The Axioms of Quantity and the Theory of Measurement

Otto Ludwig Ho lder (born, Stuttgart, 1859; died, Leipzig, 1937) was professor of mathematics at the University of Leipzig (Ho lder, 1972) when he published the paper here translated into English. Today, Ho lder is remembered in mathematics for a variety of results, but he is of significance to readers of this journal because of Ho lder's theorem. This theorem occupies a central place in modern measurement theory. Indeed, Luce reports that it has been said that ``representational measurement theory is nothing but applications of Ho lder's theorem'' (1994, p. 4). Furthermore, its applications have produced results with revolutionary implications for quantification in psychology and cognate sciences. In particular, Krantz, Luce, Suppes, and Tversky (1971) ``unified measurement theory by reducing proofs of most representation theorems to Ho lder's theorem'' (Luce, Krantz, Suppes, 6 Tversky, 1990, p. 2) and, more recently, Luce et al. (1990) used Ho lder's theorem in proofs relating to nonadditive structures. These applications obscure the fact that Ho lder's concerns were not quite those of modern theorists. His paper is a watershed in measurement theory, dividing the classical (stretching from Euclid) and the modern (stretching to Luce et al., 1990) eras. His concerns belonged, in part, to the classical era. He axiomatised the classical concept of quantity (using Dedekind's concept of continuity) in such a way that ratios of magnitudes (as understood in Book V of Euclid's Elements) could be expressed as positive real numbers (as intimated by Newton). Importantly, he achieved this result with a clarity that was not attained by others also working within the classical framework (e.g., Helmholtz, 1878; Frege, 1903; and Whitehead 6 Russell, 1913). However, other of his concerns were more modern. In Part II of his paper (a translation of which is to be published later), he showed that axioms for an apparently nonadditive structure, stretches of a straight line, entail that linear distances satisfy the axioms for quantity given in Part I. His concern to relate nonadditive structures to quantitative ones anticipates a distinctly modern interest and in doing this one of his axioms was not unrelated to the Thomson condition of conjoint measurement theory. That these results might have found direct psychophysical application is obvious and had the founder of modern experimental psychology, Wilhelm Wundt, who had established his famous laboratory at the same university, known as much of Ho lder's work as Ho lder knew of his (see Ho lder, 1900), the development of psychological measurement might have been transformed. That Ho lder's contribution eventually found its pivotal place in modern measurement theory is testimony to its enduring importance. That it was neglected within psychology until the middle of this century was psychology's loss. Contemporaneous recognition of his work within experimental psychology may have averted the criticisms of the Ferguson Committee (Ferguson, et al., 1940) and the crisis in psychological measurement theory that ensued, thereby making unnecessary Stevens' (1951) attempt at a rational reconstruction of psychological measurement and the confusions that it delivered upon the discipline. Ho lder's axioms of quantity (given in 91) have been criticised by Nagel (1932), Suppes (1951), and Luce et al. (1990), not because of any formal inadequacy leading to his main results, but because they do not match the kind of empirical structures of interest to these authors. I will not article no. 0023