Quantitative Attributes or Measurable Concepts

Should we ask whether an attribute is quantitative, or is it enough to show that its concept is measurable? This question is at the heart of the dispute between (restrictive) measurement realists and (typically permissive) operationalists.This chapter tries to disentangle this question from broader issues of scientific realism and argues in favour of restrictive realism. Quantitativeness is a feature of attributes, not concepts, and measurability of a concept is neither necessary nor sufficient to guarantee quantitativeness. Realism about quantitativeness is epistemically risky, just like realism everywhere else, since its commitments go beyond what is strictly observable.

A Structuralist View of Quantities

This chapter shows how the view arrived at in the previous chapters amounts to a structuralist understanding of quantities. It argues that the view is structuralist both in the ontology it provides for quantities and in the criterion for quantitativeness it offers. It then returns to the disputes over quantities laid out in the introduction and shows that the view presented here is a form of non-reductive restrictive realism. It is non-reductive, because even though numbers are merely dispensable representational tools for quantities, quantitativeness is not defined in terms of numbers. Dispensing with numbers is not sufficient to dispense with quantitativeness. It is restrictive, because not all attributes are quantitative and it is realist, because quantitativeness is a feature of attributes, not concepts.

Representationalism as a Basis for Metaphysics

This chapter addresses two challenges for using the representational theory of measurement (RTM) as a basis for a metaphysics of quantities. The first is the dominant interpretation of representationalism as being committed to operationalism and empiricism. The chapter argues in favour of treating RTM itself as a mathematical framework open to different interpretations and proposes a more realist understanding of RTM, which treats the mapping between represented and representing structure as an isomorphism rather than a mere homomorphism. This adjustment then enables us to address the second challenge, which is the permissivism present in standard representationalism, according to which there is no special division into quantitative and non-quantitative attributes. Based on results in abstract measurement theory, the chapter argues that, on the contrary, RTM provides the means to draw such a distinction at an intuitively plausible place: only attributes representable on ‘super-ratio scales’ are quantitative.

Fundamental Structure for Quantities

Both comparativists and sophisticated substantivalists accept that there are many equivalent numerical representations of quantities, or conversely, that no numerical representation is distinguished. But whereas comparativists seem committed to offering a reduced theory, which only trades in relational facts as fundamental facts, sophisticated substantivalism appears to offer a different kind of response. This difference in approach is best understood at the level of meta-metaphysics as a difference between those who seek out fundamental or intrinsic explanations and those who see no need to find such explanations. This chapter investigates how these strategies play out when RTM is taken to be the reduced theory for quantities and argues that RTM does not lend itself easily to fundamental or intrinsic explanations. The main difficulty is that even a retreat to purely relational facts does not avoid arbitrariness.

Questions of Priority

Are relational or monadic facts about quantities more fundamental? This question is one way of formulating the debate between comparativists and absolutists in the metaphysics of quantities. The dispute between the two positions seems to depend on responding to two challenges that pull in opposite directions. On the one hand, a suitable account needs to be able to distinguish scenarios that are observationally distinct; on the other hand a suitable account should not draw distinctions without a difference. This chapter argues that absolutism seems to have the upper hand when it comes to the first challenge, but that features of quantities uncovered in Chapter 6 make absolutism implausible as a metaphysical view of quantities in response to the second challenge. Sophisticated substantivalism is presented as a promising alternative response to these two challenges.

Quantities as Numerical Attributes

While the determinable/determinates model of quantities treats quantities as (special cases of) variable attributes, the approaches considered in this chapter focus on quantities as numerical attributes: being numerically representable is what makes quantities special, if anything does. I distinguish three different attitudes to quantities thus conceived: restrictive realism, restrictive empiricism, and permissive empiricism. Restrictive realists hold that quantitativeness is a feature of attributes, not concepts, and not all attributes are quantitative; restrictive empiricists hold that quantitativeness is a feature of concepts, not attributes, but only some concepts are quantitative; permissivists hold that there is nothing special about quantitative concepts, since any attribute can be numerically represented. This chapter argues that we should reject the idea that quantities are numerical attributes or concepts and suggest that we should focus on the uniqueness of the numerical representation instead, a claim that will be made more precise in Chapter 6.

Quantities as Determinables

On the determinable/determinate model, a quantity like mass is a determinable that requires further determination by one of its determinates, where the latter are typically thought to be its magnitudes. This chapter argues that the determinable/determinate model does not provide a good fit for quantities. Instead of focusing on the vertical relation of determination, we should pay attention to the horizontal relations between magnitudes. Such relations, in particular ordering relations, can also explain the single value principle. Similarity relations among magnitudes, by contrast, are not sufficient to establish the metric relations characteristic of quantities.

Introduction

This chapter sets out what a metaphysics of quantities is and what the key disputes in the metaphysics of quantities are. A metaphysics of quantities concerns quantitativeness, the feature that differentiates quantities from other attributes. Three disputes about quantities are distinguished: reductionism vs. non-reductionism; permissivism vs. restrictivism; and operationalism vs. realism. The view defended in this book is non-reductive, restrictive, and realist. In addition the chapter describes the metaphysical and scientific backdrop against which the novel position—substantival structuralism—is articulated and defended. The introduction concludes with a short overview of the organization of the book.

Ontologies for Quantities

This chapter presents and evaluates different proposals for an ontology of quantities. It contrasts, on the one hand, ontologies based on Platonic or Aristotelian universals with those based on particulars, and on the other hand ontologies based on finite domains with those based on infinite domains The latter difference is taken to be more decisive and the chapter argues in favour of infinite domains for quantities. Platonic universals, space-time substantivalism, and quality spaces each provide infinite domains for quantities. This chapter favours the quality spaces approach, according to which each quantity is a quality space and (determinate) quantity values are (determinate) locations in quality spaces.

The Representational Theory of Measurement

This chapter introduces the representational theory of measurement as the relevant formal framework for a metaphysics of quantities. After presenting key elements of the representational approach, axioms for different measurement structures are presented and their representation and uniqueness theorems are compared. Particular attention is given to Hölder’s theorem, which in the first instance describes conditions for quantitativeness for additive extensive structures, but which can be generalized to more abstract structures. The last section discusses the relationship between uniqueness, the hierarchy of scales, and the measurement-theoretic notion of meaningfulness. This chapter provides the basis for Chapter 6, which makes use of more abstract results in measurement theory.