Abstract

This article tells the story of the first international topological conference in Moscow (1935), an outstanding event that, for the first time, brought together the most notable American, European, and Soviet mathematicians, including those who would later play decisive roles in the mathematization of economics: John von Neumann, Leonid Kantorovich, and Albert W. Tucker. The fact that Kantorovich was in contact with von Neumann and his closest colleagues, Solomon Lefschetz and Garrett Birkhoff, is hardly appreciated in the histories of mathematics and mathematical economics. Their brief academic exchange was interrupted by the increasing international isolation of Soviet mathematics and by the wars that ensued. The article provides a historical account of the conference and traces the intellectual and personal affinities of Soviet and non-Soviet mathematicians, as well as their conceptual innovations. It argues that the conference, as a singular event linking several research communities, mattered for the development of various formal frameworks and their dissemination, contributing to the intellectual landscape in which postwar mathematical economics could emerge. The article calls for a deeper analysis of conceptual affinities and motivations in applying mathematics to economics and for a more nuanced narrative linking these motivations to social and political contexts of economic modeling.

Introduction

Between the two world wars, various formal tools were being created that, after the Second World War, would be used to reconsider the mathematical apparatus used in neoclassical economics. This reconsideration first became clear at the famous activity analysis conference in 1949 (see on that Düppe and Weintraub 2014), was less formally codified in Koopmans's (1957) Three Essays on the State of Economic Science, and acquired the most abstract axiomatic foundation in Debreu's (1959) monumental Theory of Value. General equilibrium theory has become one of the most general theoretical frames for economics, while topology—the mathematics of continuity, studying functions in much more general spaces than in standard analysis—provided the new mathematical language for it. This language was forged not only in developing the equilibrium existence proofs but also in a series of studies on linear programming and input-output models, in applying game theory to economics, and generally in formulating what Hicks (1960) called “linear theory.”

Hicks perceived these new formalisms as coming to the European continent from the United States. But the preceding transformation in mathematical tools and their gradual dissemination among economists was of a much more distributed nature. Importantly, the adoption of topological methods and associated techniques in linear optimization, convexity theory, and so on, was not solely an American phenomenon.

This article adds one important prewar episode to the (mostly postwar) story of making economics a mathematical science (Weintraub 2002). This event—the first international topological conference in Moscow in 1935—known for its significance in the history of mathematics, also involved some crucial players in the history of mathematical economics, notably John von Neumann, Leonid Kantorovich, and Albert W. Tucker. Since the mathematical backgrounds of von Neumann and Kantorovich, but also Tucker, were quite similar and since mathematics in the 1930s was (more than economics) an internationalized field, their research could often yield very similar results and lead to parallel developments.

To adequately assess the significance of this event, we need to explore its major intellectual and institutional contexts, to quickly glance at the complex webs of relations between the major protagonists, and to get a better idea of how various contacts across national and disciplinary borders, as well as across multiple local and global identities, could contribute to the postwar “tool shock” (Isaac 2010) in mathematical economics.

Moscow, 1935

By this time, Soviet mathematics had developed to a considerable degree. Based on the achievements of previous generations, the mathematical centers—located mostly in Moscow and Leningrad, but also, for example, in Kyiv and Odesa—attracted many bright minds of younger generations. Despite mathematics being a universalist and seemingly apolitical discourse, its general development was largely shaped by local contexts. Together with their country, Soviet mathematicians were relatively free in the 1920s and gradually lost most of their autonomy by the end of the 1930s. Despite its abstractness, mathematics was not a perfect refuge for Soviet academics, as the infamous “Luzin affair” of 1936—and other campaigns—demonstrated.1 In the early 1930s, however, mathematicians could still enjoy some freedoms, and, in hindsight, the conference of 1935 seems to be an utterly outstanding event, the last opportunity to directly get together for most of the scholars working in the new field. André Weil, one of the conferees, notes that the fact such a gathering could have taken place at all “appears, retrospectively, as an anomaly, and almost as a miracle” (Weil 1979: 531).

The meeting—the First International Topological Conference—took place in Moscow on September 4–10, 1935 (see the most recent historical exposition in Apushkinskaya, Nazarov, and Sinkevich 2019).2 In a sense, this was the first and the last time the great Soviet mathematicians, Andrei Kolmogorov and Lev Pontryagin,3 could meet so many of their Western colleagues (the lineup was indeed exceptional). Other prominent and influential Soviet mathematicians, such as Nikolai Luzin (from the older generation) and Israel Gelfand (from the younger generation), were also present at the conference.

As the (soon to become famous) topologist Hassler Whitney (1988: 97) later recalled, “It was the first truly international conference in a specialized part of mathematics, on a broad scale.”4 The organizer was Kolmogorov's colleague and partner, Pavel Alexandrov (whose name usually appeared as Alexandroff at that time), who was in 1935 also the president of the Moscow Mathematical Society; the site of the conference was the Institute of Mathematics at Moscow State University (headed by Kolmogorov).

The conference itself testified to the international openness of the Soviet mathematical community (as opposed to the economic one).5 By 1935, the organizer, Alexandrov, had earned an international reputation as one of the world's foremost topologists. Indeed, the book he wrote with another conferee, Heinz Hopf, describing the state of the art in the discipline, appeared in German in the same year (Alexandroff and Hopf 1935) (in 1937, von Neumann would refer to this book in his paper on general equilibrium), and in 1927 he had been lecturing on topology in Göttingen, with von Neumann being one of his students.6 Small wonder so many notable mathematicians were eager to come.

Von Neumann, as well as other participants, received a formal invitation (in English, French, and German) in April 1935.7 It stated that the conference “is supposed to be a ‘small’ congress, to which a limited number of the most eminent topologists of the world will be invited.” All of the expenses of the participants, including travel “from the boundary of the USSR to Moscow and back,” were to be covered by the Moscow State University's Institute of Mathematics.8

The American group consisted, for the most part, of what would later be called Princeton topology. Apart from von Neumann himself, there were (Moscow-born) Solomon Lefschetz (1884–1972), who had frequently visited Moscow and Leningrad before,9 his student, A. W. Tucker (1905–95), who would later become a key figure in operations research and in developing the methods of nonlinear programming, and James Waddell Alexander (1888–1971). Harvard's Garrett Birkhoff (1911–96), von Neumann's friend and coauthor and a student of Whitney (Alexanderson and Wilde 1983: 132), was also there. Most of them must have known Alexandrov through more than just his work: he had visited Princeton in the winter of 1927–28 and in the spring of 1931 (Alexandrov 1980: 322, 327).

How did the conference proceed? We have several independent accounts from the participants that give us some glimpse into the everyday of this exchange. In his memoirs, Hans Freudenthal (1987) describes some funny episodes from the event: the inequality between Western and Soviet participants—the former stayed in the luxurious Hotel National and enjoyed perfect services and food,10 while the Soviets could hardly enter this space of privilege and could only exchange with colleagues informally; some “cultural program,” as a part of which they visited a textile factory, where Freudenthal (1987: 131) recalled “von Neumann standing next to a small revolving machine, of which we tried to guess the function, until von Neumann diagnosed, ‘This is a machine whose aim and purpose is to oil itself’” (a nice illustration of a self-reproducing linear economy von Neumann had thought of in the beginning of the 1930s); von Neumann's caviar locked in the refrigerator of the dining car, which was, on the way back, disconnected from their train at the Polish border; André Weil criticizing Stalinist politics—and an embarrassed reaction to this from his Soviet interlocutor; and finally, the growing organizational chaos,11 which, however, was not perceived as something very disturbing.

Von Neumann, Hopf, Lefschetz, and Weil were also invited to speak at the special memorial session of the Moscow Mathematical Society in honor of Emmy Noether on the fifth of September.12 Noether, a great Göttingen algebraist and a good friend of both Alexandrov and Hopf, had died in April 1935.

And here is what Birkhoff (1989: 45) recollects:

Widder, Stone, and I met in Helsinki, just before the Congress, whence we took a wood-fired train to Leningrad. There we were greeted at the station by L. Kantorovich and an official Cadillac. By protocol, he took a street-car to his home, where he had invited us for tea, while we were driven there in the Cadillac. I was astonished! I would have been more astonished had I realized that within two years I would be studying the work of Kantorovich on vector lattices (and that of Freudenthal, also at the Congress); that 20 years later I would be admiring his book with V. I. Krylov on Approximation Methods of Higher Analysis;13 or that in about 30 years he would get a Nobel prize for inventing the simplex method of linear programming,14 discovered independently by George Dantzig in our country somewhat later.

Marshall Stone, infinitely more worldly wise than I, reported privately that evening Kantorovich's disaffection with the Stalin regime.

These informal accounts give us an impression of the Cold War avant la lettre—written once the borders were really closed and revealing the melancholy of isolation; the astonishment of unanticipated, and often simultaneous, discoveries; and the local particularities, in which the discussions on and around those discoveries were happening.

The Moscow conference played a decisive role in the development of topology as a distinct discipline. One could argue that modern topology—the new, more general mathematical framework for studying the problems of traditional analysis based on the idea of continuity—was profoundly shaped by this direct interaction of American and European scholars. In fact, the very term became widespread about the same time, replacing analysis situs (due to Leibniz; see James 1999 on the history of topology after Poincaré). Topology itself was evolving and differentiating, so that the second volume of Alexandroff and Hopf 1935 was never written, and the book very quickly became outdated. New concepts and techniques were being developed, some of them in parallel (Whitney 1988). The talented mathematicians quickly became inspired by each other's work, picked up the new ideas and techniques floating around, and applied them, with a remarkable success, in topology and related fields. Many knew each other from other conferences or research visits and were of course reading each other's work, but the sheer number of illustrious scholars of all ages, and the exact moment of the gathering (when topology was booming), were additional success factors.

But what did this have to do with mathematical economics? To properly assess the significance of this event, we need to look more closely at the two conferees—John von Neumann (1903–57) and Leonid Kantorovich (1912–86)—who, in 1935, were about to reconfigure mathematical economics, the former, arguably one of the most influential scholars of the past century, by formulating some of the first rigorous general equilibrium/optimal growth models and by developing game theory (together with Oskar Morgenstern) and expected utility theory;15 the latter, the only Soviet/East European Nobel Prize winner in economics, by providing one of the first full-fledged applications of linear optimization to economic problems and by creating the whole new field of mathematical economics in the post-Stalin Soviet Union.

Mathematical Prodigies and Their Conversations

Von Neumann is quite often invoked as someone whose career is similar to Kantorovich's. Thus, after Kantorovich's death in 1986, Sergei Novikov, a Fields medalist of a younger generation, mentioned this parallel in his funeral speech (Kantorovich, Kutateladze, and Fet 2002: 203). It is interesting that from the perspective of Novikov—a pure mathematician—this analogy clearly suggests the similarity of both men's concerns with mathematical economics and generally with “applied” math.

A more sustained comparison has been made by Roy Gardner (1990): both von Neumann and Kantorovich were born into bourgeois Jewish families in Eastern Europe, both were prodigies, both had been doing functional analysis and applying advanced mathematics to economic problems since the 1930s. Gardner is correct, however, to indicate further striking parallels—both were indeed involved in the atomic projects of the Cold War superpowers (Mirowski 2002; Bollard 2019),16 and both were interested in computing.

To this, I would add that they both created a new identity—that of a highbrow mathematician rethinking the boundaries of the pure and the applied (Dahan-Dalmédico 2001; Boldyrev and Düppe 2020)—and played major roles in shaping the American and Soviet cultures of expertise. After the war, Kantorovich developed another research program: he promoted functional analysis as an applied, computational field. This was a new development, and its general orientation was fully in line with von Neumann's mathematical Weltanschauung.

Von Neumann played an important role in Kantorovich's general Bildung as a mathematician. His early, pathbreaking book on the mathematical foundations of quantum mechanics (von Neumann 1927, 1932) demonstrated to the mathematical community the power of functional analysis as an applied discipline—with the most abstract mathematical objects, such as a Hilbert space, being successful models for the new physical phenomena (Kantorovich 1987).17 But Kantorovich most probably did not know—at least he does not mention this anywhere—that as early as 1928 von Neumann was able to publish a formulation and a proof of a minimax theorem for games and was involved in the conversation of formalizing the notion of economic equilibrium (by 1935, the model generalizing Brouwer's fixed-point theorem had been built). In 1935, von Neumann was already a quite famous scholar, while Kantorovich was a young and promising mathematician who had published some work in analysis and descriptive set theory in French and English but was not very well known beyond the USSR.

All in all, the careers of the two men might have proved similar—but only eventually so. The similarities noticed eventually by so many colleagues turned out to be consequential for economics only in the decades to follow. But back in 1935, the Moscow conference did leave some traces as well.

Perhaps the most obvious consequence of the Moscow meeting was the simple fact that the scholars actually met each other and learned more about each other's work. For my contexts, it is important that the conference changed academic topology not only as a field—it changed the topology of the field: the Soviets—not merely as individual visitors but as a whole and diverse community—became visible and established themselves more securely in the space of international mathematics. The exchange continued well beyond the conference, mostly by publishing the work of non-Soviet mathematicians in the USSR and vice versa.18

Curiously, however, neither von Neumann nor Kantorovich played a major role in the conference as far as topology was concerned.

What kind of material was von Neumann presenting? His papers—the one published in English in Matematicheskii Sbornik in 1936 (von Neumann 1936b), and a Russian translation of his 1934 German work in the newly founded Uspekhi matematicheskikh nauk (Russian Mathematical Surveys [von Neumann 1936a])—dealt with the uniqueness of measure in topological groups. Its title in the conference program (“Integration Theory in Continuous Groups”) also indicates that this was the same paper. Thematically, it would most closely relate to the work of André Weil (who would publish a similar result in 1936) and Lev Pontryagin. In fact, this paper was continuing von Neumann's earlier—and partly successful—attempt (von Neumann 1933) to solve the fifth problem of Hilbert.

Kantorovich, who first met Alexandrov and Kolmogorov in Kharkiv, Ukraine, in 1930, at the first All-Union Conference of mathematicians, and who by 1935 had been already a full-fledged, young member of the Soviet mathematical community, did not give a talk himself because prior to the conference he had fallen seriously ill and did not have the energy to prepare anything of a decent quality. But we know the topics on which he had been working at that time. They concerned the theory of vector lattices.

In fact, the very concept of vector lattices is attributed to Kantorovich (Birkhoff ([1940] 1948: 238), although today these objects are called Riesz spaces.19 Put simply, a vector lattice is a vector space, on which a certain notion of order is introduced—that is, its elements are ordered in some intuitive way,20 one that turned out to be helpful in functional analysis, because the function spaces could be formalized as Riesz spaces. Unlike the standard approach in functional analysis, based on the idea of a distance in vector spaces, the approach followed by Kantorovich suggested the order, the simple idea of which is known from working with numbers, as the foundation of this analysis.21

In 1936, Kantorovich wrote to Lefschetz (in Russian) to get a paper written by one of his students, Boris Vulich, published in the Annals of Mathematics, the journal of the Princeton mathematics department that soon—reinforced by the Second World War—became the most significant general mathematical outlet in the world. Kantorovich's correspondence with von Neumann belongs to the same period. Somewhere toward the end of 1935 or the beginning of 1936, Kantorovich sent to von Neumann his paper on linear partially ordered spaces—“Lineare halbgeordnete Räume I”—with an intention to publish it in the Annals of Mathematics. What we have now is the response of von Neumann, written in Paris on May 20, 1936, saying that he finds the text very interesting; that the possibility of defining topology by the notion of order, and not distance, can be “finally justified” by the analysis provided in Kantorovich's paper; and that the editors of the Annals will be happy to publish the paper—albeit in an abridged form (and merged with the second part announced by Kantorovich and obviously not received by von Neumann). The same letter announces sending back the manuscript itself (for Kantorovich to perform the necessary cuts) and further offprints by von Neumann, saying that he would be happy to receive Kantorovich's opinion on them (Kantorovich, Kutateladze, and Fet 2004: 490).

For reasons we do not know, this paper by Kantorovich, written in German, was never published in the Annals of Mathematics.22 In October 1936, he sent it to the Russian journal Matematicheskii Sbornik and it was published there at the beginning of 1937, later to be referred to by Birkhoff ([1940] 1948: 272) as one of the “works of especial originality or historical importance.” At the same time, the Annals received Vulich's paper in January 1936 and quickly accepted it (Vulich 1937: 156).23

In 1937, Birkhoff, who seems to have been an important intellectual connection between von Neumann and Kantorovich, published a paper introducing a notion of convergence that he claims to have developed together with von Neumann (Birkhoff 1937). Independently and simultaneously, the similar notion was developed in the same long paper that had been initially sent to Annals of Mathematics (Kantorovich 1937).24 Kantorovich's name then appears many times in Birkhoff [1940] 1948 as associated with some algebraic concepts and theorems in the analysis of lattice-ordered groups (that are also Riesz spaces) and ideals (219, 222, 229, 231).25

It is impossible to ascertain whether these concepts were actually discussed at the conference, but we know that the conference did facilitate the exchange, so that Kantorovich was more inclined to send his work to the American topologists, and they were more willing to engage with it in their own research and its subsequent codification (of which Birkhoff's monograph is an important example). We thus see that even beyond the conference and its topological achievements, the conversations continued—and the Moscow interlocutors were conversing on very similar, sometimes identical terms.

Von Neumann, Kantorovich, and Mathematical Economics

In 1935, topology seemed to be a field that had nothing to do with economics. But the communities of mathematicians and economists were not fully isolated from each other. Curiously, the very same Hassler Whitney who was among the prominent conferees is claimed to have formulated the famous “traveling salesman problem” between 1932 and 1934 at one of the Princeton seminars—a problem to be solved by linear programming methods in the 1950s.26 Another participant of the Moscow meeting, Eduard Čech, informed Oskar Morgenstern after his talk at the Vienna Menger colloquium in 1936 that his work had much in common with the 1928 paper of von Neumann (Morgenstern 1976: 806–7), thus initiating their collaboration on game theory.

But in 1935, topological methods still seemed to be completely outlandish to any economist (something that changed only in the 1950s). However, at the Moscow conference, there was one person—at that time perhaps the only one in the world—who could rigorously demonstrate their conceptual import. That person was von Neumann, who had had an interest in economic problems and models for some years and had first presented his model of an expanding economy in 1932, at a seminar in Princeton.27 In the model, which remained incomprehensible for most economists even in 1945, when it was first published in English, he famously used the fixed-point theorem to prove the existence of general economic equilibrium.28

Hicks (1960: 707) summarized the conceptual import of von Neumann's model and subsequent literature for economics in a particularly lucid way:

It has been made apparent, not only that a price system is inherent in the problem of maximising production from given resources—but also that something like a price system is inherent in any problem of maximisation against restraints. The imputation of prices (or “scarcities”) to the factors of production is nothing else but a measurement of the intensities of the restraints; such intensities are always implicit—the special property of a competitive system is that it brings them out and makes them visible. It is through its power of developing the intensities (in the photographic sense of developing), so that they are available for use as instruments in the process of maximisation, that the competitive system does its job.

Apart from the fixed-point theory, von Neumann's model contained some other important elements, notably the idea of duality and the system of linear inequalities (instead of equations). It is there that von Neumann actually conjectured the connection between the minimax solution of two-person games and a solution to the system of linear inequalities (Kjeldsen 2001, 2002).

Now, the use of inequalities, duality arguments, and convexity analysis (a hyperplane separation theorem) in solving optimization problems in economics—in fact, inspired not by theory but by a real planning problem—was exactly the contribution of Kantorovich (1939). At that time, neither he, nor von Neumann, however, could provide a full theory of linear inequalities and their applications to economic modeling—it was done mostly by Dantzig and others at the end of the 1940s. However, “Kantorovich's ‘resolving multipliers’ are very much like the dual variables of modern linear programming, and Kantorovich had established their essential properties some eight years previously” (Dorfman 1984: 292).

In any case, after the war, the two fields—von Neumann's game theory and Kantorovich's (and Dantzig's and Koopmans's) linear programming—finally were connected to each other, something their originators could not anticipate themselves.29 Gale (1960: 216) called this parallel development of game theory and linear programming, as well as the realization of their close relationship, “one of the most striking events” in the emergence of “linear theory.”

Perhaps the main significance of von Neumann's growth/general equilibrium model, apart from the use of the fixed-point theorem, was the clear economic interpretation of duality in terms of profit and prices, which made the linear analysis compatible with standard neoclassical notions of equilibrium. Somewhat later, convexity, too, became reinterpreted in economic terms (diminishing marginal utility or returns), so that these mathematical concepts and theorems could then have an economic content and motivation.

In the same year in which Hicks published his essay on the economic significance of “linear theory,” Kantorovich's early contribution (of which Hicks obviously did not know) appeared in English (Kantorovich [1939] 1960). In 1975, Kantorovich, together with Koopmans, would receive a Nobel Prize exactly for this achievement: for demonstrating that—even in the context of industrial planning—a linear optimization problem necessarily involves “resolving multipliers” (or shadow prices) capturing relative scarcities.

It is difficult to say whether mathematical motivations were indeed primary in von Neumann's and Kantorovich's work on economic models. While von Neumann was directly engaging with economic theories and economists, Kantorovich was closely entangled with planners, and he later became an activist for policy change that would adopt linear optimization for the whole economy (Boldyrev and Düppe 2020). But we can be sure that their interventions, while having a lot in common conceptually, contributed to changing the face of economic modeling and to reshaping the intellectual commitments of the economics profession on both sides of the Iron Curtain. We will now trace these changes in more detail.

Further Adventures of Concepts: Kantorovich and General Equilibrium

Not only did linear programming techniques emerge almost simultaneously and independently, but, as we have also seen, Soviet topology and functional analysis developed in parallel with the work done in the West, so that in some cases, Kantorovich, von Neumann, and their colleagues could be working with the same objects and developing very similar mathematical ideas.

The success of this conceptual work for the development of mathematical economics suggests that there were some deeper reasons why these particular theories were taken up. It is also quite telling that, at different stages of their careers, the famous topologists of later generations, John Milnor and Stephen Smale, who were among the most influential American mathematicians of their time, also had some interest in economics.30

The development of general equilibrium theory is, of course, the best case in point to demonstrate how deeply these advanced mathematical techniques shaped economic analysis. Again, von Neumann's contribution seems to be decisive here. He was famously followed by Nash (1950), a Princeton student of Tucker, and Arrow and Debreu (1954), who then set a standard of using fixed-point theory in the equilibrium existence proofs. This well-known story had another, less known development.

In 1981, David Kreps published a paper formalizing the notion of arbitrage in the infinite dimensional commodity spaces (Kreps 1981). This and previous paper (Harrison and Kreps 1979) essentially provided a rigorous framework for the option pricing formulas of Black and Scholes (1973) in the economies with asset markets. A more remote ancestor of this analysis was Arrow's 1953 paper (translated into English in 1964), in which a general equilibrium model was extended for the case of uncertainty, and the possibility was opened to rigorously analyze asset economies.31 It is for these economies that Kreps started using the analysis of partially ordered spaces—a broader object than a Riesz space—linear vector spaces with a set of positive vectors. This helped in formalizing the idea that a given asset x can have higher returns in some state of the world than asset y and at least not lower returns than y in all states of the world. Very soon, in 1982, a general equilibrium theorist Donald J. Brown and a finance theorist Stephen A. Ross came up with the idea that to better analyze the derivative assets on those markets, the additional structure—that of a Riesz space—is necessary (the paper was later published as Brown and Ross 1991). About the same time, Brown, together with the mathematical economists Charalambos D. Aliprantis and Owen Burkinshaw, developed a full-fledged general equilibrium theory in Riesz spaces (Aliprantis, Brown, and Burkinshaw 1989).

Thus, Riesz spaces, first explored by Kantorovich and other 1935 conferees—notably Birkhoff and Freudenthal—found their way into one of the most sophisticated domains of mainstream economic theory. While in the beginning, general equilibrium theory relied on von Neumann's idea of using fixed-point theorems to prove the existence of equilibrium, the newer approaches also took on board the Riesz-Kantorovich formalism to achieve the best representation of equilibria on asset markets.

And again, Kantorovich himself could not anticipate these developments. But he was well aware that the order structures he helped develop for functional analysis could be used in economic modeling. This idea appears continuously in his talks and notes, beginning from the 1960s.

Further Adventures of Scholars: Mathematical Economics across East and West

The intellectual giants of the 1935 topological conference and of twentieth-century mathematics, von Neumann and Kolmogorov, would meet again, when both would give plenary talks at the International Congress of Mathematicians in Amsterdam in 1954. In his correspondence with Alexandrov in the 1930s, Kolmogorov compared himself with von Neumann and indeed, the sheer breadth of their interests and some exciting parallels between their work do suggest this analogy.32 But for the history of mathematical economics, the figure of Kantorovich as von Neumann's Soviet counterpart is more important. The reconstruction of the 1935 conference suggests that these affinities were more significant than previously believed and matter for the development of postwar mathematical economics.

Von Neumann himself never visited the Soviet Union again, while Kantorovich, who had a longer life and career, devoted most of his later years—especially after 1956—to creating an academic space in which mathematical techniques could be systematically applied to the problems of Soviet planning. I would leave the comparison of their politics for another occasion. The fact that they never met again—and never could—just makes more apparent their respective roles as cold warriors: in the 1940s and 1950s, both were deeply involved in their countries’ atomic programs.

The Moscow conference of 1935 created or developed continuities even the two wars could not break. Suffice it to say that it was none other than Kantorovich who, in 1959, initiated the Russian translation of Kuhn and Tucker's pathbreaking collection Linear Inequalities and Related Systems (1956), the major source in linear and nonlinear programming. In the 1960s and 1970s, Gale and Koopmans, who were close associates of both von Neumann and Tucker and who would pay a lot of attention to the development of Soviet mathematical economics, came to the Soviet Union and exchanged with Kantorovich and his student Valery Makarov while developing optimal growth theory, which itself was an outgrowth of von Neumann's general equilibrium model.

Another conferee in 1935, Andrei Markov Jr., who was also a topologist at that time and would later actively support Kantorovich in his projects of reforming Soviet economic science in the 1950s and 1960s, was an adviser of Nikolai Vorob'ev. Vorob'ev organized the Russian translation of von Neumann and Morgenstern's Theory of Games and Economic Behavior and became the central figure in Soviet game theory. The translation appeared in 1970, preceded by Morgenstern's visit to the International Congress of Mathematicians in Moscow in 1966. Kantorovich the economist was not concerned with these developments anymore, and indeed, in 1970, game theory did not seem to be a proper formal language of economics, neither in the USSR nor in the West.

In general, after the war, almost all the other Soviet conferees had some effect on mathematical economics: Luzin was an informal mathematical teacher of the Soviet choice theorist Mark Aizerman (Boldyrev 2020); Pontryagin with his group formulated the Maximum Principle in optimal control theory that became immensely popular in economic models of growth; Gelfand, with Mikhail Tsetlin, developed a new approach in game theory (games of automata); and Andrei Tikhonov (at that time his name was spelled as Tychonoff) tried to apply his theory of “ill-posed problems” to linear programming and, in 1970, founded the Faculty of Computational Mathematics and Cybernetics at Moscow University, in which work in operations research and game theory was quite prominent.

In the West, the main “agent” of Soviet mathematical influence was Lefschetz. Indeed, he translated Soviet work on stability theory of differential equations, published various review papers and notes in Soviet mathematical journals, and generally devoted quite some time to following Soviet developments in various fields—especially in optimal control. “Solomon Lefschetz was to be responsible, in large part, for introducing Western, particularly American, mathematicians to the work of the Soviet mathematicians on differential equations and for reintegrating the traditions of Poincaré and Liapunov” (Weintraub 1991: 71). This influence seems to be crucial in the development of the postwar theory of the stability of general equilibrium (see more on Lefschetz's contribution to this process in Weintraub 1991: 90–93).

All these associations, while not being direct consequences of the 1935 conference, still involve its participants and demonstrate the diversity of transnational intellectual connections between mathematics and economics (and within them), calling for more nuanced histories.

Conclusion

Not only scholars, but concepts, too, travel in the ways their progenitors cannot foresee. This article traced both conceptual and individual travels and demonstrated, with the emphasis on the 1930s, that the history of mathematical economics—as well as the biographies and interests of its protagonists—had an important transnational dimension, in which Soviet mathematics played a crucial role. The postwar “tool shock,” along with the creation of powerful formalisms to attack difficult economic problems (such as the existence and stability of general equilibrium or the ways to conceptualize and apply to economics the techniques of linear and nonlinear optimization), was being prepared before the war. The intellectual and institutional dynamics of mathematical economics depended on the mutual awareness and collaboration of many individuals, on both sides of the Iron Curtain. Despite remarkable parallelisms and the universality of formal expression, this space was not common; local contexts mattered as well, and the divisions remained real, both in economics and in mathematics.

Preceding the Great Terror (1936–38) and decades of Soviet isolation, situated on the brink of war, the conference of 1935 turns out to be a central event in this story. In this article, I tried to look at it with the hindsight that only the history of economics could provide. For it brought together, in a singular manner, the main contributors to linear and nonlinear programming, as well as game theory. This general frame was hardly explicit at the time, but the agents of change, as well as their mathematical ideas, were real. Thus, even though economic models were not debated there, the Moscow conference must definitely be reclaimed for the history of mathematical economics.

I wish to thank the editors of this special issue, Juan Carvajalino and Thomas Mueller, for inviting me to contribute to this fascinating conversation and for their feedback. The participants of our several meetings, as well as the anonymous reviewers, have been also extremely helpful in commenting on this project. I wish to express a particular gratitude to Johanna Bockman for helping me access the necessary archival materials in the John von Neumann Papers and to Serge Lvovski for looking at my interpretations of mathematics.

Notes

1.

On the Luzin affair, see Demidov and Levshin 2016. Nikolai Luzin (1883–1950), a central figure in Russian and Soviet mathematics, might have been spared execution—unlike the economists Nikolai Kondratiev and Vladimir Bazarov, who were repressed in the 1930s—but he was denounced by the community (including, among others, Pavel Alexandrov and Andrei Kolmogorov, who are about to appear in our narrative), lost his job, and spent the rest of his life, condemned for alleged plagiarism and political mistakes, in bitter isolation. His teacher, Dmitri Egorov, after spending a year in prison, had died, in 1931, in a hospital, following a hunger strike.

2.

Gardner (1990: 638) incorrectly states that it was the conference in functional analysis held in Leningrad.

3.

The fact that Kolmogorov could make a significant contribution to topology came as a surprise to American mathematicians, who knew him mostly for his axiomatic reformulation of probability theory.

4.

Whitney’s title (“Moscow 1935: Topology Moving toward America”) suggests quite an unusual curve describing the development of topology.

5.

For example, the major Soviet economists, Nikolai Kondratiev and Eugen Slutsky, both received an invitation to participate in the meeting of the (at that time projected) Econometric Society in November 1930 (http://dev.econometricsociety.org/sites/default/files/historical/OriginalAnnouncement29%2011%2030.pdf), but Kondratiev was soon arrested and imprisoned, while Slutsky stopped doing economics (Barnett 2005).

6.

Alexandrov recollects that he and his colleague and partner Pavel Urysohn were the first Soviet mathematicians to visit the West. They earned the money themselves by giving public lectures and, thanks to the favorable conditions of the “New Economic Policy” (free exchange of rubles for foreign currency), could secure the sum necessary to go abroad—to Göttingen, in the summer of 1923 (Alexandrov 1979: 297–98), meeting Felix Klein, David Hilbert, Emmy Noether, Richard Courant, and many others.

7.

All the personal letters from Alexandrov to von Neumann are in German—the language they had spoken in Göttingen.

8.

John von Neumann (JVN) Papers, Library of Congress, box 2, folder 2.

9.

Birkhoff (1989: 45) calls him a major organizer—meaning, apparently, that it was Lefschetz who could organize the visit of such a large group of Americans to the USSR.

10.

André Weil (1992: 106) reported about this in a similar way.

11.

“Delays everywhere, always and cumulatively. Dinner gradually shifted to after midnight, we got up later and later, and by the end we had lost about half a day” (Freudenthal 1987: 132).

12.

Alexandrov’s invitation, written in German, is in the von Neumann Papers (Library of Congress, box 2, folder “Alexandroff, Paul, 1935”).

13.

In fact, this book was first published in 1936, so it was basically being written while they were drinking tea at Kantorovich’s place; still, its English translation had to wait until 1958.

14.

This is not an accurate formulation; see on that Dantzig 1963, Dorfman 1984, and Gardner 1990.

15.

Mirowski (2002: 94) even wants to see in von Neumann “the single most important figure in the development of economics in the twentieth century.” Note, however, that in Mirowski’s interpretation, von Neumann was reimagining economics as a “cyborg science,” far from the concerns of standard, mainstream economic theorists.

16.

During and after the war, not only were von Neumann and Kantorovich heavily involved in military projects but also Birkhoff, who “related vector lattices to nuclear reactor” (see Alexanderson and Wilde 1983: 141).

17.

Kantorovich is less known in the history of mathematics than von Neumann, but, curiously, his mathematical work continues to inspire the most cutting-edge scholars. At least two recent Fields medalists (Cedric Villani and Alessio Figalli; see Villani 2009 and Jackson 2018 on Figalli) explicitly draw on Kantorovich’s work in optimal transportation. The economic applications are discussed in, e.g., Ekeland 2010.

18.

Thus, in 1936, three of von Neumann’s papers were published in the Soviet Union.

19.

Riesz (1929) introduced them, announcing their use for some proofs, at the mathematical congress in Bologna in 1928 but, curiously, published the detailed proof only in 1940, when the theory of these spaces was already in full swing (Riesz 1940). Aliprantis and Burkinshaw (1978: 2) attribute the concept independently to Riesz, Freudenthal, and Kantorovich.

20.

Beyond that, Riesz space structure on the set L means that every finite nonempty subset of L has the supremum, or least upper bound, and infimum, or greatest lower bound (Aliprantis and Burkinshaw 1978).

21.

A similar idea, inspired by Riesz (1929), was suggested by Freudenthal (1936). Note, however, that Freudenthal (1936: 641) also gives priority of explicit introduction of those spaces to Kantorovich (1936).

22.

One could argue that the Luzin affair could have been a reason for this, since this led to the dramatic reduction of international publications in Soviet mathematics (Alexandrov 1996). However, Kantorovich continued to publish abroad after the Luzin case and until the USSR entered the Second World War in 1941. So it might well be that Kantorovich simply did not want to shorten his paper.

23.

Lefschetz’s acceptance letter for this paper was sent to Kantorovich (and not Vulich!) on March 26, 1936 (Kantorovich, Kutateladze, and Fet 2004: 489).

24.

See on that Birkhoff [1940] 1948: 62, 80.

25.

Birkhoff on p. 229 also refers to the first note Kantorovich published on lattice theory (Kantorovich 1935).

26.

Whitney was doing graph theory at that time and was interested in combinatorial problems. See Schrijver 2005.

27.

We cannot be sure, however, that von Neumann fully saw this connection as early as 1932 (or 1928); see on that Kjeldsen 2001, but also Weintraub 1983, which saw the elements of a fixed-point theory already in the 1928 proof, which, although not applying Brouwer’s theorem explicitly, could still be interpreted in that way.

28.

Here is, again, a radical characteristic—now from Weintraub (1983), who calls it “the single most important article in mathematical economics” (13). Note that it was not the first formalism von Neumann was using to model economic phenomena. The reconstruction provided by Carvajalino (2021) demonstrates that he was experimenting with various dynamic frameworks before developing an equilibrium model of optimal growth and applying fixed-point theory to the existence problem. This testifies to the fact that von Neumann, unlike Kantorovich, had an interest in economic modeling long before 1935.

29.

It was Ville (1938) who established the connection of game theory and the systems of linear inequalities in all clarity and proved the theorem in a much easier way. David Gale, Harold Kuhn (who did his PhD at Princeton and was the academic “grandson” of Lefschetz)—and, of course, Tucker—further developed and generalized those results (see Gale 1960; Gale, Kuhn, and Tucker 1951).

30.

Milnor’s first papers were on game theory and the axiomatization of utility theory, while Smale, in the 1970s, applied “global analysis” to general equilibrium dynamics.

31.

The assumption of infinite dimensionality comes from the fact that in the models of Black and Scholes (1973) and Merton (1973) so foundational for finance, security prices change continuously, and the infinite space of contingent claims became the best formalization for equilibrium analysis on these markets. For the genealogy of Arrow’s paper and its general significance for finance, see Mehrling 2016 and Boldyrev 2021.

32.

Boldyrev and Düppe (2020) discuss the role of Kolmogorov in supporting Kantorovich’s project of mathematizing economics after World War II and Kolmogorov’s own interest in economics.

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