Abstract
The Representational Theory of Measurement (RTM) offers a formal theory of measurement, with measurement understood as a homomorphic mapping between two types of structure: an empirical relational structure on the one hand, and a numerical structure on the other. These two types of structure are characterised axiomatically, as sets with certain relations defined on them. For a quantitative attribute like mass, for example, we find an empirical relational structure of weights with ordering and concatenation relations defined over them, and a numerical structure provided by the real numbers, less-than, and addition to represent the empirical relational structure. The numerical structure serves merely as a representational tool to capture the relationships between the weights; and the mathematical relations of ordering and addition are interpreted concretely as physical orderings and concatenations in the context of particular measurement operations. RTM has sometimes been interpreted as offering a kind of reductionist approach to quantitativeness, for two reasons: 1. RTM takes numbers to play a purely representational role in measurement 2. RTM takes a permissivist view of numerical representations: many kinds of attributes can be numerically represented, not just traditional quantities, like length or mass Insofar as we equate quantitativeness with being numerical, it would seem that RTM takes a reductionist view of quantitativeness, because it takes a deflationary view of numerical representation: the only thing you lose if you omit numerical representations is convenience. I argue here that, on the contrary, RTM not only does not commit us to a reductionist view of quantitativeness, but in fact provides us with a novel criterion for quantitativeness, which shows why reductionism about quantitativeness is so difficult. The first part of my argument rejects the view that quantitativeness is best understood as being numerical. RTM demonstrates quite clearly that numerical representation is neither necessary nor sufficient for an attribute's being quantitative. It is not sufficient, because many intuitively non-quantitative properties can be represented numerically using the tools of RTM; in general, numerical representability is pretty easy within the RTM framework. It is not necessary, because RTM itself shows how empirical relational structures can be represented non-numerical (for example geometrically). Having rejected the claim that quantitativeness means being numerical, I then show in part 2 of my argument that RTM in fact provides a novel criterion for quantitativeness. This proceeds in two steps: first I show how uniqueness theorems provide a reason for thinking that only some numerical representations are representations of quantities, and second, how we can characterise the structures amenable to such representations using the resources of RTM. This yields a criterion for quantitativeness as a feature of certain kinds of structures. Since RTM's own conception of measurement is that of a homomorphic relationship between two structures, we shouldn’t expect one of these structures to count as quantitative by this new criterion, while the other one is not. Reducing quantitativeness is harder, not easier from the perspective of RTM.
