# Workshop on Quantities, Metaphysics, and Reduction (SQuaRed-Ex)

## March 6th 2023 13:00-17:15 GMT

On March 6th, there will be an open online workshop on the metaphysics of quantity, organized by Zee Perry (Marie Curie Research Fellow, University of Birmingham) through SQuaRed-Ex (Scientific Quantitativeness Reduced and Explained, HORIZON-MSCA-2021 Project No. 101067459), titled “Quantities, Metaphysics and Reduction”.

The workshop will be held over Zoom on Monday March 6th from 1:00pm to 5:15pm GMT. The speakers are as follows;

Monday, March 6th

Zee Perry – “Against Quantitative Primitivism”

1:00pm GMT (8:00am EST) — 2:15pm GMT

Marissa Bennett (co-authored with Michael Miller) – “The Conventionality of Real-Valued Quantities”

2:30pm GMT (9:30am EST) — 3:45pm GMT

Jo Wolff – “Does the Representation Theory of Measurement offer a reductionist approach to quantitativeness?”

4:00pm GMT (11:00am EST) — 5:15pm GMT

Registration is not required, and link to the Zoom room is available here:

Topic: SQuaRed-Ex Workshop on Quantities, Metaphysics, and Reduction

Time: Mar 6, 2023 12:30 PM London

Join Zoom Meeting

https://bham-ac-uk.zoom.us/j/86727956632?pwd=ZjRzTTM3d1VrNS9CNjJWTmpGeWhwdz09

Meeting ID: 867 2795 6632

Passcode: 150875

Abstracts for the talks are available here: http://zrperry.com/squaredex-workshop-3-6-2023/

### Abstracts:

Zee Perry

Against Quantitative Primitivism

1:00pm GMT (8:00am EST) — 2:15pm GMT

**Abstract:** In this paper, I introduce a novel approach to a problem that is, in the dominant literature, often thought to admit of only a partial solution. The problem of quantity is the problem of explaining why it is that certain properties and relations that we encounter in science and in everyday life, can be best represented using mathematical entities like numbers, functions, and vectors. We use a real number and a unit to refer to determinate magnitudes of mass or length (like 2kg, 7.5m etc.), and then appeal to the arithmetical relations between those numbers to explain certain physical facts. I cannot reach the coffee on the table because the shortest path between it and me is 3ft long, while my arm is only 2.2ft long, and 2.2< 3. The pan balance scale does not tilt because one pan holds a 90g tomato while the other holds two strawberries, of 38g and 52g respectively, and 38+52=90. While they provide a convenient way to express these explanations, the arithmetical less than relation, or the `+’ and `x’ operations on the real numbers are not really part of the physical explanations of these events. They just represent explanatorily relevant features inherent in the physical systems described. To solve the problem of quantity is to provide an account of this “quantitative structure”, those physical properties and relations really doing the explaining. The vast majority of approaches in the literature have limited themselves to a much less ambitious project: Rather than explain quantitativeness in its entirety, they strive to leave “only” a small amount of quantitativeness unexplained. Primitivism about quantitativeness, or quantitative primitivism, is the position that some quantitative structure cannot be explained. I will argue that the problem of quantity, by its very nature, does not admit of any partial solutions. A reductive-explanatory account of quantitativeness is specifically one that provides an adequate explanation of quantitative struture without leaving any quantitative structure as an unexplained, primitive posit. This is done by reducing the quantitative structure to a more fundamental, non-quantitative base. Non-primitivist accounts also allow for a novel dissolution of a problem which has dominated contemporary debates about the metaphysics of quantity, the debate between “absolutists” who think that the fundamental quantitative notions are properties (like “weighs 5g” or “is 2m long”), and and “comparativists” according to whom the fundamental notions are comparative relations (like “is twice as massive as” or “is 2m shorter than”). This dispute, I argue, only makes sense from a primitivist perspective. For the non-primitivist, there is no debate to be had. There is no fundamental quantitative structure, and so there is no room for a dispute about what kind of fundamental quantitative structure we accept. The underlying intuitions which guide much of these debates (for example about whether things would be different if everything’s mass was doubled) can still be understood by the non-primitivist. Indeed, non-primitivist accounts can give a clearer and more explanatory judgement on these cases than any primitivist theory could.

Jo Wolff

Does the Representation Theory of Measurement offer a reductionist approach to quantitativeness?

4:00pm GMT (11:00am EST) — 5:15pm GMT

**Abstract:** The Representational Theory of Measurement (RTM) offers a formal theory of measurement, with measurement understood as a homomorphic mapping between two types of structure: an empirical relational structure on the one hand, and a numerical structure on the other. These two types of structure are characterised axiomatically, as sets with certain relations defined on them. For a quantitative attribute like mass, for example, we find an empirical relational structure of weights with ordering and concatenation relations defined over them, and a numerical structure provided by the real numbers, less-than, and addition to represent the empirical relational structure. The numerical structure serves merely as a representational tool to capture the relationships between the weights; and the mathematical relations of ordering and addition are interpreted concretely as physical orderings and concatenations in the context of particular measurement operations. RTM has sometimes been interpreted as offering a kind of reductionist approach to quantitativeness, for two reasons: 1. RTM takes numbers to play a purely representational role in measurement 2. RTM takes a permissivist view of numerical representations: many kinds of attributes can be numerically represented, not just traditional quantities, like length or mass Insofar as we equate quantitativeness with being numerical, it would seem that RTM takes a reductionist view of quantitativeness, because it takes a deflationary view of numerical representation: the only thing you lose if you omit numerical representations is convenience. I argue here that, on the contrary, RTM not only does not commit us to a reductionist view of quantitativeness, but in fact provides us with a novel criterion for quantitativeness, which shows why reductionism about quantitativeness is so difficult. The first part of my argument rejects the view that quantitativeness is best understood as being numerical. RTM demonstrates quite clearly that numerical representation is neither necessary nor sufficient for an attribute’s being quantitative. It is not sufficient, because many intuitively non-quantitative properties can be represented numerically using the tools of RTM; in general, numerical representability is pretty easy within the RTM framework. It is not necessary, because RTM itself shows how empirical relational structures can be represented non-numerical (for example geometrically). Having rejected the claim that quantitativeness means being numerical, I then show in part 2 of my argument that RTM in fact provides a novel criterion for quantitativeness. This proceeds in two steps: first I show how uniqueness theorems provide a reason for thinking that only some numerical representations are representations of quantities, and second, how we can characterise the structures amenable to such representations using the resources of RTM. This yields a criterion for quantitativeness as a feature of certain kinds of structures. Since RTM’s own conception of measurement is that of a homomorphic relationship between two structures, we shouldn’t expect one of these structures to count as quantitative by this new criterion, while the other one is not. Reducing quantitativeness is harder, not easier from the perspective of RTM.

Marissa Bennett (co-authored with Michael Miller)

The Conventionality of Real-Valued Quantities

2:30pm GMT (9:30am EST) — 3:45pm GMT

**Abstract:** Non-discrete quantities such as mass and length are often assumed to be real-valued. Rational-valued measurement outcomes are typically thought of as approximations of the `real’ values of their target quantity-instances. For example, the representational theory of measurement (RTM) models measurement as the construction of a function that sends a set of objects obeying certain qualitative axioms into the real numbers, such that the structure of the relations holding among the objects is preserved by the order and addition relations on the real numbers. The original architects of the modern version of RTM (Krantz et. al.) clearly acknowledge that this choice of representing mathematical structure is conventional, being influenced by pragmatic considerations related to computational simplicity, and they consider alternative representing structures that illustrate this conventionality. But whereas operations alternative to ordinary addition for additive measurement are considered, sets alternative to the real numbers are not. The formal results of RTM have recently been applied in formulating realist views of quantity, but the assumption that the real numbers are best suited for representing the structure of non-discrete quantities has not yet been examined. At the core of the standard RTM representation and uniqueness theorems is Hölder’s theorem, which Hölder originally proved from a set of axioms that he regarded as “the facts upon which the theory of measurable (absolute) quantities is based”. These axioms include Dedekind’s axiom of continuity, reflecting the close conceptual connections between the real numbers and `continuous’ quantities. Hölder regarded quantities as having magnitudes as axiomatized by Euclid, and understood Euclid’s definition of proportion in terms of Dedekind cuts. Krantz et. al. adapt Hölder’s theorem for their operationalization of quantitative concepts and construct measurement scales that are real-valued, but replace Dedekind’s axiom with the Archimedean axiom, which they see as better-suited to their empiricist interpretation. But even on a realist understanding of quantity, we argue, there are good reasons to doubt the assumption that classical physical quantities are genuinely continuous. Our paper first reproduces the results of Krantz et. al. in a realist context, where a quantity and its magnitudes are understood in terms of determinables and their determinates. Understanding a physical quantity (such as mass) as a determinable property emphasizes the metaphysical significance of representing it as having the structure of the reals. We then prove analogous representation and uniqueness theorems thus establishing that a determinable quantity constrained by the same qualitative axioms can be represented by the rational numbers. This shows that RTM methods do not inherently provide justification for representing a quantity as having the structure of the reals, and that the appearance of such justification can be attributed to stipulations of either continuity or uncountability of the non-numerical target of representation. We argue that, if the formal results of RTM are to inform a metaphysical view of quantity, then the conventionality of the choice of the real numbers as the representing structure needs to be explicitly justified.