Who Needs Magnitudes?

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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.
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McGill University

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