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History of Political Economy Annual Supplement to Volume 33 (2001) 303-312
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Measurement, and Changing Images of Mathematical Knowledge
E. Roy Weintraub
To be sure, computation invites measurement, and every easily observed regularity of certain magnitudes is an incentive to mathematical investigation.
—J. F. Herbart (in Moritz 1914, 251)
A number of essays at the conference touched on issues of mathematization, formalization, rigor, and axiomatization even as issues of measurement remained the focus of the discussions. Consequently the connection between the history of mathematics and the history of measurement remained outside the direct view of the conference program even as that history cast its shadows on histories of measurement. In this perspective I suggest how in particular the changing “image of mathematics” may provide one context for understanding the changing nature of measurement in economics.
As the historian of mathematics Leo Corry (1989, 411) has argued:
We may distinguish, broadly speaking, two sorts of questions concerning every scientific discipline. The first sort are questions about the subject matter of the discipline. The second sort are questions about the discipline qua discipline, or second-order questions. It is the aim of the discipline to answer the questions of the first sort, but usually not to answer questions of the second sort. These second-order [End Page 303] questions concern the methodology, philosophy, history, or sociology of the discipline and are usually addressed by an ancillary discipline.
The first sort of question concerns the discipline's knowledge, while the second sort concerns the image of knowledge. Corry's (1996) argument is that to speak of change in mathematics is to speak not only of change in mathematical knowledge, in the sense of new theorems proved, new definitions created, and new mathematical objects described. But change in mathematics also involves changes in the image of mathematics, in, say, changed standards for accepting proofs, changed ideas about mathematical rigor and truth, and changed ideas about the nature of the mathematical enterprise. For Corry (1989, 418), “It is precisely the task of the historian of mathematics to characterize the images of knowledge of a given period and to explain their interaction with the body of knowledge—and thus to explain the development of mathematics” (emphasis added).
I urge the view that there were several shifts in the image of mathematical knowledge over the nineteenth century—the period out of which modern economics and its concerns with measurement developed—and those shifts are one context for understanding the development of economists' ideas about measurement. To fully detail this argument is beyond the scope of this short note,1 so in what follows I will simply identify some of the issues and provide some of the references which can more fully indicate the nature of the claim.
From Geometric Certainty to Physical Representation
The first change in the image of mathematics was based on a new conception of what mathematical truth might mean. It occurred over the second third of the nineteenth century and was well incorporated in the Continental tradition in mathematics. That is, outside England there was a change in mathematics between the time of William Whewell's defense of mathematics in the educational process,2 a defense based on [End Page 304] the notion that mathematics (vide Euclid, Newton) was the paradigm of certain and secure knowledge (the time of Alfred Marshall's student days), and Marshall's later time as professor of political economy. The emergence of non-Euclidean geometries had made Whewell's argument about axiomatics and inevitable truth ring hollow long before the turn of the twentieth century. In the time of the new geometries, the difficulty of linking mathematical truth to a particular (Euclidean) geometry produced a real crisis of confidence for Victorian educational practice, a point well documented in Richards 1988. The roots of this crisis are linked to the unhealthy state of mathematics in England associated with the backward-looking mathematical tripos examination and its importance in the Cambridge institutional structures that were developed to define a fixed order of merit among honors graduates...