"Your definition of rareté [marginal utility] impresses me as legitimate. And this is how I should justify it. Can satisfaction be measured? I can say that one satisfaction is greater than another, since I prefer one to the other, but I cannot say that the first satisfaction is two or three times greater than the other. That makes no sense by itself and only some arbitrary convention can give it meaning. Satisfaction is therefore a magnitude but not a measurable magnitude. Now is a non-measurable magnitude ipso facto excluded from all mathematical speculation? By no means. Temperature, for example, was a non-measurable magnitude – at least until the advent of thermodynamics which gave meaning to the term absolute temperature. The measurement of temperature by the expansion of mercury rather than the expansion of any other substance was nothing but an arbitrary convention. One could just as well have defined temperature by any function of temperature … provided that the function was monotonically increasing. Similarly you [on your side] can define satisfaction by any arbitrary function provided the function always increases with an increase in the satisfaction it represents.Jaffé points out that some of Poincaré’s coments foreshadow later developments in economics: revealed preferences and ordinal utility.
Among your premises, there are a certain number of arbitrary functions; but once given these premises you have the right to draw consequences from them mathematically. If the arbitrary functions still appear in the conclusions, the conclusions are not false, but they are totally without interest because they depend upon the arbitrary conventions made at the start. You ought, therefore, to do your utmost to eliminate these arbitrary functions and that is what you are doing…
…I can tell whether the satisfaction experienced by the same individual is greater under one set of circumstances than under another set of circumstances; but I have no way of comparing the satisfactions experienced by two different individuals. This increases the number of arbitrary functions to be eliminated.
When I spoke of the 'proper limits', that is not all I wanted to say. What I had in mind was that every mathematical speculation begins with hypotheses, and that if such speculation is to be fruitful, it is necessary (as in applications to physics) that one be aware of these hypotheses. If one forgets this condition, one oversteps the proper limits. For example, in mechanics one often neglects friction and assumes the bodies to be infinitely smooth. You, on your side, regard men as infinitely self-seeking and infinitely clairvoyant. The first hypothesis can be admitted as a first approximation, but the second hypothesis calls, perhaps, for some reservations." -- Henri Poincaré (1901)
Jaffé published his article in 1977. He mentions that Walras initiated this correspondence after being criticized by the mathematician Hermann Laurent, but Jaffé does not explain Laurent's criticism. Since then, Mirowski (1989) has cast new light on this criticism. According to Mirowski, Laurent, in correspondence with Walras and Pareto, queried these neoclassicals about integrability and why economists felt they were justified in assuming utility was the potential of a conservative vector field.
Mirowski does not mention Poincaré's correspondence with Walras. I’d like to see an analysis of this correspondence fromm Mirowski’s viewpoint. I would not expect to see something from Mirowski. He's gone on to other aspects of the history of mainstream economics.
As I understand it, Poincaré, in addition to all of his other accomplishments, was on the verge of discovering the special theory of relativity, but Einstein arrived there first. Poincaré is also cited in the mathematics of dynamic systems. So I expect that he understood the mathematics of vector fields quite well. Did he raise any questions about integrability and conservation laws in his correspondence with Walras?
References
- William Jaffé (1977). "The Walras-Poincaré Correspondence on the Cardinal Measurability of Utility", Canadian Journal of Economics, V. 10 (May): 300-307.
- Neil De Marchi (editor) Non-Natural Social Science: Reflecting on the Enterprise of 'More Heat than Light', Duke University Press.
- Philip Mirowski (2002). Machine Dreams: Economics Becomes a Cyborg Science, Cambridge University Press.
- Philip Mirowski (1989). More Heat Than Light: Economics as Social Physics, Physics as Nature's Economics, Cambridge University Press.
- Léon Walras (1954). Elements of Pure Economics (Trans. by William Jaffé), Ricard D. Irwin.
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