# The Metaphysical Status of Quantities

## J. E. Wolff

Paradigmatic physical attributes, like energy, mass, length, charge, or temperature are quantities. That these attributes are quantitative is important for experiments (they can be measured), as well as theories (we can formulate quantitative laws that hold between them). Quantities are arguably central to science, and especially to the physical sciences.

Quantities pose peculiar epistemological and metaphysical challenges. A natural way to describe what is special about quantities is to say that quantities, in contrast to other attributes, come in degrees. Dogs may be ranked by how fast they can run or how big they are, but there is no ranking of them by how much they are dogs. Being a dog is a sortal, whereas speed and size are quantities. A quantity’s ‘coming in degrees’ can be understood as saying that quantities have (at least) one dimension of variation. For many paradigmatic physical attributes, we find a range of possible ‘amounts’ of that attribute, which we typically express as numerical values in terms of some unit. Having a range of possible amounts seems to be required by the idea that a quantity is an attribute that comes in degrees: gradations are possible in virtue of there being *different* amounts of the *same* quantity. To understand the metaphysical status of quantities, we need some account of how a gradable property like mass relates to specific amounts of mass. In the metaphysics literature, this question is often formulated in terms of determinables and determinates.^{[1][2]} But since this terminology comes with a specific understanding of the relationship between quantities and magnitudes, I will not use these terms here. In fact, I argue that the model of determinables and determinates is ultimately a poor fit for quantities, despite superficially appealing features.^{[3]}

A second intuitive way of characterizing what is different about quantities, when compared to other attributes, is that only quantities involve numbers. This characterization has a certain immediate appeal, based on the ordinary way in which we express quantitative claims. I might say that the temperature today is 10ºC, that the average speed on the tube is 20.5mph, that the flying distance between London and Leeds is about half of that between London and Edinburgh, or that the average discharge of the Danube is about three times that of the Rhine. All of these are paradigmatically quantitative claims, some of which mention numbers and units, whereas others are unit free but nonetheless contain a numerical comparison. By contrast, if I say that my mug is blue, that my office is warm, or that the birds outside my window are pigeons, the claims I’m making are paradigmatically non-quantitative and they do not contain any numerical expressions. Quantities seem to be bound up with numbers. Unlike purely mathematical entities, however, the numerical representations of quantities typically come with units, and the attributes themselves stand in causal relations to one another. Are numbers merely dispensable representational devices, or does their use in our representation of physical quantities indicate a deeper relationship between the mathematical and the physical? This question has been a major concern of Hartry Field’s work on quantities,^{[4]} and of subsequent research engaging with Field’s programme.^{[5][6]}

To make matters more complicated, while numbers and units are typically involved in the presentation of quantities, the *particular* numbers and units used to represent any given magnitude seem quite arbitrary. Establishing standard units for quantities is obviously convenient, and perhaps there are pragmatic values, such as simplicity, favouring certain systems of units over others. But it seems unlikely that there is any sense in which certain units ‘correspond better’ to the world. Accordingly, the ‘received’ view on the matter is that units, and the particular numerical assignments that come with them, are arbitrary and ultimately a matter of convention. While the conventionality of units is widely conceded, at times the suggestion has been made that quantitative claims are conventional beyond choice of units. Brian Ellis famously suggested that not only the unit, but also the ‘summation’ operation for certain quantities can be freely chosen.^{[7]} When ‘adding’ lengths, for example, nothing requires that we add measuring rods end-to-end along a single line, as opposed to at 90º angles. Both operations result in numerical representations of lengths; it’s just that the former is more familiar than the latter. If the particular numerical values and units chosen are arbitrary, this then raises the more general question of how we can tell which aspects of quantitative representations are arbitrary or conventional, and which are actually required by the phenomena represented. Call this issue the question of conventionality in the representation of quantities. Field further argued that we should aim to give intrinsic explanations of quantitative facts, namely, explanations that do not make reference to arbitrary elements. While Field’s own intrinsic explanations have been criticized,^{[8] [9]} the problem remains a live issue.^{[10]}

The most recent question raised about the status of quantities has been dubbed the ‘absolutism versus comparativism debate’ by Dasgupta.^{[11]} *Prima facie* quantities seem to have both determinate magnitudes (for example, 4kg) and determinate relations (for example, being four times as heavy as). The question is whether both are needed for quantities, and if so, which is more fundamental: the monadic properties or the relations? The resulting debate shares certain features with the absolutism–relationism debate over spacetime. In particular, it seems as though absolutists treat certain possibilities as distinct, whereas comparativists only see a difference in representation. The absolutism–comparativism debate thereby connects back to the question of arbitrariness in the representation of quantities more broadly.

The starting point for my take on these philosophical challenges is the representational theory of measurement, as canonically formulated by Krantz *et al*.^{[12]} The representational theory of measurement is the most comprehensive formal treatment of the relationship between empirical phenomena and their mathematical representation. It thereby provides a systematic framework we can employ to address the challenges articulated above.

The representational theory of measurement provides a conditional answer to the central question of how we can use numerical representations for empirical phenomena. If a phenomenon exhibits a certain kind of relational structure, then (it can be shown that) there exists a homomorphic mapping from the phenomenon to the real numbers. Representation theorems of this sort establish that a numerical representation is indeed possible for a given phenomenon.

It turns out that the existence of some numerical representation or other can be established for a wide range of different (empirical) relational structures, but only for some of those relational structures will these numerical representations actually be informative. The numerical labels on the dumbbells at my gym are informative, because the numbers on them tell me not only which dumbbells are heavier than others, but also *how much heavier* one set is compared to another. By contrast, the numerical order established by ranking the ten best movies in 2016 tells me nothing about whether the fifth ranked movie is a lot better than the sixth ranked, or whether they are instead very similar in quality. Measurement theory describes these differences between numerical representations in terms of ‘uniqueness theorems’. The more unique a numerical representation is, the more information can be gained from it about the phenomenon represented.

The representational theory of measurement establishes for a wide range of relational structures that numerical representations are possible, and how unique these representations are. In doing so, representationalism offers a formal foundation for the ‘hierarchy of scales’, originally introduced by the psychologist S. S. Stevens,^{[13]} according to which numerical representations are possible at ratio, interval, and ordinal scales, with ratio scales being the most informative. Weight is measured on a ratio scale, whereas movie rankings are an example of an ordinal scale.

Representationalism thereby answers one important question about the relationship between numbers and quantities: numbers are representational devices that represent phenomena in virtue of a shared structure between the phenomenon and the real numbers. This answer presupposes treating both numbers and quantitative attributes as *structures*, since it is otherwise unclear how they could share a structure. What are the metaphysical consequences of this explanation of the role of numbers as representational devices? This is the question I focus on in my current project.

I address the different metaphysical challenges set out above, using representationalism as the starting point. I argue in favour of a structuralist and substantivalist conception of quantities: quantities are substantival manifolds with certain relational structures on them. The account I offer is structuralist, because I argue that among the various numerical representations, none is privileged. Since all that is in common between these representations are structural features, a plausible explanation for this equivalence of representations is that quantities themselves are *just* structures. This position takes much about how we (numerically) represent quantities to be conventional. That does not mean that quantities themselves are conventional or ‘constructed’. On the contrary, I advocate a form of realism about quantities: in attributing quantitative structure to an attribute or phenomenon, we make theoretical commitments that go beyond what is observable.

*J. E. Wolff*

* King’s College London*

* jwolffphilosophy@gmail.com*

**Notes**

^{[1] }Armstrong, D. M. [1989]:* Universals: An Opinionated Introduction*, Boulder: Westview Press.

^{[2]} Denby, D. A. [2001]: ‘Determinable Nominalism’,* Philosophical Studies*, **102**, pp. 297–327.

^{[3]} Wolff, J. E. [2015]: ‘Spin as a Determinable’, *Topoi*, **34**, pp. 379–86.

^{[4]} Field, H. H. [1980]:* Science without Numbers: The Defence of Nominalism*, Princeton, NJ: Princeton University Press.

^{[5]} Burgess, J. P. and Rosen, G. [1997]:* A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics*, Oxford: Oxford University Press.

^{[6]} Arntzenius, F. [2012]:* Space, Time, and Stuff*, Oxford: Oxford University Press.

^{[7]} Ellis, B. D. [1966]:* Basic Concepts of Measurement*, Cambridge: Cambridge University Press.

^{[8]} Melia, J. [1998]: ‘Field’s Programme: Some Interference’,* Analysis*, **58**, pp. 63–71.

^{[9]} Milne, P. [1986]: ‘Hartry Field on Measurement and Intrinsic Explanation’,* British Journal for the Philosophy of Science*, **37**, pp. 340–6.

^{[10]} Eddon, M. [2014]: ‘Intrinsic Explanations and Numerical Representations’, in R. M. Francescotti (*ed.*),* Companion to Intrinsic Properties*, Berlin: Walter de Gruyter, pp. 271–90.

^{[11]} Dasgupta, S. [2013]: ‘Absolutism vs Comparativism about Quantity’,* Oxford Studies in Metaphysics*, **8**, pp. 105–48.

^{[12]} Krantz, D. H, Suppes, P., Luce, R. D. and Tversky, A. [1971]:* Foundations of Measurement*, New York: Academic Press.

^{[13]} Stevens, S. S. [1946]: ‘On the Theory of Scales of Measurement’,* Science*, **103**, pp. 677–80.