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.

This paper examines the importance of the concept of magnitude to the philosophy of measurement. Until the mid-twentieth century, magnitude was a central concept in theories of measurement, including those of Kant (1781 A162/B203), Helmholtz (1887), Hölder (1901), Russell (1903, Chapter XIX), Campbell (1920) and Nagel (1931). In the 1950s, the concept of magnitude began to fade from discussions on the foundations of measurement. The standard presentation of the Representational Theory of Measurement (Krantz et al., 1971) does not mention magnitudes. Similarly, the International Vocabulary of Metrology analyzes measurement by using the concepts of quantity and quantity value, with scarce reference to magnitudes (JCGM, 2012). This paper argues that the concept of magnitude is an important component of any satisfactory theory of measurement, and that it is not reducible to the concepts of quantity, number, and quantity value. I begin by showing that numbers cannot be assigned directly to objects or events, but only to magnitudes, which are aspects of objects or events that admit of ordering from lesser to greater. Building on Wolff (2020), I use the determinable-determinate distinction to analyze the relation between quantities and their magnitudes. I then show how the concept of magnitude can be used to resolve two ongoing debates concerning the foundations of measurement: (1) the debate concerning the nature of measurement units; (2) the debate concerning the scope and limits of the Representational Theory of Measurement. Discussions concerning the nature of units date back at least to the early nineteenth century, when they were central to the development of the analytical theory of heat (Roche, 1998, Chapter 8; de Courtenay, 2015). Recently, these debates re-emerged as part of the drafting of the ninth edition of the Brochure of the International System of Units (Mari \& Giordani, 2012; Mari et al., 2018; BIPM, 2019). The debating parties disagreed on whether units are best understood as quantities or as quantity values. I argue instead that units are best viewed as magnitudes. My proposal generalizes across different modes of unit definition (e.g. by reference to specific objects, kinds of objects, and theoretical constants), and leads to a straightforward understanding of quantity values as mathematical relations among magnitudes. The concept of magnitude similarly sheds light on debates concerning the scope and limits of the Representational Theory of Measurement (Baccelli, 2018; Heilmann, 2015). RTM axioms can be interpreted in at least two ways: as models of data gathered by empirically investigating concrete objects, or as conceptual relationships among magnitudes. The first interpretation, advanced by Patrick Suppes and Duncan Luce, runs into difficulties when applied to real data structures, which are often less well-ordered than RTM allows. I argue that once RTM axioms are reinterpreted as expressing relations among magnitudes, these problems are successfully avoided, and the most important accomplishments of RTM are preserved. These examples show the centrality of the concept of magnitude to the study of the foundations of measurement.

In this paper, I introduce a novel approach to a problem that is, in the dominant literature, often thought to admit of only a partial solution. The problem of quantity is the problem of explaining why it is that certain properties and relations that we encounter in science and in everyday life, can be best represented using mathematical entities like numbers, functions, and vectors. We use a real number and a unit to refer to determinate magnitudes of mass or length (like 2kg, 7.5m etc.), and then appeal to the arithmetical relations between those numbers to explain certain physical facts. I cannot reach the coffee on the table because the shortest path between it and me is 3ft long, while my arm is only 2.2ft long, and 2.2< 3. The pan balance scale does not tilt because one pan holds a 90g tomato while the other holds two strawberries, of 38g and 52g respectively, and 38+52=90. While they provide a convenient way to express these explanations, the arithmetical less than relation, or the `+' and `x' operations on the real numbers are not really part of the physical explanations of these events. They just represent explanatorily relevant features inherent in the physical systems described. To solve the problem of quantity is to provide an account of this ``quantitative structure'', those physical properties and relations really doing the explaining. The vast majority of approaches in the literature have limited themselves to a much less ambitious project: Rather than explain quantitativeness in its entirety, they strive to leave ``only'' a small amount of quantitativeness unexplained. Primitivism about quantitativeness, or quantitative primitivism, is the position that some quantitative structure cannot be explained. I will argue that the problem of quantity, by its very nature, does not admit of any partial solutions. A reductive-explanatory account of quantitativeness is specifically one that provides an adequate explanation of quantitative struture without leaving any quantitative structure as an unexplained, primitive posit. This is done by reducing the quantitative structure to a more fundamental, non-quantitative base. Non-primitivist accounts also allow for a novel dissolution of a problem which has dominated contemporary debates about the metaphysics of quantity, the debate between ``absolutists'' who think that the fundamental quantitative notions are properties (like ``weighs 5g'' or ``is 2m long''), and and ``comparativists'' according to whom the fundamental notions are comparative relations (like ``is twice as massive as'' or ``is 2m shorter than''). This dispute, I argue, only makes sense from a primitivist perspective. For the non-primitivist, there is no debate to be had. There is no fundamental quantitative structure, and so there is no room for a dispute about what kind of fundamental quantitative structure we accept. The underlying intuitions which guide much of these debates (for example about whether things would be different if everything’s mass was doubled) can still be understood by the non-primitivist. Indeed, non-primitivist accounts can give a clearer and more explanatory judgement on these cases than any primitivist theory could.

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.