The estimated reading time for this post is 7 minutes.

*This contribution is the second post in a four-part blog series on the history of mathematics in economics. For the first post, which also introduces the series, click here. *

**My previous contribution on Mirowski painted a grim picture of the role of mathematics in economics: Irving Fisher, one of the saints of neoclassical economics, had tried to apply physical methods and theories to economics because he thought there was a substantive analogy between the ontology of physics (particles, force, energy) and the ontology of economics (individuals, marginal utility, value). This translation was wrong, because Fisher did not understand the physical theories that he was invoking, and it was inadequate, because the physical theories that Fisher pressed on economic problems simply did not fit that domain. A different perspective on the use of physical analogies in economics is defended by Marcel Boumans (1993) in his treatment of the exchange between physicist Paul Ehrenfest and economist-to-be Jan Tinbergen. According to Boumans, this case is an example of ‘formal analogies’: Tinbergen drew on physics for his innovative macro-economic models because some aspects of the mathematical form of physical and economic systems were similar, not because of any underlying substantive analogy between the subject matters.**

**Physical analogies**

Boumans draws on the example of Ehrenfest and the early works of Tinbergen to argue that the exchange of a substantive metaphor is not the only way in which the gap between physics and economics could be bridged. He argues that the story of the interaction between Ehrenfest and Tinbergen is better told as an analogy of mathematical form, as pioneered by Maxwell. This kind of analogy is described in Klein’s biography of Ehrenfest (1970).

In his work, Maxwell took what he called ‘physical analogies’ as his starting point. To prevent confusion with the more general category of analogies with physics, which also includes Mirowski’s substantive analogies, we will call them ‘formal analogies’ here. Formal analogies were partial similarities between the forms of the laws of two sciences, and could be used to extend our knowledge of both. If two different mechanical systems can be represented in a (partially) identical mathematical form, then our understanding of one system can be enlarged by studying the other. The necessity of having two mechanical systems disappeared in Maxwell’s later work: it was sufficient to draw an analogy between a mechanical system and a dynamic system represented only in mathematics.

This idea of mechanical systems as represented by a dynamic system was further developed by Heinrich Hertz. For Hertz, analogies with dynamic models were not just *one* way to understand a mechanical system, it was the *only* way. To make this point, Hertz introduced the notion of ‘concealed masses’, i.e. masses that could not be discerned empirically, but that have a significant influence on the behavior of mechanical systems. If such concealed masses exist, limiting our studies of mechanical systems to entities that we can empirically discern would ensure that our analyses and predictions are always off. Taking dynamic models, which need not be grounded in empirical entities, prevents us from focusing only on what we can empirically ascertain.

Hertz’ innovations were gratefully received by Boltzmann, who found them, in Boumans words, “liberating” (p. 136). The idea that a model (note: not necessarily dynamic) was a sufficient explanation of a mechanical system allowed him to represent a mechanical system mathematically as having traits X and Y without ever claiming that the mechanical system, in reality, possessed those traits. Through Boltzmann, this approach to models arrived at one of his students, Paul Ehrenfest, and through him it arrived at Tinbergen.

**State of Statistics**

Before elaborating on the way in which Tinbergen applied the notion of formal analogies to economics, his own concerns should be put into their proper context. Tinbergen, at the time, was dissatisfied with the state of the economic science. Boumans mentions three general reasons. Firstly, Tinbergen felt that the static equilibrium models of Walras and Pareto wrongly represented the world as being in equilibrium. Tinbergen felt the world was continuously changing: the world might tend to equilibrium but is rarely in it, and equilibria themselves are not immune to change either. The other reasons that Boumans mentions are both concerned with Business Cycle Theory. Business cycles were one of the main topics of discussion among economists of the day, and revolved around discerning, qualifying and analyzing particular repeating patterns in macroeconomic data. Part of this effort concerned relating different patterns to each other, predominantly by relating them in time: which came first or after, and how much time was there in-between. The ‘lag’ between patterns was often established purely by observation, much to Tinbergen’s disappointment. Tinbergen felt that economists should turn to theory, rather than only to observation, to determine the lags between patterns. Finally, Tinbergen was unhappy that theories about the motivating causes behind business cycles tended to look for those causes outside of the system itself. Tinbergen’s background in physics had made him aware that this need not be the case: the same patterns could in principle be formed by dynamics internal to the system.

**Conditions for the use of Physical Tools**

It was in the light of these concerns that Tinbergen became interested in possible formal analogies between physics and economics. One of the most important innovations for economics was his attempt to represent economic (sub)systems as dynamic systems of differential equations. This kind of representation was common in physics, but was novel for economics. According to Tinbergen, applying this new language to economics was warranted if the (sub)system fulfilled the following four conditions:

- One actor: The function describes only one actor (‘only one ophelmity function’)
- Integrability: The system can be described by an integral
- Derivative in integrand: If we aim to describe the behavior of x, the function with which we describe it must include also the derivative of x.
- Limits: The function must have limits.

To clarify, let me first point out what these conditions would imply for a physical system. Most of Tinbergen’s dissertation is concerned with such systems, though his treatment is much more complex and involves more difficult cases. It should here suffice to consider the movement of a simple pendulum.

- One actor: The first condition is easily fulfilled, since there is only one pendulum.
- Integrability: Fulfilling the second condition means that the total movement of our pendulum can be described as an integral of the individual swings.
- Derivative in integrand: The third condition is met because the change in movement of the swing is the derivative of its movement.
- Limits: The final condition is met because the initial impulse we give to the pendulum constitutes clear limits.

**In the case of economics, the picture would look like this:**

- One actor: As above, there is only one actor, e.g. company, country or individual.
- Integrability: This condition is met when our function regards a total change in a variable during a particular time, such as total profit over a particular time period.
- Derivative in integrand: This means that the function describing the change in a particular variable is included in the function that describes the behavior of the variable itself. In other words, when we describe our inventory, our integrand must include a derivative which describes the change in inventory.
- Limits: This condition is met in case the dynamic system does not explode. According to Tinbergen, this condition is met in the case of business cycles.

The simplest examples of such systems, which here suffice, can be represented in a differential equation that takes the following general form:

y’’ + ay’ + by = 0

In which a > 0 would violate the fourth condition, because the function would explode.

**Formal analogies versus substantive analogies**

Returning to the main point, we can see that the kind of equation of (1) can be made to apply to physical and economic problems alike: they are formal analogies in the sense that can be traced back to Maxwell. The only kind of condition that is imposed on the use of dynamic models in economics is, correspondingly, of a formal kind: physical and economic systems can be described using the same mathematical language as long as they satisfy the four *mathematical* conditions mentioned above.

The idea of formal analogies thus captures the view that knowledge about system A can help us to understand system B if the formal structure of the systems, in this case their mathematical form, is the same. This does *not*, as we have mainly seen in the case of Boltzmann, presuppose that the two systems describe the same kind of substance.

This is the key difference between the accounts of Boumans and Mirowski with regards to the use of mathematics: while the *tertium comparationis* of the analogy between physics and economics is ‘substance’ for Mirowski’s Fisher, it is ‘formal structure’ for Boumans’ Tinbergen. A corollary is that, on Boumans’ account, there is no metaphor transferred from one science to the other.

The next contribution will be about E. Roy Weintraub’s treatment of Gerard Debreu in his ‘How Economics Became a Mathematical Science’. Debreu, who has been one of the main motors behind the mathematization of 20^{th} century economics, will radically break off from the authors that have been discussed so far: he introduces a kind of mathematics that is divorced from physics, and even shares with Mirowski a disdain towards the influence of physics in economics.

o-o-o

*I am grateful to Ivan Flis for helpful suggestions and corrections in producing this blog series. The paper from which this series is derived was written under the supervision of professor Geoffrey Hodgson (University of Hertfordshire), and has benefited greatly from his commentary.*

Manuel Buitenhuis is a graduate student at the University of Utrecht (MSc History and Philosophy of Science) and the Erasmus University Rotterdam (MA Philosophy and Economics). He holds a BA-degree from University College Utrecht, where he mainly studied economics and philosophy.

# Bibliography

Boumans, M. (1999). Built-in Justification. In M. Morgan, & M. Morrison, *Models ad Mediators: Perspecitves on Natural and Social Science* (pp. 66-96). Cambridge: Cambridge University Press.

Boumans, M. (1993). Paul Ehrenfest and Jan Tinbergen: A Case of Limited Physics Transfer. In N. de Marchi, *Non-Natural Social Science: Reflecting on the Enterprise of More Heat than Light* (pp. 131-156). Durham and London: Duke University PRess.

Hollestelle, M. (2011). *Paul Ehrenfest. **Worsteling met de moderne wetenschap, 1912-1933.* Leiden: Leiden University Press.

Klein, M. J. (1970). *Paul Ehrenfest: The Making of a Theoretical Physicist.* Amsterdam: North-Holland.

Mirowski, P. (1989). *More Heat than Light.* Cambrdige: Cambridge University Press.

Tinbergen, J. (1930). Bestimmung und Deutung von Angebotskurven: Ein Beispiel. *Zeitschrift für Nationalökonomie* , 669-79.

Tinbergen, J. (1928). *Minimumproblemen in de natuurkunde en de ekonomie.* Amsterdam: H.J. Paris.

Weintraub, E. (2002). *How Economics Became a Mathematical Science.* Durham and London: Duke University Press.

Image credits:

www.flickr.com/photos/visualsymphony/2902346130

“Jan Tinbergen 1982” by Anefo / Croes, R.C. – Nationaal Archief Fotocollectie Anefo; Nummer toegang 2.24.01.05; Bestanddeelnummer 932-3849. Licensed under CC BY-SA 3.0 nl via Wikimedia Commons – http://commons.wikimedia.org/wiki/File:Jan_Tinbergen_1982.jpg#mediaviewer/File:Jan_Tinbergen_1982.jpg

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