"T. C. Koopmans, perhaps the greatest defender of the use of the mathematical tool in economics, countered the criticism of the exaggeration of mathematical symbolism by claiming that the critics have not come forward with specific complaints. The occasion was a symposium held in 1954 around a protest by David Novick. But, by an irony of fate, some twenty years later one of the most incriminating corpora delicti of empty mathematization got into print with the direct help of none other than Koopmans. R. J. Aumann had already published in Econometrica an article dealing with the problem of a market in which there are as many traders as the real numbers, that is, as many as all the points on a continuous line. In 1972, Koopmans presented to the National Academy of Sciences a paper by Donald Brown and Abraham Robinson for publication in its official periodical. The authors assumed that there are more traders even than the elements of the continuum. Now, since the authors of both these papers and Koopmans are well versed in mathematics, they must have known the result proved long ago by George Cantor, namely, that even an infinite space can accommodate at most a denumerable infinity of three-dimensional objects (as the traders must necessarily be)." -- Nicholas Georgescu-Roegen, "Methods in Economic Science". Journal of Economic Issues, V. XIII, N. 2. June 1979. pp. 317-328.
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12 years ago
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at most a denumerable infinity of three-dimensional objects (as the traders must necessarily be)
Ummm, I think this one most definetly goes into 'missing the point' category and am a bit surprised that it would come from George. I mean, if you're ok with assuming that there's a continuum of traders, assuming that they're not 3D isn't that much of a stretch. Unless he's just joking of course.
And if he's referring to the Aumann paper on Core convergence, he's way off mark. It's supposed to be a limit result and the assumption of the continuum illustrates a particular point.
So the McCloskey critique applies: "So what?"
And Georgescu-Roegen refuted classical thermodynamics (Mirowski will be pissed off...). We cannot assume a continuum of particles if there is only a finite of them and tey have a spatiotemporal dimension.
Interestingly, one of the great things about the work of Brown and Robinson is exactly that their method implies a result for finite sets of agents.
And one can construct infinite dimensional spaces where a continuum of 3D Objects fits in (the space of functions from the reals to IR^3). Not that it would make any economic sense. Neither does Georgescu-Roegen
Actually one should use the set of funtions from IR to the powerset of IR^3.
Aumanns paper can be read here and the paper by Brown and Robinson here. That the latter paper has important implications for large but finite economies is demonstrated here.
I hope my post provided amusement. I offered it up as contrast and didn't really want to get into arguing about non-standard analysis, which I never studied.
Consider a sequence of economies with a finite number of agents. Off hand, I don't see how a limit of such a sequence can have more than a countable infinity of agents. The papers Michael reccommends are difficult reading.
I was amused by Michael's comment about Mirowski. I suppose there must be a better reading of Georgescu-Roegen.
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