This book articulates and defends an original answer to this important, insufficiently understood question through the n

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*Table of contents : DedicationContentsPreface1. Introduction2. Quantities as Determinables3. Quantities as Numerical Attributes4. Quantitative Attributes or Measurable Concepts5. The Representational Theory of Measurement6. Representationalism as a Basis for Metaphysics7. Ontologies for Quantities8. Questions of Priority: Relations or Relata?9. Fundamental Structure for Quantities10. A Structuralist View of QuantitiesReferencesIndex*

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The Metaphysics of Quantities

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The Metaphysics of Quantities J. E . WO L F F

1

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1 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © J. E. Wolff 2020 The moral rights of the author have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2019955329 ISBN 978–0–19–883708–4 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Meiner Familie, nah und fern

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Contents Preface

xi

1. Introduction

1

2. Quantities as Determinables

8

1.1. What is a Metaphysics of Quantities? 1.2. Three Cross-Cutting Disputes about Quantities 1.3. Scientific Theories of Quantities 1.4. The Metaphysical Landscape 1.5. Organization of the Book

1 2 3 5 6

Synopsis8 2.1. Quantities as Variable Attributes 8 2.1.1. Intuitive Distinctions 8 2.1.2. The Determinable/Determinate Model 10 2.1.3. Applying the Determinable/Determinate Model to Quantities

2.2. An Awkward Fit

2.2.1. Some Initial Observations 2.2.2. Horizontal vs. Vertical Relations 2.2.3. The Challenge of the Single Value Principle 2.2.4. Conclusion

3. Quantities as Numerical Attributes

13

15 15 17 20 21

22

Synopsis22 3.1. Restrictive and Permissive Views of Quantities 23 3.1.1. Restrictive Empiricism: Carnap’s Division of Scientific Concepts 3.1.2. Restrictive Realism: Numbers are Hard 3.1.3. Permissivism: Numbers are Easy

3.2. A Difference in Strength, not in Numbers

3.2.1. Numbers: Neither Necessary nor Sufficient 3.2.2. Why Draw a Distinction at all? 3.2.3. Summary and Outlook

4. Quantitative Attributes or Measurable Concepts

23 28 30

32 32 34 36

37

Synopsis37 4.1. Measurement Realism 37 4.1.1. A Naive View of Measurement 37 4.1.2. Measurement Realism Today 39

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viii Contents 4.2. Measurement Realism under Attack 4.2.1. The Line of Attack

42

4.3. Operationalism

48

4.4. Defending Quantitativeness

56

4.5. Measurement Realism as Scientific Realism

63

4.2.2. 4.2.3. 4.2.4. 4.2.5.

The Problem of Nomic Measurement Establishing Constancy Extending Scales How Far to the True Value?

4.3.1. Traditional Operationalism 4.3.2. Sophisticated Operationalism 4.4.1. Measurability vs. Quantitativeness 4.4.2. Attributes Matter: Hardness

5. The Representational Theory of Measurement

42 43 44 46 47 48 52 56 58

66

Synopsis66 5.1. Preliminaries 66 5.1.1. Origins of Representationalism 66 5.1.2. The Basic Principles of Representationalism 69 5.2. Additive Extensive Structures 72 5.2.1. Representation and Uniqueness Theorems for Additive Extensive Structures 5.2.2. Hölder’s Theorem

72 74

5.3. Other Important Types of Measurement Structures 5.3.1. Difference Structures 5.3.2. Additive Conjoint Measurement

76

5.4. Uniqueness and the Hierarchy of Scales

84

5.3.3. Types of Quantities and Types of Measurement Structures

5.4.1. Permissible Transformations and the Hierarchy of Scales 5.4.2. Uniqueness and Conventionality

6. Representationalism as a Basis for Metaphysics

76 80 83 84 87

90

Synopsis90 6.1. Representationalism and its Critics 90 6.1.1. The Representational Theory of Measurement as a Mathematical Framework 6.1.2. Is Representationalism Sufficiently Realist?

6.2. When is an Attribute Quantitative? 6.2.1. 6.2.2. 6.2.3. 6.2.4.

Uniqueness, not Representation An Intuitive Difference A Formal Difference A Criterion in Terms of the Structure Itself

90 92

97

97 99 102 106

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Contents ix 6.3. How RTM Constrains the Metaphysics of Quantities 6.3.1. The Role of Structure 6.3.2. Invariance as Meaningfulness 6.3.3. Conclusion

7. Ontologies for Quantities

109 109 111 113

115

Synopsis115 7.1. Universals as Relata 115 7.1.1. Aristotelian Universals 115 7.1.2 Platonic Universals 120 7.1.3. Hybrid Views of Quantities as Universals 122 7.2. Particulars as Relata 124 7.2.1. Objects as Relata 124 7.2.2. Space-Time Points as Relata 128 7.2.3. Points beyond Space-Time: Limitations and Modifications of Field’s Programme

7.3. Quantity Substantivalism 7.3.1. Substantivalism 7.3.2. Locationism

8. Questions of Priority: Relations or Relata?

131

135 135 137

141

Synopsis141 8.1. The Debate between Absolutists and Comparativists 141 8.1.1. Prelude 141 8.1.2. The Contenders 143 8.1.3. An Initial Victory for Absolutism 146 8.2. Why we should not be Absolutists 148 8.2.1. Scalings and Non-trivial Automorphisms 148 8.2.2. Homogeneity and Quiddities 152 8.3. Sophisticated Substantivalism as an Alternative 156 8.3.1. Sophisticated Substantivalism 156 8.3.2. Background Structure: The Response of Sophisticated Substantivalism 8.3.3. Are Sophisticated Substantivalists Cheating?

9. Fundamental Structure for Quantities

160 162

166

Synopsis166 9.1. Invariantism: Intrinsicalism or Quotienting? 166 9.1.1. Introduction 166 9.1.2. Intrinsic Explanations and Fundamental Theories 169 9.1.3. Fundamentalism and Quotienting 171 9.2. The Representational Theory as Fundamental Theory 175 9.2.1. Fundamental Mass Facts and Representational Axioms 175 9.2.2. Preferred Interpretations 178 9.2.3. Group Structure as Fundamental Structure? 181

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x Contents 9.3. Classifying Structures and Levels of Determinacy 9.3.1. The Legacy of the Erlanger Programme 9.3.2. Quantitativeness and Determinacy

184 184 187

10. A Structuralist View of Quantities

190

References Index

203 211

Synopsis190 10.1. Threads of Structure 190 10.1.1. What is Structuralism? 190 10.1.2. Homogeneous Relational Structures 191 10.1.3. The Structure of an Archimedean Ordered Group 194 10.2. Mathematical or Physical Structure? 196 10.2.1. Reductionist and Non-Reductionist Views of Quantities 196 10.2.2. Structuralism as Non-Reductionism 198 10.2.3. Final Thoughts 201

Preface Many individuals and institutions have helped to make this book possible. I gratefully acknowledge funding from the Hong Kong Research Grants Council (ECS Grant #27402714) and the Alexander von Humboldt Foundation for supporting my project on the metaphysics of quantities. I would like to thank the University of Hong Kong, King’s College London, and the University of Edinburgh for granting me leave to work on this book. The generous support by the School of Humanities at Hong Kong University allowed me to organize an inspiring conference on the metaphysics of kinds and quantities at an early point in this project; particular thanks here to Timothy O’Leary. Stephan Hartmann and the Munich Center for Mathematical Philosophy at Ludwig-Maximilians-Universität München deserve special thanks for hosting me during my Humboldt fellowship. For their extremely helpful, thoughtful, and encouraging comments on the manuscript, and for sharing drafts of their own work, I would like to thank: Anonymous, Jean Baccelli, Dave Baker, Alistair Isaac, Niels Martens, Michaela McSweeney, and Ted Sider. Numerous people have generously given their time to discuss aspects of the project at various stages in its development. My thanks go to (in alphabetical order): Jamin Asay, Jean Baccelli, Ralf Bader, Dave Baker, Alexander Bird, Andrea Bottani, Claudio Calosi, Adam Caulton, Anjan Chakravartty, Damiano Costa, Erik Curiel, Shamik Dasgupta, Neil Dewar, Julien Dutant, Uljana Feest, Hartry Field, Malcolm Forster, Steven French, Alessandro Giordani, Stephan Hartmann, Catherine Herfeld, Paul Hoyningen-Huene, Alistair Isaac, Eleanor Knox, James Ladyman, Jess Leech, Sebastian Lutz, Dan Marshall, Niels Martens, Michela Massimi, Vera Matarese, Casey McCoy, Alyssa Ney, David Papineau, Zee Perry, Oliver Pooley, James Read, Miklós Redéi, Alexander Reutlinger, Bryan Roberts, John Roberts, Darrell Rowbottom, Maria Seidl, Ted Sider, Peter Simons, Susan Sterrett, Eran Tal, Al Wilson, Jessica Wilson, Chris Wüthrich. I would also like to thank the supportive audiences in: Aberdeen, Antwerp, Arizona, Bristol, Cambridge, Cape Town, Cardiff, Durham, Edinburgh, Hanover, Helsinki, Hong Kong, London, Lugano, Munich, Oxford, Prague, Rome, Singapore, Wollongong, and Zurich. Most of all my thanks go to Alistair, who has been an exemplar of support in every conceivable way throughout the entire process.

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1 Introduction 1.1. What is a Metaphysics of Quantities? This book has two main aims. On the one hand, the book offers a defence of a novel position in the metaphysics of quantities: substantival structuralism. According to this view, quantitativeness is an irreducible feature of attributes, and quantitative attributes are best understood as relationally structured spaces. On the other hand, the book provides a thorough examination of the metaphysical questions posed by quantities as well as a careful assessment of possible answers to these questions. It thereby provides a resource for metaphysicians and philosophers of science interested in the topic of quantities, and connects quantities to a number of debates in metaphysics and philosophy of science. Quantities appear in many different scientific fields, as well as in daily life. We speak of weight, length, durations, and volumes, as well as particular weights, volumes, and lengths. Typical quantitative expressions outside of science are of the form ‘add 1/3 cup of water’ or ‘the dimensions of your suitcase must not exceed 55 cm × 40 cm × 20 cm’. In scientific contexts, we find claims like ‘the mass of the electron is 8.1871057769 × 10−14 J/c2 or 9.1093837915 × 10−31 kg’, as well as quantitative laws like E = mc2 or F = keq1q2/r2. A focus on quantities and quantitative methods is common in many sciences, so much so that quantities have sometimes been regarded as the hallmark of ‘hard’ science. Nowhere is this more apparent than in physics, where models characteristically involve the specification of quantitative parameters, and laws are presented as equations relating several quantities. But whereas the ontological status of other aspects of theorizing in physics, for instance laws, kinds, or elementary particles, has been the subject of sustained metaphysical debate, quantities have seen much more sporadic discussion by philosophers.1 Despite their ubiquity and importance, we therefore do not have a good metaphysical understanding of quantities. 1 Important exceptions are the work of Brent Mundy, especially his (1987), and David Armstrong’s writings (esp. Armstrong 1989). Classic monographs on the topic include Carnap (1966), Ellis (1966),

The Metaphysics of Quantities. J. E. Wolff, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198837084.001.0001

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2 Introduction As I will be using the term, ‘quantity’ will refer to attributes like mass, length, or temperature. Quantitative attributes are prima facie different from other properties in that they admit of variation: we want to know not only whether an object has mass, but how much mass it has. A metaphysics of quantities, as I conceive of it here, is concerned with quantitativeness. My topic is not the status of this or that quantity in a particular scientific theory, for example mass in Newtonian mechanics or temperature in thermodynamics, or even the status of particular quantities across different theories. Instead, I’m interested in the question of what it means for an attribute to be quantitative, and what metaphysical implications a commitment to quantitative attributes has. To answer the first question, we need a clear definition of when attributes are quantitative. The first half of this book aims to answer this question. This sets the stage for the question of what metaphysical and ontological consequences the existence of quantitative attributes has, which is the topic of the second half of the book.

1.2. Three Cross-Cutting Disputes about Quantities Philosophical discussions of quantities can be organized around three distinct, but often cross-cutting disputes: i. reductionism vs. non-reductionism ii. operationalism vs. realism iii. permissivism vs. restrictivism. The central question of the first dispute is whether quantitativeness is a distinct, sui generis feature, or whether it can be reduced to qualitative features. Reductive proposals take different forms, depending on where the distinction between quantities and qualities is thought to lie. When the distinctive feature of quantities is taken to be their numerical representation, then a reduction of quantities to qualities will have been achieved, if the use of numbers in the expression of quantitative facts has been shown to be dispensable. But numer ical representability is not the only candidate for a distinctive feature of quan tities. Sometimes metric relations are taken to be characteristic of quantities

Field (1980), Berka (1983), Bigelow and Pargetter (1990). Recent contributions include Arntzenius and Dorr (2012), Dasgupta (2013), Eddon (2006; 2013a; 2013b; 2014), Peacocke (2015), Perry (2015), Baker (in press), Dasgupta (in press), Sider (2020).

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Scientific Theories of Quantities 3 and accordingly a reduction of quantities to qualities takes the form of showing how metric relations, that is, distances, hold in virtue of sub-metric relations (e.g. orderings, or perhaps similarity). A third way of understanding reductionism is finitism, which seeks to reduce the apparent infinity of possible values of quantitative attributes to some finite surrogate, for example finitely many measurement results or finitely many material objects. All three reductionist projects intersect the second dispute, which concerns the question of realism and operationalism about quantitativeness. Realists about quantitativeness hold that the target of measurement operations is attributes and that quantitativeness is a feature of attributes independent of our ability to measure them. These realist theses contrast with operational ism, which holds that quantitativeness is a feature of concepts, not attributes, and that a concept’s quantitativeness or lack thereof is a matter of our ability to construct a suitable measurement procedure. The third dispute concerns the question whether quantitativeness is restricted to a certain class of attributes, or whether any attribute is (potentially) quantitative. This dispute is related to the question of realism vs. oper ationalism, since operationalists tend to be permissivists, whereas realists tend to be restrictivists. As we shall see, however, there are restrictivist oper ationalists as well. All three disputes play a role in the present investigation. The view I defend in this book is a form of restrictive realism in that I hold that the distinction between quantities and qualities is a ‘worldly’ not a conceptual matter and that not all attributes are quantitative. While numbers turn out to be dispens able in a certain sense, the view I advocate is nonetheless non-reductive, because metric structure is taken to be irreducible and the domain of vari ation for the quantities under consideration is taken to be continuous. The resulting view is hence a non-reductive, restrictive realism.

1.3. Scientific Theories of Quantities Many investigations into the metaphysics of science start with a particular scientific theory and ask how the theory is to be interpreted. While quantities occur in a wide variety of scientific theories, the closest thing we have to a scientific theory of quantities is theories of measurement. Measurement is studied by two distinct fields of enquiry: metrology and measurement theory. Metrology is mostly concerned with the establishment and maintenance of common measurement standards and their physical realization. Its work

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4 Introduction is carried out by national and international institutes working in close collaboration with one another to ensure that scientific and commercial measurements can be carried out precisely and uniformly across the globe. Measurement theory, on the other hand, is primarily concerned with the formal status of numerical representations of measurements. Accordingly, measurement theory typically provides axiom systems and proofs of the orems demonstrating how a particular concept or phenomenon can be described mathematically. Within measurement theory, the representational theory of measurement (RTM) is arguably the most developed formal theory of measurement and will be the theoretical and formal starting point for the metaphysical view developed in this book. This theory formally describes the conditions for numerical representability for a range of different, axiomatic ally characterized structures. 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 develop a metaphysical account of quantities. There are two important respects in which my approach differs from standard philosophical interpretations of the representationalist theory. The first point of difference is that representationalist theories are often taken to assume an empiricist or operationalist stance towards quantities. What are represented numerically are not, at least in the first instance, quantitative attributes, but instead ‘empirical relational structures’. The metaphysical status of these empirical structures has often been interpreted in an operationalist manner, according to which an empirical relational structure involves a specified measurement procedure, like putting objects on a beam balance. What are numerically represented would then be observations on the basis of these empirical procedures. My own view, by contrast, is realist in the sense that I take the target of our numerical representations to be quantitative attributes like charge, mass, momentum, length, temperature, and so forth, which exist independently of any particular procedure for measuring them. This modest realism about measurement is defended in Chapter 4. The second point of difference concerns the distinction between quantitative and non-quantitative attributes. As we shall see in more detail in Chapter 3, an intuitive way of distinguishing quantitative from non-quantitative attributes is to say that the former, but not the latter, are representable numerically. The representational theory of measurement seems to show that paradigmatically quantitative attributes are not special because they can be numerically represented, since this is true for many sorts of attributes. One

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The Metaphysical Landscape 5 might be tempted to adopt a permissive attitude to quantities in response. Maybe there simply is no class of ‘quantitative’ attributes distinguished from other attributes. I suggest in Chapter 3 that permissivism is not an attractive view to take. Instead, I argue that representationalism, by drawing attention to uniqueness as well as representation theorems, shows that quantitative attributes are special because they can be given particularly informative representations. The question we should ask, therefore, is no longer why are quantitative attributes numerically representable at all, but instead becomes: which numerical representations are representations of quantities? As I show in Chapter 6, the representational theory of measurement actually provides formal resources for drawing a clear distinction between quantities and other attributes.

1.4. The Metaphysical Landscape Questions regarding the metaphysical status of quantities are not posed in a philosophical vacuum. Two current trends in the metaphysics of science have been particularly important in developing the account offered in this book: First, the growing realization that Humeanism—perhaps the dominant contemporary approach in metaphysics—struggles to give a satisfactory account of quantities, and second the shift from modal to ‘post-modal’ or hyperintensional metaphysics.(Meta-)metaphysical hyperintensionalism suggests that facts must be distinguished more finely than intensionally (Nolan 2014). According to the hyperintensionalist, two facts can be necessarily equivalent, yet distinct. Such hyperintensionalist distinctions have received a lot of interest in recent years, especially in the context of metaphysical explanations. Quantities turn out to be an interesting test case for hyperintensionalism, because some numerical representations of facts about quantities are necessarily equivalent, yet apparently distinct. This raises the question of whether hyperintensionalists need to acknowledge distinctions between such numer ical representations as metaphysically substantive, or whether they can find a way of avoiding such (presumably undesirable) distinctions (Sider 2020, chapters 4 and 5). I explicitly address these issues in Chapter 9. Defining features of Humeanism are a recombination principle, according to which the fundamental entities in the ontology freely recombine, and typ ically also a preference for intrinsic, determinate properties over relations or determinable properties. The recombination principle makes good on the promise that there are no necessary connections between distinct entities. Both

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6 Introduction aspects of Humeanism are found in David Lewis’s famous ‘Humean Supervenience’, which describes the world as ‘a vast mosaic of local matters of particular fact’ (Lewis 1986b, p. ix). This mosaic consists, on the one hand, of spatiotemporal distances between points, which are understood as external relations, and on the other hand instantiations of perfectly natural intrinsic properties at these points. These are the only fundamental entities, everything else supervenes on them. Not all Humeans subscribe to Lewis’s version of Humean supervenience, and there are different ways of cashing out the recombination principle, or of specifying which properties are perfectly nat ural ones. Here I gloss over these differences and treat Humeanism as a family of views all of which hold something along the lines described above. Quantities prima facie pose a problem for Humeanism, because of the single value principle according to which particular magnitudes of the same quantity cannot be coinstantiated at the same location at the same time. A rope cannot be both 60 metres and 50 metres long. This necessary mutual exclusion of determinate magnitudes seems in conflict with the recombin ation principle together with the idea that determinate magnitudes are plaus ible candidates for perfectly natural properties. The single value principle seems to suggest that there are necessary connections between fundamental entities after all (Bricker 2017). A second way in which quantities might seem to pose a problem for Humeanism is that Humeans often characterize quan tities in terms of determinables and determinates. Humeans would seem to have to choose between determinates or determinables as the perfectly nat ural properties, since they cannot both be fundamental, yet neither choice is without costs (Hawthorne 2006). Quantities hence do not fit neatly into the most dominant metaphysical viewpoint. I argue in Chapter 2 that we should not understand quantities along the lines of the determinable/determinate model. In Chapters 7 and 8 I develop a detailed alternative ontology for quan tities in the form of sophisticated substantivalism. In Chapter 10 I argue that the resultant view is best understood as a form of structuralism, not Humeanism.

1.5. Organization of the Book The book is functionally divided into three parts. Chapters 2–4 settle two of the three disputes outlined above in favour of restrictive realism, but argue that we do not currently have a satisfactory metaphysics of quantities that delivers such a view. In particular the determinable/determinates model is

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Organization Of The Book 7 found inadequate in Chapter 2, and the idea that numerical representation makes for quantitativeness is rejected in Chapter 3. Chapter 4 establishes that quantitativeness cannot be replaced by measurability, as the operationalist would have it. This leaves us with a need for an alternative criterion for when an attribute is quantitative. Chapters 5 and 6 introduce the representational theory of measurement and show how it can be put in the service of a metaphysics of quantities. These chapters present formal results from measurement theory that will be needed in the construction of the metaphysical account. Since some of these results have not been widely discussed in philosophy, even readers familiar with the basic ideas behind RTM may benefit from the presentation, especially in Chapter 6. Chapters 7–10 develop the novel metaphysical view of quantities: substan tival structuralism. Chapter 7 establishes the need for a substantival ontology for quantities. Chapter 8 analyses the absolutism–comparativism debate, finds both views wanting, and argues in favour of sophisticated substantivalism as an alternative. Chapter 9 offers an excursion into recent meta-metaphysics and addresses the question of how we should respond to the multiplicity of equivalent numerical representations for quantities. Chapter 10 brings together the different aspects of the view developed and shows that it is best understood as a form of structuralism about quantities.

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2 Quantities as Determinables Synopsis On the determinable/determinate model, a quantity like mass is a determin able that requires further determination by one of its determinates, where the latter are typically thought to be its magnitudes. I argue in this chapter that the determinable/determinate model does not provide a good fit for quan tities. Instead of focusing on the vertical relation of determination, we should instead 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 suffi cient to establish the metric relations characteristic of quantities. I begin with a couple of intuitive distinctions between quantities and other attributes (2.1.1), before turning to the determinable/determinates model (2.1.2). After investigating its prima facie appeal for a metaphysics of quantities (2.1.3), I argue that the model does not fit quantities particularly well (2.2).

2.1. Quantities as Variable Attributes 2.1.1. Intuitive Distinctions Consider a pebble: it’s pretty, off-white, hard, cold, a little heavy, and fits nicely into your hand. These attributes are all observable qualities of the pebble. We could also measure some of the pebble’s quantitative properties: its weight, its length, height, and width, or its temperature. What makes the latter proper ties measurable quantities, in contrast to the former? Providing measure ments of these properties can be regarded as more precise determinations of our initial round of observable attributes: when measuring the pebble’s weight as 348 grams or its temperature as 10°C, for example, we seem to get more information about the pebble than if we find it to be a little heavy and cold. Not all of the pebble’s attributes are readily measured, however. There are no obvious measurement procedures for the pebble’s off-whiteness, let alone its The Metaphysics of Quantities. J. E. Wolff, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198837084.001.0001

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Quantities As Variable Attributes 9 prettiness. Even for hardness, the case is not straightforward. Provided we have a suitable test kit, we can test, for example, whether the pebble is scratched by, or scratches, apatite. If it is scratched by apatite, its number on the Mohs scale is less than 5. Have we thereby performed a measurement of its hardness and is hardness a quantity, like mass? To answer these questions, we need to know how to distinguish quantita tive attributes from other attributes. While some attributes, like mass, would seem to be paradigmatically quantitative, and other attributes, like ‘being a pebble’, seem clearly non-quantitative, there are all kinds of attributes, like hardness, for which it is far less clear whether they are quantitative or not. A first task for a metaphysics of quantities will be to give a clear characterization of what makes quantities special kinds of attributes. 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 by how big they are, but there is no rank ing of them by ‘how much they are dogs’. Being a dog is a sortal, whereas speed and size are quantities. Quantities, unlike sortals, are gradable. For any paradigmatic physical attributes, such as mass, charge, momentum, tempera ture, and so forth, we have 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 a gradable attribute: gradations are possible in virtue of there being different amounts of the same quantity. To understand what quantities are, we need some account of how a gradable attribute like mass relates to specific amounts of mass, that is, to particular magnitudes. Call this the question of the relationship between quantities and their magnitudes. Paradigmatic quantities, like mass or charge, seem to have additional fea tures that other gradable attributes, like spiciness or hardness, lack. An object of a particular amount of mass, say a dumbbell of 25 lb, stands in very specific relations to other massive objects. Dumbbells on a rack are not only ordered by weight, but we can even say, for any two dumbbells, how much more one of them weighs compared to the other. A dumbbell of 75 lb is three times as heavy as a 25 lb one. This does not hold true for all attributes which induce orderings. Some chilies are spicier than others, but while we can rank chilies according to how spicy they are, doing so does not establish how much spicier a Piri piri chili pepper is compared to a Jalapeño.1 Quantitative attributes 1 The scale on which the spiciness of peppers is measured is called the Scoville scale. The Scoville scale relates the spiciness rankings given to peppers by trained testers to the capsaicin concentration

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10 Quantities as Determinables seem to require certain characteristic relations among the objects that have them, which go beyond mere rankings. Which of these relations are necessary and sufficient for an attribute to be considered a quantity? A second intuitive way of characterizing what is different about quantities, when compared to other attributes, is that only quantities involve numbers. This characterization goes beyond the idea that quantities come in degrees, and it has a certain immediate appeal, based on the ordinary way in which we express quantitative claims. I might say that the temperature today is 10 degrees Celsius, that the average speed on the tube is 20.5 miles/hour, 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 I’m drinking tea, that my office is warm, or that the birds outside my window are pigeons, the claims I’m making are paradigmat ically non-quantitative and they do not contain any numerical expressions. Are numbers a deep feature of quantities, or are they merely representational devices, whose application may not even be limited to quantitative attributes? In this chapter, we shall look at quantities as variable attributes, before turning to quantities as numerical attributes in Chapter 3. Intuitively, quan tities are attributes that admit of variation, but perhaps not all attributes that admit of variation are quantities. The current philosophical ‘standard’ model for variable attributes is the determinable/determinates model.

2.1.2. The Determinable/Determinate Model A feature of quantities, which has been taken by some philosophers to be the characteristic feature of quantities, is that two objects can be the same in both having the quantity, yet different in having different values of the quantity (Johansson 2009; Bigelow & Pargetter 1990). For example, both an electron and a planet have mass, yet they differ greatly in how much mass they have. Philosophers who take being ‘both the same and different—in the very same respect’ (Bigelow & Pargetter 1990, p. 49) as characteristic of quantities are

in the peppers. Capsaicin is the chemical responsible for the spiciness of peppers. For a number of reasons the relationship between the rankings and the presence of capsaicin tends to be fairly imprecise.

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Quantities As Variable Attributes 11 likely to emphasize the relation between a quantity and its values as key to being a quantity. A particularly popular model for this type of relational structure has been the determinable/determinate model. In what follows I describe how this model is being applied to quantities, and I argue that it is unsuitable. The paradigmatic application of the determinables/determinate model is the logic colour predicates.2 Colour predicates seem to be subject to exclu sions that do not seem to fit into the standard understanding of logical rela tions among atomic sentences. In particular, nothing in the syntactical structure of the sentences distinguishes the sentence ‘a is red and a is square’ from the sentence ‘a is red and a is blue’. But while the first seems possible and indeed contingently true of some objects, the second would seem to be neces sarily false. Strictly speaking, of course, it is only necessarily false if we assume that what is meant is that a is red all over and blue all over, but that is perhaps a minor adjustment. How can this exclusion of colour predicates be explained, given that the logical from of the sentence does not suggest a logical contra diction in the attribution of the predicates? Today we may no longer be committed to logical atomism, but the problem has resurfaced as a problem in metaphysics. Why do colours exclude each other in a way that colours and shapes do not? Our answer need not be given in terms of the logical structure of the predicates; instead we might give it in terms of the metaphysical structure of the properties red and blue, or perhaps of phenomenal colours3 in general. The question then is: what is the (rela tional) structure of phenomenal colour properties, such that they exclude each other in this way? A widely accepted answer is that colour is a determin able, with blue and red as determinates. The relationship between determin ables and determinates can then be further specified, for example by providing formal features of the determination relation.4 A contemporary characterization of the relationship between determinables and determinates, is provided by Funkhouser (2006, pp. 548–9): 1. properties are determinable or determinate only relative to other properties 2 The terms determinable/determinate are introduced by Johnson (1921), who uses colour as his paradigmatic example. A distinction in the general vicinity may be older (see Wilson 2017 for possible origins of the distinction). 3 The determinable/determinates model is most compelling for phenomenal colour due to its ori gins in colour predicates. Once we move away from phenomenal colours, further questions arise about its adequacy even in the application to colour. 4 See again Johnson (1921) and Prior (1949) for classic accounts.

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12 Quantities as Determinables 2. determinate properties are specific ways of having determinable properties 3. no instantiation of a determinable without instantiation of one of its determinates 4. instantiation of a determinate necessitates instantiation of all relevant determinables 5. determination is transitive, asymmetric, and irreflexive 6. determinates under the same determinable admit of a special kind of comparison 7. the chain of determinables–determinates is finite 8. Instances of determinables and their determinates do not causally exclude each other. This list of characteristic relations derives from the original home of the determinable/determinate distinction as a model of the relations among colour predicates. The following may serve as a quick illustration of points 1–5: red is a determinable relative to scarlet, and a determinate relative to coloured; scarlet is a specific way of being red, crimson another. Something cannot be red without being a determinate shade of red, like crimson or scar let, and anything that is crimson is automatically also red and coloured. If scarlet determines red, and red determines coloured, then scarlet also deter mines coloured. If scarlet determines red, red does not determine scarlet. I can compare different shades of red with respect to how similar they are, and order them by similarity. Maroon is more similar to cardinal than either is to scarlet, and all three are more similar to each other than any of them is to navy. Moreover, we would not know how to compare colours and shapes in an analogous way. Is being red more similar to being triangular, or to being square? Finally, there is an end to how finely I can differentiate shades of colour: there are different shades of red, but no different shades of crimson. Especially this last point seems a lot more controversial when we think of determinates as properties than when we think of them as predicates. For example, how fine-grained our colour predicates are would seem to be a mat ter of how finely human perception can draw distinctions between colours. By contrast, whether there are further differentiations of crimson, beyond what humans can distinguish, seems to be a matter of what’s true about colour, not colour terms. It is perhaps best, then, to think of the determinates/ determinable model as a model for phenomenal colour, that is, for colour as perceived by human observers. The question of how phenomenal colours relate to their physical correlates is subtle.

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Quantities As Variable Attributes 13 Similarly, the question whether determinates and determinables causally exclude one another is highly controversial. What causes a bicyclist to stop at the traffic light is that the light is red, not that it’s coloured. Should we further say that what causes her to stop is that it is scarlet? The first suggests that determinates, not determinables are causally relevant, yet the second might suggest that determinates and determinables do not causally exclude one another, since it seems that the cyclist stopped because the light was red, and that she stopped because the light was scarlet, are both true. Fales (1990) and Shoemaker (2001) have suggested accounts on which determinables have a subset of the causal powers of (each of) their determinates, which suggests that determinates and determinables do not causally exclude one another. Against this, Gillet and Rives (2005) have argued that such a view suffers from causal overdetermination problems. Of these eight features of the model, then, we might want to regard 1–6 as core elements of the model, with 7 and 8 as more controversial add-ons.

2.1.3. Applying the Determinable/Determinate Model to Quantities Arguably the determinable/determinates model has become a standard approach to quantities in metaphysics (Johansson 2000; Bigelow & Pargetter 1990; Armstrong 1988). The determinables/determinate model looks like a promising candidate for describing the complex internal structure of quantities, because quantities share with colours something like the exclusion principle. Just as in the case of colour, there seems to be no problem in saying that a table has a mass of 10 kilograms and a length of two metres, yet it would be strangely contradictory to say that it has a mass of 10 kilograms, and a mass of 12 kilograms. While different amounts of different quantities can often be attributed to the same object at the same time, different amounts of the same quantity may not be so attributed to the same object. Of course, as in the case of colour, we need to be fairly specific about the exact nature of the attribu tion. For example, we typically do attribute two different length-measures to a table: it’s 2 metres long, and 1 metre wide, or say that its legs are 6 kilograms and its top is 4 kilograms. Just as in the case of colour we might say, at least in the first instance, that different parts of the table instantiate different masses and lengths. One important constraint on the range of possible values, then, is that for any object and any physical quantity, the object can only have one value of

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14 Quantities as Determinables that quantity at any given moment. This principle is often referred to as the single value principle.5 While (a version of) the principle is widely accepted (Ellis 1966; Chang 2004; Johansson 2009), it is not easy to say how this principle is to be justified. Hasok Chang argues that the principle can nei ther be regarded as a matter of mere logic, nor as an empirical principle. The principle tells us not to assign more than one value of a particular quantity to the same object at the same time, but to do so does not obvi ously amount to asserting a contradiction. Logic alone does not give us a reason to rule out the possibility of multiple values for a physical quantity. But neither does experience. It is not so much that we never find instru ments that yield multiple values of the same quantity at the same time; if we found one, we would discard at least some of the outcomes as erroneous.6 Our conception and practice of measurement presupposes, rather than dem onstrates, the principle of single value. Because it seems so similar to the exclusion principle at work in the case of colour, many philosophers have suggested that we should use the determinables/determinates model to describe quantities as well. Other features of determinables and determinates have encouraged the idea that determinables/determinates make for a good model of quantities. Whenever a determinate is instantiated, all of its determinables are also instantiated. A navy object instantiates blue as well as coloured. Similarly, of course, an object of exactly 4.05 kg of mass also instantiates mass. Moreover, no determinable is instantiated without being instantiated through a particu lar determinate. Anything that is coloured must be a particular colour, even though it is of course left open which colour it is. Analogously, at least in clas sical physics it would seem to be true that quantities have determinate values: an object with mass has a particular amount of mass. Prima facie, then, the determinables/determinate model seems like a good candidate for describing the nature of quantities. There are two standard ways of applying the determinables model to quantities. The more common way is to describe magnitudes as determinate monadic properties, and to describe quantities as determinables of such determinate monadic properties (Armstrong 1988; Armstrong 1989). The alternative is to start from deter minate relations, for instance, x is four times as massive as y, and to treat these 5 Strictly speaking there are several ways of formulating a principle along these lines, with slightly different implications. 6 In scientific measurement practice, of course, typically multiple measurements are taken of the same system, and the results often diverge from one another. How to construct the measurement value out of these initial data is an important concern of applied measurement (Mari & Giordani 2014).

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An Awkward Fit 15 determinate relations as fundamental, with determinables like ‘mass’ attribut able to an object in virtue of the network of determinate mass relations in which it stands (Bigelow & Pargetter 1990). Common to both proposals is the idea that determinates are fundamental, and determinables are metaphysic ally secondary, if needed at all. The emphasis on determinate properties and relations reflects the widespread preference for determinates over determina bles (Gillett & Rives 2005). Recently some philosophers have suggested that we need to focus on determinables as well, which might be as fundamental as their determinates (Wilson 2012). In the case of quantities in particular, it is far from clear that determinates are better candidates for (perfectly) natural properties (Hawthorne 2006; Bricker 2017), or that laws of nature are best understood as relations between determinates rather than determinables (Armstrong 2010). There are still open questions, then, about how exactly to apply the model to quantities. But is the determinable/determinate model really a good approach for characterizing what it means for an attribute to be quantitative in the first place?

2.2. An Awkward Fit 2.2.1. Some Initial Observations There are several reasons to be sceptical of using the determinable/deter minate model as a metaphysical model for quantities. The first is that the determinable/determinate model can be applied to many attributes, some of which, like determinable kinds, are definitely not quantities, and some of which, like colour, are at best questionable candidates for quantities. Determinable or generic kinds are (natural) kinds that admit of further specification. Iron is a determinable kind, determined by its isotopes 54Fe, 56Fe, 57Fe, 58Fe (Tobin 2011). But we would like to provide a characterization of quantities that shows how quantitative attributes differ from sortals and kinds. If the determinable/determinate model can be applied to kinds as well as quantities, it is unlikely to provide the desired distinguishing criterion. Colours are a more interesting comparison case. Colours are counted as quantities by some philosophers (Bigelow & Pargetter 1990), but they are often so counted precisely because they exhibit determinable/determinate structure. Unless we have independent arguments for thinking that colour is a quantity, we should not take this as an additional argument to think that

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16 Quantities as Determinables the determinables model is a good model for quantities.7 Given that many attributes seem to have something like a determinable/determinate struc ture, then, we should not expect this structure to be characteristic of quantities. Even if determinables/determinate structure is not what distinguishes quantities from other attributes, quantities might still fit the model. But there are some striking differences between typical quantities, like mass, tempera ture, length, or charge, and paradigmatic determinables, like phenomenal colour. One such difference is that the latter exhibit a high degree of nesting. In the case of mass, we have mass, and then we have an infinite number of possible mass-values. There is no further hierarchical structure in between the quantity and its magnitudes. By contrast, one of the appealing features of the determinable model for the case of phenomenal colour is precisely that it allows us to represent the different levels of colour determination. Initially one might simply wish to distinguish colours into blue, red, yellow, and green. But further determination of these initial distinctions is possible, for example by distinguishing different shades of blue into navy, sky, and indigo. The determinable/determinate structure replicates at different levels of determin ation or specification. Quantities, by contrast, typically lack this kind of nest ing. Instead we have just one determinable and what are sometimes called ‘super-determinates’, that is determinates which permit no further specification. Notice that further specifications leave the higher level determinable/ determinate structure intact. An object now determined to be navy is still blue, and it still excludes red, as well as other shades of blue. The same is not true for quantities. We do at times make ‘rough’ or ‘approximate’ classifica tions of objects into equivalence classes for some quantity, only to find later, through more accurate measurement, that some of these objects were in fact slightly bigger or smaller. For example, I might initially sort tiles of roughly 16 cm2 into a single pile, only to then go on to distinguish those that are 15.9 cm2 from those that are 16.1 cm2. But notice that the tiles of 15.9 cm2 do not ‘con tinue’ to be 16 cm2. Instead their previous classification simply turns out to have been inaccurate. By contrast, a tile now classified as indigo, having

7 On Funkhouser’s account, which treats colour, quantities, as well as many other properties as determinables, determinables are property spaces. I hold a similar view about quantities in general (see Chapter 7). Accordingly, colours turn out to be quantitative in my view as well, but not because they are determinables. Attributes that are metrizable spaces are quantitative, and to the extent that colour is a metric space, colour is quantitative (compare Isaac 2013). The determinables/determinate model is a detour at best.

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An Awkward Fit 17 previously merely been identified as blue, is still rightly said to be blue, and indeed coloured. One might be tempted to suggest that the intermediate determinations for quantities are not as specific as 16 cm2, but are instead something like less than or equal to 16 cm2. My initial sorting of tiles would hence be into tiles less than or equal to 16 cm2 and tiles greater than 16 cm2. These properties indeed admit of further specification, and a tile of 15.9 cm2 still instantiates the property of being less than or equal to 16 cm2. The problem with this way of conceiving of intermediate determinates for quantities is that the candi dates for determinates at this intermediate level are not unique: less than or equal to 16 cm2, less than or equal to 15 cm2, less than or equal to 14 cm2 are all plausible candidates for intermediate determinates, but this results in inter mediate determinates that do not exclude one another. Determination is a transitive relation, and hence permits the formation of determination hierarchies. The hierarchical determination structure of colours has no equivalent in the case of quantities. Less than or equal to is a transitive relation, but not a relation of greater specification. The lack of hier archical structure for quantities suggests that ‘specification’ or ‘greater deter mination’ might not be the best way to think about the relationship between quantity values and quantities.

2.2.2. Horizontal vs. Vertical Relations The best diagnosis of the problem with applying the determinable/determinate model to quantities is that the determinables/determinate model emphasizes ‘vertical’ structure, but leaves ‘horizontal’ structure largely underspecified. The main relation on the determinables model is that of determination, where ‘lower level’ entities determine or specify ‘higher level’ entities. The relations among entities ‘on the same level’ are far less clear on this model. The usual thought is that something like similarity or resemblance relations hold among them (Funkhouser 2006; Bigelow & Pargetter 1990). To use colour again as our toy example, shades of blue are all thought to be more similar to each other than any of them is to shades of red. It is this similarity relation which is supposed to account for the divisions into classes at the same level in the hierarchy. Yet the exact nature of the similarity relation is often underspecified. In some cases it is supposed to impose a kind of distance metric, for example in the case of colours many philosophers appeal to notions of ‘closeness’ between

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18 Quantities as Determinables colours (Armstrong 1989; Bigelow & Pargetter 1990). The idea seems to be that we can construct a ‘similarity metric’ among determinates of the same determinable. Whether such a metric can be constructed depends in no small part on the particular metaphysical implementation of the determinables/ determinates model. Armstrong’s version, according to which determinables and their determinates are structural universals, has been shown to be inad equate for generating a similarity metric (Eddon 2006). Funkhouser’s view, according to which determinables are property spaces, does better, but creates problems of its own. According to Funkhouser, deter minables are property spaces with different numbers of determination dimen sions. For the case of colour, these determination dimensions are hue, brightness, and saturation. The idea that properties like colour are best under stood in terms of property spaces is very plausible,8 and has been explored in philosophical psychology as well (Gardenfors 2004). However, this does not yet vindicate the account for other determinables and determinates, where it is far less clear what the determination dimensions might be. Even in the case of colour, it is not obvious that we have to construct colour spaces as the familiar colour spindle, with hue, brightness, and saturation as the relevant dimensions. Nor is the similarity metric uniform across these different ‘dimensions’ of colour space, when the space is treated as Euclidean (Isaac 2013). Most importantly, the role of the similarity relation in the ori ginal determinables/determinate model was to provide something like a qualitative ground for the quantitative relations among determinates. Once we move to a conception of determinables as property spaces with an inher ent geometry on them, it seems we are no longer providing a non-quantitative ground. We will return to this issue in Chapter 7, where I’ll argue in favour of viewing quantities as spaces. In Chapter 10 I conclude that this indeed means that quantitative relations are not grounded in purely qualitative relations. For some determinables it seems that the relationship among the deter minates is really supposed to be one of ordering, but it is also not clear how a similarity structure can be used to generate a linear order. What is required here is usually a particular ‘graded dimension’, along which determinates can be ordered. To arrange colours in a colour space, for example, dimensions of hue and brightness are introduced. With the latter we get a clear sense of gradation from brighter to darker. It is this gradation which can be used to order colours from light to dark. We can also order colour from most to least saturated. However, no similar ordering seems to occur along the dimension 8 Johnson (1921) indeed hints at it in his initial discussion of colour as a determinable.

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An Awkward Fit 19 of hue. The hues, tellingly arranged in a circle, are not ordered by ‘hueness’. Not all intuitive determination dimensions are ordered. We want to distinguish, then, between determinates whose similarities and differences produce orderings, those that give us distances, and those that yield both. Neither ordering nor distances are straightforwardly reducible to or grounded in similarity relations. This suggests that even for the paradig matic examples of determinables and determinates, more attention should be paid to the horizontal relations among the determinates, instead of focusing primarily on the vertical relation of determination. Indeed, in the case of quantities, it seems that we appeal to relations among quantity values in order to explain similarities among objects, not the other way around. Dumbbells of 12 kg and 11 kg seem very similar in weight pre cisely because they are so close in mass, not the other way around. Ordering, and, more importantly, distance metrics between magnitudes, seem to ground similarity relations among them. Objects that share the same quantity are not thereby similar to each other in very interesting ways. An electron and a planet both have mass, but that does not mean they are therefore especially similar to each other, or that we can derive many further features from this shared feature of both having mass. By contrast, being close in value with respect to a quantity can make for interesting similarity relations. Protons and neutrons are close in mass, and this similarity has important consequences for nuclear physics. To describe quantities as attributes in virtue of which objects are both similar and different in the same respect, then, seems to be misleading. ‘Having mass’ does not make for similarity in any especially inter esting sense. All of this suggests that the horizontal structure among quantity values is of more central importance than the vertical relation between the undetermined quantity and its value range. The determinables/determinate model does not fit quantities very well, because the determinables model is based around a vertical relationship, while remaining somewhat unclear about the relevant horizontal relations among the determinates. The move to understanding the determinables/determinate model in terms of property spaces points in the right direction, but actually highlights that the horizontal relations among ‘determinates’ are what matter for the structure of these properties. When we ask what the structure of phenomenal colour space is, we want to know which dimensions such a space has and what type of metric we can find on such a space. Instead of focusing on the relation between a quantity and its value range, a metaphysics of quantities would do well to look more closely at the relations among the values first.

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20 Quantities as Determinables

2.2.3. The Challenge of the Single Value Principle The main reason for introducing the determinable/determinates model in the first place, and for applying it to quantities, was that it provided a way of mod elling the manner in which different shades of the same colour and different magnitudes of the same quantity exclude one another. The determinable/ determinates model explicitly addresses this problem, albeit by writing the principle of determinate exclusion into the model ‘by hand’. It is a stipulated requirement of the model that on any given level, an object may only instanti ate one determinate of a given determinable. While this guarantees that the model includes this requirement when applied to quantities, it does not help to explain the requirement. It is worth asking whether the stipulations of the determinable/determinates model capture precisely what is required by the single value principle, and secondly, whether there might not be better ways to make sense of this principle. The single value principle, as Chang states it, says ‘a real physical property can have no more than one definite value in a given situation’ (Chang 2004, p. 90). This is a cautious formulation, since it does not require that a physical property must have a definite value in a given situation.9 By contrast, the determinable/determinates model entails just that, since it requires not only that at most one determinate of a given determinable is instantiated, but also that at least one such determinate is instantiated for the determinable to be instantiated. Call this the determinate value principle. The determinate value principle entailed by the determinate/determinable model hence looks stronger than what is strictly speaking required. The determinate value prin ciple explains the single value principle in the sense that it entails it. But the determinate/determinable model is not the only way to explain the single value principle. By focusing on the horizontal relations of ordering and distance, we also get an explanation for the requirement: ordering is only possible if each object is at most assigned a single, determinate value. The existence of an ordering explains the single value principle, because for an order to be definite an assignment of different positions in the order to different objects is required. This principle is familiar from non-scientific contexts as well. In many team 9 One reason to be cautious about claiming that physical properties must have definite values in any given situation stems from certain interpretations of quantum mechanics. Since I’m restricting the discussion in this book to classical quantities, I will set these worries aside, but see Wilson (2013), Wolff (2015), and Calosi & Wilson (2018) for discussion of the application of the determinable/deter minates model to properties in quantum mechanics.

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An Awkward Fit 21 sports, for example, a league table is essentially a ranking of teams according to a certain (set of) criteria: most points achieved so far, goal difference, goals scored. . . . These criteria are introduced and refined with the aim of producing a maximally determinate ranking of teams, so that at the end of the season, each position in the league table will be occupied by exactly one team. In measurement practice, at least when what are being ordered are objects, the ordering will not be as determinate as in the case of teams in a league table. Instead the ordering will be a weak linear order, with several objects occupy ing the same position. But even for a weak linear order to be possible, each object must not be assigned more than a single value at a given moment. For otherwise the same object would appear more than once in the ordering, and the domain would not even be weakly ordered. If an object were to occupy more than one position, features of linear orders, such as transitivity, would fail. The requirement of a weak linear order hence also explains the single value principle.

2.2.4. Conclusion The determinable/determinate model looked like a promising candidate for what it takes for an attribute to be quantitative. Upon inspection, however, several disanalogies between paradigmatic quantities and paradigmatic instances of the determinable/determinate model became apparent. First, the determinables model fits many different types of attributes, not just quan tities, which makes it unsuitable for distinguishing quantities from other attributes. Second, the determinable model seems most useful for attributes that exhibit a high degree of nesting, which is absent in the case of quantities. I argued that these disanalogies suggest that the determinables models is a less than perfect fit for quantities because it emphasizes a vertical relation— determination of a determinable by determinates—whereas quantities are characterized by horizontal relations among their magnitudes (the would-be determinates on the determinable model). To conclude, then, the horizontal relations among magnitudes, not the vertical relations among a quantitative attribute and its value range, are the structure that is characteristic of quan tities. As a result, the determinables model is not ideally suited as a model for quantities.

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3 Quantities as Numerical Attributes Synopsis In Chapter 2 we looked at a standard metaphysical approach to quantities, which treats them as a (special kind of) variable attribute: determinables. In this chapter we turn to axiomatic approaches to quantities, which typically focus on quantities as numerically representable attributes. I distinguish three different philosophical attitudes to quantities in axiomatic approaches: restrictive realism, restrictive empiricism, and permissive empiricism. Restrictive realists believe that quantitativeness is a feature of only some kinds of attributes: attributes whose magnitudes stand in ratio relations. Restrictive empiricism, which is most clearly expressed in Carnap’s writings on quan tities, assumes that quantitativeness, or measurability, resides in our concepts, not in attributes in the world. Whether a concept is quantitative depends on its axiomatization. Quantitative concepts require particularly strong axioms, in contrast to classificatory or comparative concepts. It is in virtue of satisfy ing strong axioms that these concepts can be numerically represented and it is numerical representability that makes them quantitative concepts. Permissivism goes a step further and suggests that there is nothing peculiar about quantitative concepts. Weaker axioms can also be used to assign num bers to objects, so quantitative concepts are not special numerical concepts. Empiricism and permissivism each lead to the view that quantities are not special kinds of attributes deserving of their own metaphysical account. I argue in the second half of the chapter that we should not take numerical representability as the defining characteristic of quantitative attributes. This opens a route for an axiomatic approach that restricts quantitativeness to only some attributes.

The Metaphysics of Quantities. J. E. Wolff, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198837084.001.0001

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Restrictive And Permissive Views Of Quantities 23

3.1. Restrictive and Permissive Views of Quantities 3.1.1. Restrictive Empiricism: Carnap’s Division of Scientific Concepts To understand better what makes quantitative attributes special, it is helpful to situate them in relation to other types of attributes. A useful point of entry is provided by Carnap’s distinction between different types of scientific con cepts (Carnap 1966), because it captures several features of empiricism about quantities: quantitativeness is a feature of concepts, not attributes, only some concepts are quantitative, and concepts are quantitative in virtue of satisfying operational axioms. Carnap’s view exhibits many of the characteristic flaws of empiricist approaches to quantitativeness, yet the criterion for quantitative ness defended in Chapter 6 rehabilitates some features of Carnap’s distinction. Carnap draws a threefold distinction1 among concepts: classificatory concepts, comparative concepts, and quantitative concepts (Carnap 1966, chapter 5). Examples of classificatory concepts are old, large, hot, as well as dog, tiger, tree. . . These concepts can be used to sort objects into classes, hence the label classificatory concepts. Unlike comparative or quantitative concepts, classificatory concepts are of limited scientific value, according to Carnap, although he concedes that they ‘vary widely in the amount of information they give us about an object’ (Carnap 1966, p. 51). Classifying an object as an African elephant tells us more about the object than classifying it as grey. Carnap attributes this difference solely to the fact that the class of African elephants is smaller than the class of grey things (ibid.). But while it is correct to say, formally, that both taxonomic concepts and concepts like cold or warm place objects into classes, the concepts are other wise quite different. For species and other taxonomic concepts are typically sortal kinds. Kind membership is not just classification, since kinds themselves stand in hierarchical relations to one another (see Hawley & Bird 2011 for a recent discussion). For example, in classifying an animal as an African ele phant, we thereby also classify it as belonging to the family of Elephantidae, the order of Proboscidea, the class mammalia, and so forth. That is to say, clas sification into kinds imposes a hierarchical order on the domain thus classi fied: members of lower kinds are automatically also members of higher kinds. 1 For a closely related distinction between qualitative, topological, and metric concepts, see Berka (1983, ch. 1).

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24 Quantities as Numerical Attributes At the same time, membership of kinds at the same level is exclusive: if an animal is an African elephant, it cannot also be a zebra or a lion. Kind mem bership, at least traditionally, is taken to be all or nothing: an animal either is an African elephant, or it is not. Whether current scientific kinds, that is, those employed in the taxonomies of zoology or chemistry, indeed satisfy these criteria is a slightly different matter. We might aim for exclusive, all-ornothing classification schemes, only to find that the domain in question resists our efforts. This may be true in biology, for example, where species have turned out not to be the unchanging exclusive classes early taxonomists took them to be. Instead, both empirical evidence and evolutionary theory suggest that we do find intermediates between different species, which has been taken by some philosophers and biologists as a sign that species are not (natural) kinds (Dupré 1981).2 Be that as it may, the reason species concepts, and kind concepts in general, are so useful in science is not merely, as Carnap suggests, that the class of animals belonging to the kind African elephant is smaller than the class of objects falling under the concept grey. Instead, much of their usefulness derives from the relational hierarchy into which an animal is placed when it is classified as an African elephant. Membership in a given kind K depends on meeting certain conditions, where these conditions specify a requisite set of properties that are jointly necessary and sufficient for membership in K. The challenge of course lies in the specification of such necessary and sufficient conditions. The other type of classificatory concept Carnap considers is concepts like cold and hot, or old and young. They may be classificatory, but they are not kinds.3 Instead, they are qualities. Qualities differ from kinds in several respects. First, they lack a hierarchical embedding of the sort just described, making them much less informative for scientific purposes than kinds. If I say that my mug is hot, I am not thereby placing it in a hierarchy of kinds; there are no higher kinds in which my mug is a member in virtue of being hot. This classification therefore does not enable many additional inferences, unlike a classification into kinds. Closely related to this is another point of difference, namely that whereas kinds of zoology, botany, or chemistry come with par ticular criteria for kind membership, cold, hard, or heavy do not have criteria 2 Boyd (1991) introduces the possibility of indeterminate kind membership to deal with atypical members of a kind. This seems sensible as an approach to biological species, which turn out to be neither as static nor as exclusive as the traditional taxonomic enterprise would suggest. 3 This point was made very clear by John Stuart Mill in his initial discussion of kinds (Mill 1974, p. 159).

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Restrictive And Permissive Views Of Quantities 25 by which to judge whether an object should be so classified. As a result ‘clas sifications’ into hot/cold or young/old are more easily challenged and appear to be highly context sensitive: my tea is hot compared to its immediate sur roundings, but not compared to a steel furnace or to the surface of the sun. This context sensitivity also makes these concepts less useful for science. Despite being ‘classificatory’, then, old and cold are quite different from dog and gold. Carnap’s characterization of classificatory concepts is inadequate, because it conflates kinds and qualities under the label ‘classificatory concepts’. Comparative concepts,4 as the name suggests, can be used to compare two objects with respect to the same attribute. For example, heavier than is a com parative concept in that it relates two objects with respect to their weight. Carnap describes these comparative concepts as being governed by two rules: a rule for judging that two objects are equal with respect to a comparative concept, and a rule for judging that one object is less than the other object with respect to this comparative concept. For Carnap, these rules will be based on empirical (observable) relations, for example, putting objects on a beam balance to see which pan, if any, goes down (Carnap 1966, p. 54). Such comparative concepts induce an ordering or ranking over a domain of objects, which makes them significantly more useful for science. Since the relations of equality and less than are transitive, and the former symmetric and the latter asymmetric, they are not purely conventional or arbitrary, in contrast to the extensional specification of classes. Unfortunately, the alleged purely empirical ‘operational definitions’ for the relations governing comparative concepts are insufficient to establish the transitivity and asymmetry or symmetry of these relations. It is possible to devise an observational series such that we might pairwise judge all objects to be equivalent, but judge the first and last object to be inequivalent. This is possible when, for any two objects in the series, their difference in weight is just below the perceptible threshold, yet the difference between the first and last objects is just big enough to be noticeable. That means transitivity is not a feature we can ascribe to these comparative relations based on observations alone; instead attributing transitivity reflects theoretical commitments we

4 Recall that Carnap takes these distinctions to be applicable to our concepts only, not to their metaphysical counterparts. From a metaphysical point of view, comparatives are relations, which raises the question whether they can be said to hold in virtue of monadic properties instead (we shall return to this question in Chapter 8). Since many comparative relations are asymmetrical, it might seem particularly problematic to have both ‘longer than’ and ‘shorter than’ as elements of a fundamen tal ontology for comparative relations.

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26 Quantities as Numerical Attributes have regarding attributes like weight, which lead us to dismiss such failures of transitivity as measurement errors.5 Accordingly we should conclude that the empiricist goal of providing a purely operational analysis of comparatives fails, if the latter are understood to establish a total or even partial order over a domain. Finally, quantitative concepts permit attributions of specific amounts of an attribute to an object, for instance, being four feet long, or having a mass of 4 kg, as well as quantitative comparisons between different objects, for instance, being twice as massive as. For Carnap, the tell-tale sign of a quantitative con cept is the use of numerical expressions (see for example Carnap 1966, p. 53). This use of numbers to talk about (empirical) attributes needs to be justified. Carnap introduces rules specifying empirical relations that will constrain how these concepts can be applied. He specifies different sets of rules for ‘intensive’ quantities, like temperature, and ‘extensive’ quantities, like mass.6 From the perspective of contemporary measurement theory, his account is defective in particular aspects of the rules he specifies. I will set aside these concerns and focus instead on the contrast between the observational axioms and the identification of quantitative concepts with numerical concepts. As in the case of transitivity for comparative concepts, the use of numbers and the numerical claims made by using quantitative concepts goes beyond what is observable. Carnap may have shown that the operations and relations in the empirical phenomenon are embeddable into a numerical structure. But as Vadim Batitsky (Batitsky 2000) has argued, this ‘physical-to-mathematical’ direction, while necessary for measurement, is not sufficient to permit the confirmation of mathematical theories by empirical experiments. For the lat ter we also need the reverse direction, from mathematics to the world. This means that not only do we need to show that certain empirical rules can be represented mathematically; we in fact need to show that the mathematical relations and the empirical relations are equivalent, so as to permit inferences in both directions (Batitsky 2000, p. 94). Without inferences in both directions, ‘predictions’ from the mathematical relations that are not satisfied empirically would not count against the theory in question, since we would have no reason to expect that such inferences are legitimate to begin with. While representational measurement theory (see Chapter 5) proceeds to provide theorems demonstrating this equivalence for a wide range of 5 See Kyburg Jr (1997) for the general point, and Batitsky (2000) for the application to Carnap’s philosophy of measurement. 6 The distinction between intensive and extensive quantities in this form is somewhat misleading, as we shall see in Chapter 5.

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Restrictive And Permissive Views Of Quantities 27 empirical and mathematical structures, Batitsky argues that this commitment to equivalence is in tension with Carnap’s empiricism. The mathematical structures in question have properties like closure under addition that do not hold true for empirical structures of the sort Carnap considers. Carnap, like many other empiricists, believes that qualitative concepts are epistemologically prior to quantities, and more importantly, that quantitative concepts can be analysed purely in terms of qualitative concepts. If Batitsky is right, however, this analysis of quantitative concepts in terms of qualitative concepts fails, since qualitative observations are ultimately insufficient to support the required rela tionship between mathematical and physical or empirical structures.7 Carnap’s approach to the distinction between scientific concepts is helpful, because it captures many intuitive differences between concepts. In emphasiz ing these differences, Carnap advocates what I will call a restrictive approach to quantities. On a restrictive view, only some concepts are genuinely quanti tative. The challenge for any restrictive view is to say what makes quantitative concepts special. Carnap’s approach is also useful, because it is an example of an axiomatic approach to quantitativeness. A concept is quantitative if and only if it satisfies particular axioms. As we have seen, Carnap’s empiricist approach to the quantitative/qualita tive distinction is problematic in several respects: it underestimates the con straints placed on kinds, and it overestimates the extent to which orderings and numerical relations can be based on purely observational axioms. Implicit in many empiricist accounts of quantities is a twofold contrast between quan tities and qualities. On the one hand, the distinction is supposed to track the difference between numerical and non-numerical concepts, which sets up the problem of justifying the use of numbers. On the other hand qualities, unlike quantities, are thought to be observable, which seems plausible insofar as both classifications like ‘warm’ and ‘cold’ and comparisons of ‘warmer’ and ‘colder’ are made in the first instance on the basis of perception.8 As a result, empiricists often propose to justify the use of quantitative concepts on the basis of qualitative-observational relations. However, perceptual judgements alone are not sufficient to justify the claim that an attribute satisfies the strin gent axioms governing quantities.

7 Batitsky acknowledges that attempts have been made to solve the difficulties arising from the stronger features of mathematical structures compared to empirical structures, but argues that the solutions themselves carry further commitments beyond the empiricist’s comfort zone. 8 Compare Ismael and van Fraassen (2003).

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28 Quantities as Numerical Attributes One strategy in response to these problems is to opt for realism in addition to a restrictive view on quantities. Alternatively, empiricists might give up on a restrictive view of quantities. We shall now turn to these two responses.

3.1.2. Restrictive Realism: Numbers are Hard Restrictive realists share Carnap’s view that numbers are what make quan tities special, but argue that numbers are not merely representational tools. Restrictive realism means that measurement requires the existence of quanti tative attributes (that’s the realism part), and that not all attributes are quanti tative (that’s the restrictive part). Restrictive realism has recently been defended by Joel Michell and, as he points out, realism has historically been the more common position (Michell 2004).9 Much in keeping with traditional conceptions of number and measurement, Michell takes ratios between mag nitudes to be key to both quantities and numbers: If attributes are measurable, then the world contains continuous quantities, that is, attributes which, by virtue of their additive structure, sustain numer ical relations of the appropriate kind between magnitudes. The appropriate kind of numerical relation is this: if a and b are any magnitudes of the same quantity, then a:b = r, (where r is a real number), That is, the existence of measurement not only presumes the existence of quantities, it presumes the existence of real numbers, as well. (Michell 1999, p. 59)

The line of thought seems to be the following: measurement is the discovery of numerical relations in the world, not the assignment of numbers to objects. While numerals may be representational devices, numbers are not. Unlike most contemporary philosophers, Michell thinks of numbers not as abstract particulars, but as relations between magnitudes of quantities. Since magni tudes and the relations between them are in some sense ‘in the world’, numbers are not mysterious, abstract entities, but simply dependent upon the existence of magnitudes. Michell’s view, then, is realist in three ways: first, he assumes that attributes of a certain type need to exist to make measurement possible, in contrast to operationalism according to which measurement only requires the specification of measurement procedure, but not the existence of 9 Michell calls the view that all attributes are quantitative ‘Pythagoreanism’, which is somewhat different from common philosophical uses of the term.

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Restrictive And Permissive Views Of Quantities 29 particular attributes. It is realist in the further sense that it requires the existence of numbers, although numbers are clearly dependent entities on Michell’s view, since he takes numbers to be definable in terms of relations among magnitudes. Finally, he seems to assume that magnitudes are best understood as universals, in contrast to a nominalist understanding of properties.10 All three aspects of his realism have far-reaching metaphysical implications, and even philosophers who share Michell’s anti-operationalist impulse might hesi tate to embrace the further metaphysical commitment to numbers and univer sals. In Chapter 4 I shall argue that only anti-operationalism is required to make the quest for a metaphysics of quantities worthwhile. We shall return to the ontological commitments that arise from realism about quantities in Chapter 7. Here I am more interested in the restrictive aspect of Michell’s realism. Since Michell requires of measurable attributes that they sustain ratios between their magnitudes, comparatives will not count as quantitative. On this point, Michell agrees with Carnap. But where Carnap takes quantitative ness to be a matter of whether or not a concept satisfies certain (empirical) axioms, Michell takes quantitativeness to be a feature of attributes. Michell and other restrictive realists see a clear division between attributes that are quantitative, and can hence be numerically represented, and those that fail to be quantitative and shouldn’t be represented numerically. To represent nonquantitative attributes numerically would be a misrepresentation, because only ratio-relations, but not mere orders, imply the existence of numbers. Michell, like many restrictive realists before him, goes a step further and suggests that the ability to sustain ratios is grounded in the additivity of an attribute. Following empiricists, one might think of the additivity of attributes as manifest in a concrete, empirical operation of concatenating objects instan tiating magnitudes of the quantity. Michell rejects this conception of additivity, which he call ‘extensivity’, as too narrow (Michell 1999, p. 54). An attribute is additive in virtue of satisfying a set of axioms, not in virtue of having an obvi ous empirical concatenation operation. For Michell, ‘intensive’ quantities like density can turn out to be additive, in the sense of sustaining ratios among magnitudes, but the test for additivity may be ‘indirect’. Michell thereby com bines a realist metaphysics with elements of empiricist views of quantities: the axioms that characterize quantities are still supposed to be empirically testable. 10 Since Michell’s primary concern is with what he argues are illegitimate notions of ‘measurement’ in psychology, his position doesn’t neatly fit into the extant philosophical debate and his consider ations don’t speak to all philosophical considerations in the vicinity. I will return to some of these issues in more detail in Chapter 7, where I consider Aristotelian, Platonist, and nominalist ontologies for quantities.

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30 Quantities as Numerical Attributes In rejecting an empirical concatenation operation as a necessary condition for additivity of an attribute, Michell avoids a problem that has plagued trad itional forms of restrictive realism: how to account for bona fide physical quantities, like density, temperature, and temporal duration, that lack obvious concatenation operations. If tests for quantitativeness can be indirect, the lack of an empirical concatenation operation does not immediately speak against the status of an attribute as a quantity. Restrictive realism provides a compelling answer to why quantitative attributes are numerical: they are numerical because they involve ratios of magnitudes, which in turn imply, on Michell’s view, the existence of numbers. Since not all attributes permit ratios among their magnitudes, this view of quantities is highly restrictive. At the same time, it comes at considerable ontological cost: it requires the existence of real numbers, and the treatment of attributes as universals.

3.1.3. Permissivism: Numbers are Easy Carnap’s division of scientific concepts had shortcomings, but the underlying empiricism is shared by many other views on quantities as well. In particular, the core idea of representationalism—the view that numbers are merely tools for representing measurement—is motivated in part by an empiricist outlook. Representationalism is based on drawing a distinction between two kinds of structures: numerical structures and ‘qualitative-empirical’ structures. The former are used to represent the latter, and the question a theory of measure ment sets out to answer is whether a numerical representation is possible and how unique it is. The motivating idea behind this conception of (a theory of) measurement seems to be that ‘qualitative’ or ‘empirical’ structures are observationally accessible,11 whereas numerical structures are not. We do not perceive quantitative relations, if by quantitative we mean ‘numerical’. Representationalism hence seems to offer the following distinction between qualitative and quantitative structures: quantitative simply means numerical, qualitative seems to mean ‘empirical’ or perhaps ‘observable’. This echoes Carnap’s distinction discussed above, but there is an important difference. On Carnap’s account, quantitative concepts are characterized by 11 This is not the full story, however. Patrick Suppes at least makes it clear that the mapping takes place not between a somehow directly perceived empirical structure and a numerical representation of it. Instead the relationship between experimental data and numerical representation is mediated by theory at every level on an entire ‘hierarchy of models’, where the models differ in their level of abstraction (Suppes 1969).

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Restrictive And Permissive Views Of Quantities 31 the use of numbers, which in turn is made possible because the concepts are governed by more stringent axioms than either comparative or classificatory concepts. Carnap, unlike standard representationalists, was a restrictive empiri cist. For Carnap, the use of numbers is restricted to concepts that are gov erned by axioms of a certain strength. Permissive empiricism, by contrast, follows Stevens (Stevens 1946) in argu ing that numbers can be used to represent concepts of various types, yielding a ‘hierarchy’ of scale types: These scales share much with the Carnapian distinctions among the differ ent kinds of scientific concepts we encountered earlier. Very much unlike Carnap, however, permissive empiricists hold that the use of numbers in the representation of attributes is not restricted to concepts characterized by Carnap’s more stringent axioms, but can be applied to pretty much any kind of concept, including classificatory ones. The question is only how such a numerical representation may be transformed to yield another, equally good numerical representation of the same concept. Stevens identifies these ‘per missible transformations’ by what is preserved under them. As Table 3.1 shows, more is preserved as we go up the hierarchy of scales, with ratio scales being the strongest type of scale and classificatory scales being the weakest. Which scale type is appropriate depends on the rule by which numbers are assigned to objects. The one point to bear in mind is that the numerical repre sentations on weaker scales will not be strong enough to support all the infer ences made possible by representations on stronger scales. Not all features of such weaker representations are ‘meaningful’ (we’ll return to the question of meaningfulness in more detail in Chapter 5). For example, if we have an ordinal scale of 1–5 stars for ranking restaurants, we cannot conclude that a restaurant with 2 stars is half as good as one with 4 stars in the ranking. All we can say is that the 4-star restaurant is better than the 2-star restaurant. If quantities were thought to be characterized by their special relationship to numbers, Stevens’s permissivism seems to show that a relationship to num bers is possible for many different types of concepts. The relationship will be different for different concepts: some concepts will have ‘stronger’ numerical Table 3.1. Scale types by Stevens and Carnap Scale Type

Carnapian Concepts

Transformations

ratio interval ordinal classificatory

extensive quantities intensive quantities comparative classificatory

preserve ratios preserve equal distances preserve order preserve difference in kind

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32 Quantities as Numerical Attributes representations than others. But being representable by numbers is not limited to phenomena we would intuitively describe as quantitative. Even mere clas sification, for example, putting all the cats in one class, and all the dogs in another class, can be represented numerically, by labelling the class of cats ‘1’ and the class of dogs ‘2’. For permissivism, the difference between this representation and a map of all massive objects into the real numbers is a matter of the degree of ‘uniqueness’ of the numerical representation, not a difference between quantitative and qualitative phenomena. In adopting this permissivist outlook, representationalists in general seem to embrace the idea that numerical relations are not special—they are just one representational tool among many. Yet at the same time the representationalist programme relies on drawing a seemingly sharp distinction between qualita tive and quantitative structures, where the former are ‘observable’ and the latter are numerical. But why should numerical representations be particularly interesting or important, if numbers can be used to represent anything from classification to measurement of extensive quantities? The quantitative/ qualitative distinction, which seemed so important in order to motivate the representationalist programme, appears to have been rendered trivial by the very same programme. As we shall see in Chapter 6, this apparent puzzle is solved if we take the representationalist turn as a way of reframing the question of the border between qualities and quantities. Since it demonstrates how different—ostensibly qualitative—structures can be represented numerically, the interesting dis tinction occurs not between structures that are numerically representable and those that are not so representable, but between structures of different strength. Quantities are no longer characterized by a use of numbers per se, but only by certain uses of numbers. Two questions arise in light of this. First, why should we draw a distinction between quantities and qualities at all? Second, if we do draw such a distinction, what sorts of numerically represent able structures should count as quantitative? In section 3.2 I will suggest a strategy for drawing a distinction between quantities and qualities, which will be implemented and defended in Chapter 6.

3.2. A Difference in Strength, not in Numbers 3.2.1. Numbers: Neither Necessary nor Sufficient The view that what is peculiar about quantities is that they involve numbers was shared among both realist and empiricist restrictivists. Permissivism,

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A Difference In Strength, Not In Numbers 33 however, seems to show that there is little reason to hold a restrictive view of numerical representation. If we want to follow representationalism in viewing numbers as representational tools, instead of adopting Michell’s more substan tive realism about numbers, do we have to give up on a distinction between quantities and qualities? Giving up this distinction is neither desirable nor mandated by representa tionalism. As we shall see in Chapter 6, we can, on the contrary, use represen tationalist tools to articulate a distinction between qualities and quantities in terms of the strength of the numerical representation available for them. A first step in this direction is to give up the idea that numerical representation is a necessary or a sufficient condition for quantitativeness. There are good reasons to drop this assumption, especially from a representationalist standpoint. We’ve already seen that when numbers are merely regarded as a means of representing certain relationships, nothing forces us to restrict this representa tion to attributes we would intuitively characterize as quantitative. Numerical representability is hence not a sufficient condition for quantitativeness. Representationalists acknowledge that if we permit a wide range of numerical representations, we have to be careful in specifying which representations are meaningful. In particular, we must avoid the trap of mistaking peculiarities of one numerical representation for features of the represented attribute. Nor is numerical representability necessary for quantitativeness. If numbers are merely a representational tool and represent in virtue of the structure they exhibit, then other structures might do just as well. Geometrical structures in particular have indeed been discussed in the context of repre sentational theory of measurement (Suppes et al. 2007). If numerical rep resentability is neither necessary nor sufficient for quantitativeness, we could conclude, as representationalists have traditionally done, that there is no important distinction between quantities and qualities. Alternatively we might conclude that the distinction is not grounded in numerical representability. Realists will need different reasons to be persuaded to drop the assumption that numbers are the defining feature of quantities. For a realist like Michell, numerical representability is neither here nor there, since it is a mistake to begin with to think that numbers have merely representational function. Even for realists, though, there are reasons to think that numbers are not the key. On Michell’s view, numbers are secondary, dependent entities. What sus tains numbers, and what makes quantitative attributes quantitative, are not numbers but ratios between magnitudes. This implies that there are relations that we consider quantitative, but which are not numerical. Ratios can be defined without the use of numbers (and were indeed traditionally defined

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34 Quantities as Numerical Attributes without the use of numbers),12 but nonetheless seem like paradigmatically quantitative relations. Different attempts (e.g. Koslow 1992) have been made to articulate exactly what it is about such relations that makes them quantita tive, but for now we can note that if this is right, then it is wrong to equate quantitative relations with numerical relations. Another good reason for not doing so is that the numerical/non-numerical distinction is closely aligned with the distinction between analytic and syn thetic methods in mathematics. The former are familiar from Descartes’s analytic treatment of geometry: we assign n-tupels of numbers (coordinates) to points in a space, thereby opening up geometry to the methods of algebra, most importantly the calculus. Conversely, Hartry Field’s attempt at ‘science without numbers’ goes in the opposite direction, replacing the numerical relations standardly used in physics by synthetic relations.13 Field uses axioms involving the relations of betweenness and congruence to characterize quan tities. While betweenness arguably isn’t quantitative, congruence looks as if it might well be a quantitative relation, even though it isn’t numerical. Once again we find that not all quantitative relations are numerical, and synthetic methods in mathematics furnish a wide array of such relations. Neither representationalists nor realists should insist on numbers as the defining feature of quantities. As we shall see in Chapter 6 representationalist tools can be used to articulate what is special about quantities. But first let’s see why we should draw a distinction at all.

3.2.2. Why Draw a Distinction at all? In light of the discussion in section 3.1, it is tempting to conclude, with the permissivists, that there is no deep or important distinction between quanti tative and non-quantitative attributes. If numbers are neither necessary nor sufficient for quantitative attributes, then what, if anything, remains special or noteworthy about these attributes? I shall argue here that there are still reasons to distinguish quantitative from non-quantitative attributes, but that we need to employ a criterion other than numerical representability. In contrast to the representationalists’ deflationary understanding of the difference between qualitative and quantitative attributes, many scientists continue to draw a distinction between qualitative and quantitative methods and 12 This is how Eudoxos defined them (see Mueller 2006 for details). 13 Field’s view will be discussed in detail in Chapters 7 and 9.

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A Difference In Strength, Not In Numbers 35 results. Why might it seem useful to draw a distinction between quantities and qualities even though representationalism shows that numbers can be applied to all kinds of attributes? First, being able to give a quantitative description of a phenomenon is typic ally a major advance in the investigation of the phenomenon. This is especially clear in the social sciences, where such descriptions appear to be harder to come by than in the physical sciences. Quantitative descriptions are prized because they are rare and hard to achieve. Equating quantitative with numerical repre sentation is only of interest if numerical representation is rare. On the permissiv ist representationalist view, however, numerical representation is common. Highly informative numerical representations of the ratio or interval scale variety, by contrast, continue to be rare. It is precisely because scientists wish to have informative representations that they seek quantitative representations, and it is precisely for this reason that they continue to draw a distinction between ‘merely qualitative’ and ‘quantitative’ descriptions of phenomena. Second, the permissivist account suggests that there is nothing remarkable about attributes that can be represented numerically, since it is in virtue of having specified a suitable rule that numerical representation is possible, not in virtue of features of the attribute. Yet it seems that there is a difference between attributes like mass, which come ‘in different amounts’, and attri butes like being a cat, which are an all-or-nothing matter. In his attempt to make ‘measurement’ available to psychology, Stevens rejected both the idea that measurement is confined to cases of particularly informative representa tions and that some attributes are special because they are measurable. Representationalism supports the claim that numerical representation is not confined to particularly informative numerical representations, but this does not entail that there is nothing to distinguish attributes representable on the different scales. Only if we follow Stevens’s anti-realist conception of meas urement as ‘assigning numbers to objects’ does the hierarchy of scales lead to the view that there are no special quantitative attributes. On Stevens’s view, the assignment of numbers to objects only requires that a rule according to which this assignment takes place is specified, but it is not required that the objects in question instantiate a quantitative attribute. Restrictivists who want to maintain a distinction between quantitative and non-quantitative attributes would do well to grant that numbers are represen tational devices, while insisting that not all numerical representations are representations of quantities. What makes quantities different from other attributes is not that they can be numerically represented, but how they can be represented numerically.

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36 Quantities as Numerical Attributes

3.2.3. Summary and Outlook In this chapter we’ve looked at three different views of the distinction between quantities and qualities: restrictive empiricism, restrictive realism, and per missive empiricism. Restrictive empiricists hold that quantitativeness is a matter of the axioms governing our concepts: if the axioms are strong enough to support numerical representation, then the concept is quantitative. This combination of empiricism about quantities, where quantitativeness is a matter of concepts satisfying operational axioms, and a restrictive view of quantities, according to which quantitative concepts are a special kind of concept, is challenged by permissivism. Permissivists share the empiricist outlook, but argue that many different rules can lead to numerical representation. Numerical representability does not create a division of concepts into quantitative and non-quantitative concepts. Permissivists go further in suggesting that there is no deep distinction between quantities and non-quantities. Against both forms of empiricism, restrictive realists hold that whether a concept is quantitative depends on whether it refers to a quantitative attribute. An attribute is quantitative when its magnitudes admit of ratios. Admitting of ratios will also mean that the attribute satisfies the axioms in question, but where the empiricist axiomatic approach took a concept satisfying particular axioms to be a matter of the availability of a suitable axiomatization, restrictive realists hold that it is ultimately a matter of the structure of the attribute whether an attribute is quantitative, not a matter of whether we can find a suitable operationalization. Restrictive realism, however, might both be too restrictive and too realist. In Chapter 4, I will look at just how realist of a view we need to assume to get the metaphysics of quantities off the ground. After introducing the representational theory of measurement in detail in Chapter 5, I will then return to the question of how we can have a restrictive view of quantities on the basis of an axiomatic approach to quantities.

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4 Quantitative Attributes or Measurable Concepts Synopsis Are there quantitative attributes, or is the question of quantitativeness really a question whether our concepts are measurable? This question is at the heart of the dispute between (restrictive) measurement realists and (typically per missive) operationalists. In this chapter I try to disentangle this question from broader issues of scientific realism and argue in favour of restrictive realism. In section 4.1, I argue that not all commitments of measurement realism are required for the metaphysics of quantities. All that is required is that quan tities are independent of measurement procedures. In section 4.2, I discuss challenges to this aspect of measurement realism, in particular to the idea that we can empirically find out whether an attribute is quantitative. In sec tion 4.3, I consider operationalism as an approach to measurement that seems to undermine the project of a metaphysics of quantities. In section 4.4, I defend the claim that attributes are quantitative independent of any particu lar procedure to measure them. In section 4.5, I show how this claim interacts with scientific realism when applied to quantities.

4.1. Measurement Realism 4.1.1. A Naive View of Measurement Naively, one might think of measurement as an extension of observation by means of instruments. We can estimate the size of a room by looking, but if we want to make sure, we have to measure it using a metre stick. Similarly, we can gauge which of several fluids is the warmest by putting our hands in them, but in case of doubt we need to use a thermometer. Examples like these sug gest that measurement is simply a sophisticated form of observation, which provides more accurate estimations of the true size, weight, or temperature of The Metaphysics of Quantities. J. E. Wolff, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198837084.001.0001

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38 Quantitative Attributes or Measurable Concepts a system, and which permits the extension of observation beyond the range of magnitudes directly accessible to human perception (it’s not a good idea to put your hand in boiling water, let alone molten rock). This naive view of measurement combines an empiricist epistemology with a realist view of quantities. It is realist in its assumption that there are quantities like length or temperature and that physical objects, like rooms or fluids, have determinate magnitudes of such quantities. It is these determinate magnitudes—the true values of the quantities in question—that measurements aim to estimate as accurately as possible. The view is empiricist in that it assumes that observations, or perhaps instrument aided observations, are enough to estimate the magnitudes of quantities at a given time and place. It is easy to see why the combination of realism and empiricism is appealing. Measurement is key to testing quantitative hypotheses for the purpose of con firming or refuting theories. Even if we concede that the process of confirm ation involves more than a single measurement, and that the measurement operation in question might be quite far removed from any ordinary observa tion, we are still inclined to think of measurement as an empirical test of a theory. The measurements involved in the discovery of the Higgs boson were themselves nothing like observations, yet we are inclined to think of them as empirical confirmation for the Standard Model.1 Closer to home, measuring the air pressure using a barometer is an empirical test of the weather forecast, even though it detects changes in pressure we do not notice through observa tion alone. We can make sense of treating measurements as empirical tests, if we assume that even measurements performed using more or less complicated instruments are somehow ultimately based on observations. Foundationalism, developed in detail in Campbell (1920), claims precisely this. Foundationalists begin with quantities like length, which they suggest can be measured directly, that is, without making use of their relationships to other quantities. We can establish the quantitative nature of length, for example, by observing that we can concatenate lengths in a way analogous to the addition of numbers: two rods laid end to end yield a total length that is the combined length of the two rods individually. We can determine various lengths by choosing one rod as our unit and comparing how often this unit fits into the length to be meas ured. At least a certain range of lengths seems to be fairly well covered using this method. 1 For a careful study of the epistemology of experiments at the LHC, compare Mättig and Stöltzner (2018).

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Measurement Realism 39 Not all quantities can be measured in this manner. We do not have concatenation operations for density or temperature, nor is there a way to use an object of a given temperature or density to establish the density or tem perature of a system of interest. Instead we measure density and temperature indirectly, through their nomic connections with volume and mass or pressure respectively. These indirect measures presuppose that the measurement of the other quantities can somehow be established independently, and that suitable laws hold between them and the quantity measured indirectly. As we shall see in sections 4.2.2 and 4.2.3, both presuppositions turn out to be problematic. The distinction between direct and indirect forms of measurement sug gests a corresponding distinction into fundamental and derived quantities. Fundamental quantities are the ones that can be measured directly, that is, independently of their relationship to other quantities. If we hold, as many traditional measurement realists did, that the only form of direct measurement is measurement by concatenation, this yields the result that only additive exten sive quantities,2 like length or mass, are fundamental quantities. The quantitative character of attributes like density or temperature is only established indirectly through their relationship to such extensive quantities, which is why we should treat them as derived quantities.

4.1.2. Measurement Realism Today Measurement realists today are not committed to this naive picture of measurement. They are nonetheless committed to the idea that measurement is empirical and that quantities are special kinds of attributes that exist in dependently of our measurement procedures and conventions. Measurement is not just the assignment of conventionally agreed upon numbers to objects, as the permissivist views in Chapter 3 seemed to suggest. Measurement is instead an empirical procedure that yields information about the properties of the system under measurement. Michell goes even further and holds that it is a matter of empirical investigation whether an attribute is quantitative in the first place, just as it is a matter of empirical investigation to estimate magnitudes, or ratios between them (Michell 2004). Measurement realists are typically motivated by the thought that measurable attributes are special—not just any attribute can be measured—that is they 2 We shall discuss different types of quantities and their structure in Chapter 5.

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40 Quantitative Attributes or Measurable Concepts are restrictive realists in the sense of Chapter 3 (Michell 1999). Moreover, measurement is an achievement, not something we can succeed at by conven tion or stipulation alone. This contrasts with instrumentalism or convention alism about measurement, which suggests that our numerical representations of measurement outcomes are largely based on convention and that we should not interpret them as representations of quantitative attributes. Rather than a single thesis, measurement realism is an assortment of inter related claims. Mari and Giordano (Mari & Giordani 2014) list the following features as characteristic for the measurement realist: (1) ‘[B]oth general and individual quantities exist independently of meas urement’ (Mari & Giordani 2014, p. 2). (2) ‘[I]ndividual quantities are actually related by numerical ratios’ (Mari & Giordani 2014, p. 2). (3) ‘[O]nce an individual quantity is selected as a unit, all other individual quantities of the same general quantity are determined by a number’ (Mari & Giordani 2014, pp. 2–3). (4) ‘[M]easurement is a process aimed at discovering the measurand, where the quantity value states the result of such a discovery’ (Mari & Giordani 2014, p. 3). A slightly different statement of measurement realism is provided by Michell (Michell 2004; Michell 2005). According to Michell, measurement realism consists of three aspects: First, it distinguishes what is measured from how it is measured. Second, it holds that what is measured are attributes of things, rather than things themselves. Third, it claims that in measurement, numbers are discovered rather than assigned (Michell 2004, p. 3).

Michell’s third claim corresponds to Mari and Giordano’s last feature of realism: measurement is a discovery, not an assignment of numbers. The formulation offered by Mari and Giordano is slightly more subtle, because they distinguish measured parameter—the measurand—from the (numerical) quantity value that is used to report the outcome of the measurement. The first claim is similar to (1) in the list above in that it distinguishes between the act of measuring and what is being measured. This element of realism contrasts with operation alist approaches, which hold that quantities are dependent on measurement procedures. I will ultimately argue that this is the most important feature of realism for the purposes of a metaphysics of quantities.

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Measurement Realism 41 In his second clause, Michell specifies what he takes to be the target of measurement: attributes of things rather than things. Once again this is in line with Mari and Giordano, who seem to think of magnitudes (‘individual quantities’) and quantities (‘general quantities’) as attributes. The opponent, for both, is the operationist claim that measurement is ‘the assignment of numerals to objects or events’ (Stevens 1946, p. 677). Numbers are not assigned, because attributes themselves stand in ratio relationships, but since these ratios are ratios of attributes, measurement must target attributes rather than objects. The thought seems to be that the ratio relationship between the mag nitudes is itself in a sense a numerical relationship, and hence numbers are not merely a representational tool. Numbers are ‘implied’ independent of our representation. In order for attributes to be numerically representable, their magnitudes must be able to stand in ratios to one another. As stated, both Michell’s second claim and (1) above make very strong claims concerning the ontological commitments of measurement. Both seem to rule out nominalism in the sense of avoiding commitments to universals. Moreover, realists seem to have a particular view of the role of numbers in measurement: they are indispensably connected with the attributes measured, instead of being mere representational devices. Realists hence seem to reject representationalism about measurement. Measurement realism, then, contains two types of theses: on the one hand an epistemic thesis regarding the process of measurement, and on the other hand several metaphysical theses regarding the target of measurement. The epistemic thesis says that measurement is a discovery of some sort, which is typically understood along the lines of an empiricist epistemology. The meta physical claims are not exactly the same for the two statements of measure ment realism presented above, but in both cases they seem surprisingly strong. They include: (i) realism about universals, in contrast to nominalism, (ii) realism about the target of measurement, in contrast to operationalism, (iii) realism about numbers, in contrast to representationalism. Of these three metaphysical claims, only a version of (ii) is actually presup posed by a metaphysics of quantities. A metaphysics of quantities presupposes that quantities are special kinds of attributes that are quantitative independently of our measurement procedures and conventions. Otherwise a metaphysics of quantities would seem to lack a subject matter and should instead be replaced by an epistemology and semantics of measurement. More specifically, what is at stake is whether quantitativeness is a feature of attributes, which we might discover through the construction of measurement procedures, but which is not brought about by measurement procedures. As we shall see, operational ists deny this, and replace the question of the quantitativeness of particular

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42 Quantitative Attributes or Measurable Concepts attributes with the question of whether certain concepts are measurable, where measurability is a feature a concept only acquires through the construction of a measurement procedure. The question of the independence of quantitative ness from measurement procedures is often intermingled with the question of the independence of the attribute from the measurement procedure. It is vital for the project of providing a metaphysics of quantities that measurability of concepts depends at least in part on whether the attribute to which the con cept purports to refer is quantitative. This aspect of measurement realism is indeed presupposed by the metaphysics of quantities. The first and third claims above, by contrast, are particular positions within the metaphysics of quantities. A nominalist might agree that the targets of measurement are quantitative attributes, but offer a nominalistic account of properties, instead of accepting universals. And a representationalist, as we shall see in Chapters 6 and 7, can accept that quantities are special kinds of attributes, without thinking that this requires or implies the existence of numbers. In the remainder of this chapter, we will look at challenges specific ally to the claim that the quantitativeness of attributes is independent of our measurement procedures and conventions.

4.2. Measurement Realism under Attack 4.2.1. The Line of Attack Since we do not need the full realist package to motivate the project of a meta physics of quantities, we will focus here on attacks on the claim that the quan titativeness of an attribute is independent of procedures devised to measure the attribute. This independence has been called into question by operational ists, conventionalists, and coherentists on the basis of a number of challenges we shall look at in this section. These challenges focus on the claim that we can empirically find out whether an attribute is quantitative, a claim some realists have explicitly endorsed (e.g. Michell 1999). Realists make strong metaphysical commitments to the existence and nature of quantities, while traditionally maintaining a picture of measurement as an empirical process. This combination suggests that if some or all of the metaphysical claims made by realists cannot be justified empirically, measurement realism fails. Recent attacks on measurement realism have tried to show that even basic realist commitments are beyond empiricist justification. Observations alone do not seem sufficient to establish indirect measurement. Indeed, observations

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Measurement Realism under Attack 43 alone might not even suffice to establish direct measurement. Nor can observations guarantee that the value we arrive at on the basis of measurement indeed corresponds to the true value of the magnitude. Finally, ascertaining that an attribute measured by one procedure in one regime is indeed the same attribute as the one measured by a different procedure or in a different regime is by no means trivial and typically goes beyond empirical evidence. In each case the critics of measurement realism allege that empiricist justification turns circular: the claim we are trying to justify is already being presupposed by the method we are using to justify it. A particularly detailed and sophisti cated articulation of these problems has been provided by Hasok Chang (Chang 2004) in the form of a history of the measurement of temperature. In my discussion of these problems below I will frequently use his examples as particularly clear illustrations of the difficulties. While these difficulties reveal, in the first instance, that empiricism is unsuitable as an epistemology of measurement, they have sometimes been taken to have implications for the metaphysics of measurement as well. In particular, both Chang and van Fraassen use these difficulties to undermine realism about quantitative attri butes, as we shall see in section 4.3.

4.2.2. The Problem of Nomic Measurement The most obvious difficulty for empiricist foundationalism is indirect meas urement. For quantities that are measured indirectly, a law needs to be found that relates them to known quantities. Since we cannot compare the temperature of two systems directly, we need to detour through another quantity, volume, to measure temperature. The volume of a suitable substance like alcohol or mercury in an enclosed vessel varies as we place it in or near objects of different temperature. Variations in volume seem to track variations in temperature, which means we can use volume as a proxy measure for temperature: as the temperature of a fluid or object increases, the column of mercury (or alcohol) rises inside the vessel. But how do we know that the rate of increase in volume is proportional to the rate of increase in temperature? Following Hasok Chang (Chang 2004, pp. 89f.) we might call this process ‘nomic measurement’, because it relies on an empirical law connecting tem perature to other, ‘directly’ measurable quantities like volume and pressure. The problem of nomic measurement arises because we need to know the form of the law (e.g. whether it is linear or quadratic) in order to determine the values for the quantity to be measured indirectly; yet in order to discover the

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44 Quantitative Attributes or Measurable Concepts form of the law empirically, we would need to know at least some of the values of the quantity to be measured indirectly. But the values of this quantity are precisely what we were trying to find using the law! More generally, nomic measurement involves finding a functional relation ship between two (or more) quantities, such that X = f(Y). To establish what that relationship is, it seems we would need at least some values for both X and Y, that is to say, we need to measure X and Y independently. In our example, volume (Y) may indeed be measured independently, but since the measurement of temperature (X) depends on the measurement of volume, the values for X in the above equation cannot be measured without relying on the relationship between X and Y. Chang does not mean to cast doubt on the possibility of measuring tem perature using its relationship to pressure and volume. His point is that such measurements require more than just observation. Nomic measurement can be used to show indirectly that an attribute is quantitative through the attribute’s relationship with other quantitative attributes. But doing so does not amount to an empirical demonstration of the quantitativeness of the attribute, because of the justificatory circle described above. The law we need to assume to dem onstrate the quantitativeness of one attribute cannot be tested without assum ing that the attribute in question is indeed quantitative.

4.2.3. Establishing Constancy Following the attack on indirect measurement, foundationalists might be tempted to respond that at least direct measurement is firmly grounded in experience and that the problem of nomic measurement really only applies to a comparatively small number of ‘derived’ quantities. Anti-foundationalists have accordingly tried to show that even in the case of so-called direct meas urement we run into circularities akin to the ones found in the case of indir ect measurement. In order to measure anything at all, including quantities measurable directly, we need to pick something as a fixed standard for com parison. But how do we know that a purported fixed standard actually has constant temperature, length, or pressure? To give a complete answer to this question, it seems we would need to know all the different circumstances under which a proposed measurement standard changes. Not only are length comparisons using rigid rods problem atic when carried out at varying temperatures, we now also know that length cannot be straightforwardly compared between systems moving with respect

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Measurement Realism under Attack 45 to one another. Even seemingly unproblematic candidates for constant length, therefore, are not based purely on observation. This case is superficially similar to the one arising from nomic measure ment, since it once again seems as though we need to have knowledge of at least two quantities to get measurement started. To measure length, a para digmatic ‘fundamental’ quantity, we use the metre, and in (older) definitions of the metre we make reference to temperature: ‘the meter is defined to be the distance between the midpoints of the ends of the metre des archives at the temperature of melting ice’ (van Fraassen 2008, p. 135). The problem is nonetheless different from, and arguably less severe than, the problem of nomic measurement. For in the case of finding a constant standard for length, we can begin with what looks to be a system of constant length, say a rigid rod, and compare other systems to it, and to each other. We would expect that if our candidate standard gives the same result for two of the test objects, these objects themselves should also agree with each other in length. If we find that they diverge, we might suspect that at least one of them is not of constant length, or that our comparison was faulty in some other way. Of course we do not thereby know which of them might have changed length, or where the mistake might be found. Sequential comparisons might help us to narrow down the problem.3 It may also be worth noting, that the most recent attempt to solve the problem of finding constancy involves not material prototypes, but physical constants. Instead of using a particular metal rod as the standard metre, the definition of the metre now invokes the speed of light, c, fixed to a particular numerical value in the units of the International System of Units (BIPM 2019, p. 131): ‘The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit ms−1, where the second is defined in terms of the cae sium frequency ΔνCs.’ Unlike the old definition in terms of a physical prototype, this definition needs to make no reference to temperature. If light is indeed a universal con stant, then the path light travels in a given amount of time is always the same length. There is no longer a need to keep other quantities fixed and temperature is no longer part of the definition of the metre. A similar strategy has now been implemented to replace the definitions of the other SI-base units with definitions involving physical constants. 3 A more sophisticated comparison of this sort has revealed that the standard kilogram is ‘drifting’, for example.

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46 Quantitative Attributes or Measurable Concepts This strategy shows that the search for constancy has two different parts. On the one hand, there is the question of being able to maintain a particular physical object at constant length, which might require keeping other quan tities, such as temperature, stable as well. On the other hand, there is the ques tion what reason we have to think that anything has constant length, mass, or temperature. The former is something we can begin to establish from obser vations, with the assumption that if we do not observe a difference in length, mass, or temperature, this gives us defeasible reason to believe that we in fact have a constant standard.4 The latter, by contrast, cannot be so established. We believe that there is a constant speed on the basis of theory. In general, that there are universal constants and what they are is a matter of theory, and more specifically of the laws that hold between quantities (Tal 2018). There is no purely observational basis for selecting what to take to be constant.

4.2.4. Extending Scales Realists should believe that we can measure the same quantity using a number of different procedures. We assume that placing an object on a beam balance and suspending that object from a spring balance are both measures of mass, even though the procedures are distinct. One reason we have for this is that the two procedures yield the same measurement results. Other reasons are ‘theoretical’, deriving from our theories of statics, springs, and gravity. Only the former reason is a candidate for being a reason based on observation alone. The situation becomes more complicated once we try to establish measure ments of a quantity in different domains or regimes. To measure the diameter of a barrel, a galaxy, or an atom, different methods for measuring length are required. Why should we conclude that local, astronomical, and microscopic length measurements are all measurements of a single attribute, length? As Chang has shown in great detail, establishing measurements for tem perature in very cold or very hot conditions, where mercury thermometers no longer worked, proved tremendously difficult (Chang 2004, ch. 3). The question was how to establish that the scale developed on the basis of one measuring standard—the mercury thermometer—and the scale developed on the basis

4 This idea is akin to Chang’s ‘Principle of Respect’ (Chang 2004, p. 43), which invites us to give some authority to prior standards, such as sensations, even in the light of later corrections on the basis of later standards.

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Measurement Realism under Attack 47 of a different measurement standard—Wedgwood’s pyrometer5—could be combined into a single measurement scale. Wedgwood attempted to connect his scale to the mercury-based Fahrenheit scale by comparing both to an ‘intermediate’ measure: an expanding silver piece. Since the silver piece expansion covered part of the pyrometer scale and part of the mercury scale, it could be used to compare the mercury to the pyrometer scale to allow Wedgwood to say at what temperature in Fahrenheit a particular shrinkage of clay had occurred. This strategy of finding an intermediate is not without problems, however. It requires getting the various fixed points right and having the correct conversion factor of degrees Fahrenheit to ‘degrees Wedgwood’. Most importantly, the entire comparison presupposed that clay pieces contracted linearly with temperature. The latter assumption cannot be tested empirically, for lack of an independent measure of the temperature range in question. Once again, observation alone does not seem to suffice to establish that the two scales can be combined, since the scales are based on observations in different domains. Where there is at least some overlap in the domains of the two measurement procedures, it may of course be possible to extrapolate from the overlap to the rest of the domain, or to use a third-party intermedi ate, as Wedgwood did, although not without encountering difficulties. Additionally, one might seek theoretical reasons for thinking that the scales can be combined. A particular theory of what temperature is, for instance, might help to establish what the temperature scale should look like. Both of these strategies go beyond strict observation. In general, once we recognize that two procedures for measuring the same quantity are different, we are confronted with the question of why we should take them to measure the same quantity and how the resulting scales can be combined (Domotor 1992). As we shall see in 4.3.1, this problem was a major motivation for Bridgman’s operationalism.

4.2.5. How Far to the True Value? Setting aside these fundamental problems with arriving at a measurement procedure to begin with, even once a method for measurement has been established, measurement realism faces problems. Recall that a realist takes

5 Wedgwood’s pyrometer was essentially a standardized piece of clay whose shrinkage at high tem perature was used by Wedgwood to determine temperature at levels of heat far beyond those measur able with a mercury thermometer. For a detailed account see Chang (2004, pp. 118–34).

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48 Quantitative Attributes or Measurable Concepts measurement to be an estimation of the ratio of two magnitudes, or the true value of the magnitude. In calling measurement an ‘estimation’, realists con cede that any measurement process is subject to error and uncertainty. In estimating the value of a magnitude, one typically takes several measurements when enough is at stake, be it science or home improvement. The results of these measurements will typically diverge ever so slightly. Which of the results should be taken to be the true value? At first pass, this does not undermine realism, provided one accepts that any measurement is fallible and might be improved in the future. Indeed, measure ment seems especially suitable for such improvements if its target is the true value of a magnitude, and if successive measurements can be understood as approximations of the true value. Instead of taking one measured value as the true value, one might instead take any convergence of measured values as an indication that these values are closer to the true value than any outliers. This conception of measurement is appealing, but faces two epistemological challenges (Tal 2016). First, we need a reason to believe that convergence of measured values is indicative of closeness to the true value. After all, if there is a systematic error in our measurement instrument or procedure, measured values might converge somewhere other than close to the true value. Second, even granting the first point, it seems that we have no way of knowing how far the value on which measurements converge is from the true value. The latter point is important for metrologists, because it means that there is no way of providing a quantitative estimate of the uncertainty resulting from this difference. Whereas other sources of uncertainty are the result of known defi ciencies in the measuring apparatus, and can often be given a quantitative value, distance from the true value cannot be estimated even in principle, because it would require knowing what the true value is independently. Once again we seem to face a circle in the foundations of measurement that cannot be broken by observation alone.

4.3. Operationalism 4.3.1. Traditional Operationalism The problems presented in section 4.2 show that the quantitativeness of an attribute cannot be established on the basis of observation alone. It is not clear that this means we should give up on the idea that quantitativeness is a feature of attributes independent of our measurement procedures. Operationalists,

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Operationalism 49 however, have done just that. Operationalists retreat from the claim that attributes are quantitative independent of measurement procedures in two respects: instead of asking about the nature of attributes, operationalists fre quently prefer to talk about our concepts, and instead of asking about quanti tativeness they ask about measurability, where the latter is often equated with numerical representability. The problem of measurement, from an oper ationalist point of view, is how to ensure that a particular theoretical concept (a ‘construct’) is measurable, that is, to ensure that we can construct a meas urement procedure for the concept that will permit a numerical representa tion of the concept. At least among philosophers, operationalism is often taken to be an unten able position, an assessment that may indeed be applicable to traditional forms of operationalism. More recently, however, sophisticated forms of operationalism have seen a bit of a resurgence in philosophy of science (Hardcastle 1995; Feest 2005; Chang 2009). It is worth discussing both forms of operationalism in some detail. Traditional operationalism is commonly associated with the name of Percy Bridgman, whose influential 1927 book The Logic of Modern Physics (Bridgman 1958) seemed to fit nicely with the then dominant positivistic outlook on science, only to be condemned together with positivism in the following decades (Gillies 1972; Byerly & Lazara 1974). It is fairly clear that both the adoption and later the condemnation of operationalist ideas contain serious misunderstandings of Bridgman’s own views, although Bridgman himself arguably invites at least some of these readings. The central feature of Bridgman’s position picked up by philosophers was the claim that a quantity (concept) could be defined by a set of operations: ‘In general, we mean by any concept nothing more than a set of operations; the concept is synonymous with the corresponding set of operations’ (Bridgman 1958). This seemingly semantic thesis has sometimes been understood to imply a view of quantities as existentially dependent on measurement procedures; there are no quantities independent of particular measurement procedures, and perhaps indeed no quantities at all (Dingle 1950). Operationalism thereby gives up the key metaphysical commitment of measurement realism: the independent existence of quantitative attributes. Giving up this commitment clearly undermines the project of providing a metaphysics of quantities. There are two main objections to operationalism thus conceived. First, it would seem that the semantic thesis yields a very large number of different quantities, since whenever a new procedure is used to estimate the magnitude of a quantity, we have to assume that we are actually measuring a new quantity.

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50 Quantitative Attributes or Measurable Concepts Length as measured by a metre stick and length as measured by the travel time of a light ray are not the same quantity. To make matters worse, it is unclear how we can ascertain that any two procedures are in fact the same in all relevant respects, since doing so presupposes knowing just which respects are in fact relevant to the measurement procedure in question. Even in the case of two metre stick measurements, we might worry that we’ve failed to keep all relevant aspects fixed. Second if the meaning of a term like ‘length’ is exhausted by a small set of operations, then how can a measurement carried out using these operations ever be found to be inadequate? That is, as we introduce a new, ‘more accu rate’ measurement of the quantity in question, we find that from an oper ationalist point of view, this new measurement operation cannot be regarded as an improved measurement of the original quantity (e.g. metre-stick length), but instead must be regarded as a measurement of a different quantity (lightray length). Both objections correctly identify consequences of the semantic thesis, but both of them might also seem somewhat question-begging. For Bridgman, both the question of how to extend measurement of a quantity beyond its known range, and the question of what happens when we increase the accur acy of our measurements, were central motivations in his development of operationalism (Bridgman 1938). It was precisely because he worried that extending the range of measurement of a quantity, or improving the accuracy of measurement, might lead to conceptual revolutions and upheavals that he wanted to restrict the concepts themselves to their initial operations of meas urement and range of application. Bridgman’s quest for certainty drove him to a kind of semantic defeatism: if we never meant ‘more’ by our concepts than what had in fact been established by operational methods, there was no longer a risk of conceptual revision. Recognizing the essential unpredictability of experiment beyond our pre sent range, the physicist, if he is to escape continually revising his attitude, must use in describing and correlating nature concepts of such a character that our present experience does not exact hostages of the future. (Bridgman 1958)

The defeatist approach implicit in the semantic version of operationalism negates the epistemic risk-taking and fallibility involved in developing scien tific theories. Bridgman’s motivation is clear. His own work on pressure, for which he extended the measurement of pressure far beyond the range

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Operationalism 51 previously achieved, and his experience of the revolution in physics brought on by Einstein’s relativity theory, led to a desire to tie physical theories back to immediate experience, to ensure their validity in the future. The goal was to regain firm foundations for physical theorizing, after Einstein had shown that the old foundations were anything but secure. After decades of philosophical picking apart of Bridgman’s views, it is clear that, as a semantic thesis, operationalism has little to offer. The problem is not just that his views are overly restrictive, avoiding epistemic risk at all cost. In addition, the firm empirical foundations he sought to erect turn out to be less firm and less empirical than Bridgman realized. In many of his examples, Bridgman seems to assume that some set of basic or simple operations are available for a given quantity, and that complex operations are reducible to such simple operations, in principle and perhaps indeed in practice. But even seemingly straightforward measurements of length through alignment of metre sticks can be refined and improved through knowledge about the relationship between temperature and volume. The length of a metre stick changes as the temperature changes, and this may have to be corrected for in measurements. Our ability to carry out such cor rections, and to understand when such corrections might become necessary, depends upon having a theory about the relationship between temperature and volume. This just invites back the problem of nomic measurement: any attempts to measure temperature presuppose measurements of length. What we find here is a circle of operations none of which can be isolated as more basic than the others. So if the goal was to come up with a completely theoryfree set of fundamental operations, Bridgman’s account fails. Bridgman also struggled to define permissible operations precisely. While laboratory operations seem to have held a special status in his account, thereby making it amenable to an empiricist interpretation, he acknowledged the importance of ‘paper and pencil’ operations as well. These are permissible at least for ‘intermediate constructs’ (Bridgman 1938). But if such paper and pencil operations are sufficient to confer meaning on terms, then the connec tion to empiricism turns out to be much weaker than it might have initially appeared. A paper and pencil operation does not seem to rely on empirical input in the way in which a laboratory operation does. Without acknowledge ment of such operations, however, Bridgman’s account once again seems too restrictive.6 Not only do we in fact take epistemic risks and mean more by our 6 This problem bears many similarities to the sophisticated accounts of theoretical terms offered by later logical positivists. For there, too, we find that more tenuous connections to experience needed to

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52 Quantitative Attributes or Measurable Concepts theoretical terms than empiricism permits, it turns out that empiricist restrictions would in fact not help to restore certainty. Operationalists reject some elements of measurement realism, most notably the claim that quantities and magnitudes exist independently of measurement procedures. On the other hand, they retain empiricist foundationalism. We can now see that this won’t do as a response to the problems for measurement realism. Serious epistemological worries remain, while the cost of giving up independent quantities seems too high to warrant operationalist restrictions.

4.3.2. Sophisticated Operationalism Despite the bad press operationalism has received in mainstream philosophy of science, recent work on measurement has succeeded in rehabilitating oper ationalism to some degree. Chang in particular has not only helped us to see Bridgman’s work in a new light, but has also hinted at a view I will be calling ‘sophisticated operationalism’. Sophisticated operationalists differ from trad itional operationalists in two key respects: (i) they are coherentists, not empiricist foundationalists, and (ii) they do not claim that the meaning of a concept is exhausted by the procedure used to measure it, but merely that its operationalization contributes part of its meaning. Let’s look at these two steps more closely. Replacing empiricist foundationalism by coherentism helps to solve the problems discussed in section 4.2. That we regard temperature as quantitative is justified not through a series of empirical tests that conclusively show that the attribute to which ‘temperature’ refers has the requisite structure. The epistemological lesson Chang wishes to draw from his description of the his tory of temperature measurement is that we should give up on empiricist foundationalism as our strategy for justifying measurement, and adopt a form of iterative coherentism instead. We should adopt coherentism, because no observation will break the circles encountered in 4.2. Instead, we support our methods and assumptions through comparisons of several different standards or methods with each other. It is agreement or convergence among different standards or methods that provides justification, not agreement between the standard or method and ‘the world’.

be introduced to make the accounts rich enough, but at the cost of making the criterion of verifiability much weaker and less capable of ruling out dubious concepts.

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Operationalism 53 This coherentism should be thought of as iterative to make sense of progress in science. It seems clear that we are now in a much better position to measure temperature than we used to be, so this progress in our measure ment of temperature needs to be accounted for. Chang’s suggestion is that at various points in the history of measurement, methods were improved by starting from an unjustified assumption, developing a measurement operation on the basis of it, and then correcting the assumption on the basis of the measurements arrived at using that assumption. Similarly, once a standard had come into use and was reasonably successful, this standard itself could be the object of scientific investigation and could be improved on the basis of results achieved using it. The starting point of the investigation is not itself beyond question, nor does the world force a particuar starting point upon us. Justification does not rest on any decisive observation or unquestionable first principle, but instead accrues gradually as different methods and assumptions begin to converge. The justificatory circles in the history of measurement unveiled by Chang might seem to undermine the realists’ insistence that quantities and their magnitudes are independent of measurement procedures or that there is a true value for a magnitude at any given time and place. The iterative nature of the process and our inability to test empirically whether a given attribute is quantitative seem to invite a form of operationalism. Van Fraassen concludes from Chang’s history of temperature measure ment: ‘Somewhat hesitantly one might say that the measured parameter—or at the very least, its concept—is constituted in the course of this historical development’ (van Fraassen 2008, p. 138). Similarly, the title of Chang’s book ‘Inventing temperature’ suggests a form of antirealism. While Chang acknowledges that many of the scientists in his historical narrative were in some sense realists about temperature (Chang 2004, p. 59), he himself cau tions against the idea that our theoretical concept of temperature corresponds to a particular attribute in the world. Any such ‘correspondence’ is mediated by an operationalization of the concept. It’s worth quoting Chang at length on this point: It is very tempting to think that the ultimate basis on which to judge the validity of an operationalization should be whether measurements made on its basis yield values that correspond to the real values. But what are ‘the real values’? Why do we assume that unoperationalized abstract concepts, in themselves, possess any concrete values at all? . . . Once an operationalization is made, the abstract concept possesses values in concrete situations. But we

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54 Quantitative Attributes or Measurable Concepts need to keep in mind that those values are products of the operationalization in question, not independent standards against which we can judge the correctness of the operationalization itself. That is the root of the circularity that we have encountered time and again in the attempt to justify measure ment methods. (Chang 2004, pp. 206–8)

A few pages later he adds: But we must keep firmly in mind that the existence of such ‘real values’ hinges on the success of the iterative procedure, and the successful oper ationalization is constitutive of the ‘reality’. (Chang 2004, p. 217)

In saying that temperature, or our concept of temperature, or its values, are ‘constituted’, van Fraassen and Chang seem to reject the idea that quantities and their magnitudes exist independently. Similarly, when Chang suggests that abstract concepts do not possess ‘concrete values’ independent of an operationalization, it sounds as though he is denying the realists’ commit ment to the idea that measurement is the estimation of the true value of a magnitude. How do Chang and van Fraassen arrive at these apparently anti realist conclusions? Talk of ‘constitution’ is notoriously ambiguous between a semantic claim about the definition of a concept and a metaphysical claim regarding the existential dependence of an attribute. It seems that both van Fraassen and Chang have in mind the semantic claim first and foremost, although there is more than a hint of ontological antirealism as well. What is under attack, in the first instance, is the idea that theoretical terms, like ‘temperature’, correspond ‘directly’ to anything in the world. Both van Fraassen and Chang insist that any correspondence between our theoretical concept temperature and the values we determine through measurement requires operationalization and coordination. Even if we are willing to say that such concepts have meaning independently of operationalization (as Chang, unlike van Fraassen, seems willing to grant (Chang 2004, p. 208 note 53)), the operationalization contributes an important aspect of their meaning, namely the connection to the empirical world. Without specifying standards and measurement procedures, these abstract concepts do not have any meas urable values in world, Chang seems to suggest. One way to interpret this dependence on measurement procedures is that temperature is only quantita tive in the context of a particular operationalization. Without the latter, talk of the ‘values’ of temperature makes no sense. But this is not the only available

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Operationalism 55 interpretation. A more realism friendly reading might suggest that only once we’ve specified standards and measurement procedures can we claim particu lar outcomes as measurements of temperature. Unlike the more operational ist reading, this reading allows for temperature to be quantitative independent of our measurement procedures. Chang and van Fraassen seem to reject this more realism friendly reading, because the operationalization and coordination involved in connecting the abstract concept to any empirical results could be, and indeed have been done in more than one way. If a particular operationalization is part of our concept of temperature, and if it were indeed not possible to ask what the true values of temperature are independently of such an operationalization, then it would seem that there is a sense in which our concept of temperature is constituted by our efforts to operationalize and coordinate, and that the measured values resulting from these efforts are not independent. Operationalism is hence paired with pluralism to undermine realism about quantitative attributes. There is no antecedent referent of ‘temperature’ that could determine, which operationalization of ‘temperature’ is correct. Only through the operationali zations is anything like reference achieved. The referent can hence not serve as basis on which to decide between different possible operationalizations. Chang is somewhat hesitant to draw a fully antirealist conclusion from these considerations, and rightly so. For what the need for coordination and operationalization shows is not so much that quantities or their magnitudes do not exist, or that there is no fact of the matter as to the value of a quantity in a given situation. Instead, it seems to show that a particular picture of the (semantic) relationship between theoretical concept and physical property is mistaken. Theoretical concepts do not simply refer to physical properties; instead we need to operationalize (and thereby coordinate and standardize) to connect theoretical concepts to empirical evidence and to make them testable. It is not entirely clear whether Chang takes this to undermine, in the first instance, the semantics of scientific realism, that is, the view that theoretical concepts of successful theories refer to theory independent properties, or an aspect of measurement realism, namely that attributes are quantitative independent of the procedures by which we measure them. The latter view would be undermined insofar as the realists insist that the quantitativeness of the referred to attribute is a necessary condition for the measurability of a concept. But if a concept does not refer to any attributes independently of an operationalization, then it becomes difficult to see how the adequacy of an operationalization could depend on any features of the referent of the concept

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56 Quantitative Attributes or Measurable Concepts at all. The question of the quantitativeness of attributes is replaced by the question of the measurability of concepts. Two problems seem to get conflated in this discussion. On the one hand, we have the general question of scientific realism, which includes the question of whether and how our theoretical concepts refer to unobservable entities in the world. On the other hand, there is the measurement specific question whether a particular attribute of interest is measurable and whether measur ability is a feature an attribute has in virtue of being quantitative, or just in virtue of us being able to devise a measurement procedure for it.

4.4. Defending Quantitativeness 4.4.1. Measurability vs. Quantitativeness From the discussion so far emerge three competing viewpoints regarding the question of measurability: empiricist operationalists, empiricist realists, and sophisticated operationalists. For empiricist operationalists, the starting point for the question of meas urability is a particular operation, for instance, putting objects on a beam balance or placing a glass tube filled with mercury into various liquids. If the operation satisfies the axioms required for numerical representability, then the operation is a measurement procedure and the concept defined by the operation, for example, ‘hardness’ or ‘temperature’, is found to be measurable. For the empiricist realist, by contrast, the starting point for the question of measurability is an attribute of interest, for instance, mass or temperature. If that attribute has the right sort of structure (e.g. its magnitudes stand in ratio rela tions to one another), it will satisfy (strong) axioms for numerical representabil ity and will hence be measurable. Whether the axioms hold true can be empirically tested (directly or indirectly). If an attribute lacks the appropriate structure, it fails to be quantitative and hence fails to be measurable. Our (theor etical) concepts are quantitative insofar as they refer to quantitative attributes. Sophisticated operationalists take as their starting point a concept, like ‘mass’ or ‘temperature’, which they treat as a theoretical concept. They are operationalists insofar as they take particular measurement procedures to be (partially) constitutive of the concept in question. Unlike either empiricist operationalists or empiricist realists, they do not believe that it is possible to test empirically whether the axioms required for numerical representability hold. Any operationalization presupposes particular laws and standards,

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Defending Quantitativeness 57 which in turn have theoretical commitments that are not shared by the pro ponents of competing theoretical concepts. Rejecting empiricism in favour of coherentism is offered as a solution to this (and other) epistemic circles in the establishment of measurability. Both varieties of operationalism threaten to undermine the metaphysics of quantities, because they suggest that whether an attribute is quantitative reduces to the question of whether or not its concept is measurable. Together with a weak conception of measurement, like the one proposed by Stevens, this yields the view that whether an attribute is quantitative depends only on whether or not we can come up with a suitable operationalization, that is, a rule for assigning numbers to objects. From this perspective, it seems there isn’t anything ‘the world’ has to contribute to the success of measurement. Whether a concept is measurable depends on the availability of a measure ment procedure, not on whether it refers to an attribute with a special kind of structure. Even if we restrict the rules to only those that permit measurement on an interval or ratio scale, the process is still one of starting from a concept and finding a measurement procedure, with no particular role for the world to determine whether such a procedure might apply. Operationalism is appealing because empiricist realism faces severe chal lenges. In particular, the claim that we can establish purely on the basis of observations whether the axioms required for numerical representability hold, must be rejected in light of the problems discussed in section 4.2. Realists are drawn to the claim that axioms can be empirically tested, because this promises a route to establishing the quantitativeness of attributes that seems more objective and less open to misuse by scientists eager to shore up the ‘hard science’ credentials of their discipline. If it can be empirically tested whether a given concept satisfies the (strong) axioms of numerical represent ability, then it is a mistake to represent numerically a concept that fails to sat isfy them and it is scientifically irresponsible not to test whether the axioms hold for a given concept. This explains why Michell is keen to insist that it is possible to test the relevant axioms, even though he rejects the strict empiri cist doctrine of ‘fundamental measurement’ (i.e. the idea that we can always test directly whether the axioms hold). But as we’ve seen above, the problems for a purely empirical test of measurability go far beyond the problem of indirect or nomic measurement. So conceding that tests might be indirect does not really solve the difficulties for empiricism. Perhaps Michell means to include not only indirect measurements, but any measurements based (partly) on theory. This would amount to accepting a much broader notion of empir ical testing than is typically assumed by critics of measurement realism, but

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58 Quantitative Attributes or Measurable Concepts perhaps it is defensible in light of the fact that most theory testing is quite far removed from immediate observation and involves substantial reliance on theory. Nonetheless, realists have a point, too. The problem with operationalism is that whether a concept is measurable seems to depend entirely on whether we are able to devise a procedure that satisfies the conditions for numerical assign ment, and for permissivists, these conditions tend to be quite weak. Suppose, by contrast, that we restrict the type of numerical assignment we are looking for to particularly informative numerical representations, however, it is nowhere near as easy to find measurement procedures. In section 4.4.2 we shall look at an example of an attribute, hardness, which has resisted stronger forms of measurement. This example suggests that perhaps realists are not mistaken in thinking that attributes play a role in making measurement possible.

4.4.2. Attributes Matter: Hardness Among the features of the pebble introduced in 2.1 was its hardness. Hardness is something with which we are sensorily acquainted: we can feel the differ ence between hard and soft. It is somewhat more difficult to characterize hardness in the abstract. Hardness typically refers to a material’s ability to resist localized, plastic deformations due to scratching or indentation. The difficulty of finding a clear ‘definition’ is by no means unique to hardness, but is indeed characteristic for qualitative attributes. We similarly struggle to pro vide definitions of hot and cold, even though we know perfectly well what hot or cold objects feel like. For the difference between hot and cold, we’ve been able to develop an abstract concept, temperature, that we can measure. Can we measure hardness as we measure temperature? In both cases, we need to make theoretical assumptions about the nature of the attribute to decide which phenomena to include in the operationaliza tions. Mohs decided that hardness, at least as far as minerals were concerned, was in the first instance a matter of resistance to scratching, as opposed to resistance to indentation. This choice suggests a certain kind of measurement procedure for the (relative) hardness of minerals: does mineral A scratch mineral B, or vice versa? If A scratches B, A is harder, if B scratches A, B is harder. If A and B scratch each other, they are of equal hardness. Does this procedure satisfy the axioms for numerical representability? In order to make sure that a procedure satisfies at least some of the axioms of numerical representability, very similar steps need to be taken in the

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Defending Quantitativeness 59 context of very different sciences: (i) establishing constancy, (ii) establishing congruence/equality of intervals, (iii) establishing a zero point or finding a concatenation operation. Each of these steps adds strength to the numerical representation. In setting up his scratch test scale for hardness, for example, Mohs needed to assume that each mineral has a characteristic hardness that remains stable under a wide range of conditions. By selecting minerals of different hard nesses as reference minerals for particular steps on the scale, Mohs estab lished hardness numbers for minerals, ranging from 1–10, with 10 being the hardest (reference mineral: diamond). Some minerals, notably kyanite and garnet, vary in hardness, so the assumption that there is a single hardness for each mineral is empirically violated. This corresponds to using the freezing and boiling points of water as fixed points for the temperature scale, which Celsius assumed to be of constant temperature, despite empirical evidence to the contrary in the form of superheating and supercooling (Chang 2004, ch. 1). For both hardness and temperature, then, constancy can be established, but doing so goes beyond immediate observation. The Mohs scale satisfies axioms for ordering, which are relatively weak axioms for numerical representation. When it comes to establishing equal intervals and meaningful zero points, on the other hand, temperature and hardness diverge. For temperature, the congruence of intervals between different ‘degrees Celsius’ on a scale from boiling to freezing can be established using the relationship between pressure, temperature, and volume. Chang rightly points out that establishing this rela tionship and using this relationship to establish the congruence of intervals between degrees Celsius can only be mutually coherent, but cannot be in dependently tested. That such equal degrees can be established, on the other hand, makes the Celsius scale a stronger numerical representation of tem perature than the Mohs scale is for hardness. Suppose three minerals A, B, and C are such that A and B scratch C, and B also scratches A. From this we can conclude that B is the hardest mineral, C the softest, and A in between. Mohs’s scratch test does not tell us whether the difference in hardness between A and B is the same as that between C and A.7 Why was it that in the case of temperature, the Celsius scale could be strengthened to a scale with equal intervals, but the Mohs scale could not be so strengthened? 7 Other tests for hardness, notably the diamond indentation method, suggest that the intervals on the Mohs scale are in fact unequal. From a realist perspective, indentation methods for testing hard ness can, in principle, count as measures of the same attribute, hardness, as Mohs’s scratch test. The ordering of minerals by hardness is the same on the scratch and on the indentation methods.

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60 Quantitative Attributes or Measurable Concepts In the first instance, there is a relatively simple law relating temperature to other, fairly easily measurable quantities like volume and pressure. This law was exploited for nomic measurement, which established the equality of intervals between degrees on single scales. Moreover, because various differ ent temperature scales, such as Fahrenheit and Celsius, relied on the same law, these scales could be compared to each other, despite the fact that the ‘size’ of the difference between degrees on the respective scales was not the same. Both the Celsius and Fahrenheit scales are interval scales, whereas the Mohs scale is an ordinal scale. Notice that in moving from fixed points to equal intervals, the law assumed in the case of temperature is an ‘empirical law’, that is, the terms involved are not ‘abstract’ in Chang’s sense. The difference between hardness and tempera ture appears already at the level of empirical phenomena, independent of the introduction of a more abstract concept of temperature. It is only for the third step, the question of a meaningful zero point, that we seem forced to intro duce a more abstract concept of temperature, on Chang’s reconstruction of the history of temperature measurement. Since the Mohs scale cannot be turned into an interval scale, one might choose to look for resistance to indentation, as opposed to scratching, to develop a (quantitative) measure of hardness. There are several different indentation scales for hardness, like the Vickers, Brinell, and Rockwell scales.8 On these scales, hardness is probed by indenting a mineral surface with steel or diamond tips of particular shapes under a known load. The area (or depth) of the indentation is then measured and indentations produced by this method on different mineral surfaces can be compared. Because area is a quantity, comparing areas is a quantitative comparison, which has sometimes been taken to show that indentation tests provide quantitative measures for hardness, in contrast to the Mohs test, which simply asked, whether a mineral was scratched by another. But unlike in the case of temperature, there is no empirical law relating hardness to area of indentation, and as a result, even different indentation scales cannot be easily converted into one another. Temperature, unlike hardness, can be measured on a quantitative scale, and conversions between different temperature scales are relatively straightfor ward. As Chang’s history of temperature measurement shows, this is not because we can simply observe that temperature is quantitative. It took a lot of ingenuity, careful experimentation, and theory to invent and improve 8 For detailed descriptions of each procedure as well as tables comparing their results, see Chandler (2004).

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Defending Quantitativeness 61 measurements for temperature. But mineralogists and metallurgists were no less inventive and resourceful when it came to measuring hardness. The problem, a realist might say, is that hardness is not quantitative, whereas temperature is.9 Operationalists would be inclined to interpret the problem with hardness differently. To them, there are simply different measurement procedures, and we haven’t figured out how to combine them into a single scale of measure ment. But this doesn’t answer the question why it was possible to do so in the case of temperature, but not in the case of hardness. Realism, by contrast, readily explains this difference. The problem seems to be that the cause of variation in heat and cold turned out to be a single quantitative attribute— temperature—whereas the cause of the variation in hardness and softness of materials turns out not to be a single quantitative attribute, but instead is the (complexly) combined effect of different attributes of materials.10 In both cases it made perfect sense to attempt to find measurement procedures, but because the attributes in question are different, it worked out well in one case, and less so in the other case. Whether we succeed at finding strong numerical representations of attributes depends not just on our ingenuity, but also on the nature of the attribute. Weak numerical representations can be found for almost any attribute (including hardness), but strong forms of measurement do depend on whether the attribute has the right sort of structure (we shall see in Chapter 6 which structure is required). The key to defending the claim that quantities are independent of measure ment procedures is to distinguish quantitativeness from measurability. An attribute can be quantitative without its concept being measurable; conversely we may believe a concept to be measurable without there being a quantitative attribute to which it refers. The realist idea is that for measurement to be suc cessful, two things have to come together: we need to be able to construct a measurement procedure that yields empirical data that can be interpreted as satisfying the axioms for numerical representability and we need to have a concept whose (purported) referent has the right sort of structure. Without the former, we simply cannot find out whether an attribute has the right sort of structure, but without the latter our efforts at representing the attribute

9 The idea that successful measurement requires a ‘worldly’ element in addition to experimental and conceptual ingenuity has also been defended, in somewhat different contexts, by Smith (2001) and Isaac (2019). 10 Wikipedia lists: ductility, elastic stiffness, plasticity, strain, strength, toughness, viscoelasticity, and viscosity (Wikipedia contributors).

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62 Quantitative Attributes or Measurable Concepts numerically are doomed to fail. Our best way of finding out whether an attribute is indeed quantitative is to attempt to construct a measurement pro cedure for its concept. The trouble is that all such efforts may be in vain, if there is no quantitative attribute that could serve as the referent for our concept. Epistemically, however, there is no shortcut to testing the quantitativeness of an attribute: we have to attempt to construct measurement procedures. What about Chang’s worry regarding the referents of our unoperational ized theoretical concepts? Here we need to distinguish two problems. The first is whether some sensory or otherwise observable variation, for instance, vari ation in hot and cold or hard and soft, is due to a single attribute that has the structure of a quantity. The second is whether a particular theoretical concept of temperature (or mass, or hardness) can be empirically tested using the measurement procedures developed in the context of solving the first prob lem. Measurement realists are concerned primarily with the first problem, since they wish to restrict numerical representation to strong measurement. Any variation can be (weakly) numerically represented, but only if the vari ation is due to a quantitative attribute is such a representation truly a meas urement. The question whether a particular theoretical concept is empirically significant and has a (quantitative) referent, on the other hand, is a question for scientific realism more broadly (see section 4.5). Realists are right, then, that the world has a part to play in making meas urement possible: in order for a procedure to amount to genuine measure ment, the targeted attribute must be quantitative. We may of course be mistaken about whether we have in fact constructed a successful measure ment procedure in any given case. It is possible to mistake a quantitative attribute for a non-quantitative one or to mistake a non-quantitative attribute for a quantitative one. But only on a realist view is it possible to make such mistakes. From an operationalist point of view, the question of measurability and the question of quantitativeness of the attribute cannot come apart. For the realist, they can.11 Realists are wrong to think, though, that quantitativeness can be tested for prior to, or independently of, constructing a measurement procedure. Whether we can measure a quantitative attribute depends on our ability to develop a suitable theoretical concept and to devise suitable measurement procedures for the concept in question. Neither whether a procedure is ‘suit able’ for the concept, nor whether it can be said to satisfy the axioms of numerical representability, are questions that can be answered purely 11 I would like to thank an anonymous reader for helping me to clarify this point.

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Measurement Realism as Scientific Realism 63 empirically, as coherentists rightly emphasize. Both depend on theoretical assumptions. But the manner in which theory is involved in the two cases is not exactly the same. The conditions for numerical representability are math ematical conditions that hold across different sciences. Which axioms need to be satisfied for a measurement procedure (or the attribute measured by the procedure) to be numerically representable is independent of the particular scientific theory under discussion. Whether a particular operationalization is appropriate for a given theoretical concept purporting to refer to the quanti tative attribute of interest, by contrast, is a question that can only be answered in the context of a scientific theory concerning that attribute. The metaphysics of quantities is concerned not with the question of the ontological commitments of a particular theoretical concept, like tempera ture, but with the ontological commitments required for quantitativeness. Provided that successful operationalization requires quantitative structure ‘in the world’, this means that the metaphysics of quantities is relevant even if our theoretical concepts must be operationalized. What matters from the per spective of the metaphysics of quantities is not whether any particular (theor etical) quantity exists, but only that quantitativeness is a feature of attributes, not merely of concepts or measurement procedures.

4.5. Measurement Realism as Scientific Realism A realist about measurement should take quantities to be the target of meas urement. That involves treating quantities as metaphysically and semantically independent of measurement practices; quantities are ‘out there’ and meas urement is our strategy for finding out about them. Nonetheless, a responsible realist needs to acknowledge the epistemic challenges discussed. What this means is that a realist about measurement needs to accept that measurement is not simply sophisticated observation, but instead requires a good deal of theory (Morrison 2009; Tal 2011; Tal 2016). Theory dependence, unlike dependence on particular conventions or dependence on particular measurement pro cedures, is not devastating to basic realist commitments. Conventions are something entirely dependent on us in the sense that we can choose to adhere to them or not. The world, as it were, has no say. Measurement procedures, while not up to us in quite the same way as conventions, nonetheless seem more accidental than a realist can allow. Without a good criterion for when two measurement procedures count as the same operation, measurement practice ends up too fragmented and subjective to permit intersubjectivity

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64 Quantitative Attributes or Measurable Concepts and progress. Any sufficiently robust criterion for sameness of measurement procedure, however, involves theoretical commitments not unlike those that come with realism about quantities. By contrast, if quantities depend epistemically on the theory used to meas ure them, realism is not similarly undermined. Instead, how realist we can be with respect to quantities will depend on the stance we take towards the the ories in which they occur. If we are realists about the theory in question, there is no particular reason to treat quantities differently from any other commit ments of the theory. Quantities do not pose particular problems for the scien tific realist in virtue of being theory dependent in the sense that we come to know them not through observation alone. Realism about quantities, then, is simply an aspect of scientific realism in general (pace Isaac 2019). Quantities are unobservable entities in two different senses. On the one hand, positing the existence of an attribute like mass or temperature to explain variation in lightness/heaviness, cold/warmth, is to make a theoretical posit not unlike the positing of viruses to explain symptoms of disease or the positing of positrons to explain characteristic cloud chamber tracks. To claim that there is a single attribute whose variation explains our sensations of both heat and cold goes beyond those sensations themselves. As in the case of other theoretical terms, no amount of empirical evidence can prove conclusively that tempera ture exists and indeed explains the variation in warmth we experience. On the other hand, quantities are unobservable entities in a more peculiar sense: they are quantitative, which realists interpret as saying that they are attributes with a certain structure. Since this structure is not directly observ able, and since it cannot be tested directly whether a given attribute satisfies the axioms required for quantitativeness, positing quantitative structure itself is a theoretical commitment. Moreover, the quantitative structure of a (pos ited) attribute is itself explanatory. It is the variation in temperature that explains the variation in our sensation of hot and cold, and it is the quantita tive character of temperature that makes it possible for temperature to vary in a certain way. Yet quantitativeness is more than mere variation, as the contrast between hardness and temperature already indicated, and as we shall see in more formal detail in Chapter 6. For a concept to be operationalizable, it has to refer to an attribute that is in fact quantitative. As with other theoretical terms, this reference is deeply theory-laden and may be imperfect. For example, a particular concept of tem perature, or mass, may be rejected by subsequent theories. In this regard quantities are no different from other theoretical concepts and we are dealing with the familiar phenomenon of reference failure. But even if we reject the

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Measurement Realism as Scientific Realism 65 caloric theory of heat, we still agree with the calorics’ assessment of temperature as a quantitative attribute. In this regard, quantities are somewhat different from other theoretical concepts. Just as in the case of other unobservable entities, merely being a realist does not settle all metaphysical disputes about the entities in question. This is espe cially true in the case of physical quantities, since we are interested in what it means for an attribute to be quantitative and the ontological commitments that come along with that claim. No physical theory addresses this question. The measurement realists I quoted at the beginning of this chapter each offered at least a rough answer: quantities are ratios, they require the existence of universals, and their numerical values are in some sense ‘out there’ inde pendent of our attempts to measure them. We shall see in the rest of the book whether we should really accept these particular commitments or whether it is possible to be a realist in the sense of taking quantitative attributes to be independent of measurement without committing to the particular claims bundled together under the heading of measurement realism.

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5 The Representational Theory of Measurement Synopsis 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. In section 5.4.2 I discuss 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.

5.1. Preliminaries 5.1.1. Origins of Representationalism The topic of this chapter is measurement theory, and in particular representationalism in measurement theory. Today representationalism is tightly bound up with measurement theory, largely due to the influential work of David Krantz, Duncan Luce, Patrick Suppes, and Amos Tversky and their three volume Foundations of Measurement (Krantz et al. 1971–90).1 The representational

1 Subsequent citations of the work follow the page numbers of individual volumes of the 2007 Dover Edition: Krantz, D. H., Suppes, P., Luce, R. D., & Tversky, A., 2007, Foundations of Measurement Volume 1: Additive and Polynomial Representations, Dover, Mineola, NY; Suppes, P., Krantz, D. H., Luce, R. D., & Tversky, A., 2007, Foundations of Measurement Volume 2: Geometrical, Threshold, and Probabilistic Representations, Dover, Mineola, NY; Luce, R. D., Krantz, D. H., Suppes, P., & Tversky, A.,

The Metaphysics of Quantities. J. E. Wolff, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198837084.001.0001

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Preliminaries 67 theory of measurement will be the main focus of our discussion, but before turning to the formal details it helps to isolate three aspects of measurement theory that provided key influences for the contemporary representationalist framework.2 The first concerns framing the question of measurement. For representationalists, the problem of measurement is to show how it is possible to use numbers to represent empirical phenomena. Framing the question this way suggests that the target of measurement is not itself a numerical or mathematical attribute; instead numbers are merely used as a representational device to permit certain inferences about the empirical phenomenon. This contrasts sharply with more traditional views of quantities, according to which quantities are special kinds of attributes that stand in relations of proportion to one another.3 Whether an attribute is measurable, on the traditional view, is a question of whether it admits of relations of proportion. While numbers can be used to represent such relationships, the possibility of representing attributes with numbers is treated as secondary. By contrast, for the representationalist the question of numerical representation is the primary question about measurement. The second important aspect is the idea of axiomatization, which is closely tied to the idea of representation. Starting in the nineteenth century, (proto-) representationalists used the strategy of axiomatization to show how numbers could be used to represent particular phenomena. Instead of characterizing quantitative attributes in terms of relations of proportion, axiomatic approaches specify a list of axioms an attribute must satisfy to be quantitative. Many of these traditional axiomatizations focused on providing axioms for extensive quantities. Extensive quantities were thought to be quantitative in virtue of satisfying axioms analogous to axioms for addition over the (real) numbers. As we shall see, unlike traditional axiomatizations, the representational theory of measurement does not restrict itself to extensive quantities, but considers axioms for a wide range of possible measurement structures. This openness towards measurement beyond additivity reflects the third influence on the representational theory of measurement. In a now famous paper, ‘On the theory of scales of measurement’ (1946), psychologist S. S. Stevens proposed that if measurement was indeed the 2007, Foundations of Measurement Volume 3: Representation, Axiomatization, and Invariance, Dover, Mineola, NY. 2 For a detailed history of the development of the representationalist point of view see Díez (1997a) and Díez (1997b). 3 Michell (2004) provides a contemporary account along these lines.

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68 The Representational Theory of Measurement assignment of numbers to objects according to a rule, then there was no reason to restrict oneself to rules that made use of addition. Instead, he thought, measurement scales could be constructed for many different attributes, only some of which would rely on the availability of concatenation relations. Stevens observed that these scales would be of different strength, with concatenation operations yielding the strongest representations in the form of ratio scales. But other scales, such as interval or ordinal scales, were also legitimate measurement scales for Stevens. Recall from Chapter 3 that Stevens’s hierarchy seemed to make measurement too easy to achieve: if measurement was simply the assignment of numbers to objects according to any rule whatsoever, then anything from weight measurement to ordering a group of people by their shoe size or postcode would count as measurement. At the same time, researchers in physics, and even more so in newer sciences like psychology and economics, were confronted with attributes that seemed measurable, but for which no concatenation relation could be specified. Even if Stevens’s hierarchy was not fully satisfactory, he had put his finger on a significant problem. The representationalist theory of measurement as we now know it was developed to resolve this tension between an overly restrictive and an overly permissive view of measurement. The key to resolving the problem, as I shall show below, was to adopt a more abstract, structuralist approach to the representation relation between empirical phenomena and numerical representations. Instead of assigning numbers to objects, both the empirical phenomenon and the numbers are treated as structures, related by homomorphisms, that is, structure-preserving mappings. This move enables the representationalist theory to abstract away from the constraints of the concatenation relation, but, as we shall see, it also imposes clearer formal constraints on the hierarchy of scales proposed by Stevens. The representationalist theory of measurement therefore marks the culmination of a research programme that started in the nineteenth century, and provides a flexible framework for measurement that permits the application of measurement techniques to a wide range of scientific fields. If we want to understand what quantities are, the representationalist framework provides a natural starting point. Nonetheless representationalism is not without critics: it is perceived as insufficiently realist, and to some it still seems too permissive in its account of measurement. I take up these criticisms in Chapter 6. We will find that an appropriate metaphysical account of quantities requires abandoning some of the features the representationalist theory of measurement inherited from its precursors.

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Preliminaries 69

5.1.2. The Basic Principles of Representationalism According to the representational theory of measurement (RTM), measurement is ‘the construction of homomorphisms (scales) from empirical relational structures of interest into numerical relational structures that are useful’ (Krantz et al. 2007, p. 9). This point of view marks a significant departure both from theories of measurement that characterized measurement as the assignment of numbers to objects, and from earlier theories that characterized measurement as the estimate of ratios. RTM, by contrast, focuses on structure preserving mappings (homomorphisms) between relational structures. Schematically, the basic idea is as follows: an empirical structure, typically understood as consisting of empirical objects and empirically testable relations and operations, is mapped into a numerical structure, typically the real numbers under suitable relations (e.g. ordering and addition) (Figure 5.1). Three steps are needed to construct such a mapping, one ‘conceptual’, the other two mathematical (Luce et al. 2007, p. 201). The mathematical steps involve proving a representation and a uniqueness theorem, which are at the heart of the representationalist approach. Before such theorems can be proved, however, the first step is the description of the ‘empirical relational structure of interest’. This description is to be given in the form of axioms characterizing the relations and operations among elements of the structure. There are different types of such structures; for our purposes the most important types of structures will be extensive structures, difference structures, and additive conjoint measurement structures. While Krantz et al. often provide particular empirical examples of such structures, these structures should be distinguished from any particular empirical realization or interpretation. Extensive structures, for example, can be realized by a set of weights, an ordering relation among weights (e.g. from heaviest to least heavy), and a ɸ

〈A, ≳, o〉

Figure 5.1. Representation theorem schematic

〈R, ≥, +〉

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70 The Representational Theory of Measurement concatenation operation such that the combination of two weights results in the ‘sum’ of the weights for the combined objects. A standard operationalization for ordering and concatenation can be provided by a beam balance: we order weights through pairwise comparisons (if one pan is lower than the other, the weight in the lower pan is heavier), and we produce concatenations by placing two objects in the same pan. The particular realizations of an extensive structure should not be confused with extensive structures in general, which are abstract. The identity of a measurement structure is determined by the axioms that govern the relations and operations among the elements, not by the particular empirical operations we might use to implement or test such a structure. A (closed) extensive structure, for instance, is described by the following axioms: Let A be a non-empty set, ≿ a binary relation on A, and ∘ a closed binary operation on A. The triple ⟨A, ≿, ∘⟩ is a closed extensive structure iff the following four axioms are satisfied for all a, b, c, d, ∈ A:

1. Weak order: ⟨A, ≿⟩ is a weak order, that is, ≿ is a transitive and connected relation. 2. Weak associativity: a∘(b∘c)∼(a∘b)∘c. 3. Monotonicity: a≿b iff a∘c≿b∘c iff c∘a≿c∘b. 4. Archimedean: If a≻b, then for any c,d ∈ A, there exists a positive integer n such that na∘c≿nb∘d, where na is defined inductively as: 1a=a, (n+1)a=na∘a. (Krantz et al. 2007, p. 73) These axioms can be applied to empirical as well as to numerical structures, with different interpretations given for the elements of the structure and the binary relation and operation. The relation ≿ can be understood as ‘succeeds or is equivalent to’, whereas the ring operator ∘ is often interpreted as concatenation on the empirical structure and as addition (or multiplication) on the numerical structure. The weak order axiom says that for any two elements in the structure, either a≿b or b≿a (connectedness), and if a≿b and b≿c, then a≿c (transitivity). The binary operation ∘ is characterized by axioms 2 and 3. For numerical structures, addition as well as multiplication satisfy the two axioms: (i) adding a to the sum of b and c is equivalent to adding the sum of a and b to c and (ii), if a is greater than or equal to b, then adding a and c will be greater than or equal to adding c and b, regardless of the order in which the operation is carried out. For empirical structures, the operation in question is typically a kind of concatenation operation, that is, some manner of combining objects that satisfies weak associativity and monotonicity.

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Preliminaries 71 The Archimedean axiom ensures that the ratio of any two elements is finite. The requirement that the structure be Archimedean is less intuitive than the conditions imposed by the other axioms, but will turn out to be quite important as we move to more abstract aspects of measurement theory. Unlike the other axioms, the Archimedean axiom explicitly makes an existence claim. Another existence claim is hidden in the description of ∘ as a closed binary operation. This means that for any a,b ∈A, a∘b also ∈A. While these axioms are mathematically very convenient and unproblematic when applied to numerical structures, they are ontologically rich, which is potentially problematic when it comes to characterizing non-numerical structures. The numerical structure in question will often be a subset of the real numbers with the standard ordering and a suitable binary operation (addition and multiplication are common, but not the only choices). It is important to realize that sets of actual observations will typically not satisfy these axioms, since the most interesting axiomatically described measurement structures are not finite, whereas all actual collections of data are. As Krantz et al. initially conceive of it, the empirical structure will typically be an abstraction or idealization from some actual data (Krantz et al. 2007, p. 13; Suppes 1969). This echoes Carnap’s empiricist operationalist conception of the testability of axioms. Some of the illustrative examples of measurement structures suggest that we should think of the ‘empirical relational structure’ as a collection of (material) objects among which certain relations (e.g. ordering, concatenation) hold, regardless of whether we’ve carried out an experiment and collected data. Whether the latter way of understanding measurement structures is sufficient for a metaphysics of quantities will be addressed in Chapter 7. Instead of describing the measurement structure as an empirical structure, representationalists also often speak of the ‘qualitative relational structure’, in contrast to the numerical relational structure. I will use both in what follows, with the caveat that both are mere placeholders until we have a better sense of what’s required for a metaphysics of quantities. How do we know whether a particular phenomenon or data set satisfies the axioms for an extensive structure? From a representationalist perspective, this is not a problem for the theory of measurement per se, but a problem of theorizing and experimental design in the different scientific fields.4 If we have empirical and/or theoretical reasons to believe that a particular phenomenon satisfies the axioms set out above, we have reasons to believe that it can be described as an extensive measurement structure. 4 This is clear from the discussion in Luce et al. (2007, pp. 323–5).

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72 The Representational Theory of Measurement

5.2. Additive Extensive Structures 5.2.1. Representation and Uniqueness Theorems for Additive Extensive Structures The main objective of representationalism is to determine which numerical representations are possible for a given empirical relational structure. Two mathematical steps are necessary to accomplish this: first, a representation theorem to demonstrate that a mapping from the empirical relational structure to a specified numerical structure is possible; second a uniqueness theorem to show how unique the mapping is. The Foundations of Measurement is to a large extent a collection of proofs of representation and uniqueness theorems for a wide variety of qualitative and numerical structures. The function of the representation theorem is fairly obvious: it establishes that a particular qualitative empirical structure can be represented by a particular numerical structure. This is usually done by constructing a homomorphism from the empirical structure to the numerical structure.5 Since a homomorphism is a structure preserving map, this procedure effectively shows that the set with qualitative relations and the numbers share a common structure. A representation of the ‘qualitative relational structure’ by means of a numerical structure is possible, because the qualitative relational structure shares relevant structure with numbers, typically the real numbers under some operation. Which structure is relevant? RTM on its own does not say. What it shows instead is that different structures (e.g. extensive, difference, or conjoint) can be represented by numerical structures, and how unique this representation is. For this last point we need to turn to the second step in the representationalist agenda, the uniqueness theorem. If one numerical representation of a qualitative structure is possible, typically many such representations are possible. There are two ways in which a structure can have multiple numerical representations.6 First, for some phenomena we are able to find more than one way in which the phenomenon satisfies the axioms for extensive, difference, or conjoint structures. This case will be discussed in more detail in Chapter 9. The second, more familiar case concerns the fact that once a mapping to a numerical structure has been

5 There are some instances of non-constructive proofs in measurement theory. For discussion see Baccelli (in press). 6 See (Luce et al. 2007, p. 329ff.) for a comprehensive discussion of this point. The problem resurfaces throughout the Foundations of Measurement.

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Additive Extensive Structures 73 established, transformations of this mapping are possible that retain the structure thus mapped. A familiar example is changes of units for quantities like mass or length. Starting from a set of weights we might use one of them as the ‘unit’ weight and assign it the number ‘1’, which will fix the numerical assignments for the other weights. But we could have chosen a different particular weight as our unit weight, and had we done so, the numerical assignments would once again have been fixed, but this assignment would have differed from the first. How would it have been different? That’s what the uniqueness theorem tells us. The uniqueness theorem is needed to show which of these different numerical representations are equivalent to one another. Whereas the representation theorem establishes the existence of a homomorphism between two structures, the uniqueness theorem shows which alternative mappings lead to an equivalent representation of the qualitative structure in the same numerical structure. The way to find these alternative mappings is to consider transformations of the original homomorphism, such that the homomorphically p reserved structure is still represented by the numerical structure in question. It turns out that different qualitative structures have different characteristic levels of uniqueness when they are mapped to the real numbers. To make this more concrete, let’s look back at our example of the closed extensive structure from above.7 The representation theorem establishes that closed extensive structures can be represented in the real numbers (Re) by a function φ satisfying the following two conditions (Krantz et al. 2007, p. 74):

(i) a b iff ϕ (a) ≥ϕ (b) (ii) ϕ (ab) = ϕ (a) + ϕ (b)

The first clause establishes that the ordering relation ≿ is mapped into the real numbers in such a way that the order among the elements of the empirical structure is reflected in the order of the real numbers assigned to them by the homomorphism. The second clause establishes that the concatenated object a∘b is mapped to the sum of the numbers representing a and b respectively. We might say that the homomorphism maps the ordering relation among the elements of the empirical structure to the ordering among the real numbers, and the concatenation operation among the elements of the qualitative 7 Statements and proofs of the following representation and uniqueness theorems for extensive structures are provided in Krantz et al. (2007, ch. 3).

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74 The Representational Theory of Measurement structure to the addition operation among the real numbers. We can see why such a homomorphism produces a suitable representation of the empirical structure in question: if the qualitative structure is appropriately described as an extensive structure, then the axioms for the binary relation ≿ and the binary operation ∘ among the elements of the empirical structure fit the axioms for weak order and addition among the real numbers. Since their axioms are the same, the relations and operations characteristic of the two structures form the same type of algebraic structure. The uniqueness theorem for this representation states: ‘Another function φ´ satisfies (i) and (ii) iff there exists α > 0 such that φ´ = αφ’ (Krantz et al. 2007, p. 74). The homomorphism by which an extensive structure is mapped to the real numbers is therefore unique up to multiplication by a positive constant, a transformation that is typically interpreted as a ‘change of unit’. The resulting representation is often called a ratio scale representation. Instead of speaking of the homomorphism we should instead perhaps speak of the family of homomorphisms related by this transformation.

5.2.2. Hölder’s Theorem The basis for the representation and uniqueness theorems for additive extensive structures was laid by Otto Hölder in his 1901 axiomatization of quantities (Hölder 1901).8 Hölder formulates axioms for quantities to show under what conditions a ratio representation of quantities is possible. These axioms are somewhat different from the ones used by Krantz et al. above, but Hölder’s results are nonetheless of ongoing importance to theories of measurement, including representational theories.9 Hölder showed how ratio representations are possible on the basis of additive extensive structures very similar to the ones discussed above, but using somewhat different axioms: I. Given any two magnitudes, a and b, one and only one of the following is true: a is identical to b (a=b, b=a), a is greater than b and b is less

8 ‘Hölder’s theorem’ refers to the following result only in the context of measurement theory. In general mathematics, a different result is known as ‘Hölder’s theorem’, namely the theorem that the gamma function does not satisfy any algebraic differential equations whose coefficients are rational functions, which Hölder first proved in 1887. 9 In addition to the result concerning extensive structures, he also showed how ratio representations are possible on the basis of difference structures (compare section 5.3.1).

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Additive Extensive Structures 75 than a (a>b, ba, a0. (Krantz et al. 2007, p. 53)

The theorem and definition given above correspond to Hölder’s 1901 results, but are formulated in terms of more abstract axioms. Instead of Hölder’s axioms for magnitudes, we find the characterization of an abstract structure: an Archimedean simply ordered group. A group is a set of elements with a binary operation such that the set is closed under the binary operation, the operation is associative, there is an identity element, and there is an inverse element. The operation specified for such a group is not limited to operations presumed to hold among numbers. However, unlike most of the definitions and theorems used by Krantz et al., this theorem states that there is an isomorphic mapping between the non-numerical structure and the numerical structure, not just a homomorphic mapping. This will become important for the results used in Chapter 6, which rely on this abstract isomorphic version of Hölder’s theorem to arrive at very general results concerning the features of structures that can be represented on ratio scales. Before we can turn to these exciting developments in more recent measurement theory, however, we need to have a look at the representation and uniqueness theorems for other important structures and see how this plays out in terms of the hierarchy of scales introduced by Stevens.

5.3. Other Important Types of Measurement Structures 5.3.1. Difference Structures Additive extensive measurement structures were well known and had already been successfully axiomatized before the representational theory of measurement. Since additivity was initially presumed to be a necessary condition

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Other Important Types of Measurement Structures 77 for quantitativeness, most axiomatizations of quantities focused exclusively on additive extensive structures. Attributes that could be (empirically) shown to satisfy the axioms for additivity were thought to be quantitative, in contrast to attributes that failed to satisfy them. Even these traditional accounts had to acknowledge that some physical attributes, for example density, lacked an empirical concatenation operation, despite seeming to be representable on ratio scales. A common move was to assume that physical attributes lacking an empirical concatenation relation were secondary or derivative compared to the primary, extensive attributes.10 The representational theory of measurement, by contrast, treats additive extensive structures as one type of measurement structure among many. For our purposes, two of these other measurement structures are particularly relevant: absolute difference structures and additive conjoint structures. Many of the attributes that could not easily be described in terms of extensive structures turn out to be easily described using difference or conjoint structures. Just as in the case of extensive structures, RTM proceeds by providing representation and uniqueness theorems for difference and conjoint measurement structures. While difference structures and their representations had been previously known, conjoint measurement was a key development in the RTM literature (Luce & Tukey 1964; Krantz 1964). Let’s first have a look at difference structures. The basic idea behind difference structures is that instead of comparing objects (or their magnitudes) directly, we compare differences between objects (or their magnitudes). Contour lines on maps, which indicate elevation changes in the local topography, are a familiar example. Points on each contour line have the same elevation, and the contour interval—the difference in elevation between successive contour lines—is equal across the entire map. Notice that contour lines by themselves do not provide information about the elevation of a point compared to normal sea level. Instead of an ordering relation and a concatenation operation, difference structures are characterized by relations of congruence (e.g. points on a contour line have the same elevation) and betweenness (e.g. points on counter line l, placed between contour lines l´ and l´´ have an elevation in between the points of l´ and l´´). For contour maps, it matters in which direction we cross the contour lines: in one direction, elevation increases, in the 10 Helmholtz (Helmholtz 2010) and Hölder (Hölder 1901) both provided axiomatizations of quantities in terms of additive operations. The aim in both cases was to provide axiomatizations of quantitativeness, not to show that or how physical attributes satisfied these requirements. Tolman (1917) and Campbell (1920), by contrast, provided influential treatments of physical attributes that seemed to make empirical concatenation operations a necessary requirement of primary physical quantities.

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78 The Representational Theory of Measurement other it decreases. Other difference structures are indifferent with regard to direction of interval. Consider an absolute difference structure11 for a single dimension, which is defined as follows: Suppose A is a set with at least two elements and ≳ is a binary relation on A × A. The pair ⟨A × A, ≳⟩ is an absolute-difference structure iff, for a, b, c, d, a´, b´, c´ ∈ A, and all sequences a1, a2, . . . ai, . . . of elements of A, the following six axioms hold:

1. ⟨A × A, ≳⟩ is a weak order. 2. If a ≠ b, then ab ∼ ba ≻ aa ∼ bb. 3. (i) If b ≠ c, ac ≿ ab, bc, and bd ≿ bc, cd, then ad ≿ ac, bd. (ii) If ac ≿ ab, bc and ad ≿ ac, cd, then ad ≿ bd. 4. Suppose that ac ≿ ab, bc. If ab ≿ a´b´ and bc ≿ b´c´, then ac ≿ a´c´; moreover if either ab ≻ a´b´, or bc ≻ b´c´, then ac ≻ a´c´. 5. If ab ≿ cd, then there exists d´ ∈ A such that ab ≿ d´b and ad´ ∼ cd. 6. If a1, a2, . . . ai, . . . is a strictly bounded standard sequence (that is, there exist d´, d˝ ∈ A, such that for all i = 1,2, . . . , d´d˝ ≻ ai+1a1 ≿ aia1 and ai+1ai∼a2a1 ≿ a1a1), then the sequence is finite. (Krantz et al. 2007, p. 172)

The relations defined by the axioms for the structure are betweenness and congruence. Axiom (1) is very similar to the first axiom for extensive structures, except that here the ordering applies to pairs of elements, not single elements. Axiom (2) says that if element a is different from element b, then ab and ba are equivalent and greater than the difference between a and itself or b and itself respectively, which are in turn equivalent. This holds for absolute difference structures, which make no distinction between going from a to b or going from b to a along a single dimension. Axiom (3) is basically about betweenness: b is between a and c, if ac ≿ ab, bc. So what axiom (3) ensures is (i) that if b is between a and c, and c is between b and d, then b and c are both between a and d; and (ii) that if b is between a and c, and c between a and d, then b is between a and d. Axiom (4) ensures that if b is between a and c, any a´, b´, c´ such that the stretches between a´b´ and b´c´ respectively are less than or equal to ab and bc respectively, fall in between a and c. Axioms (5) and (6) are once more existence claims, with (6) corresponding to the Archimedean 11 Absolute difference structures are worth presenting in detail, since they are used by Hartry Field in his famous nominalistic treatment of quantities (Field 1980). However, outside of philosophy, nonabsolute difference structures are often more attractive to use, precisely because they pay attention to differences ‘in sign’. Thanks to Jean Baccelli for pointing this out to me.

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Other Important Types of Measurement Structures 79 axiom for extensive structures. Axiom (5) ensures that for any two equal intervals ab≿cd, a point d’ between a and b can be found, such that it divides ab into two parts, at least one of which is congruent to cd. A physical realization of an absolute difference structure can be provided by a measuring rod with unlabelled interval markings (the markings do not have to be numerical, nor do the intervals have to be equal). The endpoints of intervals form the elements of the set A; the ordering relation obtains among intervals along the rod, regardless of direction (since we are only interested in the length of the interval in absolute difference structures, we treat going from a to b as equivalent to going from b to a). The relations of equivalence in length and betweenness on the rod are then shown to satisfy the axioms for absolute difference structures set out above. The representation theorem for absolute difference structures is as follows: ‘If ⟨A × A, ≿⟩ is an absolute-difference structure, then there exists a function φ: A → Re such that for all a, b, c, d ∈ A, ab≿cd iff │φ(a)−φ(b)│≥│φ(c)− φ(d)│’ (Krantz et al. 2007, p. 173). Unlike the representation theorem for additive extensive structures, this homomorphism does not focus on ordering relations and concatenation operations, but instead says that the interval ab is greater than the interval cd if and only if the distance on the real line between φ(a) and φ(b) is greater than the distance between φ(c) and φ(d). The uniqueness theorem for such homomorphisms says: If φ´ is any other function with the same property, then φ´ = αφ + β, where α, β are real, α ≠ 0. It is worth stressing that this uniqueness result will only hold true for absolute difference structures and that it requires a rich domain. A common interpretation of this uniqueness property is to say that the representation permits both changes of units and moveable zero points.12 The family of homomorphisms used to map absolute difference structures to the real numbers is different from the homomorphisms that map extensive structures to the real numbers, as the representation theorem shows. The uniqueness theorem further shows that the resulting representation is less unique than the one resulting from mappings of extensive structures: there are two types of transformations under which the structure is invariant, which means that fewer features of the representation are preserved across different mappings. Difference structures are typically not represented on ratio scales, but on interval scales in Stevens’s classification of scale types. This means they are weaker measurement structures than extensive additive structures, but still stronger than mere orderings. 12 As we shall see in section 5.4, this informal characterization is not quite right.

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80 The Representational Theory of Measurement

5.3.2. Additive Conjoint Measurement The idea behind conjoint measurement is that the quantitativeness of several attributes is assessed jointly, instead of separately. Luce and Tukey (1964) suggest that, since concatenation operations cannot be found for all attributes of interest, we might instead look for cases where two (or more) attributes can be found whose combined effects result in a structure not entirely unlike the structure suitable for describing additive structures. Where in the case of concatenating masses we had a case of combining two objects with (magnitudes of) the same attribute, and compared the joint effect to that of the effect of each separately, in simultaneous conjoint measurement we compare the joint ‘effect’ of two attributes, each of which is assumed to contribute to the ‘effect’ independently of the other. Examples of conjoint measurement from physics include kinetic energy in relation to mass and velocity, and momentum in relation to gravitational potential and mass. The latter example is used by Luce and Tukey in their original 1964 paper. They describe a set-up—a kind of double armed ballistic pendulum—in which the combined effects of mass and gravitational potential for different objects can be qualitatively compared by looking at the direction of the momentum produced in the pendulum bob. The set-up is as follows: pebbles of a range of masses are dropped simultaneously from a range of heights; for each pair of pebbles the height from which they are dropped and their ‘identity’ are recorded. The observation to be recorded is the direction of the first swing of the bob of the pendulum. In this way we will arrive at an ordering of pairs of pebbles and heights according to which combination of pebble mass (m) and height (h) produces a greater momentum in the bob, for example: h1m2≿h2m1. For such a conjoint measurement to be an additive conjoint measurement, the two components have to be independent of one another. A necessary condition for this is the following: if one component is shared between two pairs, the ordering remains unaffected by a switch to another shared component. For example, if we have found that: h1m1≿h2m1, that is, we’ve dropped the same pebble from two different heights and the momentum was greater when dropped from height h1, then h1mi≿h2mi will hold for any other pebble mi. This feature is sometimes called independence. In addition to independence, an additive conjoint structure must also satisfy a condition called double cancellation. Double cancellation can be illustrated as follows. If we have two pairs of pebbles and heights, such that h1m2≿h2m4 and h2m5≿h3m2, then h1m5≿h3m4. The idea is that by comparing

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Other Important Types of Measurement Structures 81 the different pairs, we can cancel out h2 and m2. A slightly weaker condition, the Thomsen condition, which requires that if h1m2∼h2m4 and h2m5∼h3m2, then h1m5∼h3m4 can be illustrated using indifference curves: If h1m2 and h2m4 are on an indifference curve, and so are h2m5 and h3m2, then h1m5 and h3m4 are on an indifference curve in between. The final important feature of conjoint measurement is that for some measurement structures, such as the one described above, we can furthermore note that for any combination of pebble and height, we can find a combination of a different pebble dropped from a different height such that the effect of the second combination is equivalent to the effect of the first combination (solvability). Note that this axiom is an existence axiom, which makes strict operational interpretations of these axioms implausible. The requirement is that there is a combination of pebble and height to match our first combination, not that we’ve actually carried out a measurement involving such a pair. Solvability is needed for a strong uniqueness theorem for conjoint structures. An additive conjoint structure is a structure ⟨A1, A2, ≿⟩ such that ≿ is a weak ordering that satisfies independence, double cancellation, solvability and that is Archimedean and in which the components both contribute to the effect. For such a structure, the following representation and uniqueness theorems hold: Suppose ⟨A1, A2, ≿⟩ is an additive conjoint structure. Then there exist functions φi from Ai, i=1, 2 into the real numbers such that, for all a,b ∈A1 and p,q ∈A2, ap≿bq iff φ1(a)+ φ2(p) ≥ φ1(b)+φ2(q). If φ´i are two other functions with the same property, then there exist constants α>0 and β1, and β2 such that φ1´=αφ1+β1 and φ2´= αφ2´+β2. (Krantz et al. 2007, p. 257)

The remarkable feature of additive conjoint measurement is that from a mere ordering of pairs of components, a kind of additive representation can be generated that is unique at interval scale level. In our example this is unsurprising, for we think that heights, masses, and momenta have an extensive structure, so we might expect to be able to arrange them in such a way as to generate a combined extensive structure. But the important point about conjoint measurement is that this representation theorem can be established independent of knowing that either component is extensive, provided the above conditions are satisfied. What this shows is that conjoint measurement offers yet another means for arriving at an interval scale representation. A further important point to note is that the interval scales have consistent

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82 The Representational Theory of Measurement units: the ‘change in unit’ for φ1 and φ2 is not independent, since α is the same in the uniqueness theorem above. Conjoint measurement provides an alternative test13 for quantitativeness that is independent of finding a concatenation operation for a single attribute. If a structure satisfies the axioms of a conjoint measurement structure, then a numerical representation of at least interval type is possible for each component attribute. In some cases, a ratio scale representation is possible. Luce and Tukey observe: ‘If either (a) we have a distinguished pair of elements F0 in A and X0 in P for which it is reasonable to require φ(F0)=0=ψ(X0), or (b) we find it reasonable to exponentiate φ(F) and ψ(X), using Φ(F)=eφ(F) and Ψ(X)=eψ(X) as measures of F and X, we will have measured the elements of Α and P on ratio scales’ (Luce & Tukey 1964, p. 10). For the examples from physics, it is especially the second condition that will be used to move from interval scales to ratio scales. Note that the conditions under which such a move will be ‘reasonable’ remain unspecified. One reason they might seem reasonable in the physics examples is of course that we already believe that the relevant quantities can be measured on ratio scales, based on concatenation, and that there are laws (in classical physics), relating the quantities in question. It is a much more difficult question how we determine the reasonableness of such moves in the absence of pre-existing laws. Conjoint measurement provides a great expansion of possible measurement structures that can lead to interval and ratio type representations. While the physical examples of conjoint measurement seem somewhat contrived, since we in many cases antecedently believe that the attributes have extensive structure, conjoint measurement is quite important conceptually in psychophysics, where it is not possible to establish extensive structure independently. More importantly, conjoint measurement connects quantitativeness to the possibility of formulating law-like relations among attributes. For in the examples from physics, we notice that the attributes related in conjoint measurement are often related by known multiplicative relationships, for example kinetic energy EK=1/2mv2. Historically we didn’t come by these relationships through conjoint measurement, so using them in conjoint measurement examples shouldn’t suggest the best route to discovering these laws or the quantitativeness of the involved attributes. Instead, characterizing them as involving a conjoint measurement structure provides a clue to the formal structure required for quantitativeness and laws, and thereby provides a first 13 As we’ve already seen in Chapter 4, tests for axioms for any measurement structure cannot be regarded as straightforwardly empirical tests.

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Other Important Types of Measurement Structures 83 stab at systematizing the features an attribute needs to have to be quantitative. To see this more clearly, we need to turn, in Chapter 6, to the formal features of the different scale types, which reveal a close affinity between interval and ratio type scales.

5.3.3. Types of Quantities and Types of Measurement Structures As we observed earlier, additive extensive structures are not plausibly ascribed to all quantitative attributes. In particular, only ‘extensive quantities’, but not ‘intensive quantities’ seem to be characterizable as additive extensive structures. The former are attributes like length and mass that are clearly ‘additive’, whereas the latter are attributes like temperature and density, for which we do not have an obvious concatenation operation to demonstrate their additivity.14 Intensive quantities have traditionally posed a challenge for restrictive realists, since they seem to be ‘in between’ paradigmatic quantities like length and mass, and mere orders, like hardness or movie rankings. Because they are not additive in the sense of having an empirical concatenation operation, they undermine the claim that only additive attributes are quantities and have ratio scale representations. Instead, the requirement of additivity looks too strong even for some physical quantities, thereby opening the door to claim quantitativeness for attributes in other fields where an empirical concatenation operation seemed harder to come by. From the perspective of RTM, an empirical concatenation operation is no longer the defining feature of additive extensive structures, which are instead characterized by a specific set of axioms. While these axioms include an axiom for a binary, associative operation, this operation does not have to be addition, nor does it have to be an empirical concatenation operation. RTM nonetheless offers new insight into when strong numerical representations of non-extensive attributes are possible. First, RTM shows how both difference structures and conjoint measurement structures can give rise to interval or log-interval scale representations, and in some cases even to ratio scale representations. Second, RTM ultimately provides formal resources to characterize 14 The distinction between extensive and intensive quantities has been drawn in different ways at different times and in different disciplines. Here I’m interested in the distinction between attributes with an explicit concatenation operation and attributes that lack such an operation. Using the term ‘extensive’ to refer to attributes with an empirical concatenation operation is common in contemporary measurement theory (Michell 2004, p. 54), but does not reflect all past uses of the term.

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84 The Representational Theory of Measurement what is special about structures that can be represented on interval or ratio scales, as we shall see in Chapter 6, and indeed does better than traditional restrictivists. The plurality of measurement structures poses a challenge for an approach that takes the represented entity to be an attribute rather than a specific measurement procedure. When extensive measurement structures were first introduced in the context of the representational theory of measurement, concatenation operations and ordering relations were illustrated using examples of particular measurement procedures. Two weights would have been concatenated if they had been placed in the same pan of a beam balance, and two rods would have been concatenated if they had been placed end to end in a straight line. But if we take the represented entity to be an attribute, the interpretation of the axiomatically defined relations becomes unclear. It is one thing to say that a particular measurement structure (approximately) satisfies the axioms for betweenness and congruence, or concatenation and ordering, and quite another to claim that the attributes measured by the described procedure have a structure involving these relations. Extensive, difference, and conjoint measurement are different ways of coming to know about a quantity and its relational structure, but we should not mistake a distinction in our ways of measuring quantities for a distinction of types of quantities (Michell 1999, pp. 54ff.). This is all the more important as quantitative attributes can be measured using different measurement set-ups, which are in turn best described as different measurement structures. We usually measure mass extensively, but as we’ve seen above, we could construct a conjoint measurement set-up involving mass. Similarly, procedures for measuring length can be extensive or difference structures. The distinctions between extensive, difference, and conjoint structures are, in the first instance, distinctions between different ways of setting up measurements of quantitative attributes, not differences between attributes. As a result, we should not classify quantities as ‘intensive’ or ‘extensive’ based either on the procedures most commonly used to measure them, or on the type of scale used to represent them.

5.4. Uniqueness and the Hierarchy of Scales 5.4.1. Permissible Transformations and the Hierarchy of Scales Different measurement structures can be shown to have numerical representations via representation theorems. As the uniqueness theorems for the

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Uniqueness and the Hierarchy of Scales 85 respective representations show, the resulting representations are not equally unique. The uniqueness of a representation is expressed by comparing the relationship of different ‘permissible’ homomorphic mappings to each other: if φ is a homomorphism from the qualitative structure into the numerical structure, then what homomorphisms φ´, φ˝ . . . are also homomorphic maps of the qualitative structure into the same numerical structure? For the structures we’ve looked at, the relationships between the homomorphisms can be expressed as φ´ = αφ, for α>0 and φ´ = αφ+β for α, β are real, α ≠ 0, respectively. The first family of homomorphisms is more restrictive than the second family of homomorphism. The different families of homomorphisms (roughly) correspond to the different scale types introduced by Stevens. Ratio scales are scales whose family of homomorphisms is unique up to φ´ = αφ, whereas interval scales are scales whose family of homomorphisms is unique up to an affine transformation φ´ = αφ+β. The contribution of the representational theory of measurement is not limited to this observation, however. Unlike Stevens, Krantz et al. aim to provide a formal treatment of the different scale types, and to explain how the different scales arise (Narens 1985; Luce et al. 2007, pp. 111–12). To this end, Krantz et al. move from a description of the particular transformations to descriptions of the groups formed by the transformations characteristic for a particular scale type under function composition operations. We can ask whether the transformations for a particular scale type themselves form a group, that is, whether: (i) two homomorphisms compose to form another homomorphism; (ii) whether this composition is associative; (iii) whether for each homomorphism there is an inverse, and (iv) whether there is a homomorphism that leaves everything unchanged. The transformations of different scale types form groups with different numbers of free parameters, where a free parameter indicates, at first pass, how much choice there is in a mapping. The fewer free parameters a transformation group has, the stronger the associated scale. The group of the relevant transformations is often called the scale group for a particular type of scale. This feature of the transformations turns out to be extremely important. Instead of comparing the particular transformations, we can compare the groups characteristic of the different scale types in terms of their free parameters. It turns out that the scale groups of interval and ratio scales differ in that the former has two free parameters and the latter has only one. That should not come as a surprise: in the characterizations of the permissible homomorphic mappings above, there were two elements we could adjust in the case of interval scale representations and only one in the case of ratio scale

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86 The Representational Theory of Measurement representations. The more abstract viewpoint taken here helps us to abstract away from the particular changes one might make to the mapping, which turn out not to define the scale type. The essential fact about the uniqueness of a representation is not the particular group of admissible transformations, but that all groups are isomorphic and, in the case of extensive measurement, are all one-parameter groups; that is, there is exactly one degree of freedom in any particular representation. (Krantz et al. 2007, p. 102)

What is necessary for determining scale type is not a particular type of transformation, but instead that the transformations themselves form a more abstract structure. In the case of ratio scales this abstract structure is the structure of a one-parameter group. Different scale types differ with respect to how many free parameters the transformations groups have, with stronger scales having groups of transformations of fewer degrees of freedom. Notice that on this account, neither the fact that paradigmatic ‘extensive’ quantities like mass have a concatenation relation, nor the fact that one standard way in which different numerical representations of such quantities are related to one another is through multiplication by a positive constant, is in any way decisive for the classification of these structures as representable on ratio scales. The resulting hierarchy of scales is an expansion of Stevens’s original classification (Luce et al. 2007, p. 113) (Table 5.1). The expansion of this classification of Stevens’s hierarchy of scales distinguishes among different numerical domains to which the qualitative structure might be mapped; in particular the reals and the positive reals are distinguished. This matters, because the respective domains have different amounts of ‘built-in structure’: the positive reals do not have a structurally singled out element, whereas the reals do, because they include 0. Whereas Stevens’s classification assumed that it was intuitively obvious how the scale types differed from one another and perhaps also, which scale type would be appropriate in a given case, the augmented table reveals that the relationship between the different scale types is more complex. For example, ratio scales on Re+ turn out not to be a special case of interval scales on R, with s=0. Instead they are a special case of the log-interval scale on Re+, with r=1. This fact turns out to be important in Chapter 8, because r=1 means that the transformation in question is a translation. A translation, in this context, is mapping with no fixed points, in contrast to a dilation, which is a mapping that maps some points

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Uniqueness and the Hierarchy of Scales 87 Table 5.1. Expanded hierarchy of scales

Re+

Re

function type

f: Re→Re f: Re+→Re+ strictly increasing f: x→rx+s, translation if r=1; r>0 dilation if r≠1 or if the identity log-interval f: txr, r,t >0 difference f: x→rx+s f: x→tx, t>0

group

parameters

ordinal interval

homeomorphism countable positive affine 2

ratio absolute

power translation similarity similarity identity

f: x→rx f: x→x

f: x→x

2 1 1 0

onto themselves. For now we simply note that in the context of the representational theory of measurement, we arrive at an expanded and more abstractly grounded account of the different scale types. What is the significance of the different scale types thus classified?

5.4.2. Uniqueness and Conventionality Since any qualitative structure except for absolute ones can be given more than one numerical representation, any one such representation contains conventional elements. We can think of the uniqueness theorem as a means by which to distinguish the conventional from the non-conventional aspects of a numerical representation. In the context of measurement theory, the question of how to distinguish conventional or arbitrary features of representations from non-conventional features is often called the question of ‘meaningfulness’ (e.g. Narens 1981a; Mundy 1986; Luce et al. 2007; Narens 2001). Intuitively, a meaningful relation or feature of a representation is one that corresponds to a relation or feature of the represented phenomenon, whereas a non-meaningful relation or feature is one that is an artefact of the particular representation. The question of meaningfulness is closely tied to the hierarchy of scales. Which inferences from a given numerical representation are appropriate depends on the scale type of the representation. For example, to conclude that a baby who weighs 6 kg has doubled her birthweight of 3 kg is a legitimate inference, whereas to conclude that it is half as warm today, at 3°C compared to 6°C yesterday, is not. The reason is that temperature in Celsius, unlike weight, is measured on an interval scale, not a ratio scale, and only for ratio

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88 The Representational Theory of Measurement scales are numerical ratios meaningful. Crucially, without knowing the scale type we can’t tell which inferences are meaningful, because the numbers by themselves would seem to permit the inference in both cases. Scale types are hence important for ensuring that we limit ourselves to appropriate inferences. In light of the more formal account of scale types provided by the representational theory of measurement, we can say a bit more about the concept of meaningfulness involved. The answer given by the representational theory is that being preserved across representations linked by a particular family of homomorphisms is a necessary condition for being ‘meaningful’ (Luce et al. 2007, ch. 22, esp. 22.9 develops this idea in formal detail). By contrast, features or relations that vary as the representations belonging to a single family of homomorphisms vary are to be regarded as pertaining only to particular representations of the phenomenon, not to the phenomenon represented. Since the representations are achieved by homomorphic mappings, and since the families of homomorphisms are found by considering the permissible transformations, the features of the representations regarded as non-conventional are those that remain invariant under the permissible transformations of the homomorphisms. Invariance under permissible transformations is hence a necessary condition for being meaningful. If fewer transformations are permissible, more relations are preserved across representations of the same family. This is why it makes sense to treat representations with few permissible transformations as more informative: more features of the representation may be treated as features of the represented phenomenon. Intuitively, representations that are invariant under fewer transformations are more informative than representations that permit a wider range of transformations. The key question is, of course, which transformations are permissible. In providing uniqueness theorems for different types of structures, the representational theory of measurement shows which transformations are permissible for the structure in question. This move relies crucially on the structural characterization of the phenomenon to be measured. In characterizing the phenomenon as a certain type of structure, we know which relations and operations must be preserved. The uniqueness theorem shows which features a homomorphic mapping must have in order to preserve the structure in question. The representationalist theory of measurement thus provides a systematic approach to the question of how and why we can use numbers to represent empirical phenomena. In first describing phenomena as ‘qualitative’ or ‘empirical’ relational structures, and then demonstrating the availability of

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Uniqueness and the Hierarchy of Scales 89 homomorphic mappings from these structures to suitable numerical structures, representationalism answers the initial question of how we can use numbers to represent attributes. In demonstrating how unique a particular assignment is, representationalism tells us how informative a given numerical representation is. Moreover, it turns out that the use of uniqueness theorems leads to a systematic account of the different scale types introduced by Stevens. Representationalism prima facie offers two ways of classifying relational structures. On the one hand, relational structures can be classified by looking at the different axioms characterizing the relations and operations for the structure in question. This yields distinctions between extensive, difference, conjoint, and other structures. The distinction picks out different structures in terms of the relations characteristic of them. On the other hand, relational structures can be classified by the strength of the representations that are appropriate for them. This yields a hierarchical classification in terms of scale types, with ratio, log-interval, and interval scales as the most relevant scale types for our purposes. While the different structures of the first classification are often associated with characteristic scale types appropriate for representing them, it is not the case that the two classifications coincide in all cases. Ratio scale representations are often associated with (additive) extensive structures, but difference structures and conjoint structures can, under certain circumstances, also give rise to ratio scale representations. This last point is one of the key insights of the representational theory of measurement, which helped to broaden the notion of measurement beyond the narrow focus on additive extensive structure, which had been central to theories of measurement in much of the earlier literature. It also raises an interesting problem for the question of quantitativeness. If quantitativeness is restricted to certain types of numerical representation, yet different types of structures can give rise to numerical representations of the same level of uniqueness, then how are we to characterize quantitative structures? Traditional restrictive realists demanded that the attributes’ magnitudes must stand in ratio relations, whereas restrictive empiricists had demanded that instantiations of them must be concatenable, yet neither of these seems required in the case of conjoint measurement. As we shall see in 6.2, the representational theory of measurement answers this question by providing a more abstract characterization of the relevant structures.

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6 Representationalism as a Basis for Metaphysics Synopsis 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 oper ationalism and empiricism. I argue in favour of treating RTM itself as a mathematical framework open to different interpretations and propose a more realist understanding of RTM, which treats the mapping between repre sented and representing structure as an isomorphism rather than a mere homo morphism. 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, I argue that, on the contrary, RTM provides the means to draw such a distinction at an intui tively plausible place: only attributes representable on ‘super-ratio scales’ are quantitative (6.2).

6.1. Representationalism and its Critics 6.1.1. The Representational Theory of Measurement as a Mathematical Framework The representational theory of measurement (RTM) is the most thoroughly developed theory of measurement, but it is neither the only such theory, nor is it without problems. In the first part of this chapter, I will address two important concerns about the representational theory: (a) it offers an incom plete account of measurement and (b) it is insufficiently realist for a meta physics of quantities. My response to both problems is to treat RTM as a mathematical framework rather than a complete theory of measurement The Metaphysics of Quantities. J. E. Wolff, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198837084.001.0001

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Representationalism and its Critics 91 (also compare Heilmann 2015; Vessonen in press). Like other mathematical frameworks found in science, formal measurement theory is open to inter pretation. Interpretation plays a role in the application of measurement the ory to actual physical phenomena and, more pertinent to the project at hand, in ter pret ation is involved in extracting metaphysical consequences from mathematical formalisms. Critics of the representational theory of measurement have sometimes sug gested the theory fails to offer a full account of measurement as practiced in the sciences (Savage & Ehrlich 1992). In particular, RTM has little to say about metrological concerns, error in measurement, and in general on the question of how to ensure that a given empirical phenomenon indeed has the requisite structure to be represented numerically. Perhaps because of RTM’s silence on this last question, some empirical scientists have found the repre sentational theory unsatisfying as a theory of measurement. After all, this first step often proves extremely difficult. Before we can prove or apply any repre sentation or uniqueness theorems, we need to find out whether the attribute of interest satisfies the axioms for extensive (or difference, or conjoint) struc tures. But that is no easy task. So how can we have a theory of measurement without some prescription as to how to accomplish the first step? The operationalist outlook implicit in much of the Foundations of Measurement suggests that we could somehow observe or empirically test, whether the axioms of, say, difference measurement, apply to a particular phe nomenon. As we’ve seen in Chapter 4, there are many reasons for being scep tical about this possibility. Representationalists concede that any actual data set is finite, whereas many of the structures under consideration require infin ite sets. But even granting some amount of idealization in the characterization of the relevant empirical structure does not solve some of the deeper problems with empirical tests of abstract axioms. To apply the axioms to any empirical phenomenon at all, we have to characterize the phenomenon structurally, that is, we have to choose relations and operations as particularly relevant, as well as finding ways to ‘operationalize’ them (van Fraassen 2008). This step, I sug gest, can be understood as a form of interpretation: it can be understood as interpreting the phenomenon of interest as a measurement structure of a cer tain kind, or conversely it can be understood as offering the phenomenon as an empirical interpretation of an abstract measurement structure. RTM does not immediately contribute to this task; instead it merely provides a wide range of possible structures, with some illustrative examples from different sciences. The main reason we do not find adequate accounts of this aspect of the epistemology of measurement in representationalist writings is that this step

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92 Representationalism as a Basis for Metaphysics is not considered part of the foundations of measurement, but part of the different empirical sciences in which measurement takes place. It seems to me that RTM gets the division of labour exactly right in this case. How to charac terize a phenomenon or whether a particular attribute satisfies a set of meas urement axioms are questions that require both empirical and theoretical input from the relevant sciences. A theory of measurement cannot declare a priori, what sorts of reasons might be appropriate for concluding that a particu lar set of axioms is satisfied by a given phenomenon or attribute. What meas urement theory can say, a priori, is which sorts of numerical representations are possible, if a particular type of structure is given. The theory of measure ment, understood in this way, is a particular branch of mathematics or perhaps better, of the ‘logic of science’, but it is not itself an empirical science. For this reason it does not seem illegitimate to divorce the mathematical framework provided by representationalism from the particular philosoph ical motivations of its defenders (Heilmann 2015). Doing so does not mean giving up on philosophical uses of representationalism, however. The math ematical framework, once divorced from the particular empiricist concerns of its founders, is open to all kinds of metaphysical interpretation. This situation is familiar from the philosophy of physics, where we also find partially empir ically interpreted mathematical frameworks, of which we can then ask what their metaphysical or ontological implications are. This view of measurement theory contrasts with a more naturalistic per spective, which suggests that measurement theory should not only provide mathematical tools for describing the representation relation, but also provide a substantive account of when numerical representations are appropriate.1 The representational theory of measurement, as it stands, provides only a con ditional account of appropriateness: given a particular type of structure, RTM says, which numerical representations are possible and how unique they are.

6.1.2. Is Representationalism Sufficiently Realist? The deeply engrained empiricism in classic treatments of representationalism is not just a problem for the application of abstract measurement theory to con crete situations. It is also a potential problem for the metaphysics of quantities.

1 Michell (1999), for example, aims to provide a more substantive, restrictive account of measurement.

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Representationalism and its Critics 93 This comes out clearly when we compare the key commitments of restrictive realism to the philosophical approach implicit in representationalism: i. Where realists see numbers as ‘implied’ by ratios between magnitudes, representationalists think of numbers as being merely representational tools. This contrast is clear in the representationalists distinction between the qualitative structure and the numerical structure, where the latter is used to represent the former. ii. Where realists see a sharp division between quantitative and qualitative attributes, representationalists seem to follow Stevens’s permissivism. iii. Realists are committed to the idea that quantities are attributes and often suggest that these attributes should be understood as universals. Representationalists, on the other hand, lean towards operationalism, and present their formalism as a first-order theory, which suggests a nominalistic ontology. I will address (ii) in section 6.2 of this chapter. Here I will say a little more about (iii) first, with obvious echoes of our discussion of operationalism in Chapter 4. There are several problems with treating qualitative structures as empirical structures in the context of measurement theory. According to rep resentationalism, such qualitative empirical structures are homomorphically embedded in suitable numerical structures. These numerical structures, as Domotor (1992) has argued, are in many respects richer than the empirical structures that are embedded in them. So we need to ask not only whether the empirical structure can be represented using the numerical structure, but also to what extent we can actually project the numerical structure (back) onto the empirical structures thus represented. The question of projection goes beyond the question of uniqueness of the representation. The real numbers are closed under a variety of operations (such as addition), but it seems that, strictly speaking, no finite structures are closed in this way. This suggests an ontological commitment beyond what is strictly observed and perhaps even beyond what is observable, if the latter is understood to be finite. Similarly, the ordering relations among numbers are transitive, yet for any ordering based on observations, there will be a threshold past which we can no longer discriminate between objects very close in mass, length, or any other attribute of interest. This permits the construction of sequences of com parisons, where we judge a1 to be of the same length as a2, and a2 to be of the same length as a3 and so forth until judging that an-1 is the same length as an, yet judge that a1 is not the same length as an, thereby violating the transitivity

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94 Representationalism as a Basis for Metaphysics of the ordering. While each pair is empirically indistinguishable, the first and last elements of the sequence are distinguishable and are not judged to be the same. Transitivity thereby presents an epistemological challenge to a strictly empiricist interpretation of measurement structures. It seems that the axioms for extensive structures do not, strictly speaking, hold of empirical structures. Moreover, if empirical structures are what is represented numerically, then the question arises as to how representations of closely related empirical struc tures are to be understood. Measurements of standard physical quantities, like length, are carried out at many different regimes, using a wide variety of techniques. I might use a measuring tape to measure different lengths around my flat, yet when measuring interstellar distances, different techniques are needed. On standard representationalist accounts, each technique will gener ate its own empirical structure. How can empirical structures generated from these different measurements be compared and combined? We tend to assume that different techniques for measuring length at different scales are compatible, and are indeed measurements of the same quantity, with the same relational structure. However, if each data structure is given its own represen tation, and each representation goes beyond the (finite) data structure to make claims about length measured at all scales, then we need to make sure that the different representations ‘fit together’ in a consistent manner. We need a way of combining the numerical representations of the different data structures into a single numerical representation, and we need to decide how to handle the ‘overlap’ between any representations, that is, that part of the scale where two or more measurement methods are usable. The problem is both technical and philosophical. The technical part consists in checking how comparisons of such representations are possible in given cases and what their formal properties are.2 The philosophical problem is to provide justifications for taking some data structures to be measurements ‘of the same quantity’ in the first place. Arguably this problem is not purely philosophical, but depends on scientific theorizing as well. Our reasons, in any given case, for treating diverse measurements as measurements of a single quantity will be provided by the scientific investigation in question. What philosophy needs to add, minimally, is a metaphysical account of quantities that makes sense of this practice. If quantities were just functions from particular (qualitative) empirical structures to particular numerical represen tations, as seemingly suggested by standard readings of representationalism, 2 See (Domotor 1992) for some suggestions on why we should not assume that this will be unprob lematic or obvious in all cases.

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Representationalism and its Critics 95 then there is nothing ‘in the world’, as it were, to answer to the concept of length measured in different data structures. A realist account, by contrast, would suggest that there is a feature of the world, length, which is epistemically accessed through (theory-mediated) measurement, and which is represented by numerical structures. For this is what we ordinarily try to do with length measurements on dif ferent regimes, although other cases of measurement, such as measurements of probability, might be more doubtful. The proposal made by Domotor and others is that the empiricism present in much of the representationalist approach needs to be replaced by a more realist construal of what is measured (Domotor & Batitsky 2008; Michell 2005). Instead of taking the represented structure to be confined to a particular data structure, we should interpret the represented structure as the structure of an attribute, like mass or length. A more realist take on measurement might provide a philosophical justifica tion for interpreting different data structures and their numerical representa tions as measurements of a single quantity. This realist criticism echoes some of the worries we already encountered in the context of the empiricist approaches to quantities in Chapter 3. The prob lem is that the axiomatic approach to quantitativeness traditionally goes hand in hand with empiricist expectations that the relevant axioms are empirically testable. Even when it is conceded that the axioms are idealizations that any given empirical structure will fail to satisfy exactly, the thought seems to be that axioms describe measurement operations that we need to be able to carry out, at least in principle.3 Realism, by contrast, suggests that what makes for quantitativeness is not operationalizability, but features of attributes. A ques tion we need to address, then, is how the axiomatic approach central to rep resentationalism can be combined with a more realist outlook that avoids the operationalism inherent in the (standard) representational theory of measurement. Earlier metaphysically inclined responses to representationalism have focused on offering alternative formalizations of measurement in order to end up with a different ontology for measurement. Mundy (1987), for example, provides a formalization using second-order variables to make his formal framework conform to the idea that we need (Platonic) universals to account 3 This is at least the viewpoint of early work on RTM, especially (Krantz et al. 1971, pp. 13f). More recent work in the representationalist tradition, especially (Narens 1985) and (Luce et al. 2007), goes significantly beyond this early operationalist outlook. Instead of proving representation and unique ness theorems for known or constructible measurement procedures, more abstract questions of meas urement theory are addressed (see Baccelli (in press) for discussion).

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96 Representationalism as a Basis for Metaphysics for quantities, not just collections of objects. Zoltan Domotor and Vadim Batitsky have proposed Analytic Measurement Theory (ATM) as an alternative to standard RTM (Domotor & Batitsky 2008). Like Mundy, they advocate a more realist framework for quantities and measurement: not only can quan tities be instantiated by various systems, they also have ‘a life of their own’ (Domotor & Batitsky 2008, p. 134). This suggests a Platonic outlook not unlike Mundy’s. Like Mundy, Domotor and Batitsky aim to provide a formal theory that reflects their realist commitments, which include not only a more Platonic conception of quantities, but also a treatment of quantities as theoretical entities and a rejection of the empiricist distinction between fundamental and derived measurement. While I’m sympathetic to these philosophical aims, I am less convinced that the mathematical framework we adopt for the representation of quan tities is the locus for working out the disagreements between realists and anti realists about quantities, or for determining the right ontological picture for quantities. To capture the commitment that the rich structure of the numerical representation is projected back to the physical world, we do not need to move to a representation involving second-order predicates. We can instead insist that the homomorphism between the two structures must be an isomorphism. This approach has been taken by Narens, who proceeds very much in the tradition of RTM, but assumes that the mappings be isomorphisms instead of homomorphisms (Narens 2001, p. 211). Aside from technical advantages, the isomorphism approach permits the kind of mathematical-to-physical reason ing demanded by Domotor, without departing from the first-order formalism used by RTM. As we shall see in 6.2.3, focusing on the case of isomorphic representation has further advantages for the project at hand. More importantly, rejecting operationalism—the view that what is measured depends for its existence or identity on the procedure by which it is measured— does not require rejecting nominalism—the view that what is measured is concrete particulars, rather than abstract universals. There are good reasons to think that the attributes we measure should be treated as independent of any one procedure by which we measure them (recall the discussion in Chapter 4). But rejecting operationalism leaves open the possibility of providing a nominalist treatment of quantities, that is, of offering an ontology for quanti tative attributes in terms of (classes of) concrete particulars instead of univer sals. I will return to the latter question in Chapter 7. These ‘realist’ critics of representationalism have a point, although I think it ultimately does not undermine representationalism as a theory of measure ment. Instead it shows that representationalism alone does not provide a

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When is an Attribute Quantitative? 97 sufficient metaphysical account of quantities. As we have seen in 6.1.1, it makes sense to treat representationalism as a mathematical framework instead, which still allows us to be informed by representationalism, without committing ourselves to the empiricist and operationalist views commonly associated with it.

6.2. When is an Attribute Quantitative? 6.2.1. Uniqueness, not Representation Critics like Michell point out a second concern regarding representationalism in measurement: it is highly permissive, because it does not seem to put any restrictions on which attributes are numerically representable. Michell argues that such a permissive theory of measurement has been detrimental to the development of measurement in psychology. But the permissivism implicit in representationalism also threatens the project of using RTM as a basis for a metaphysics of quantities. At least on the standard representationalist account, many phenomena and attributes are numerically representable. If numerical representability is enough for an attribute to be quantitative, then the representational theory leads to a form of permissivism. It is now time to articulate and defend an alternative way of drawing the distinction between quantitative and qualitative structures. The distinction will be based on representationalist tools, but it is more substantive than the permissivist view typically associated with representationalism. What we need is an account that tells us what is special about quantitative attributes in terms expressible from a representationalist point of view. A key innovation of the representational theory of measurement is to dis tinguish the question of whether a structure can be numerically represented, from the question of how unique that representation is. This distinction points us in the right direction for drawing a distinction between quantitative and non-quantitative attributes. Being numerically representable is not by itself necessary or sufficient for being a quantity, but being representable at a certain level of uniqueness, is. The proposal, then, is to draw a distinction between quantitative and non-quantitative attributes in terms of how uniquely the attribute is numerically representable. Only attributes with a numerical repre sentation of a certain level of uniqueness will count as quantitative. This proposal is situated between permissivism and restrictive realism. Like restrictivism it suggests that there is an important distinction between

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98 Representationalism as a Basis for Metaphysics quantities and other attributes. Like permissivism it suggests that numerical representability as such is not the decisive factor. Instead it takes on board the permissivist idea that numerical representations come at different levels of uniqueness, but suggests that a particular difference in the uniqueness of the representation marks the difference between quantities and non-quantities. To make this work, we need to identify a sharp divide in the uniqueness of different scale types. As we’ve seen above, representationalism gives us a general method for assessing the uniqueness of numerical representations. For a metaphysics of quantities we need to know in addition which level of uniqueness should count as sufficient for quantitativeness and what makes an attribute suitable for being represented at this level of uniqueness. I will address these questions in sec tions 6.2.2 and 6.2.3, but before doing so it is important to address an obvious worry about this strategy for distinguishing quantities from other attributes. My proposal is that the uniqueness of the numerical representation available for a given attribute distinguishes quantities from non-quantitative attributes. But this might seem to have things backwards. After all, realists suggested that it is something about the attributes in question that makes them numer ically representable and representable at a certain level of uniqueness. While representability at a certain level of uniqueness may be indicative of an attribute being a quantity, surely such representability isn’t what makes it the case that an attribute is quantitative. Surely quantitative attributes have a feature in virtue of which they are numerically representable with a high degree of uniqueness, and it is this feature, whatever it may be, that sets apart quan tities from other attributes, with the uniqueness of the representation being a mere consequence. The objection puts pressure on the idea that a middle ground between real ism and permissivism is possible. Permissivists treat numbers as mere repre sentational tools, not as in any way essential to the nature of quantities. The proposal at hand shares this representational understanding of numbers, while maintaining that quantities are special kinds of attributes. Is it possible to hold this view without collapsing into restrictive realism, which holds that quantities are in some sense ‘numerical’ attributes? A middle ground position will need to identify a non-numerical feature in virtue of which attributes are quantitative. I will argue in section 6.3.1 that there is indeed such a feature, which is revealed in more abstract treatments of the representational theory of meas urement. It turns out that attributes are quantitative in virtue of their structure, and that this structure is best characterized in terms of the automorphisms of

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When is an Attribute Quantitative? 99 the structure. In order to identify what is special about quantities, we first need to identify which representations are representations of quantities. We can then ask which features such quantities must have to permit the relevant types of numerical representations.

6.2.2. An Intuitive Difference If we take the widespread idea that quantitative structures must be unique at least up to interval scale representation as our starting point, we might ask why it makes sense to draw the distinction there. What is it we can do at this level of uniqueness that we can’t do on an ordinal scale? An intuitive, and I think fairly plausible, answer suggests that scales at least as unique as interval scales permit quantitative comparisons, whereas ordinal scales do not. This perhaps circular sounding explanation means that merely ordered items may be compared as to their relative positions, but not as to their relative distances, whereas elements of structures representable by at least interval scales may be compared with respect to their relative distances as well. For a concrete example consider the difference between movie rankings and contour lines. Most movie ratings will agree in ranking Alien higher than Alien vs. Predator, but they cannot tell you how much better the original Alien movie was than any of its successors. By contrast, contour lines on a map tell us not only whether points are at the same or different altitude, but also how much the altitude changes between two points on the map—a very useful fea ture when plotting a route across a mountain range. Comparisons of contour lines are quantitative, whereas comparisons of rankings are comparative in the sense discussed in 3.2.1. The first provides an answer to questions of the form: how much bigger/better/higher . . .? The latter provides an answer to questions of the form: which is bigger/better/higher . . .? The position an item has in a ranking depends on its competitors: whether a football team is in third or fourth place in the league table depends not just on the total number of points achieved and goals scored, but also on how other teams have been doing.4 Whether the difference in elevation between A and B is greater than that between B and C, depends on A, B, and C, not on anything else.

4 In 2016, Leicester City managed to win the Premier League at 81 points. In 2018, Manchester United also gained 81 points, but failed to win the league because Manchester City finished top at 100 points.

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100 Representationalism as a Basis for Metaphysics While we often find it convenient to express answers to questions of the form how much numerically, we do not have to do so. For instance we might say instead that the Great Chilean Earthquake was as much bigger than the 2011 Tōhoku earthquake as the 2004 earthquake off the coast of Sumatra was bigger than the 2005 northern Sumatra earthquake. This cumbersome expres sion does not make use of numbers, but it quantitatively compares the differ ences in size between the pairs of earthquakes. That we are able to draw these quantitative comparisons is reflected in the uniqueness properties of the scales in question, as understood by representa tionalism. A ranking is an ordinal scale, which is invariant under a family of homomorphisms that preserve monotonicity, but which does not have to preserve any further relations, such as ratios, among items in the ranking. The Richter scale, by contrast, is a log-interval scale, which means that the (loga rithmic) interval between steps on the scale is the same across the scale. The interval from 6 to 7 and the interval from 8 to 9 each mark a tenfold increase in strength, which means the interval from 7 to 9 marks a hundredfold increase in magnitude. Quantitative comparisons are made possible by additional equivalence rela tions that hold among items represented on interval scales. One such equiva lence relation holds among earthquakes of the same magnitude, for example, the 2011 Tōhoku earthquake had the same magnitude as the 1952 Kamchatka earthquake. But a second equivalence relation holds among the intervals between the equivalence classes of earthquake sizes: the difference between the equivalence class of magnitude 8.5 earthquakes (to which the 1938 Banda Sea and the 1963 Kuril Islands earthquakes belong) and the equivalence class of magnitude 8 earthquakes (El Salvador 1862, Myanmar 1946) is equivalent to the difference between the equivalence class of magnitude 8.5 earthquakes and the equivalence class of magnitude 9 earthquakes (e.g. Tōhoku 2011). This second kind of equivalence relation is absent in the case of ranking such as movie ratings. While there may be equivalence classes of equally good movies (2001: A Space Odyssey, Solaris) vs. (Alien, Matrix) vs. (Alien vs. Predator, Pixels) and rankings among these equivalence classes, there are no second equivalence relations that hold between the different classes. Is the interval in goodness between the first and second equivalence classes the same as the interval between the second and third? Not only do we not know the answer, it seems doubtful that there is an answer to this question. One reason there does not seem to be an answer to the quantitative com parison question in the case of movie rankings is that the quality of movies simply isn’t as determinate as the size of earthquakes. Because quality in

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When is an Attribute Quantitative? 101 ovies is not as determinate as the size of earthquakes, we can only rank m movies, but not measure the goodness of movies on an interval scale. The question then becomes, what is required for an attribute to have the requisite determinacy? Before turning to this question, we need to address an important concern that the proposed distinction between qualitative and quantitative inevitably raises. The worry is that there are differences in determinacy not only between rankings and interval measurement, but also between interval scales and ratio scales, and between ‘classificatory scales’ and ordinal scales. If so, why should the difference in determinacy between ordinal and interval scales be more significant than these other ones? In response it helps to distinguish the case of classificatory vs. ordinal scales from the case of interval vs. ratio scales. The latter, I shall argue, is indeed a difference in degree of determinacy, yet structures represented on either type of scale will count as quantitative. The former two, by contrast, differ in more than just their ‘degree of determinacy’. Indeed, determinacy is not even the best way of capturing the difference in structure between classificatory and ordinal scales. As we’ve seen in section 3.2, kinds are characterized not only by the classification of entities into equivalence classes, but also, and more importantly, by the hierarchies in which these equivalence classes stand. By contrast, rankings lack the hierarchical structure of kinds: a ranking provides a horizontal ordering of items of the same type. The difference between clas sifications and rankings is not a difference in determinacy, it is a difference in the sorts of relations characteristic of the structures in question. Rankings are characterized by ‘horizontal’ ordering relations—relations that hold among objects of the same domain, whereas classifications are characterized by hori zontal equivalence relations and vertical containment or subset relations. It is not clear that we can compare these structures for determinacy. The reason classifications have sometimes been regarded as less determinate than rank ings is that the hierarchical embedding of kinds has been ignored. Accordingly, rankings have been treated as classifications with an additional ordering rela tion, which would indeed make them more determinate than classifications. But once we take into account the hierarchical embedding of kinds, rankings can no longer be thus understood, and hence the difference between kinds and comparatives, or classifications and rankings, can no longer be viewed as a difference in determinacy. If, as per my proposal, the distinction between quantitative and qualitative structures is one of determinacy, with quantitative structures meeting a threshold of determinacy not met by qualitative structures, then it seems clear

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102 Representationalism as a Basis for Metaphysics that the difference between classification and ranking is not a difference between quantitative and qualitative structures. This leaves us with the question of why to draw the distinction between ordinal rankings and interval scales, as opposed to between interval and ratio scales. The difference between both the former two scales and the latter two scales is a difference in determinacy. Either division is therefore a legitimate candidate for where we might draw the distinction between quantitative and qualitative structures, if the latter distinction is to be drawn in terms of deter minacy. Nonetheless I think that much speaks in favour of drawing the dis tinction at the border between ordinal and interval scales instead of drawing it between interval and ratio scales. The reason, in short, is that the difference in determinacy between interval and ratio scales is much less significant than the difference in determinacy between interval and ordinal scales. This can be shown formally.

6.2.3. A Formal Difference Recall that Stevens initially introduced his hierarchy of scales by asking which transformations of the numerical representation are ‘permissible’, that is, which transformations of the numerical representation result in equivalent, equally adequate numerical representations. The more transformations are permissible, the weaker the scale. Stevens never provided any systematic for mal grounding for this classification, but in the context of RTM, this informal characterization has been used to develop a much deeper understanding of the different scale types. In Stevens’s hierarchy, scales can be characterized by the number of free parameters in the group of transformations. The transformations of a scale are the (homomorphic) functions that map the measurement structure into a subset of the reals. Recall that the transformations of different scale types form groups with different numbers of free parameters, where a free param eter indicates, at first pass, how much choice there is in a mapping. The fewer free parameters a transformation group has, the stronger the associated scale. The group of the relevant transformations is often called the scale group for a particular type of scale (Narens 2001, p. 52). Formally speaking the difference in uniqueness between ratio scale repre sentations and interval-scale representations is that between scales whose group has one free parameter in the case of ratio scales, and two free param eters in the case of interval scales. By contrast, the difference between interval

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When is an Attribute Quantitative? 103 scales and ordinal scales is that of a scale group with two free parameters and a scale group with countably infinite free parameters (Luce et al. 2007, p. 113). This radical difference in the number of free parameters for the groups of homomorphisms suggests that the difference between ratio and interval scales is less significant than that between interval and ordinal scales. This impression of a sharp formal difference between interval and ratio scales on the one hand, and ordinal scales on the other, is confirmed by a fur ther result in measurement theory, which shows that, given certain assump tions, all scales fall into one of three types: ratio, interval, and a ‘hybrid category’ (Baccelli, in press) in between ratio and interval scales. This result, known as the Alper–Narens theorem, is relevant for our purposes here, because it shows that while there is a scale type in between ratio and interval scales, there are no scale types in between interval and ordinal scales.5 The statement of this theorem and the characterization of the hybrid scale type requires a distinction between two ways of understanding the free parameters of a transformation group. The first, more familiar way of charac terizing the free parameters of the transformation group is (n-point) uniqueness, which describes on how many points two transformations can agree before they are identical. By this measure, ratio scales are 1-point unique: any homomorphism that maps a single point into itself is simply the identity map, that is, it maps every point into itself. This also means that two homomorph isms that agree in one point are identical. Interval scales are 2-point unique: a homomorphism that maps two points into themselves becomes the identity, and any two homomorphisms that agree on two points are identical. Ordinal scales, by contrast, are not finitely point unique. Consider the centimetre and inches scales for length. The two scales assign different numerical values to all lengths: if the length of the side of a table is represented by ‘50’ on the inches scale, it will be represented by ‘127’ on the centimetre scale. In general, no length will be represented by the same number on the two scales. For an attribute like length, which is represented on a ratio scale, any two scales that assign the same number to a particular length would also have to agree on all other numerical assignments. A ‘schmentimetre’ scale, for instance, which agrees with the centimetre scale in assigning ‘127’ to the side of our table, would also have to agree in the numbers assigned to all other lengths. Since the two scales agree on each point, they are simply the

5 The result was originally developed in Narens (1981a); Narens (1981b); Alper (1985); and Alper (1987). The importance of the theorem for our understanding of quantities has already been recog nized by Michell (1999) and more recently been discussed by Baccelli (in press).

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104 Representationalism as a Basis for Metaphysics

Figure 6.1. Homogeneity

same mapping. By contrast, there is a temperature to which the Celsius and Fahrenheit scales assign the same number: −40°C = −40°F. The scales are nonetheless distinct, because they disagree on all other numerical assign ments: 32°C is a warm day, 32°F is not. This is possible because they are inter val scales and hence 2-point unique, not 1-point unique. Uniqueness is primarily of interest for scales that are homogeneous, in the sense that for each element in the structure, there is a function in the scale group such that it maps that element to another element in the structure. This criterion is satisfied by any structure that does not have a structurally distin guished element, such as a minimum or maximum. Here’s a toy geometrical example to illustrate the point. An infinite line is homogeneous, whereas a line that has a definite starting or end point is not. In the latter case, mapping the end point to any other point in the structure constitutes a change in the structure, which means the mapping is not a homomorphism. Figure 6.1 is a simple geometric example to illustrate the point: The top line is homogeneous, whereas the bottom two lines are not, since either contains a distinguished point that cannot be mapped to the other points without changing the structure. In the measurement theoretic case, the question of the homogeneity of a numerical representation takes into account both the structure of the numerical domain and the scale type of the repre sentation. The real numbers by themselves are not homogeneous, because 0 is a structurally distinguished element. But for many scale types, 0 is not a dis tinguished element. Ordinal and interval scales on the reals are both homoge neous in R, as are ratio scales for representations in R+ and R–. The former are homogeneous in virtue of the scale type, since neither ordinal scales nor inter val scales use a distinguished zero point. The latter are homogeneous, because neither R+ nor R– contains a distinguished element. Ratio scales on R, on the other hand, are not homogeneous, although they are not very inhomogeneous; they are homogeneous except for one point: 0. Many scales of interest for the purposes of measurement, and especially measurement in physics, are largely homogeneous. Quantities, like Newtonian mass, that are represented on R+ are homogeneous, as are quantities that are representable on interval scales, like the gravitational potential. Exceptions

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When is an Attribute Quantitative? 105 are quantities like speed, which have a distinguished element in the form of a maximum (the speed of light in vacuum), and quantities like electric charge that have ratio scale representations in R rather than R+. We can distinguish degrees of homogeneity in much the same way in which we distinguished degrees of uniqueness. To do so, we ask how large an ordered subset of the structure may be such that it can be mapped, by a member of the scale group, into an arbitrary set of points that preserves the order in the subset. Following Narens, this feature has been called (m-point) homogeneity. Ratio scales are 1-point homogeneous, interval scales are 2-point homoge neous, and ordinal scales are not finitely point homogeneous. Both ratio and interval scales are finitely n-point unique and finitely m-point homogeneous. In fact, parameters n and m agree for these scale types. The hybrid category between ratio scales and interval scales is charac terized by a mixture of free parameters, depending on whether these are taken to measure uniqueness or homogeneity. If scale types are given as (M,N), then ratio scales are of type (1,1), interval scales of type (2,2), and the hybrid category as of type (1,2), that is, it is 1-point homogeneous and 2-point unique.6 The Alper–Narens theorem states that any scale, with an image on the reals, that is ordered, homogeneous, and finitely point unique is either an interval scale (2,2), a ratio scale (1,1), or a hybrid between the two (1,2).7 A homoge neous scale that is finitely point unique is at most 2-point unique, that is, of interval type. Following Narens we might call scales that are of these three types super-ratio scales (Narens 2001, p. 53). The upshot of the characterization of scales in terms of uniqueness and homogeneity is that any finitely point unique scale is at least an interval scale; there are no scale types in between interval and ordinal scales for homoge neous scales with images in the reals. This further supports the hypothesis that interval and ratio scales are more closely related to each other than either is to ordinal scales and hence supports the claim that representability on interval and ratio scales should be taken as indicative of quantitativeness, whereas representability on ordinal scales is not a sign that an attribute is quantitative. The cost of accepting this criterion is to follow Narens’s isomorphic approach and with it the idea that the structures mapped into the reals are continua.

6 There are no known examples of scientific measurement structures of this type. 7 A formal statement of the theorem can be found in (Luce et al. 2007, p. 120, theorem 5). (Narens 2001, pp. 50–5) provides a clear contemporary presentation of the result.

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106 Representationalism as a Basis for Metaphysics

6.2.4. A Criterion in Terms of the Structure Itself We can now see that certain types of numerical representations are significantly more unique than others, which I suggested makes them indicative of the quantitativeness of the represented attribute. But what makes attributes repre sentable on such scales special? Once again RTM can be of use. Formal differences between scale types are investigated initially by studying the algebraic features of the homomorphisms of the different relational struc tures to suitable numerical structures. Instead of looking at the transform ations of the numerical representations, we can also look at the automorphisms of the relational structure as the relevant transformations to study for the purpose of determining the uniqueness of scale types. In particular, we will be interested in whether the automorphisms of the structures form (algebraic) groups of certain types. An automorphism is a mapping of a structure into itself. Some such map pings will change the structure, whereas others leave the structure invariant. The appropriate scale type for a relational structure can be found by asking which automorphisms leave the structure invariant. Every structure has the identity as an automorphism, which maps each element of the structure to itself. The structure is ‘trivially’ preserved in this case, since the automorphism simply leaves everything unchanged. The structures of interest for measure ment theory typically have automorphisms beyond the identity.8 We can dis tinguish between automorphisms that leave some points fixed and those that lack any fixed points, where a fixed point is a point that is mapped to itself under the automorphism. As we shall see, a particularly important class of automorphisms is translations, which in the context of RTM are understood as automorphisms without any fixed points (Luce et al. 2007, p. 118, definition 2). We know from the Alper–Narens theorem that any homogeneous, finitely point unique, ordered scale with an image in the reals will be at least of interval type strength. A theorem due to Luce (Luce 2001, theorem 5) provides an abstract characterization of structures that satisfy the antecedent of the Alper–Narens theorem. According to the result by Luce, a relational structure whose translations form an Archimedean ordered group supports a super ratio scale representation. This result thereby formulates a sufficient condition, in terms of a subset of the automorphisms of the relational structure, for when numerical representations of the super ratio scale type are possible. That the

8 An exception are so-called absolute scales.

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When is an Attribute Quantitative? 107 group is Archimedean means that no positive member is so large that it cannot be ‘reached’ by other positive members through multiplication by an integer. That the translations, 𝔗, form an ordered group means that for any two translations t and s ∈ 𝔗, t ≾ s or s ≾ t, and the translations form a group, which means, in particular, that any composition of two translations, s and t, is another translation: s ∘ t ∈ 𝕿. This permits the application of a version of Hölder’s theorem to the transla tions of the structure. Recall that Hölder’s theorem shows that a structure that is an Archimedean ordered group has a representation in the real numbers that is unique up to multiplication by a constant, that is, that has ratio scale uniqueness. When we first encountered Hölder’s theorem, it was applied to measurement structures and the uniqueness of their representations. Now we are applying it to (a subgroup of) the automorphisms of a measurement struc ture, namely the group of translations. Luce’s result shows that the translations of structures that permit at least interval type representations themselves have something like ‘extensive’ structure: they can be ordered and ‘added’. The Alper–Narens theorem suggested that super ratio scales are a special class of closely related scale types, thereby providing a reason for drawing a sharp distinction between these scales and the much weaker ordinal scale type. The result due to Luce provides a very general and abstract characteriza tion of the kinds of structures that satisfy the criteria of the Alper–Narens theorem (homogeneity and finite point uniqueness). What are the philosoph ical, and more specifically the metaphysical implications of this criterion for quantitativeness? Restrictive realists like Michell have seized on this result to argue that ratios are still ultimately involved in measurement and see these results as a vindica tion of realism over ‘representationalism’ (Michell 1999, p. 211).9 I agree with Michell that the formal results discussed here, as well as the more intuitive differences between ordinal and other scales discussed in 6.2.2, speak in favour of taking quantitative attributes to be restricted to those that can be repre sented at super ratio scales. But this neither signals a return to the traditional concept of quantity, nor does it settle the question of the role of numbers in measurement. Instead it shows that restrictive views of quantities can be defended from within the representationalist framework. The representational theory of measurement succeeds where earlier re strict ive views of quantities had failed, precisely because it adopts an 9 Michell’s use of the term ‘representationalism’ is not restricted to RTM, but refers to any theory that claims that numbers have a merely representational role in measurement.

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108 Representationalism as a Basis for Metaphysics increasingly abstract view of the relevant structure. Earlier restrictive views, such as Campbell’s, had tried to identify criteria of quantitativeness by using surface features of the attributes they counted as paradigmatic quantities. This proved to be unworkable, since the surface structures of paradigmatic quan tities are not all alike (in particular, not all of them come with empirical con catenation operations). The much more abstract perspective of the representational theory of measurement avoids the problems of traditional restrictive views. The ques tion is not whether elements of the measurement structure can be concaten ated, or even whether the structure is extensive in the axiomatic sense. Instead it is features of the automorphisms of the structure (indeed, features of a sub set of the automorphisms—the translations) that make for quantitativeness. A wide range of different measurement structures can all have translations that form Archimedean ordered groups. This explains why the surface features of characteristic physical quantities can differ, even though they are represent able at the relevantly strong scale types. It is possible to arrive at this result in the context of the representational theory of measurement, precisely because RTM adopts a more abstract view point. Abstracting away from concrete measurement procedures and particu lar numerical representations made visible the common group theoretical structure in virtue of which certain types of numerical representation are possible. This suggests that what makes for quantitativeness is not ratios, or ‘numbers in the world’, but the determinacy of certain types of structure. The automorphisms of a structure characterize its determinacy, because they show how much symmetry there is in a structure. Structures that make for quanti tativeness are sufficiently determinate to be finitely point unique, while not being so rigid as to permit only the identity automorphism. This puts inter esting bounds on quantitativeness. My main focus in this chapter has been to argue that in one direction, quantities must be representable at least on interval scales, since they have to be at least 2-point unique if they are to be finitely point unique at all. But an interesting point to note is that they should also be at least 1-point homogeneous, which rules out absolute scales as scales for physical quantities. Absolute scales are not homogeneous, since each element is treated as structurally distinguished. Since the criterion offered by Luce’s theorem is presented in terms of the automorphisms of the structure of the attribute, rather than in terms of the permissible transformations of the numerical representations of the attribute, we now have a criterion for quantitativeness in terms of features of the attri bute, not in terms of its representations.

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How RTM Constrains the Metaphysics of Quantities 109 Carnap’s division of scientific concepts, with which we started the discussion of axiomatic approaches in Chapter 3, turns out to be not too far off, even though there are problems with the details. Like Carnap, I conclude that both ‘intensive’ and ‘extensive’ attributes are genuinely quantitative, while exclud ing both qualities and comparatives. Unlike Carnap, I side with the realists in taking quantitativeness to be a matter of the attributes, not a feature of con cepts. Unlike standard realists, however, I hold that an attribute is quantita tive in virtue of the determinacy of its structure, not in virtue of it ‘implying’ numbers. To draw the distinction between quantities and qualities, we needed to characterize both in structural terms, by specifying axioms. The distinc tion between quantities and qualities was then initially drawn in terms of the uniqueness of their numerical representations. This allowed us to discover that the structures of attributes representable on super ratio scales are closer to each other in uniqueness than they are to ordinal scales or absolute scales. We finally saw that it is possible to characterize what is special about attributes representable in this way purely in terms of their internal structure, by looking at the structure of their automorphisms. This suggests that it is the level of determinacy in a structure that makes for quantitativeness.

6.3. How RTM Constrains the Metaphysics of Quantities 6.3.1. The Role of Structure So far I’ve argued that the representational theory of measurement can be used as a basis for the metaphysics of quantities, because it can be regarded as a formal framework open to interpretation. From this perspective, we can largely divorce the mathematics of RTM from the operationalist and empiri cist proclivities of some of its proponents. It is moreover possible to use RTM to formulate a restrictive view of quantities, in contrast to the usual permis sivism of representationalists. In this final section of this chapter, we will look at two related ways in which RTM nonetheless seems to constrain the meta physical account we give of quantities. The two aspects are the implicit structuralism of RTM, and using invariance as a criterion for meaningfulness. Krantz et al. treat mathematics in a broadly structuralist fashion. This structuralist methodology is apparent, first of all, in the focus on homomorphic maps as the key to how numbers can be used to represent objects and their properties. Since a homomorphism, by definition, is a structure-preserving map, using homomorphisms to relate objects and numbers requires treating

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110 Representationalism as a Basis for Metaphysics both as structured sets. In order for the mapping to take place at all, we have to consider collections of objects and their relations, not just an individual object. Similarly, we have to think of the (real) numbers as a set with relations and operations defined on it. Only once we have ‘structures’ on both sides can we ask whether or not they have structure in common. This commitment to structures is at the heart of the formulation of representation theorems, which means that it cannot be separated from the formal framework of RTM in the manner in which the operationalist and empiricist sympathies of authors of the Foundations of Measurement can be set aside when interpreting the repre sentational theory. While this implicit structuralism does not amount to a full-blown structur alist metaphysics of mathematics, it is important to keep in mind that even this methodological structuralism is not the view behind most older theories of quantity or measurement. Many of these older views treat numbers as par ticulars (albeit typically abstract particulars) and not primarily as elements in a structure. The structuralist approach, therefore, constitutes an important presupposition of representationalism, which must be taken into account when interpreting the metaphysical commitments of the representational theory of measurement. Moreover, from a representational point of view, structuralism applies not just to mathematics, but to empirical phenomena as well, at least if they are to be represented mathematically. The (algebraic) structure attributed to both numbers and empirical phenomena by the axioms for extensive or difference structures is quite rich. In order for the axioms to be satisfied in a non-trivial manner, we need not just one or two, but many objects to stand in the pre scribed relations. For representationalism, then, mathematical representations of empirical phenomena presuppose treating these phenomena as a plurality of entities standing in certain types of relations. This imposes a constraint on metaphysical accounts of physical quantities thus represented. As we shall see in Chapter 7, it does not force us to adopt the view that these collections of entities are (ordinary) objects, but it does require a plurality of entities. Indeed, formal alternatives to the representationalist theory—for example Mundy’s second-order theory and Domotor and Batitsky’s Analytical Measurement Theory—share a similar commitment to structuralism, although less obviously so. Mundy’s account requires both fundamental relata (magnitudes, understood as Platonic universals) and fundamental relations (sum of and less than) to provide an ontology for quantities. This suggests that his account is also structuralist in the minimal sense that quantities are thought of as relational structures, where a relational structure is a set of

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How RTM Constrains the Metaphysics of Quantities 111 elements with relations defined on them. The dispute between Mundy and (empiricist) representationalism is over whether the relata are concrete par ticulars or abstract universals, and similarly, whether the relations are first or second order. Domotor and Batitsky seem to go further than representationalism in that we need to take into account not only the relations among different masses or lengths, as it were, but also the relationships among different quantities, such as any laws connecting quantities like volume, temperature, and pressure, for instance. For the former Domotor and Batitsky seem to be thinking of quan tities as topological spaces, for the latter they invoke algebras of quantities. All three views are structuralist in the sense that quantities ontologically involve both relata and relations. That is, we need both a plurality of magnitudes and a plurality of numbers, plus relations among them, to get any (interesting, non-trivial) mapping relations between them. Representationalism thereby con strains the kinds of answers we can give to the question of what the relationship between numbers and magnitudes is: it is a mapping relationship between two structures. Representationalism does not tell us what structures of magnitudes or structures of numbers are, metaphysically speaking. This is not yet a form of philosophical structuralism. Developing the latter requires a specification of an ontological interpretation of quantities that has characteristically structuralist features, for example a rejection of haecceities in the relata, or a prioritizing of relations over relata. I shall develop such an account in Chapters 7–10.

6.3.2. Invariance as Meaningfulness The same implicit structuralist approach is behind the resolution the repre sentational theory offers to the problem of the multiplicity of numerical rep resentations. The idea that we should take as meaningful only those aspects of any one representation that are invariant across all (equivalent) representations is a common structuralist move in the philosophy of science. While treating invariance as a necessary condition for meaningfulness does not obviously require a structuralist approach, it nonetheless seems especially suitable for it. The reason is that invariance (or lack thereof) is especially clearly characteriz able for structural representations. Structural representations are related by homomorphisms, and as a result we can ask which transformations of these homomorphisms leave which aspects of the structure unchanged. This provides a clear criterion for invariance between different representations. By contrast,

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112 Representationalism as a Basis for Metaphysics representations that are not purely structural stand in less well-defined relations to one another, and accordingly it is much more difficult to say in what sense features are invariant. Just as the structuralist treatment of numbers and empirical structures is required for proving representation theorems, the invariance criterion of meaningfulness is behind the uniqueness theorems of (standard) RTM. Representation theorems describe which relations are mapped from one structure to the other and how, whereas uniqueness theorems say which other mappings preserve the same relations. This approach provides explanations for several of the features we ascribed to quantities earlier. The conventionality of measurement is explained by the non-uniqueness of numerical representations. Extensive structures are typically represented by ratio scales, which are unique up to multiplication by a positive constant, which can in turn be interpreted as a choice of unit. Numbers nonetheless serve as an important representational device, because of the shared structure between quantitative attributes and numerical struc tures like the real line. This shared structure explains both that we can use numbers to represent quantitative attributes and which inferences are sup ported by such representations. The structuralist element of the representa tional theory of measurement thereby helps to explain why numbers are a suitable representational device for quantities in the first place, despite the specific numbers typically being conventional and not informative. The structuralist approach implicit in representationalism therefore shows how representationalists can respond to some of the worries set out earlier. Representationalism gives a clear answer to the problem of units, and the problem of conventional elements in numerical representations more generally. By looking at what is invariant under permissible transformations, we can determine how unique a given representation is, that is, how many degrees of freedom we find in any given representation. The more degrees of freedom, the less determinate the representation, that is, the more con ventionally chosen elements we’ll find in any one representation. The conven tionality of units is thereby explained as just one of the ways in which numerical representations of empirical phenomena might contain conven tional elements. A consequence of the combination of structuralism and the criterion of invariance is that, as far as measurement theory is concerned, only structural features can be regarded as meaningful, since only they can be homomorphically mapped and hence be invariant under changes of representations. This perspec tive is not unique to the representational theory of measurement; something

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How RTM Constrains the Metaphysics of Quantities 113 along these lines is often defended by structural realists in the philosophy of science. It is not an uncontroversial point of view, however, as we shall see in Chapters 9 and 10. A form of structuralism seems integral to the representational theory of measurement on two counts, then. On the one hand, the mathematical approach chosen requires treating both the qualitative and the numerical side of representation theorems as structures, that is, as consisting of relations and relata. On the other hand, the use of invariance as a necessary condition for meaningfulness seems to commit RTM to the claim that only structural features of the qualitative phenomenon are meaningful. The latter form of structuralism seems to pose a more serious and controversial constraint on a metaphysics of quantities. We will return to it in chapter 9.

6.3.3. Conclusion In this chapter I’ve argued that the representational theory of measurement provides a promising starting point for a metaphysics of quantities. Using representationalist tools, we can provide a clear formal distinction between quantitative and non-quantitative attributes. RTM hence provides a formal criterion for quantitativeness. Moreover, we do not need to take on board the operationalist and empiricist outlook of many of the founders of the represen tationalist theory of measurement. Instead, RTM can be treated as a math ematical framework in need of a metaphysical interpretation, not unlike mathematical theories in physics. The axiomatic approach of RTM can be interpreted as providing conditions for the ‘internal’ structure of quantitative attributes, not merely as providing conditions that empirical measurement structures must operationally satisfy. This combines an axiomatic approach to measurement with a restrictive view of quantitativeness, which is what we had set out to provide at the beginning. The resulting view is nonetheless distinct from restrictive realism as pro posed by Michell. While Michell also draws attention to the results by Alper, Narens, and Luce, he is eager to conclude from this that the newer results in the representationalist approach in fact vindicate the restricted realism based on ratios. One reason for this is the role played by Hölder’s theorem in the characterization of homogeneous structures. Since Hölder’s original discussion seems to provide a means for defining ratios of magnitudes and real numbers, the implication of Hölder’s theorem in the characterization of quantities, for Michell, means numbers are also ‘implied’.

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114 Representationalism as a Basis for Metaphysics My own conclusion from these results is quite different. While it is true that an abstract version of Hölder’s theorem is indeed involved, the theorem shows that the relevant structure does not depend on the particular oper ations and relations appropriate for numbers, but rather that what makes for quantitative structure can be found through greater abstraction. These devel opments point away from the close analogy between numbers and magni tudes and instead suggest that numbers and quantitative attributes are related through more abstract structure and not because quantitative attributes are straightforwardly numerical.

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7 Ontologies for Quantities Synopsis With the formal framework for quantitativeness on the table, we can now turn to the question of what kind of ontology we need to adopt for quantitative attributes. We know from the discussion of the representational theory of measurement, that we are looking for relational structures. I begin by contrasting views favouring universals as the relata in such relational structures and views that favour particulars. I discuss two approaches under each heading: among the friends of universals we can distinguish Aristotelians and Platonists, with the former restricting universals to instantiated universals only; among the nominalistic approaches we find defenders of ordinary material objects as relata, in contrast to those who argue for space-time points as relata. I argue that Platonic universals or space-time points make for better relata than Aristotelian universals or material objects. The two victorious positions are both forms of substantivalism about quantities, since they do not require material objects as relata, but instead suggest that material objects stand in quantitative relations in virtue of a rich ‘background structure’. As a result there isn’t much to choose between them when it comes to ontological parsimony: both succeed precisely because they posit an infinite domain of (non-material) relata. This result raises a further question: does quantity substantivalism require a form of absolutism about quantities as well? This question will be taken up in Chapter 8.

7.1. Universals as Relata 7.1.1. Aristotelian Universals A very natural approach to the metaphysics of quantities is to take magnitudes— particular amounts of a quantity—as universals. If magnitudes are universals they can be wholly present at two different instances; two (or more) particulars can share the same universal. Universals can be either ‘Aristotelian’ or The Metaphysics of Quantities. J. E. Wolff, Oxford University Press (2020). © J. E. Wolff. DOI: 10.1093/oso/9780198837084.001.0001

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116 Ontologies for Quantities ‘Platonic’. Platonism about universals suggests that universals exist whether they are instantiated or not, that is, even a world without any red objects might contain the universal redness. Aristotelian views of universals, by contrast, require that a universal must be instantiated at least once to exist in a world. Universals are typically invoked to explain resemblance among distinct entities: two entities are similar in virtue of instantiating the same universal. This is indeed the role they play in Armstrong’s Aristotelian realism discussed first. Mundy’s view, as we shall see in section 7.1.2, is somewhat different. The main function of universals there is to account for uninstantiated magnitudes, which are needed for the formal requirements placed on quantities. Of these two views, Aristotelian realism is prima facie less ontologically costly, since it only permits universals that are in fact instantiated. As we shall see, however, Platonism is far better suited to address the problems arising from the metaphysics of quantities in particular. Aristotelian universals prima facie offer an intuitive and metaphysically not too costly ontology for quantities. When we say that a desk has a mass of 12 kg, we seem to attribute a certain property to it. Moreover, quantitative relations, such as being twice as massive as or being two kilograms more massive than something else, seem to hold among material objects in virtue of these objects having particular quantitative properties, for instance, having a mass of 4 kg and 2 kg respectively. This intuitive view has been developed and defended in detail by David Armstrong.1 Armstrong (Armstrong 1988; Armstrong 1989) takes magnitudes to be determinate monadic properties. Quantities like mass are determinables for collections of such magnitudes.2 Magnitudes are special kinds of universals: not only are two objects similar to each other in virtue of having the same mass, for example both having 12 kg of mass, properties like 12 kg of mass are themselves similar to other properties, for example to the property of having 13 kg of mass. To explain this resemblance among properties, Armstrong introduces the notion of ‘complex’ or ‘structural’ properties (Armstrong 1989, p. 107). Structural properties are universals which have other universals as constituents, much in the way in which concrete objects can have other objects as parts. Armstrong’s conception of structural universals is not exactly mereological, though. Complex universals have constituents, but since two universals can share a constituent while remaining distinct from one another,

1 Chris Swoyer (1987) similarly advocates Aristotelian universals as an ontology for quantities. 2 For more discussion on the determinable/determinate distinction and why I ultimately think it is inadequate for quantities, recall the discussion in Chapter 2.

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Universals as Relata 117 their mode of composition is not mereological.3 There is nonetheless a close relationship between universals and their constituents on the one hand, and objects and their parts on the other. For an object to instantiate a structural universal, it has to have parts instantiating the constituents of the structural universal. Consider the property of being just five kilograms in mass. For something to have that property the thing must consist of two parts, parts with no overlap between them, such that one part is just four kilos in mass, the other just one. It is a simple form of structural property, simple because no special relations are needed between the two parts. (Armstrong 1989, p. 106)

Armstrong’s realism is strictly Aristotelian, and so there cannot be uninstantiated universals. Accordingly, if the constituents of structural universals are to be instantiated, they have to be instantiated in objects that are parts of the object instantiating the structural universal. If having a mass of 2 kg is a structural universal, with constituents 1.5 kg and 0.5 kg, then any object instantiating 2 kg, say a container of orange juice, must also have parts of 1.5 kg and 0.5 kg respectively. Maya Eddon has helpfully formulated the principle behind structural universals as the constituency principle: [A] universal x is a constituent of universal y iff every object in every possible world that instantiates y has some proper part that instantiates x. This principle links the structure of universals to the structure of objects—universals have constituents when the objects that instantiate them have parts. (Eddon 2006, p. 387)

Magnitudes, for Armstrong, are structural universals, understood along the lines of the quasi-mereological model just sketched. Armstrong hopes to use this account to explain the resemblance features of quantities, namely that objects instantiating different amounts of the same quantity can resemble each other to varying degree. For example, 2 kg is thought to be more similar to 4 kg than to 20 kg. This can be explained on the model of quantities as structural universals by appeal to what Maya Eddon calls the ‘Resemblance Principle’: ‘a is more similar to b than to c, and c is more similar to b than to a 3 This departure from a purely mereological conception marks a change in Armstrong’s position; initially his conception of structural universals was mereological, but in light of serious objections (Lewis 1986a) he moved to the less-mereological position. The main problem is that two particulars that share a part will overlap, whereas two universals that share a constituent remain distinct.

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118 Ontologies for Quantities iff a