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As it is known that math as a branch that is disputed whether it is a science or a language, I am just curious whether there is an economist that contributed to mathematics as a branch (I am not talking about the implementation of mathematics to economics), as Newton invented calculus (I know Leibniz as well) to solve the physical problems and several instances of physicist contributed to Math can be observed as well.

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  • $\begingroup$ Hi @Tunay. Probably someone polymathematical like von Neumann who made important contributions to economics (and is often called an economist, accordingly). Most of his contributions are in pure math, but he of course would draw on these for computer science, economics, etc. There are potentially more scholars than vNM. But I couldn't think of anyone else off the top of my head $\endgroup$
    – EB3112
    Commented Aug 31, 2023 at 11:15
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    $\begingroup$ Depending on how big a contribution you count, the answer is obviously yes. There are a lot of economists with PhDs in economics working in economics departments who have publications in mathematics. $\endgroup$ Commented Sep 3, 2023 at 8:23

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We can mention John Maynard Keynes and Frank Plumpton Ramsey, in the field of the theory of probability. In particular, their contributions are considered important with regard to the subjective theory of probability.

Subjective theory of probability was developed mainly by the mathematicians Bruno De Finetti and Leonard J. Savage, as opposed to the frequentist and to the classical theories of probability.

Keynes published, in 1921, his A Treatise on Probability.

Even if Keynes' theory of probability is important also with regard to economics, and the relevant matter of expectations in economic theory, the Treatise on probability is a true text of pure mathematics, or better, Keynes would have said, of logic, as he considered the theory of probability as a part of logic.

Keynes is an advocate of the subjective view of probability and criticized the frequentist approach which, neglecting the unique event as extraneous to science, and in general situations that cannot be traced back to classes of events, excludes from the domain of probability problems that are typical of situations of uncertainty:

“The identification of the probability with statistical frequency is a very grave departure from the established use of the word; for it clearly excludes a great number of judgments, which are generally believed to deal with probability.$^1$

Distancing himself from frequentism, Keynes outlined a ‘logicist’ conception of probability, considered as ‘rational belief’ and expressed by a logical relation between two statements:

We seek to justify some degree of rationale belief about all sort of conclusions. We do this by perceiving certain logical relations between the premises and the conclusions. The kind of rational belief we infer in this manner is termed probable (or in the limit certain), and the logical relations, by the perception of which it is obtained, we term relation of probability.$^2$

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Ramsey also, who was a mathematician, a philosopher and an economist, is considered a proposer of the subjective theory of probability. His most well-known work about probability is Truth and Probability (1926)$^3$.

Ramsey proposed a subjective theory of probability as a measure of the degree of partial belief of an individual before alternatives options. $^4$

But he maintained a different perspective with respect to Keynes , and he wrote in 1922 a review of Keynes’ Treatise, ‘MR Keynes on probability’, where he criticizes Keynes’ concepts of logical probability-relations and non-numerical probabilities.$^5$

Ramsey is well-known also as a pioneer of the theory of decision in conditions of uncertainty, and for his studies in combinatorics, in particular he is famous for the so-called Ramsey Theory, and a generalization of the Pigeonhole Principle.

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The French mathematician Émile Borel also devoted an article to Keynes’ Treatise, in his Traité du calcul de probabilité et ses applications (1924-34), distancing himself from Keynes:

Borel and Keynes were the contemporaries who crossing over the English channel, greatly influenced each other in the twentieth century. In his influential book, Borel (1938) harshly criticized Keynes' position on probability theory. In plain English, Borel regarded probability as a measurable object, thus constituting one important branch of mathematics. In contrast, Keynes thought of probability as a non-measurable item, thereby belonging rather to one branch of logic. Their controversies were rather well-known in the academic world, producing so many papers even after their deaths until today.$^6$


$^1$ Keynes, John M., A Treatise on Probability, Wildside Press, 2010, p. 105

$^2$ ibid. , p. 123.

$^3$ Ramsey F.P. (1926) Truth and Probability, lecture at Moral Science Club, Cambridge; published for the first time in Ramsey F.P. (edited by Braithwaite R.B.) (1931),The Foundations of Mathematics and other Logical Essays, Kegan Paul, Trench, Trubner & Co., London.

$^4$ Ramsey F.P. , ‘Probability and partial belief’, https://www.repository.cam.ac.uk/items/74959b5f-c87f-4bfe-b723-96877693f882

$^5$Ramsey, F. P.1922, [1989], ‘Mr. Keynes on Probability’, The Cambridge Magazine, vol. 11 , 3–5. Reprinted in British Journal for the Philosophy of Science, vol. 40, 219–222. https://www.journals.uchicago.edu/doi/abs/10.1093/bjps/40.2.219

$^6$ Yasuhiro Sakai, ‘Émile Borel Versus John Maynard Keynes: The Two Opposing Views on Probability

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The following are some contributions.

  • Game Theory: Perhaps the most well known field of mathematics to which economists have made significant contributions is Game Theory (if it is not considered a branch of Economics itself.... so much is the value of Economists' contributions....). The founders of this field are John von Neumann and Oskar Morgenstern. von Neumann was a polymath and is considered an economist as well. Morgenstern was primarily an economist by profession. They founded the discipline with their analysis of zero sum games. Many concepts of choice within the set of equilibria, such as Schelling points were developed by economists. A lot of equilibrium concepts were developed by economists, in fact, too many to mention. The entire sub-field of studying games of incomplete information was pioneered by economist John Harsanyi. Reinhard Selten developed the concept of sub-game perfect equilibria and other concepts to study extended form games. The development of Mechanism Design also owes a lot to economists like Leonid Hurwicz, Eric Maskin and Roger Myerson, who received their Nobel Memorial Prize for the same.

  • Linear Programming: Leonid Kantorovich and Tjalling Koopmans made significant contributions to linear programming. Both received the 1975 Nobel Memorial Prize in Economics. While Kantorovich had formal training in mathematics, he is widely credited as an economist as well. Kantorovich's approach to resource planning is foundational and he is credited as one of the founders of Linear Programming. He discovered that coefficients in linear programming problems can be essentially interpreted as prices. You can watch Fields Medalist Cedric Villani's lecture where he also discussed Kantorovich's work. Koopmans had formal training in Economics under Jan Tinbergen, first winner of the Economics Nobel, in addition to training in mathematics and physics. One of Koopman's major contributions include the determination of optimality conditions in linear programming problems.

  • Statistics and Probability Theory: Economists have obviously contributed majorly to Econometrics, and thus, to many parts of Statistics and Probability Theory that are relevant to Econometrics. The Matrix formulation of the Linear Regression Model was developed by economists like Ragnar Frisch, in addition to statisticians like Fisher. Frisch, along with Frederick Waugh and Michael Lovell, also contributed to a geometric understanding of regression, for example, in the Frisch-Waugh-Lovell theorem. Trygve Haavelmo has contributed to the development of the theory of maximum likelihood, structural models etc. Economists have contributed to the development of numerous estimators. For example, the development of the theory of the Generalised Method of Moments owes much to economists like Lars Peter Hansen. Economists like Lawrence Klein l, Robert Engle and Granger have made highly influential contributions to times series analysis such as cointegration and Granger causality. Statistical approaches to the study of causality owes a lot to economists such as Josh Angrist and Guido Imbens, who used Rubin's model to go beyond the "correlation is not causation" idea.

  • Decision Theory: Decision theory is closely related to microeconomics and has applications to philosophy and logic,statistics, computer science and even quantum mechanics. Several foundational results in this field have been given by economists. While the concept of utility was propounded by philosophers like Bentham and Mill, economist Alfred Marshall formalised it. Later, Ragnar Frisch laid the foundations of the Theory of Ordinal Utility using binary preference relations. This was developed further by economists like Arrow and Debreu who developed the Theory of Rational Choice and Social Choice. Paul Samuelson, an American economist, developed the logically equivalent theory of Revealed Preference. In a parallel stream, von Neumann and Morgen Stern's work led to the development of the Theory of Cardinal Utility. All of these are foundational to the field, in addition to the work of statisticians and mathematicians, like the Savage Axioms.

Many of these contributions are often considered parts of Economics itself, while in other contexts, they will be considered distinct. For example, Game Theory is often taught as part of Microeconomics, but it has applications in Computer Science, Biology, Political Science etc. as well. Similarly, Decision Theory is almost inseparable from Rational Choice Theory, which is at the heart of Microeconomics. But you would see this topic discussed in detail in other places independent of Economics, such as in Philosophy or Statistics books. Take the example of this entry from the Stanford Encyclopedia of Philosophy on Expected Utility or this entry on Decision Theory. The topics mentioned for Statistics and Probability are also often rightly taught as part of Econometrics, but they are also taught in other academic fields.

Thus, it is often difficult to separate results by discipline, especially when two of the disciplines concerned are as diverse and wide a field as Economics and Mathematics. For an interesting conversation on this, you can see this interview of Economics Nobel Laureate Amartya Sen, where he describes how many a times what is considered interdisciplinary can be thought of as a part of one's own discipline or vice versa, when asked why he chose to combine Economics and Philosophy. This happens in the cases of other disciplines as well. For example, as you mentioned Physics, Newton considered geometry and calculus to be parts of Physics (see pages 51 to 59 here).

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