Burn the Mathematics

I am teaching history of economic thought this semester (a joint undergraduate/graduate course) and at the moment we are discussing the contributions of Marshall. Marshall has this great advise about burning the mathematics. (1) Use mathematics as a shorthand language rather than as an engine of inquiry; (2) Keep to them till you have done; (3) Translate into English; (4) Then illustrate by examples that are important in real life; (5) Burn the mathematics; (6) If you cannot succeed in (4) then burn (3).

In thinking through this warning that we make sure that math is a servant to economic reasoning rather than the master of it, it is striking how resistant the scientistic impulse is to such practical advise.  As we will discuss later in the class, Samuelson promoted mathematical economics with the argument that confusion results in economics when we use the same words to mean different things or different words to mean the same thing.  He further argued that mathematical reasoning forces one to make explicit the assumptions underlying the analysis that remain hidden or implicit in literary analysis.

But Kenneth Boulding wrote an absolutely brilliant review essay of Samuelson's Foundations in the JPE and concluded that the flawless precision of mathematical economics may prove less productive in understanding the processes of human interaction than the literary vagueness of economic sociology and political economy.  Much of the history of modern economics has proved Boulding right and Samuelson wrong.  One of the main reasons for this is that mathematical reasoning can only ensure syntactic clarity and not semantic clarity, but economic reasoning requires both form of clarity to advance our understanding of the world.

Comments

[Jüri Saar]

In case someone wants to read Boulding's essay you can find it in the Journal of Political Economy, issue 3, volume 56, from 1948. The text is also availbale from JSTOR.

[Rafe]

A few comments on this. First of all, there is a good, though not recent article by Schwartz, 'The pernicious influence of mathematics in science' in "Logic, Methodology and Philosophy of Science" eds Nagel, Suppes and Tarski, 1962. To summarise, he argued that even in physics the maths can often obscure as much as it illuminates, and he suspected that the problem is even more intense in the social sciences. Perhaps the main reason for the success of physics, at least celestial mechanics, was less the use of maths than the incredible good fortune to have a stable and isolated system to work on, with a convenient scale, so it only took earth a year to do a circuit and not several decades a la Halley's comet.

Second, there is a fascinating book by two Italians that describes the origins and evolution of mathematic equilibrium theory in economics. There is a summary here.
http://oysterium.blogspot.com/2006/02/maths-invades-economics.html

This is an extract: In the chapter on developments in the US the authors pause in their narrative to provide an overview of four developing lines of research that can be discerned.

First, von Neuman’s decisive step towards the strongest possible mathematical approach. Developments in this line mostly employed convex analysis and fixed point theorems.

Second, the application of game theory to economic behaviour.

Third, adherence to the “ideal” Walrasian model, “the boldest and most consistent program of axiomatization, which was carried out by G Debreu and achieved the greatest successes of GET.” (259)

Fourth, Paul Samuelson’s work, which assimilated the Hicks approach but provided a more sophisticated mathematical apparatus and “a deeper understanding of physicomathematical culture.” (259)

Samuelson was a “child prodigy” in economic theory. His move from Chicago to Harvard, in his own words ‘put me right in the forefront of the three great waves of modern economics: the Keynesian revolution…the monopolistic or imperfect-competition revolution, and finally, the fruitful clarification of the analysis of economic reality resulting from the mathematical and econometric handling of the subject – including an elucidation for the first time of the welfare economic issues that had concerned economists from the days of Adam Smith and Karl Marx to the present’. (quoted on page 260)

“Samuelson’s work [during the 1930s and 1940s] marks a decided step forward in the mathematization of the discipline. It is the first treatise of economic theory in which the formal apparatus is not confined to the appendices but is one with the main argument”. (262)

Finally, Talcott Parsons thought that a closed system of equations represented the ultimate in theoretical development, even for sociology. He also spent most of his career at Harvard and I wonder if he was subjected to any direct influence from the Cowles Commission people and the Samuelson circle or whether they were all just drowning in the same current.

[Rafe]

For a number of criticisms of the abuse of maths in economics and sources of other critical commentary, see the Appendix to Chapter 4 in Bill Hutt's book "The Keynesian Episode".

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