Is “Mathematical Science” An Oxymoron?

Paul Davidson Is “Mathematical Science” An Oxymoron When Used to Describe Economics? Journal of Post Keynesian Economics, Vol. 25, No. 4 (Summer, 2003), pp. 527-545.

This is a post-Keynesian review of Roy Weintraub’s How Economics Became a Mathematical Science. It’s main point is that 

[T]he mathematical scientist emperor of mainstream economics is without clothes.

It will clearly be of the greatest signifiance if mathematics could say nothing about economics, and we would want to know why. But, as a review by Mirowski makes clear, the problem is not that the ‘mathematical scientist’ economist is mathematical, but that he employs an inappropriate kind of mathematics (and science). Mirowski notes that von Neumann, for example, didn’t follow either of Weintraub’s inappropriate mathematical method, and Davidson notes that Keynes didn’t either.

Weintraub is critical of Cambridge mathematicians, but from a British mathematical perspective mathematics only really started (or was at least reinvented) by Whitehead and Russell at Cambridge, and it is this ‘mainstream British’ approach that Keynes followed, as distinct from the other schools of economics whose limitations Weintraub, correctly, draws attention to. From this perspective, the ‘emperor’ was not so much properly scientific as scientistic: more concerned with appearing scientific than in using appropriate methods.

The main point, then, might be more properly expressed as:

The reductionist, formulaic, scientistic emperor of main-stream economics is without clothes, having rejected the appropriate scientific and mathematical clothing.

Thus, while others might think ‘mathematical science’ to be an oxymoron when applied to economics, I conjecture that it should be an aspiration which has yet to be achieved.

Details 

The following quote by Weintraub is insightful:

[E]conomists [were] deflected from their appropriate concerns by a mathematics that did not permit continued expression of a number of important ideas.

This can be compared with physicists who, before Einstein, found that Euclidean Geometry did not allow them to express some important ideas about the motion of, for example, Mercury. It was not that the Geometry was ‘wrong’ in itself, but that it was not appropriate to the intended application. Weintraub seems to confuse the two, as do more than a few  mainstream  economists. 

Davidson notes:

By the middle of the twentieth century, as far as mathematicians were concerned, rigor meant “derivable from an axiomatization in a formal or formally consistent manner”. In economics, rigor did not require quantification and the testing via empirical evidence to support the theoretical conclusions drawn from its axioms. Accordingly, there exists today an unbridgeable gulf between modelers and theorists. Weintraub argues that econometricians or applied economists today are modelers who “insist that the assumptions and conclusions of an economic model, a model constructed and developed mathematically, must be measurable or quantifiable”.

There seems to be some confusion here. The mathematical view of rigor is appropriate for mathematics, but not outside of it. Economics is not a field of mathematics. Rather, economics ought to be an empirical subject whose development should be ‘covered’ by the mathematics of Keynes, Russell and Good, for example.

Davidson notes:

Alternatively, is “truth” obtained by an axiomatic theory based on the least number of assumptions (“a general theory of employment, interest and money”) that is descriptive and applicable to reality? This alternative was Keynes’s vision – as suggested in his analogy of comparing classical economists with Euclidean geometers in a non-Euclidean world. … The onus is on those who add such restrictive axioms, such as Debreu, to demonstrate the relevance to the real world of their additional restrictive axioms of their specific case analysis.

On my reading, much of the problem in mainstream economics seems to be as a result of the kind of group-think that mathematics – if taken seriously – ought to be able to challenge. There is a lot of criticism of Bourbaki and Debreu yet they seem to be quite open in advertising their own limitations:

[T]heory, in the strict sense, is logically disconnected from its interpretation.

Debreu believed his work to be “the definitive analytic mother-structure from which all further work in economics would depart, primarily by weakening its assumptions or else superimposing new interpretations upon the existing formalism.”

It seems to me that economics become more like a craft than a science: with only slight exaggeration, it developed rigid interpretations that could not be challenged on pain of expulsion from the guild.

Davidson notes:

There is a good deal of evidence, only slightly alluded to by Weintraub, that those mathematical economists who understood the irrelevant nature of the Arrow-Debreu axiomatic system and tried to warn their colleagues that this system was not applicable to the real world, were ignored.

Why has the not obscure John Maynard Keynes been ignored and dismissed as irrelevant in this era when “economics became a mathematical science”? My answer to this question turns on the fact that the axiomatic basis of the prewar classical economics of Keynes’s time had not been defined in the rigorous manner of Debreu. This made it more difficult for Keynes (especially in chapter 2 of  The General Theory) to correctly identify the classical axioms (similar to the Euclidean axiom of parallels) that he insisted had to be overthrown if a non-Euclidean economics were to be worked out.

 Thus the problem is not in a failure of mathematics, but in the failure of economists to appreciate the mathematics of Keynes, Whitehead, Russell et al. 

See Also

Mirowksi (as above) notes that Weintraub only attacks the kinds of mathematics used by mainstream mathematics, leaving open the possibility of using different mathematics. He notes

[T]his stance effectively permits a residual commitment to the technological determinism prevalent amongst economists, … .

This suggests to this reader the following questions:

  1. Did economists select the mathematics that they did because it implied determinism?
  2. Would alternative mathematics, such as Keynes’, also imply determinism?

Such questions seem highly pertinent.

Another review is Roy J. Epstein and David Colander The Journal of Economic History Vol. 63, No. 2 (Jun., 2003), pp. 514-516.

Dave Marsay

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