Fisher's Instrumental Approach to Index Numbers

If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist.

-David Hilbert, "Mathematical Problems" (1902)

Sometimes control with a single lens is impossible since some incompatible features are required and a compromise becomes necessary calling for further judgement on the part of the designer as to which error should be reduced and to what degree.

-R. J. Bracey, The Technique of Optical Instrument Design (1960)

Since the beginning of the nineteenth century, a large number of price index number formulae have been developed, mostly named after their inventors, such as the Paasche and Layspeyres indexes. Parallel with the invention of new index formulae, criteria were developed for distinguishing between them. These parallel developments culminated in Irving Fisher's two classics on index numbers, The Purchasing Power of Money (1911) and The Making of Index Numbers (1922). In these, Fisher evaluated index formulae in a systematic way with respect to a number of "tests." Although these two volumes are considered the "Old and New Testament" of Axiomatic Index Theory (Vogt and Barta 1997, viii), the axiomatic approach originated from challenges to Fisher's system of tests on grounds of their inconsistency and the seeming arbitrariness of the choice of tests. That debate started with Ragnar Frisch in 1930, but the Axiomatic Index Theory only got its current name and shape-based on functional equation analysis-from Wolfgang Eichhorn in 1973. In Axiomatic Index Theory, the tests are considered as requirements on the functional form of the index number from which the index formula can be derived. If these requirements are inconsistent, no formula can be constructed. So although Fisher's work is seen as the forerunner of the Axiomatic Index Theory, his system of tests was much criticized because of its apparent internal inconsistency. The aim of this essay is to show that evaluating Fisher's work from an axiomatic perspective leads to a misconception of his empirically inclined approach to the assessment of index numbers. For a better understanding of his work on index numbers, his background in mathematics, his philosophical thinking, and his inventions of measuring instruments will all be examined.

The crucial problem behind the assessment of measurement formulae is that they are not theories and thus cannot be assessed as such. They cannot be tested in the usual way by comparing quantitative data generated by the formula with the quantitative data of the phenomenon to see whether they differ significantly. The reason is quite obvious: to obtain quantitative data of the phenomenon, we need to possess a measuring device. Of course such devices can involve theories, in a strong or loose sense; however, they are not built to explain or to make predictions about a phenomenon but to generate numbers about it. Nevertheless, we are not satisfied with any number; for example, we care about its representing quality. To attain this quality, measurement formulae should fulfill the relevant theoretical and empirical requirements with respect to the phenomenon concerned.

The assessment of the formulae depends not only on whether they fulfill certain theoretical and empirical requirements but also on how these requirements are fulfilled. That is, formulae are also assessed on the basis of how they are constructed, whether this construction is done in a rigorous way or not. And what is taken as rigor depends on the kinds of objects formulae are assumed to be, whether they are considered as formal concepts or as instruments. Although one usually associates instruments with physical devices, like a thermometer or a ruler, in economics they are immaterial. Despite their immateriality they still function as if they were empirical objects.¹ This treatment contrasts with the received or standard account of mathematical objects in which they are considered as formal axiomatic abstractions.² Both accounts will be considered to see whether they enable us to understand specific developments in the history of index number theory. It will be shown that the two quotations presented at the beginning of this essay, which characterize these two opposite accounts-from here on to be labeled as the axiomatic and the instrumental approach...