Quantities are ubiquitous in science and especially in the physical sciences. My project aims to provide a comprehensive metaphysical account of quantities. A metaphysical account of quantities must address three key questions:

- What distinguishes quantities from other attributes like qualities and kinds?
- What ontology is needed for quantities?
- What is the relationship between quantities, theory, and measurement?

In this project I develop and defend a systematic answer to all three questions. The resulting view of quantities is (cautiously) realist in the sense of scientific realism, structuralist, substantivalist, and non-reductionist.

Select chapters are available here — please ask me for the password. A table of contents with chapter abstracts can be found below.

## Table of Contents

*Preface*

#### 1. Introduction

Here I briefly explain why a metaphysics of quantities is of interest: physical quantities are ambiguous between physical attributes and mathematical entities. They are moreover a key to an important debate in metaphysics and the philosophy of science: which aspects of our scientific representation of the physical world give us access to the intrinsic features of the world?

I then go in more detail through several challenges quantities pose for metaphysics: What distinguishes quantities from other attributes? What role do numbers play in relation to quantities? What should we make of apparently conventional features of quantities, like units? Finally I lay out the structure of the book.

#### 2. Realism about Quantities

This chapter articulates and defends realism about quantities against operationalist and coherentist challenges. If our understanding of measurement did not require, or indeed did not permit realism about quantities, then the project of providing a metaphysics of quantities might seem misplaced. A realist about quantities, roughly, is somebody who believes that quantities exist independently of procedures to measure them or theories that describe them.

This realism is challenged by operationalists (Bridgman 1927; 1949; Stevens 1935), who claim that each measurement procedure defines its own quantity. While operationalism is treated by most philosophers with suspicion, it has recently seen some renewed interest (Hardcastle 1995; Feest 2005). Coherentists, on the other hand, argue that measurement is always theory-laden and that quantities are always inter-defined—a circle that can only be broken by conventional choice. For coherentists, quantities are hence not independent of our theories about them (Chang 2004; Tal 2016).

I defend realism about quantities against these challenges by showing, first that operationalism is untenable as a position, and second, that coherentists mistake the causal interdependence of quantities for definitional dependence. While neither operationalism nor coherentism are plausible alternatives to realism about quantities, their challenges are nonetheless instructive. They show us that quantities are to some extent theoretical posits — as such they pose problems for purely empiricist theories of science.

The final section of the chapter investigates the sense in which quantities are theoretical posits more closely. I conclude that realism about quantities is an aspect of scientific realism.

#### 3. The representationalist theory of measurement

This chapter presents the representationalist theory of measurement as the formal framework I will be using in the book. The representational theory of measurement provides a comprehensive formal theory of measurement, which was developed in detail in the second half of the 20th century, with major contributions by Krantz, Luce, Suppes, Tversky (1971- 1990) and Narens (1984; 2002). A theory of measurement in the relevant sense is concerned with the status of numerical representations of measurement result and their conditions of meaningfulness. The main questions a theory of measurement aims to answer are: when can a given “measurement structure” be represented numerically, and when is a numerical representation meaningful?

Representationalism answers these questions in the form of representation and uniqueness theorems. The former show that a given “empirical relational structure” can be mapped to a numerical structure by a structure preserving map (a homomorphism). The latter shows, how unique that mapping is, that is, how many other functions also map the empirical structure to this numerical structure.

After explaining the basic features of representationalism, I turn to the philosophical implications of representationalism. Traditionally representationalism has been strongly associated with operationalism and empiricism. I argue that the formal framework can separated from this association and is open to metaphysical interpretation. Representationalism does, however, place one important constraint on the metaphysics of quantities: representations of measurement are structural representations.

#### 4. When is an attribute quantitative?

With the representational theory of measurement on the table, we can ask what it takes for an attribute to be quantitative. A common way to contrast quantitative and qualitative attributes is to suggest that the former, but not the later, involve numbers. In the first part of this chapter I argue that numbers are neither necessary nor sufficient for an attribute to be quantitative. The representational theory of measurement in fact clearly indicates why that is: many attributes can be given numerical representations of some sort, but only in some cases are these representations informative (or meaningful). Numbers are not necessary, because some quantitative relations, like ratios, can be formulated without the use of numbers. Since numbers are nonetheless commonly used to express quantitative claims, the question we need to ask instead is: which numerical representations are representations of a quantity?

The answer, I suggest, lies in the difference of strength of the representation. An attribute is quantitative, as opposed to qualitative or sortal, when it can be meaningfully represented on at least an interval scale, or to put it differently, when distances between values are meaningful. If an attribute is meaningfully representable on a (stronger) ratio scale, it is also quantitative. Attributes representable on a merely ordinal scale (a ranking) are not quantitative. I justify drawing the line between quantitative and non-quantitative attributes at this point, by arguing, first, that formally interval and ratio scales are closer to one another than either is to ordinal scales, and secondly, by arguing that there is a close relationship between the availability of these two types of scales and the possibility of formulating quantitative laws.

#### 5. Counting and measuring

This chapter addresses the question, whether counting is a form of measuring, and answers in the negative. The first part of the chapter looks at the relationship between quantities, units, and dimensions. Numerical values of quantities typically come with units that are in turn linked to the dimensions of the quantity. Pure numbers, by contrast, lack units and dimensions. Sometimes, however, numerical values for quantities also seem to lack dimensions. A particularly interesting example of such quantities are so called “counting quantities”: quantities that give the number of entities in a collection.

In a detailed case study, I contrast such counting quantities with a quantity that, at first glance, also seems to be a counting quantity: amount of substance. Amount of substance is an important quantity in chemistry, it is used to specify the amount of a particular chemical substance in a given sample. Amount of substance is a base quantity in the International System of units, and its unit, the mole, is a base unit. As such the quantity should be expected to have its own dimension, just like the other base quantities of the SI. Yet, amount of substance is controversial, because it seems almost identical to another “quantity”: the number of entities in a given sample. The latter, however, is a counting quantity, and hence lacks dimensions. I argue that there are reasons to accord amount of substance its own dimension, and to treat it as a distinct quantity from number of entities.

After this detailed case study, I draw some general lessons from the observations made in this chapter. Counting must be distinguished from measuring: the former is directed at discrete aggregates (e.g. the population of a city) and requires the specification of a sortal (e.g. unmarried adult). Measuring, by contrast, is directed at quantities and requires the specification of dimensions, and possibly units. Measurement is not just a sophisticated form of counting, nor is counting a species of measurement.

#### 6. Against the determinable/determinate model

A standard metaphysical approach to quantities is to model them on the determinables/determinate relationship. A determinable is an attribute that admits of further specification by a determinate, in the way in which red can be specified further by distinguishing crimson from scarlet. Red would here be the determinable, with scarlet, crimson, coral etc. being its determinates.

Quantities like mass or charge might seem to follow a similar pattern; mass can be determined further by the different magnitudes of mass, expressible as 1kg, 2kg, 3kg, etc., and similarly charge might be specified further into positive and negative charges, and charges of different magnitudes. The determinate/determinable model is here supposed to capture the “vertical” relationship between a quantity, like mass, and its magnitudes (1kg of mass), which are said to “determine” it.

I argue that while the model has some prima facie attractions, a closer look shows that the determinable/determinate model does not fit quantities particularly well. Moreover, as I show through a case study of quantum mechanical spin, not all quantities always have determinate values. Hence a key requirement for determination—that any given object instantiates only one determinate of the same determinable at a time—is violated. I conclude that the problem with the determinable/determinate model is its emphasis on the vertical relation between quantity and magnitudes, whereas what is crucial for understanding quantities are the “horizontal” relations between the magnitudes.

#### 7. Ontologies for quantities

After showing in the previous chapter, that the horizontal relationships among magnitudes are what matters, I now turn to the question of how we should understand the relata of these relations. I begin with a broad distinction between those approaches that take the relata to be universals and those that take the relata to be particulars. For each of these approaches we can distinguish two variants. Universals can be either Aristotelian (Armstrong 1989), or Platonic (Mundy 1987). I show that of these two, the Platonic approach is much more promising as an ontology for quantities, but at the cost of greater ontological commitment.

Among nominalist approaches, we can distinguish between those who take the relata to be material objects (Carnap 1966), and those, like Hartry Field (1980), who argue that spacetime points make for nominalistically acceptable relata. While material objects are more obviously nominalistically acceptable, spacetime points ensure that the supply of relata is large enough to yield sufficiently determinate representations.

The key element contributing to the success of Platonic universals and spacetime points as relata is that both provide an infinite domain of possible relata, which is required by the formal representations of quantities discussed in the earlier chapters. Upon closer inspection, there is indeed little difference between the most successful Platonic account and a recent variant of Field’s approach (Arntzenius and Dorr 2012), which takes the relata to be points of “mass-space” and “charge-space”, instead of points of spacetime. I conclude that the key to a suitable set of relata is a form of substantivalism: the relata must stand in quantitative relations independently of any material objects that may or may not occupy them.

#### 8. Comparative and absolute quantities

This chapter turns to a particular hot topic in the metaphysics of quantities: the question whether we should be absolutists or comparativists about quantities. This question has recently seen a lot of debate, although it isn’t entirely clear how best to frame the issue. One way of doing so is to ask whether quantitative relations, like being twice as massive as, hold in virtue of the relata (or intrinsic properties of the relata), or whether these relations themselves are to be treated as primary. Defenders of comparativism are often motivated by the thought that absolute values of quantities do not seem to make an empirical difference—all that matters empirically are quantitative comparisons (Dasgupta 2013). Against this, defenders of absolutism have devised scenarios in which absolute values of quantities do seem to make an empirical difference (Baker ms). A close examination of these scenarios reveals that whether absolute values matter depends heavily on how we interpret laws of nature, and especially the role of natural constants in laws of nature.

After a careful discussion of several sample scenarios, including a discussion of relationism about space-time, which seems like a close cousin of comparativism about quantities, I conclude that for some quantities, absolute values make a difference, whereas for other quantities, they do not. The former are quantities with extensive structures, whereas the latter are quantities with difference structures, in the sense discussed in chapter 4.

#### 9. Intrinsicalism vs. Structuralism

In this chapter, I am concerned with two interrelated questions. On the one hand the question whether we can provide a purely intrinsic explanation of quantities, in the sense of Field (1980; 2016), and on the other hand the question what we should conclude from the widespread possibility of different numerical representations for the same quantitative structure. The two questions are related, since at least one aspect of Field’s search for intrinsic explanations is expressed as the desire to avoid arbitrary or conventional elements in explanations.

The question of how to deal with a plurality of apparently equivalent representations is a concern for much of metaphysics, and I will briefly survey the issue in section 9.2. Two general strategies can be found in the literature. The first insists that one of these representations is in some sense privileged, even if we might not be in a position to determine, which particular representation that is (Sider ms). At the other extreme one might hold that all relevantly equivalent representations are equally good and that there is nothing ‘in the world’ to make any of them better than the rest (van Fraassen 2008).

I then ask, how these positions play out in the special case of quantities and their representations. A desirable middle ground position would be one according to which there is some flexibility in the representation of quantities, without opening up the possible representation too much. I argue that such middle ground is unlikely to be defensible in the case of quantities, and argue in favour of structuralism.

#### 10. So what are quantities, then?

In this chapter I bring together the results from the different threads of the book. We should be realists about quantities, which prompts the question of what quantities are and what ontological commitments come with their acceptance. Quantities are relational structures rich enough to support at least interval scale representations. Neither ordinal scales or counts are quantities in the proper sense. The relata in these structures are best understood as points in abstract spaces. Depending on whether the structure is an extensive or a difference structure, quantities have to be treated as absolute or comparative. Representationalism about quantities strongly suggests that there is no single preferred numerical representation of any physical quantity; for any such quantity, multiple numerical representations are equally good. A plausible conclusion from this is structuralism, which suggests that we take as objective (or in Field’s parlance: as intrinsic) only those features of quantities that are invariant across their different numerical representations.

With this picture in hand I address one of the key motivating questions about quantities: the role of mathematics vis-à-vis physical quantities. Traditionally this question has often been framed as the problem of the relationship between mathematical entities, like numbers, and physical quantities. Once we understand quantities as structures of a certain type, however, this question needs to be reframed. While numbers indeed play only a representational role when it comes to quantities, it nonetheless seems odd to suggest that quantities thereby are non- mathematical or do not involve mathematics. They are structures satisfying strong axioms— axioms that are also satisfied by certain numerical structures. It is because both quantities and numbers satisfy these axioms, that numbers can be used to represent quantities as successfully as they do.

If what makes for mathematics is structure (Chihara 2004; Shapiro 1997), rather than the presence of certain designated “mathematical” entities, then quantities are indeed mathematical, even though they are non-numerical. This somewhat surprising conclusion suggests that the quest for an account of quantities free of “mathematics” is misplaced. Instead we need to embrace quantities as theoretical posits, which commit us ontologically beyond what we can observe. In describing the world quantitatively, we posit that there are features of the physical world structurally akin to the real numbers. Positing anything less fails to make sense of our actual scientific practice and falls short of realism about quantities.

*Bibliography *

*Index*