spring of 1937, in a series of papers all published in 1938, Samuelson, who was then twenty-three years old, claimed to have established new foundations for consumer theory by developing its empirical implications (1938a, 1938c, 1938d). When Samuelson sent to Wilson the last of the three cited papers for suggestions, the latter responded explaining that he had refereed positively the work for publication in Econometrica. Wilson believed,

There is no evidence in the style in which the paper is written that you have taken anything other than an intellectual attitude toward any of the questions. If however, there are any particular points where you yourself have any doubt or think other people might have some which you want to take up with me I shall be glad to discuss the matter with you.^{Footnote 30}

In the thesis, Samuelson elaborated on the Evolution of the Utility Concept (1941a, pp. 111–134), which eventually culminated, he hinted, at his operationally meaningful theorems, deducible, he argued, however, from the standard analysis.

Samuelson regarded utility theory as a convenient convention, which did not yet reflect “the factual behavior of consumers” (1941a, p. 114). Its relevance, “for better or worse,” was due to the fact that it “has occupied an important position in economic thought for the last half century. This alone makes it highly desirable that its meaning be clearly understood” (1941a, pp. 113–114). ^{Footnote 31} The notion of utility in economics represented, therefore, one of those invariants in science that Wilson regarded as necessary for the applicability of mathematics; determining its operational meaningfulness required, then, properly connecting it with some sort of data.

In his course on Mathematical Economics, Wilson presented consumer theory analogically to thermodynamics by explaining that certain discrete inequalities, which he called the “Gibbs conditions,” characterized the static and stable equilibrium position of thermodynamics and economics systems.^{Footnote 32} Such analysis did not imply the use of calculus, Wilson argued, but corresponded, in the discrete, to the conditions of stability of equilibrium of standard economic problems of optimization under constraint, in a static world. Wilson’s consumer analysis was indeed time-independent: “With time introduced, everyone recognizes that preferences change.”^{Footnote 33}

In this Wilsonian manner, in the thesis, Samuelson framed his Meaningful Theorems (1941a, pp. 134–144) on consumer analysis in a time-independent and static idealized world. He rephrased something that he had called in his doctoral papers the postulate of “consistency in idealized individual’s behavior,” with which he had connected utility analysis with observable data, by establishing certain correspondences between observable expenditure, the preference-field, and the demand function.^{Footnote 34}

With his approach, which consisted of playing with his skills in logical and arithmetical operations and his knowledge of the economic theory of index numbers, Samuelson was able to infer certain relations in the preference-field from observable expenditure. On this basis, Samuelson was then able to deduce a specific correspondence between such relation and demanded quantity behavior, expressed by the demand function. To accomplish this, and building on his theorem, Samuelson deduced the following relationships:

$$\mathop \sum \limits_{i = 1}^n {p_i}{\rm{\Delta }}{x_i} \le 0implies\mathop \sum \limits_{i = 1}^n \left( {{p_i} + {\rm{\Delta }}{p_i}} \right){\rm{\Delta }}{x_i} < 0$$

Following Wilson’s lead, Samuelson showed that a discrete inequality relationship, the second one, corresponded to the necessary and sufficient conditions of stability of an extremum position, as found in standard procedures of consumer-constrained optimization defined at the margin. The second inequality, Samuelson argued, “contained almost all the meaningful empirical implications of the whole pure theory of consumer’s choice” (Samuelson Reference Samuelson1941a, pp. 138–139); it corresponded to the well-established negative-slope and stability-concavity restrictions in maximization procedures upon (Marshallian) demand functions.^{Footnote 35} In this way, he connected his consistency postulate, grounded on observable data—not observed data—and the notion of equilibrium, with some structural characteristics of optimization under constraint.^{Footnote 36}

In the standard continuous analysis, however, there was an empirical restriction, which Samuelson did not succeed in deriving from his discrete formula: the integrability conditions.^{Footnote 37} In his words, integrability conditions

reflect differential properties of our demand functions which are hard to visualize and hard to refute.… I have tried, but thus far with no success, to deduce implications of our integrability conditions which can be expressed in finite forms; i.e., be conceivably refutable merely by a finite number or point observations. (Samuelson Reference Samuelson1941a, p. 134n13)

In spite of the difficulties that he encountered, Samuelson remained optimistic about his approach and hoped that “a proof may still be forthcoming by which [his approach] may be slightly generalized to include the question of integrability” (1941a, p. 139n14).

All in all, in consumer theory, Samuelson felt that he had developed something new, based on the old. Because he believed that he had translated abstract formulas defined at the margin into a discrete form, Samuelson felt that he had developed the empirical implications of the abstract utility and Marshallian demand theories. He thought that he had failed to encompass integrability precisely because he had not been able to establish such a continuous-discrete connection. From our Wilsonian perspective, it can be argued that the novelty of Samuelson’s consumer theory appears in the emphasis that he gave to the working hypothesis of a stable individual’s equilibrium, which, following Wilson, had to be defined in the discrete. From this point, Samuelson connected such a definition of the stable equilibrium with certain mathematical structures of optimization and with some sort of data. In this way, he could present his work as operationally meaningful—namely, as mathematically, theoretically, and empirically well founded, emphasizing more one aspect or the other, depending on the part of the thesis. This amalgamation of these three different elements had, in Samuelson’s thesis, the consequence that the notion of stable equilibrium could simultaneously be regarded as mathematically constructed, theoretically well founded, and empirically intuitive. His work, however, was not based on actual empirical data, but only on discrete formulas, which he presented as having an empirical nature.

Production and Cost Theory

Even though Wilson did not cover the theory of the firm in his courses, he significantly influenced Samuelson’s production and cost theory.^{Footnote 38} His relevance emerged as Samuelson found some difficulties when dealing with optimizing problems in which “certain costs [were] regarded as completely fixed,” or when a “firm [was constrained] to employ the same total of labor.”^{Footnote 39} These problems raise new questions about stability when dealing with systems, the equilibrium of which depended on “prescribed values of … ‘conjugate variables’,”^{Footnote 40} or parameters. They led Samuelson to study thermodynamics, where, he claimed, analogical problems were found, and which implied optimizing with a greater number of constraints.^{Footnote 41} But as the system had more constraints, Samuelson was concerned about the implication for the stability of equilibrium when the system faced changes of a parameter.

In his course on Mathematical Economics, Wilson had treated, in passing, the Le Chatelier Principle as a principle of stability of equilibrium in the case of infinitesimal changes of a parameter. Following Wilson, Samuelson interpreted this principle as implying, in the case of infinitesimal changes, that the greater the number of constraints the system had, the more stable the equilibrium position was in response to the marginal change of a parameter. The question remained to be established whether the principle could be generalized to the case of discrete finite changes.

During his fellowship years, when he was dealing with these issues, Samuelson even wrote a paper on the Le Chatelier Principle, which he sent to Wilson; in Samuelson’s words, his “manuscript represents a dangerous excursion … into a field about which I know very little. It was inspired partly by some remarks of yours in class some time ago, [and] partly by some work I have been doing in the field of economic theory.”^{Footnote 42}

In his response, Wilson wrote as follows:

[G]eneral as the treatment is I think that there is a possibility that it is not so general in some respects as Willard Gibbs would have desired. … I remember Gibbs used to talk about non-negative quadratic forms meaning those which never had negative values though they might take zero values for values of the variables which weren’t zero. Moreover, in discussing equilibrium and displacements from one position of equilibrium to another position he laid great stress on the fact that one had to remain within the limits of stability. Now if one wishes to postulate the derivatives including the second derivatives in an absolutely definite quadratic form one doesn’t need to talk about limits of stability because the definiteness of the quadratic form means that one has stability.…

I wonder whether you can’t make it clearer or can’t come nearer following the general line of ideas of Willard Gibbs as given in his Equilibrium of Heterogeneous Substances, equation 133. He doesn’t use derivatives but introduces a condition which is equivalent to saying that his function has to be on one side or in a tangent plane to it. He doesn’t even assume that there is a definite tangent plane but merely that at each point of his surface it is possible to draw some plane such that the surface lies except for that point and some other points entirely to one side of the plane.^{Footnote 43}

Following Wilson’s disciplining comments, Samuelson acknowledged that his paper “relates to instantaneous rates of change and does not approach the generality of the Gibbs formulation which makes no continuity or differentiability assumptions but only requires certain arithmetic inequalities (‘single concavity conditions’) to hold.”^{Footnote 44} Assuming that he remained in the limits of stability, Samuelson then came to the conclusion, as he wrote to Wilson again, that as a matter of formal definition the Le Chatelier Principle did not hold in the discrete case of finite changes, when several constraints were taken into account. In his words:

Implicitly assuming that we remain within ‘the limits of stability’, I was able through the Gibbs approach to show that

$${\rm{\Delta \alpha }}\Delta {x_1}{|_{(n\,constraints)}} \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} 0$$

This corresponded to the theorems on partial derivatives:

$${\left. {{{d{x_1}} \over {d\alpha }}} \right|_{(n\,constraints)}} \ge 0$$

Intuitively, I had expected that the generalized theorem on the partial derivatives of the form

$${\left. {{{d{x_1}} \over {d\alpha }}} \right|_{\left( {no\,c.} \right)}} \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} {\left. {{{d{x_1}} \over {d\alpha }}} \right|_{\left( {1\,c.} \right)}} \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} \ldots \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} {\left. {{{d{x_1}} \over {d\alpha }}} \right|_{\left( {n - 1\,c.} \right)}} \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} 0$$

would have an analogous theorem of the Gibbs type of the form

$${\rm{\Delta \alpha \Delta }}{x_1}{|_{\left( {no\,c.} \right)}} \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} {\left. {{\rm{\Delta \alpha \Delta }}{x_1}} \right|_{\left( {1\,c.} \right)}} \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} \ldots \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} {\left. {{\rm{\Delta \alpha \Delta }}{x_1}} \right|_{\left( {n - 1\,c.} \right)}} \mathbin{\lower.3ex\hbox{$\buildrel>\ove{\smash{\scriptstyle=}\vphantom{_x}}$}} 0$$

Unfortunately, I was not able to develop a proof of this, and in trying to do so, became aware that such a theorem is not true, at least on the basis of the very general Gibbs curvature assumptions.^{Footnote 45}

In the thesis, however, “[b]y making use of Professor E. B. Wilson’s suggestion that [the Le Chatelier Principle] is essentially a mathematical theorem applicable to economics” (Samuelson Reference Samuelson1941a, p. 98), Samuelson claimed that it held for finite as well as for marginal changes, as long as the system remained at the limits of equilibrium (Samuelson Reference Samuelson1941a, p. 43n12). It corresponded to the economic intuition according to which, for a firm in equilibrium, there was no possible movement that would improve its profits, no matter the number of constraints it had to face.

Samuelson used Wilson as a rhetorical figure of authority in order to introduce, as a general principle, his Le Chatelier Principle. To some extent, Samuelson was not persuaded that the formal analogy embodied in the existence of certain inequalities was formally consistent relative to all the cases that he analyzed; there were substantial differences in the treatment of discrete and continuous cases. By filling the mathematical gaps with meaning and intuition, Samuelson followed his master’s reassuring suggestion and the intuitive economics insight, which, however, led him to take the Le Chatelier Principle seriously. He also presented his cost and production theory as being operationally meaningful.

The Aggregate Level

In the thesis, with the second working hypothesis, which consisted of assuming a dynamical stable equilibrium, Samuelson aimed at establishing consensus and offering operationally meaningful theorems. He analyzed how equilibrium of the aggregate system was determined through time by studying the “stability conditions relating to the interaction between economic units”—namely, between aggregate variables (Samuelson Reference Samuelson1941a, p. 193), through time.^{Footnote 46}

Such interactions were often studied by analyzing the dynamics and stability conditions of Marshallian or Walrasian aggregate supply-and-demand systems when confronted with changes of prices.^{Footnote 47} But in Wilson’s spirit, Samuelson thought that “the economist would be truly vulnerable to the gibe that he is only a parrot taught to say ‘supply and demand’” (Samuelson Reference Samuelson1941a, p. 192). For Samuelson, Wilson’s “great virtue was [precisely] his contempt for social scientists who aped the more exact sciences in a parrot-like way” (Samuelson Reference Samuelson1998, p. 1376).

Samuelson’s “mathematical dynamics reflects in large measure the beliefs and prejudices of E. B. Wilson” (Weintraub Reference Weintraub1991, p. 58) on dynamical systems. Samuelson supposed that his second working hypothesis implied a correspondence between comparative statics and dynamics, as a way of connecting, while keeping separated, optimizing behavior of individuals, a static problem, and the evolution through time of the aggregate system. With such a correspondence, Samuelson presented comparative statics as a special case of dynamics; this intuitively implied that individual’s optimizing behavior was a special case, related to discrete moments in time, of the continuous evolution over time of the aggregate system at large. In his dynamics, Samuelson suggested, individuals were necessarily optimizing at every discrete moment in time, not over time. Further, at discrete moments in time, their optimizing behavior gave rise to the aggregates of the system, and hence individual optimizing behavior, he argued, “affords an unified approach” in economics. Comparative statics lay thus at the basis of his treatment of dynamical systems.^{Footnote 48}

These ideas about the correspondence between comparative statics and dynamics seemed to have been directly related to Wilson’s lectures in Mathematical Economics, where he discussed thermodynamical systems.

In the early 1920s, in correspondence with Francis Edgeworth, Wilson had claimed that there were two main working hypotheses in quantum theory regarding the treatment of dynamical systems. In the first working hypothesis, it was assumed that atomic nature was dynamical in essence and studied statistically only to ease the analysis. In the second working hypothesis, “the dynamical is a consequence of the statistical”: it was assumed that atomic nature was essentially discrete and that dynamics resulted from arbitrary manipulations with the theory of probability through which the discrete elements (quanta) were averaged and put into aggregates “to develop dynamics on the statistical basis.” Aggregates did not result from a sampling and taxonomical statistical analysis; they and their dynamics, he thought, were freely constructed. He believed that the two approaches were legitimate, depending on the problem in hand. However, he remained skeptical about using probability to freely construct aggregates and their dynamics.^{Footnote 49}

In 1936, A Commentary on the Scientific Writings of J. Willard Gibbs, in two volumes, was published. In the first volume, Wilson discussed Gibbs’ Lectures on Theormodynamics. In the second volume, Paul Epstein, a mathematical physicist at the California Institute of Technology, commented on Gibbs’ Methods in Statistics. Epstein’s argument resonated with Wilson’s comment on the different working hypotheses in physics to deal with dynamics. He explained, indeed, that in old quantum theory, there was equivalence between dynamical systems and integrable systems. He also pointed out that in the new quantum theory, based on wave theory, such was not necessarily the case because quanta could jump from one stationary equilibrium state to another and there was no way of determining the probability of a specific trajectory. Epstein then argued that such “probability could only be inferred indirectly and approximately, by classical analogies known under the name of ‘principle of correspondence’” (1936, p. 530). Based on the principle of correspondence, Epstein suggested, modern physicists connected and clarified the relationship between the old and the new quantum theory.

In the thesis, Samuelson did not (yet) call his correspondence between comparative statics and dynamics the Correspondence Principle. He thought, however, that with it the relation between old and modern economics, as had been the case regarding the relation between classical and modern quantum mechanics, according to Epstein, could be clarified. At the same time, following Wilson, Samuelson’s emphasis lay on comparative statics rather than on dynamics, as he focused on the (discrete) properties characterizing stationary equilibrium, more cogent with data, and not on moving equilibrium.

Samuelson’s dynamics was also informed by his personal research on business cycles (1939a, 1939a, 1940) and population dynamics.^{Footnote 50} In all these investigations, Samuelson encountered a similar formal difficulty when facing series and polynomials that did not converge. Eventually, he defined “dynamics” as the study of behavior through time of all variables of a system from arbitrary conditions, and referred to “stability”—as perfect stability of the first kind—as the cases in which “from any initial conditions all the variables approach their equilibrium values in the limit as time becomes infinite” (Samuelson Reference Samuelson1941a, p. 198). He used the general and mathematical formulation of functionals to map a great number of variables themselves functions of time.^{Footnote 51}

Within his general and mathematical framework, Samuelson used some examples of business cycles and of aggregate supply-and-demand dynamical systems to illustrate his general ideas about stability. He was able to show the correspondence between John Hicks’s difference equation-system, related to the dynamics of a multimarket system, with a differential equation system. He also showed, in the Keynes–Hansen business cycles case, that there were important correspondences between the static and dynamical cases, studied either with difference equations or with differential equations systems. In all these cases, certain inequalities represented the necessary and sufficient conditions for stability. Also, in all these cases, the correspondence between difference equations and differential equations embodied the ideal of possible translations between continuous and discrete mathematical formulas, while the correspondence between static and dynamical systems showed, Samuelson thought, that the study of dynamics shed light on comparative statics problems, and vice versa.

In the last paragraph of the thesis, the mathematical appendix excluded, Samuelson concluded, pointing out that the study of dynamics and stability had led him “into the most difficult problems in higher mathematics” (Samuelson Reference Samuelson1941a, p. 250), some of which he had shown in the thesis, and for some of which he did not yet have finite results.

***

After the defense of the thesis, Wilson advised Samuelson to translate the mathematics into English. In his words,

Wilson liked the thesis; it embodied his program for mathematical economics. Notwithstanding this, Wilson believed that, in its too-mathematical form, the thesis would not play the pedagogical role among economists that he wanted it to play.

Two years after the defense, Samuelson communicated to Wilson that he was revising the thesis and would love to have his suggestions. In response, Wilson wrote: “The thesis is so good and you are so busy [with war work and instruction] that I wonder whether you ought to put your time in revising it at all unless there is something really rather important in the way of improvements which you think you can make.”^{Footnote 53}

Eventually, Samuelson did not follow Wilson’s advice and kept working on the highly mathematical problems that he had encountered.

Samuelson and Wilson remained in close contact, as Samuelson was working on a manuscript based on his thesis, which he would submit for publication to the Harvard University Press at the beginning of 1945. Foundations of Economic Analysis, as he titled the extended version of his thesis, wasn’t published until 1947, due to publishing delays.