Abstract
Non-discrete quantities such as mass and length are often assumed to be real-valued. Rational-valued measurement outcomes are typically thought of as approximations of the `real' values of their target quantity-instances. For example, the representational theory of measurement (RTM) models measurement as the construction of a function that sends a set of objects obeying certain qualitative axioms into the real numbers, such that the structure of the relations holding among the objects is preserved by the order and addition relations on the real numbers. The original architects of the modern version of RTM (Krantz et. al.) clearly acknowledge that this choice of representing mathematical structure is conventional, being influenced by pragmatic considerations related to computational simplicity, and they consider alternative representing structures that illustrate this conventionality. But whereas operations alternative to ordinary addition for additive measurement are considered, sets alternative to the real numbers are not. The formal results of RTM have recently been applied in formulating realist views of quantity, but the assumption that the real numbers are best suited for representing the structure of non-discrete quantities has not yet been examined. At the core of the standard RTM representation and uniqueness theorems is Hölder's theorem, which Hölder originally proved from a set of axioms that he regarded as ``the facts upon which the theory of measurable (absolute) quantities is based''. These axioms include Dedekind's axiom of continuity, reflecting the close conceptual connections between the real numbers and `continuous' quantities. Hölder regarded quantities as having magnitudes as axiomatized by Euclid, and understood Euclid's definition of proportion in terms of Dedekind cuts. Krantz et. al. adapt Hölder's theorem for their operationalization of quantitative concepts and construct measurement scales that are real-valued, but replace Dedekind's axiom with the Archimedean axiom, which they see as better-suited to their empiricist interpretation. But even on a realist understanding of quantity, we argue, there are good reasons to doubt the assumption that classical physical quantities are genuinely continuous. Our paper first reproduces the results of Krantz et. al. in a realist context, where a quantity and its magnitudes are understood in terms of determinables and their determinates. Understanding a physical quantity (such as mass) as a determinable property emphasizes the metaphysical significance of representing it as having the structure of the reals. We then prove analogous representation and uniqueness theorems thus establishing that a determinable quantity constrained by the same qualitative axioms can be represented by the rational numbers. This shows that RTM methods do not inherently provide justification for representing a quantity as having the structure of the reals, and that the appearance of such justification can be attributed to stipulations of either continuity or uncountability of the non-numerical target of representation. We argue that, if the formal results of RTM are to inform a metaphysical view of quantity, then the conventionality of the choice of the real numbers as the representing structure needs to be explicitly justified.
