David Hilbert's Influence on Economics

Left: Colourized photograph of David Hilbert. 1932 (Photo: Getty Images). Right: Cover of Hilbert and Ackermann’s influential book Grundzüge der Theoretischen Logik (1928)

Eminent mathematician David Hilbert (1862-1943) is not known for his interest or works in economics. In fact, Hilbert never did any work in economics. His primary obsession was mathematics, and sometimes—such as when he raced Einstein to find the field equations of general relativity—mathematical physics. He primarily did work in geometry, algebra and logic. Indeed, he is credited as one of the founders of proof theory and the discipline of mathematical logic.

Yet still, Hilbert’s influence on economics has been a topic of debate among historians of economics. How so? Well, from Hilbert’s emphasis on axiomatization in mathematics—working backwards from statements towards the fundamental axioms such statements rest on—one can present a narrative of influence from his work, to his pupil John von Neumann (1903-57), to Karl Menger’s Vienna Colloquium in the 1930s, to Menger’s student Abraham Wald (1902-50) and so to the proof of the existence of competitive equilibrium and the theory of general equilibrium proofs by Kenneth Arrow (1921-2017), Gérard Debrau (1921-2004) and Lionel W. McKenzie (1919-2010) in the 1950s. Weintraub (2002) emphasizes three works from the history of economics to make this point. These are:

• Ingrao, B. & Israel, G. (1990). The Invisible Hand: Economic Equilibrium in the History of Science*. MIT Press. Cambridge, MA.

• Punzo, L.F. (1991). The School of Mathematical Formalism and the Viennese Circle of Mathematical Economists. Journal of the History of Economic Thought, 13(1), 1-18

• Mirowski, P. (1992). ‘What were von Neumann and Morgenstern trying to accomplish?’ In: Toward a History of Game Theory. Duke University Press.

This essay is about David Hilbert and his unintentional yet important influence on the development of modern economics.

Hilbert’s Program (1921-31)

I suppose the beginning of this story is Hilbert’s proposed solution to the foundational crisis of mathematics introduced by the works of Georg Cantor (1845-1918) in the 1800s. Cantor had shown that set theory, his own invention, was inherently and provably contradictary. By the early 20th century, in the philosophy of mathematics, Bertrand Russell (1872-1970) had summarized the problem of a lacking foundation in mathematics by formulating his Russell Paradox which goes as follows:

Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition entails that it is a member of itself; If it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves.

That is: Let R = { x | x ∉ x}, then R ∈ R ⟺ R ∉ R

A more accessible version of Russell’s paradox is the Liar paradox ‘I am lying’, or a more technical version, ‘This sentence is false’. Is the sentence false? If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on.

Hilbert’s program was a proposed solution to this epistemic crisis, formulated in 1921, but which Hilbert had been working on since the early 1900s. In short, Hilbert proposed to ground all existing theories to a finite, complete set of axioms and provide proofs that these axioms were consistent. Thus, Hilbert believed, the consistency of more complicated systems (such as, say, first-order logic or general equilibrium) could be proven in terms of simpler systems.

The prototypical work in the vain of Hilbert’s program is Russell and Alfred North Whitehead (1861-1947)’s herculean three-volume work Principia Mathematica which, among other results, grounded the theory of addition to logic by proving—in no less than thirty pages—the validity of the proposition 1+1 = 2. In 1928, Hilbert and his student Wilhelm Ackermann (1896-1962) published a similar, highly technical book on elementary mathematical logic, grounded in first-order logic entitled Grundzüge der theoretischen Logik (“Principles of Mathematical Logic”). The book was based on Hilbert’s 1917-22 lectures and contained the first ever exposition of first-order logic, posing the problems of completeness and decidability—the later famous Entscheidungsproblem of uncomputability, eventually resolved by Alan Turing (1912-54) in 1936.

Turing Uncomputability

Hilbert’s Disciple (1926-31)

A young and ambitious John von Neumann (1903-1957) left Berlin for Zürich in 1923 in order to further his studies of Hilbert’s theory of consistency with German mathematician (and future colleague at the Institute for Advanced Study, Hermann Weyl (1885-1955). About three years later, financed by a Rockefeller fellowship, von Neumann travelled to the University of Göttingen to study with the man himself, Hilbert. According to Leonard (2010), von Neumann was initially attracted to Hilbert’s stance in the debate over so-called metamathematics, also known as formalism. In particular, in his fellowship application, he wrote of his wish to conduct:

"Research over the bases of mathematics and of the general theory of sets, especially Hilbert's theory of uncontradictoriness [...], [investigations which] have the purpose of clearing up the nature of antinomies of the general theory of sets, and thereby to securely establish the classical foundations of mathematics. Such research render it possible to explain critically the doubts which have arisen in mathematics"

Very much both in the vein and language of Hilbert, von Neumann was likely referring to Cantor’s work regarding the nature of infinite sets. von Neumann, along with Ackermann and Paul Bernays (1888-1977) would eventually become Hilbert’s key assistants in the elaboration of his Entscheidungsproblem (“decision problem”). By the time he arrived in Göttingen, von Neumann was already well acquainted with the topic, in addition to his Ph.D. dissertation having already published two related papers while at ETH. More about von Neumann’s work and contributions to set theory can be found in the May 19th Privatdozent newsletter ‘The Unparalleled Genius of John von Neumann’.

The Unparalleled Genius of John von Neumann

Around the same time as he was working with Hilbert on set theory, von Neumann also published his first paper in economics—a proof that two-person zero-sum games with finitely many pure strategies (or a continuum of pure strategies and continuous convex payoffs) have minimax solutions. Still very much a mathematician, von Neumann in the paper—which was published in Mathematische Annalen—gives no reference to any economic ideas or implications.

Gödel’s Theorem (1931)

Famously, Hilbert’s program abruptly ended in 1931 when an unknown young Austrian logician by the name of Kurt Gödel (1906-78) published a paper showing its fundamental limitations, his incompleteness theorem. von Neumann happened to be in the audience when Gödel first presented it:

"At a mathematical conference preceding Hilbert's address, a quiet, obscure young man, Kurt Gödel, only a year beyond his PhD, announced a result which would forever change the foundations of mathematics. He formalized the liar paradox, "This statement is false" to prove roughly that for any effectively axiomatized consistent extension T of number theory (Peano arithmetic) there is a sentence σ which asserts its own unprovability in T.

John von Neumann, who was in the audience immediately understood the importance of Gödel's incompleteness theorem. He was at the conference representing Hilbert's proof theory program and recognized that Hilbert's program was over.

In the next few weeks von Neumann realized that by arithmetizing the proof of Gödel's first theorem, one could prove an even better one, that no such formal system T could prove its own consistency. A few weeks later he brought his proof to Gödel, who thanked him and informed him politely that he had already submitted the second incompleteness theorem for publication."

— Excerpt, Computability. Turing, Gödel, Church and Beyond* by Copeland et al. (2015)

Gödel’s proof effectively ended Hilbert’s program of establishing a rigerous foundation for mathematics. von Neumann discontinued work on set theory around the same time, instead shifting his focus to his work on quantum mechanics, ergodic theory, operator theory, geometry and, eventually, economics.

The Morgenstern Paradox (1928)

Oskar Morgenstern (1902-77) was an economist interested in forecasting. In particular, he was interested in the limitations of economic forecasting. His 1925 dissertation Wirtschaftsprognose presented the argument that economic forecasts are fundamentally impossible. His circular argument went as follows (simplified):

Any intelligent economic agent will react to a forecast. That reaction will have to be anticipated when the forecast is made. The agent would take this into account, which would also have to be anticipated, and so on.

In the book version, Morgenstern uses the plot and characters from Doyle’s The Memoirs of Sherlock Holmes: The Final Problem (1893) as a narrative device to illustrate this dynamic:

The Morgenstern Paradox (1928)
“Sherlock Holmes, pursued by his opponent, Moriarty, leaves London for Dover. The train stops at a station on the way, and he alights there rather than traveling on to Dover. He has seen Moriarty at the railway station, recognizes that he is very clever and expects that Moriarty will take a faster special train in order to catch him in Dover. Holmes' anticipation turns out to be correct.

BBut what if Moriarty had been still more clever, had estimated Holmes' mental abilities better and had foreseen his actions accordingly? Then, obviously, he would have travelled to the intermediate station. Holmes, again, would have had to calculate that, and he himself would have decided to go on to Dover. Whereupon, Moriarty would again have "reacted" differently.

Because of so much thinking they might not have been able to act at all or the intellectually weaker of the two would have surrendered to the other in the Victoria Station, since the whole fight would have become unnecessary.”

Unaware of von Neumann’s 1928 proof of the minimax theorem, Morgenstern at this point seems interested in, but incapable of resolving the paradox. In his diary at the time, Morgenstern even writes about travelling to a mathematical conference attended by Hilbert and of reading his and Ackermann’s book, writing later that:

“Hilbert was a great mathematician but not a pleasant person. I remember the Congress in Bologna in 1928 & how he, his wife & I happened to drive from Milan to Zurich, ate together in the dining car etc. I was of course “overawed” at the time (26!) But asked a few things about axioms & had just spent a few weeks beforehand, partly in Cannes, studying Hilbert-Ackermann’s logic, which was crucial for me.”

Oskar Morgenstern's Transformation (1925-38)

Abraham Wald’s Existence Proofs (1934)

During a meeting of Menger’s Vienna Colloqium in 1934, a Viennese banker by the name of Karl Schlesinger (1889-1938) presented a paper suggesting a modification to Léon Walras (1834-1910)’ set of equations describing a general economic equilibrium, which he (Walras) had never proved to have solutions except arguing that they should have based on the number of equations and unknowns. Schlesinger argued that in such a model, the supply of free commodities (of which demand is smaller than supply) should not be assumed have a number a priori, but rather that their supply (like all other supply) should be determined by the system of equations of general equilibrium, each free commodity having a price of zero.

This (seemingly) trivial insight inspired Abraham Wald (1902-50) (who was attending the Colloquium where Schlesinger presented his claims) to, that same day, prove the existence of the general equilibrium which had evaded Walras. His first attempt, given somewhat strict conditions was presented on March 19th. He soon after also presented a proof given much weaker conditions, on November 6th. His papers—one using a model based on a system of exchange and the other based on a model of production and exchange—were published in the proceedings of the Colloquium in 1934. Although the existence of the Walrasian equilibrium had been proposed as early as in the 1870s, the mathematical tools available to Walras never permitted him to show that such solutions must exist, given reasonable assumptions. Although the assumptions of Wald’s model are not very reasonable according to modern standards (including the assumption that there was only one consumer in the market), he was the first to solve the equilibrium problem that had plagued economists up until that point (Düppe & Weintraub, 2014).

Like von Neumann, Wald was trained as a mathematician and had little appreciation (or even awareness) of the significance of his findings for economics. To him, he had solved a mathematical problem by providing a mathematical solution with limited mathematical implications. Indeed, his boss and pupil Morgenstern reportedly had to ‘badger’ Wald to write a summary of his two papers, which were later published for an audience of economists in the prestigious Zeitschrift für Nationalökonomie (where Morgenstern served as editor from 1930-38) in 1936.

• Wald, A. 1936. Über einige Gleichungssysteme der mathematischen Ökonomie(“On some systems of Equations in Mathematical Economics”), Zeitschrift für Nationalökonomie 7(5). p. 637-670.

The paper was also later translated to English by Otto Eckstein and published in the prestigious economics journal Econometrica.

von Neumann’s Return to Economics (1937)

Although von Neumann as early as 1926 had done work in economics with his proof of the minimax theorem (published in 1928), he appears to have been focused on other matters for most of the 1930s. In 1937— utilizing L. E. J. Brouwer (1881–1966)’s fixed-point theorem on continuous mappings into compact convex sets — von Neumann however provided a purely topological proof of the existence of general competitive equilibrium, a much clearer and more elegant proof than that included in his 1928 paper, which emphasized strategic behavior rather than competitive markets. The paper was entitled:

• von Neumann, J. (1937). ‘Über ein Oikonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes’ in Menger, K. (ed). Ergebnisse eines Mathematischen Seminars. Vienna.

Although he had given no reference to any economic tradition in the 1928 paper, in the 1937 paper von Neumann refers to ‘a typical economic equation system’ and focuses on competition, rather than ‘strategic behavior’. He too presented the main results of the paper at Menger’s colloqium in 1936, a lecture which was attended by Menger’s student Wald but not by Morgenstern who was travelling. Another two years would pass before von Neumann and Morgenstern met in Princeton. They would collaborate in the years 1940-44 on the book Theory of Games and Economic Behavior which is considered the first coherent work on cooperateive game theory. Inspired by the work, John Forbes Nash, Jr. (1928-2015) in 1949 proved the existence of equilibrium points also for non-cooperative games.

The Arrow-Debreu Model (1954)

In 1954, inspired by the work of Wald and von Neumann, economists Kenneth Arrow (1921-2017) and Gérard Debreu (1921-2004) independently proved the existence of a market clearing equilibrium. Their model, the Arrow-Debreu model shows that under certain economic assumptions (convex preferences, perfect competition and demand independence) there must be a set of prices such that supply will equal demand for every commodity in the economy. Based on the Kakutani fixed-point theorem (a generalization of the Brouwer fixed-point theorem employed by von Neumann in 1937, also used by Nash in 1951), the Arrow-Debreu model is the most general model of competitive economies and remains a crucial part of the general equilibrium theory that underlies most of contemporary economic theory.

Hilbert’s Influence

The line of influence from Hilbert to modern economic theory thus might go as follows:

• Göttingen, 1921. Hilbert sets out to ground mathematics to logic, emphasizing the axiomatization and mathematical formalism.

• Göttingen, 1926. von Neumann apprentices with Hilbert, adopting the view that theories should rest on formally stated axioms.

• Vienna, 1928. Morgenstern publishes the Morgenstern Paradox but is unable both to state it formally and resolve it.

• Göttingen, 1928. von Neumann publishes his first paper in economics on the minimax theorem, (inadvertantly) addressing the Morgenstern Paradox in a formal, axiomatic manner.

• Princeton, 1932. von Neumann presents a draft of a proof of the existence of general competitive equilibria during a mathematical seminar in Princeton.

• Vienna, 1934. Wald, student of Menger and attendee at his colloqium develops the first proof of the existence of general equilibrium. Badgered by Morgenstern, he publishes the finding in an Austrian economic journal.

• Vienna, 1936. von Neumann at Menger’s colloquium presents a completed proof of the existence of general competitive equilibria based on Brouwer’s fixed-point theorem.

• Princeton, 1944. von Neumann and Morgenstern publish the first book on cooperative game theory, emphasizing, in a formal and axiomatic manner, cooperation and the existence of fixed-point solutions in competitive games.

• Princeton, 1950. Nash, inspired by von Neumann and Morgenstern’s book publishes a proof of the existence of equilibrium points in non-cooperative games.

• Stanford, 1954. Inspired by both Wald and von Neumann, Arrow and Debrau publish the first rigerous proof of the existence of a market clearing equilibrium.

• Stockholm, 1972. Kenneth Arrow is awarded the Nobel Memorial Prize in Economic Sciences for his “pioneering contributions to general economic equilibrium theory […]”.

• Stockholm, 1983. Gérard Debreu is awarded the Nobel Memorial Prize in Economic Sciences for his ‘“…] rigerous reformulation of the theory of general equilibrium”.

• Stockholm, 1994. John F. Nash Jr. is awarded the Nobel Memorial Prize in Economic Sciences for his “pioneering analysis of equilibria in the theory of non-cooperative games”.

Thus, as Punzo (1991) argues, if it hadn’t been for von Neumann’s apprenticeship with Hilbert, the foundations of economics may indeed have evolved quite differently. Had Gödel been born ten years prior, for instance, perhaps Hilbert’s program would never have been developed and von Neumann would instead have focused on his many other passions rather than set theory—and so never come to influence economics in such a fundamental way.

---

Thank you for reading this edition of Privatdozent. Those interested in reading more about the foundations of mathematical economics are encouraged to acquire the books:

• Düppe, T. & Weintraub, E.R. 2014. Finding Equilibrium*. Princeton University Press.

• Weintraub, E. R. 2002. How Economics Became a Mathematical Science*. Duke University Press.

Sincerely,

Jørgen

Related Stories

• Oskar Morgenstern’s Transformation (1925-38), August 27th 2021

• Karl Menger’s Vienna Colloquium (1928-36), October 22nd 2021

• John von Neumann’s Minimax Theorem, March 26th 2021

• The Unparalleled Genius of John von Neumann, May 19th 2021

• Cantor and Dedekind’s Early Correspondence (1873-74), June 28th 2021

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References

• Düppe, T. & Weintraub, E.R. 2014. Finding Equilibrium*. Princeton University Press.

• Ingrao, B. & Israel, G. 1990. The Invisible Hand: Economic Equilibrium in the History of Science. MIT Press. Cambridge, MA.

• Leonard, R. 2010. Von Neumann, Morgenstern, and the Creation of Game Theory*. Cambridge University Press. pp. 42

• Punzo, L.F. 1991. The School of Mathematical Formalism and the Viennese Circle of Mathematical Economists. Journal of the History of Economic Thought, 13(1), 1-18

• Mirowski, P. 1992. ‘What were von Neumann and Morgenstern trying to accomplish?’ In: Toward a History of Game Theory. Duke University Press.

• Weintraub, E.R. 2002. “Whose Hilbert?” In: ‘How Economics Became a Mathematical Science’ (ed.) Duke University Press. pp. 72-100.

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