Mathematical Analogies: An Engine for Understanding the Transfers between Economics and Physics

Abstract

The influence of physics on economics has been largely analysed; the opposite influence also exists even if it has been less studied. In the last decades the relation between these two disciplines has increased. Economic models are more often used in physics (minority game, GARCH model, etc.). The aim of this paper is to explore the role of mathematical analogies in the evolution of the relation between physics and economics. We show how these analogies have contributed to make the disciplinary boundaries of economics and physics more permeable. We investigate three examples: Frisch’s PPIP model (1933); the use of the Ising model for creating econophysics in the 1990s; and the minority game, created by econophysicists in 1997 for solving an economic problem and nowadays used in physics.

Keywords:

• Ragnar Frisch
• Le Corbeiller
• mathematical analogies
• economics and physics
• history of business cycle theory
• econophysics
• minority game

JEL CLASSIFICATION:

• B16
• B23
• B26
• B3
• B4

Acknowledgements

The authors are grateful to John B. Davis, Jean-Marc Ginoux and Guy Numa, for their helpful remarks, and the editor for his constructive advice.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 PPIP: Propagation Problems and Impulse Problems.

2 Ménard’s paper is the translation of a study written in French and published in 1981 in a two-volume book, Analogie et connaissance. This book is the sequel of interdisciplinary conferences on analogical reasoning held at the Collège de France.

3 Giorgio Israel (1945–2015), trained in mathematics, was a historian and philosopher of sciences. This book, written in French and only translated into Italian, remains little known by English-speaking scholars.

4 All translations from the French are ours, except when specified otherwise.

5 See Ménard (1978) and Le Gall (2007).

6 Translation by Mirowski and Cook (1990, 208).

7 On Regnault’s contributions to financial economics, see Jovanovic and Le Gall (2001).

8 Knuuttila and Loettgers (2016, 378) clarify that ‘Vito Volterra cast his theorizing on biological associations – the most famous example of which is the Lotka–Volterra model – in terms of “mathematical analogies” drawn from mechanics (Volterra 1901)’.

9 On the phenomenological approach in physics, see Cartwright (1983).

10 Louça (2007, 118–120) discussed some of Frisch’s positions from the perspective of Israel’s analysis. He compared for instance some key positions of Bourbaki (particularly causality) with Frisch’s ones, and concluded that Frisch’s approach was not compatible with the mathematical analogies. However, as section 1 explained, mathematical analogies evolved during the whole twentieth century. Remember that Van der Pol did not drop the causality, while such a dropping is a major characteristic of mathematical analogies.

11 However, it is worth reminding that at this time oscillation theory was still in its infancy and its formalism was not completely fixed. Therefore, the definitions and terminologies used by Frisch are confusing and have generated many misinterpretations in the literature (Jovanovic and Ginoux 2021).

12 Another reason for Van der Pol’s regular presence in France is the connection with Poincaré’s work on self-sustained oscillations and limit cycles (Ginoux 2017). In other words, this research topic was extremely en vogue in France at this time. Moreover, Van der Pol discovered Poincaré’s work thanks to Le Corbeiller.

13 Several economists who worked on business cycles acknowledged the influence of Van der Pol and Le Corbeiller. Richard M. Goodwin is a telling example. As he explained, Le Corbeiller had a great influence on his work, in particular for stressing that nonlinearity was necessary for explaining how macroeconomic oscillation could be a recurrent phenomenon (Goodwin 1951, 2). During the Second World War, when he was in Harvard, Goodwin taught physics. It was during this period that he met Le Corbeiller in his office and asked him ‘whether he would teach [him] the theory of nonlinear dynamics. He, then, literally took [him] “by the hand” and taught [him] nonlinear dynamics’ (Velupillai 2017, fn 4). See also Velupillai (1998).

14 Hamburger was working between 1928 and 1931 on the application of Van der Pol’s relaxation oscillations to economic cycles by using an analogical reasoning.

15 Frisch’s PPIP led to one of the most fruitful debates with another key econometrician, Michał Kalecki (1935). This debate started during the third Econometric Society meeting held in Leyde in 1933 and was published in Econometrica from 1935 onwards.

16 The birth and the early development of econophysics is today well-known (Jovanovic and Schinckus 2013; Kutner et al. 2019; Rosser 2006, 2018; Schinckus 2021). Over the past two decades, econophysics has carved out a place in the scientific analysis of financial markets, macroeconomics, international economics, market microstructure, labor productivity, etc., providing new theoretical models, methods, and results (Aoyama et al. 2017; Gabaix 2009; Jovanovic and Schinckus 2017; Potters and Bouchaud 2003).

17 A critical phenomenon is a phenomenon for which the passage from one phase to another one is continuous. At the critical point, the system appears the same at all scales of analysis. This property is called ‘scale invariance’, which means that no matter how closely one looks, one sees the same properties. The dynamics of critical states can be characterized by a power law that deserves special attention, because this law is a key element in econophysics literature. See Jovanovic and Schinckus (2017, ch. 3) for an extended definition of phase transitions and critical point in economic terms.

18 The Ising model was invented by the physicist Wilhelm Lenz in 1920, and named after one of his PhD students, Ernst Ising, who solved it in 1925.

19 A power law distribution is a special kind of probability distribution, such as $p\left(x\right)=C{x}^{-\alpha }.$ Such distribution is leptokurtic, and it is widely used for studying variables that have extreme values. In economics, a well-known power law is the Pareto law. The main property of power laws is their scale invariance. Moreover, from the viewpoint of statistical physicists, power laws are synonymous with complex systems, which makes their study particularly valuable.

20 As Knuuttila and Loettgers (2016) remind, ‘The Ising model is one of the most simplified and successful models in physics’, and it is used as an example of a minimal model by Weisberg (2007).

21 In the same vein, econophysicists introduced truncation techniques for power laws in order to fit the observations of the target domain with their mathematical model (Jovanovic and Schinckus 2017, ch. 3).

22 In fact, many of the power laws that econophysicists have been trying to explain are not power laws at all (Clauset et al. 2009).

23 Game theory was founded by the mathematician von Neumann thanks to the publication of his On the Theory of Games of Strategy in 1928, but it was his book published with the economist Oskar Morgenstern that initiated research in this field (Leonard 1995).

24 ‘Self-organized criticality’ is another key econophysics concept (Jovanovic and Schinckus 2017, ch. 5) Self-organized criticality is the property of dynamic systems that naturally self-organize into a critical point while obeying a power law.

25 This observation is compatible with Claveau (2019): articles published in economics journals cited fewer and fewer articles published in mathematics journals, suggesting that mathematical models used by economists are more and more created within economics.

26 We detailed the example of the minority game. However, it is not the only one. For instance, physicists nowadays are using GARCH models (Modarres and Ouarda 2014).

27 This expression is borrowed from Sornette (2014, 1).