What are physical quantities, and in particular, what makes them quantitative? This book presents an original answer to this question through the novel position of substantival structuralism, arguing that quantitativeness is an irreducible feature of attributes, and quantitative attributes are best understood as substantival structured spaces.

A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...) unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world. (shrink)

Quantities like mass and temperature are properties that come in degrees. And those degrees (e.g. 5 kg) are properties that are called the magnitudes of the quantities. Some philosophers (e.g., Byrne 2003; Byrne & Hilbert 2003; Schroer 2010) talk about magnitudes of phenomenal qualities as if some of our phenomenal qualities are quantities. The goal of this essay is to explore the anti-physicalist implication of this apparently innocent way of conceptualizing phenomenal quantities. I will first argue (...) for a metaphysical thesis about the nature of magnitudes based on Yablo’s proportionality requirement of causation. Then, I will show that, if some phenomenal qualities are indeed quantities, there can be no demonstrative concepts about some of our phenomenal feelings. That presents a significant restriction on the way physicalists can account for the epistemic gap between the phenomenal and the physical. I’ll illustrate the restriction by showing how that rules out a popular physicalist response to the Knowledge Argument. (shrink)

According to comparativist theories of quantities, their intrinsic values are not fundamental. Instead, all the quantity facts are grounded in scale-independent relations like "twice as massive as" or "more massive than." I show that this sort of scale independence is best understood as a sort of metaphysical symmetry--a principle about which transformations of the non-fundamental ontology leave the fundamental ontology unchanged. Determinism--a core scientific concept easily formulated in absolutist terms--is more difficult for the comparativist to define. After settling on (...) the most plausible comparativist understanding of determinism, I offer some examples of physical systems that the comparativist must count as indeterministic although the relevant physical theory gives deterministic predictions. Several morals are drawn. In particular: comparativism is metaphysically contingent if true, and it is most natural for a comparativist to accept an at-at theory of motion. (shrink)

A simple interpretation of quantity calculus is given. Quantities are described as two-place functions from objects, states or processes (or some combination of them) into numbers that satisfy the mutual measurability property. Quantity calculus is based on a notational simplification of the concept of quantity. A key element of the simplification is that we consider units to be intentionally unspecified numbers that are measures of exactly specified objects, states or processes. This interpretation of quantity calculus combines all the advantages (...) of calculating with numerical values (since the values of quantities are numbers, we can do with them everything we do with numbers) and all the advantages of calculating with standardly conceived quantities (calculus is invariant to the choice of units and has built-in dimensional analysis). This also shows that the standard metaphysics and mathematics of quantities and their magnitudes is not needed for quantity calculus. At the end of the article, arguments are given that the concept of quantity as defined here is a pivotal concept in understanding the quantitative approach to nature. As an application of this interpretation of quantity calculus, an easy proof of dimensional homogeneity of physical laws is given. (shrink)

This dissertation draws upon historical studies of scientific practice and contemporary issues in the metaphysics and epistemology of science to account for the nature of physical quantities. My dissertation applies this integrated HPS approach to dimensional analysis—a logic for quantitative physical equations which respects the distinct dimensions of quantities (e.g. mass, length, charge). Dimensional analysis and its historical development serve both as subjects of study and as a sources for solutions to contemporary problems. The dissertation consists primarily (...) of three related papers on: (1) the methodological and metaphysical foundations of dimensional analysis, (2) the use of dimensional analysis in determining physical symmetries, (3) the use of dimensional analysis in securing metrological extension. (shrink)

My thesis is that there are good reasons for a philosophical account of measurement to deal primarily with the properties or magnitudes of objects measured, rather than with the objects themselves. The account I present here embodies both a realism about measurement and a realism about the existence of the properties involved in measurement. It thus provides an alternative to most current treatments of measurement, many of which are operationalistic or conventionalistic, and nearly all of which are nominalistic.1 This enables (...) the present account to give better explanations of a number of features of measurement and other aspects of science than competing accounts of measurement can, and to be more readily integrated into a realist account of natural laws and causation. It also illustrates a general strategy for combining a familiar and powerful approach to representation with intensional entities like properties, which I think can be useful for dealing with a number of philosophical problems. (shrink)

Newton's use of mathematics in mechanics was justified by him from his neo-platonician conception of the physical world that was going along with his «absolute, true and mathematical concepts» such as space, time, motion, force, etc. But physics, afterwards, although it was based on newtonian dynamics, meant differently the legitimacy of being mathematized, and this difference can be seen already in the works of eighteenth century «Geometers» such as Euler, Clairaut and d'Alembert (and later on Lagrange, Laplace and others). Despite (...) their inheritance of Newton's achievements, they understood differently the meaning and use of mathematical quantities for physics, in a way that was more neutral to metaphysics. The continental reception and assimilation of Newton's Principia had indeed occured as its budding onto Leibniz' calculus and a cartesian conception of rationality (spread in particular by the malebranchist disciples of Leibniz). This new thought of the legitimacy of mathematization is clearly at variance with Descartes' identification of physics with geometry, but it nevertheless can be traced back to Descartes' conception of magnitudes, as it was developed and analyzed from the notion of dimension in his Regulæ ad directionem ingenii (in particular, rule 14). This idea can be followed afterwards with further philosophical or mathematical specifications through authors such as Kant, Riemann and others. This inquiry into the original thought of magnitudes, and of physical magnitudes conceived through mathematization, leads us to suggest an extension of meaning for the concept of physical magnitude that puts emphasis on its relational and structural aspects rather than restraining it to a simple «numerically valued» acception. Such a broadening would have immediate implications on our comprehension of «non classical» aspects of contemporary physics in the quantum area and in dynamical systems. (shrink)

Metaphysics is sensitive to the conceptual tools we choose to articulate metaphysical problems. Those tools are a lens through which we view metaphysical problems; the same problems look different when we change the lens. There has recently been a shift to "postmodal" conceptual tools: concepts of ground, essence, and fundamentality. This shift transforms the debate over structuralism in the metaphysics of science and philosophy of mathematics. Structuralist theses say that patterns are "prior" to the nodes in the patterns. (...) In modal terms this would mean that the nodes cannot vary independently of the pattern. But its postmodal meaning is unclear. Does it mean, e.g., that a fundamental account of reality somehow speaks only of patterns? What would that mean? Three structuralist positions are examined through a postmodal lens: nomic/causal/dispositional essentialism, structuralism about individuals (e.g., structural realism, e.g., ante rem mathematical structuralism), comparativism about quantities. The question of when theories are equivalent, and how that impacts the debate over structuralism, is also discussed. (shrink)

The widely held picture of dynamical symmetry as surplus structure in a physical theory has many metaphysical applications. Here, I focus on its relevance to the question of which quantities in a theory represent fundamental natural properties.

In this paper, I propose to resolve the controversy over the derivation of the categories of quantity by spelling out three claims: (1) the three quantitative functions/forms of judgment (universal-particular-singular), qua synthetic categories of quantity (unity-plurality-totality), lawfully direct the determination of sensible manifolds as singular totalities, which (2) brings to light a specifically categorial type of judgmental activity, distinguishable from but presupposed by empirical judgments. This calls for (3) pairing the category of totality with the singular judgment and the category (...) of unity with the universal judgment (‘Order B’), rather than pairing the singular judgment with unity and the universal judgment with totality (‘Order A’). At stake in this controversy is our understanding of the categories’ normativity, their scope, and the division of labor between intuitions and categories in individuating objects. (shrink)

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in (...) Posterior analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers. (shrink)

This paper aims to discuss two realist conceptions about causation in the light of the general theory of relativity. I first consider the conserved quantity of causation, which explicitly relies on the energy conservation principle. Such principle is however problematic within GTR, mainly because of the dynamical nature of the spacetime structure itself. I then turn to the causal theory of properties, according to which properties are such that insofar as they are certain qualities, they are powers to produce certain (...) effects. In order to be compatible with GTR, such theory has to assume non-trivial global conditions on the spacetime structure; such assumptions seem to deprive the „singularist‟ non-Humean feature of this theory of causation. The question of the possible causal nature of spacetime properties is addressed in the conclusion. (shrink)

Physicists use different theories to describe the world on different scales. In particular, they use the standard model of particle physics at very high energies, but move to various effective field theories, such as quantum electrodynamics, when modelling lower energy scattering processes. One way to explain this methodological fact is pragmatic in spirit. According to this view, physicists move to an effective field theory at lower energies in order to extract predictions and qualitative understanding which would be difficult or impossible (...) to extract directly from a more fundamental theory. By contrast, another way of accounting for the methodological data is metaphysical in spirit. On this view, the reason that physicists use different theories at different scales is that the world actually exhibits different nomic structure on different scales. These two positions have recently been thrown into sharp relief in a debate between Woodward (2016) and Batterman (2021), with Woodward taking the pragmatist line and Batterman the metaphysical line. One difficulty with the metaphysical answer is that its content is unclear: in what sense exactly could an effective field theory provide a better representation of the nomic structure of the world in its domain of applicability than our world’s fundamental theory? This talk attempts to provide an answer to this question by developing a simple metaphysical model. I start by assuming a set of possible worlds which are nomically possible according to some fundamental theory (I suggest that one is free to adopt either a Humean or Non-Humean view of these fundamental nomic necessities). I then introduce an equivalence relation between possible worlds: two worlds are scale-E equivalent if they assign the same values to physical quantities below energy scale E (up to some finite level of precision P). The key question now is whether this relation induces equivalence classes on the set of nomically possible worlds. I argue that in the case of current quantum field theories this does indeed occur, as evidenced by the possibility of “integrating out” high energy degrees of freedom in the renormalisation group framework. The formation of equivalence classes in the fundamental theory's possibility space indicates that some of its high energy features do not make a difference to its low energy dynamical behaviour. This suggests the following reading of the claim that an effective field theory provides a better representation of the nomic structure of the world in its domain of applicability: the solutions of the effective field theory stand in one-to-one correspondence with equivalence classes induced by the scale-E equivalence relation, whereas the fundamental theory overcounts the physical possibilities which are relevant at scale E. In this sense, the effective field theory provides a more natural representation of low energy physics than the fundamental theory. I conclude by explaining how this metaphysical model offers a reduction compatible understanding of multiple realisability, once again pointing to renormalisation group results in quantum field theory to support this. (shrink)

Immanuel Kant's Metaphysical Foundations of Natural Science (1786) provides metaphysical foundations for the application of mathematics to empirically given nature. The application that Kant primarily has in mind is that achieved in Isaac Newton's Principia (1687). Thus, Kant's first chapter, the Phoronomy, concerns the mathematization of speed or velocity, and his fourth chapter, the Phenomenology, concerns the empirical application of the Newtonian notions of true or absolute space, time, and motion. This paper concentrates on Kant's second and third chapters—the Dynamics (...) and the Mechanics, respectively—and argues that they are best read as providing a transcendental explanation of the conditions for the possibility of applying the (mathematical) concept of quantity of matter to experience. Kant again has in mind the empirical measures of this quantity that Newton fashions in the Principia, and he aims to make clear, in particular, how Newton achieves a universal measure for all bodies whatsoever by projecting the static quantity of terrestrial weight into the heavens by means of the theory of universal gravitation. Kant is not attempting to prove a priori what Newton has established empirically but, rather, to clarify the character of Newton's mathematization by building Newton's empirical measures into the very concept of matter that is articulated in the Metaphysical Foundations. (shrink)

The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after (...) measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question. Number and being (by the meditation of time), the natural and artificial turn out to be not more than different hypostases of a single common essence. This implies some kind of neo-Pythagorean ontology making related mathematics, physics, and technics immediately, by an explicit mathematical structure. (shrink)

According to one prominent view, current metaphysics is hopelessly disconnected from the implications of modern science and as a result should be abandoned forthwith (Ladyman and Ross 2007). Others have taken a more conciliatory stance, suggesting that the metaphysicians’ toolbox may yet yield devices that could prove useful to the philosopher of science (French and McKenzie 2012). In this book, Sider aims to contribute to the metaphysics of science by setting out an array of such tools and indicating (...) which are up to the job. More particularly, he wants to demonstrate the usefulness of certain ‘postmodal’ concepts, such as fundamentality, which is a promising tack, given that (Callender 2011), for example, has argued that it was the modal turn initiated by the likes of Kripke that is ultimately responsible for the disconnect. To illustrate his overall claim that the tools we deploy may help frame the issues that we take to be central to the metaphysics of science, Sider chooses to analyse various forms of structuralism, the view that we should take structure to be primary and the relevant entities to be secondary : nomic essentialism, which takes properties to be ‘bound up’ with laws (ibid., p. 23); structural realism, which holds, in its epistemic form, that all that we know about the world is its structure and, in the ontic alternative, that the world is, in some sense, structure; and comparativism about quantities, whereby quantities are ascribed to objects comparatively or relationally (ibid., p. 122). His conclusions are that no postmodal sense can be made of the first two, and so much the worse for them, whereas comparativism about quantities can unproblematically be stated in such terms (ibid., p. 119), and indeed, when it comes to ‘fundamentality’, at least, may well be this concept’s ‘Waterloo’ (ibid., p. 166). The worry some philosophers of science will have is that Sider’s arguments are driven more by his metaphysics than the relevant science and the issues that he identifies don’t quite map onto those they take to be central. (shrink)

This paper deals with Suárez's theory of extension and continuous quantity, as it is discussed in the Metaphysical Disputations and as a possible source for Descartes's concept of res extensa. In a first part of the paper, I analyse Suárez' account of divisibility and extension in a comparison with the Dominicans', Scotus and Fonseca's, and Ockham's. In the light of this analysis, Suárez's most original contribution seems being the claim that material composites have integral parts 'entitatively' extended (partem extra partem) (...) independently from categorial quantity. Such a theory allows Suárez to merge the Dominicans', the Scotists' and the Ockhamists' account, as the integral parts fund the internal but pre-quantitative divisibility of body, which is the source of quantity. In a second part of the paper, I deal with the reception of Suárez’s theory (especially in Rubio, Arriaga, Araujo, Eustachius and Abra de Raçonis), as well as with Descartes’ concept of res extensa. I conclude that Suárez’s account has been mainly criticized by the seventeenth Century Aristotelianism, as well as by Descartes. Despite that, Suárez’s idea of non-quantitatively extended parts had an interesting reception and effect in the debate, and especially in the (dominant) Scotistic context. (shrink)

An unpublished manuscript on eucharistic metaphysics by Thomas White (1593–1676) supplies new information about his contributions to philosophy and theology—especially his irenic efforts to find middle ground between traditional Aristotelian views and challenges from the new mechanical philosophy. The work by White studied here, “A Discourse Concerning the Eucharist,” sheds light on his other writings and is illuminated by them. Substance, quantity, place, and accident were the main philosophical issues at stake in White's attempt to give a reasoned account (...) of controversial religious and theological issues, especially transubstantiation and the real presence. (shrink)

Ever since David Lewis argued for the indispensibility of natural properties, they have become a staple of mainstream metaphysics. This dissertation is a critical examination of natural properties. What roles can natural properties play in metaphysics, and what structure do natural properties have? In the first half of the dissertation, I argue that natural properties cannot do all the work they are advertised to do. In the second half of the dissertation, I look at questions relating to the (...) structure of natural properties. I argue that the metric structure of fundamental quantitative properties cannot be reduced to mereological structure, and I argue that the simplistic picture of natural properties as monadic must be abandoned in light of theories of fundamental physics. (shrink)

Despite its position in an introductory work to logic, the account of continuity presented by Abū Naṣr Al-Fārābī in his paraphrase of Aristotle’s Categories is apparently even less accessible to the beginner than Aristotle’s original. This is in part because Al-Fārābī integrated elements of the accounts of continuity in Aristotle’s Physics and Metaphysics into an account mainly derived from Aristotle’s Categories. While Al-Fārābī’s account chiefly follows Aristotle’s Categories 6 in describing a continuous object that can be divided into parts (...) with a shared boundary, it borrows the terminology of limits from Physics 5 and the notion of quantity as divisible into its parts from Metaphysics 5.13. In doing so, Al-Fārābī defines continuity using a notion of a limit without determining whether or not it was part of the continuous quantity. This definition could also accommodate atomists, since it would work even if the limit were an atom or several atoms that were part of the continuous object. The broadness of this definition is probably intended to allow students of logic who may have atomist tendencies to accept the account of quantity in the categories tradition, even though they may not be ready to reject atomism until after studying physics. (shrink)

Quantities are properties and relations which exhibit "quantitative structure". For physical quantities, this structure can impact the non-quantitative world in different ways. In this paper I introduce and motivate a novel distinction between quantities based on the way their quantitative structure constrains the possible mereological structure of their instances. Specifically, I identify a category of “properly extensive” quantities, which are a proper sub-class of the additive or extensive quantities. I present and motivate this distinction using (...) two case studies of successful physical measurements.. I argue that the best explanation for the success of the length measurement requires us to adopt a notion of proper extensiveness, which is distinct from mere additivity, which is exhibited by inertial mass. I further discuss the consequences of this distinction, sketching an application of proper extensiveness for the project of producing a non-mathematical and non-metrical reductive metaphysics of quantity. (shrink)

I analyze the meaning of mass in Newtonian mechanics. First, I explain the notion of primitive ontology, which was originally introduced in the philosophy of quantum mechanics. Then I examine the two common interpretations of mass: mass as a measure of the quantity of matter and mass as a dynamical property. I claim that the former is ill-defined, and the latter is only plausible with respect to a metaphysical interpretation of laws of nature. I explore the following options for the (...) status of laws: Humeanism, primitivism about laws, dispositionalism, and ontic structural realism. (shrink)

Pt. I. Zeno and the metaphysics of quantity. Zeno's paradox of measure -- Tractarian nominalism -- Logical atoms and combinatorial possibility -- Strict coherence, sigma coherence, and the metaphysics of quantity -- pt. II. Coherent degrees of belief. Higher-order degrees of belief -- A mistake in dynamic coherence arguments? -- Dynamic coherence and probability kinematics -- Updating, supposing, and MAXENT -- The structure of radical probabilism -- Diachronic coherence and radical probabilism -- pt. III. Induction. Carnapian inductive logic (...) for Markov chains -- Carnapian inductive logic and Bayesian statistics -- Bayesian projectability. (shrink)

This paper deals with the metaphysics of the notion of quantity in the philosophy of Leibniz, and its aim is to defend the following bi-conditional: for any object x, x has a certain quantity if and only if x has a (metaphysical) limit or a bound. The direction from left to right is justified in §3, while in §4 I develop an argument to justify the direction from right to left. Since the bi-conditional links the metaphysical notion of limit (...) to the mathematical notion of quantity (and in this way it links Leibniz’s metaphysics with his conception of Mathesis Universalis), it allows the use of metaphysics to clarify the features of the mathematical notion of quantity. This task is accomplished in §5 and §6. Finally, §7 discusses a possible objection. (shrink)

This book explores and compares the reflections on space and quantity found in the works of five philosophers: Spinoza, Leibniz, Bergson, Whitehead, and Deleuze. What unites these philosophers is a series of metaphysical concerns rooted in 17th-century rationalism and embraced in 20th-century philosophies of process and difference. At the heart of these concerns is the need for a comprehensive metaphysical account of the diversity and individuality of things. This demand leads to a shared critique of Cartesian and Newtonian conceptions of (...) space. The most problematic aspect of those notions of space is homogeneity. In essence, uniform space fails to explain the differences between locations, thus violating the principle of sufficient reason. Cartesian and Newtonian theories of space thereby fail to meet the metaphysical requirement for explaining diversity and individuality. The traditional concept of quantity faces similar issues. Motivated by these problems, these five philosophers develop an alternative conception of space and quantity. By examining these theories, the book sheds new light on an unexplored relation between rationalism and 20th-century Continental philosophy. A Geometry of Sufficient Reason will appeal to scholars and graduate students working in Continental philosophy, history of philosophy, metaphysics, and the history and philosophy of science. (shrink)

This article introduces and motivates the notion of a “properly extensive” quantity by means of a puzzle about the reliability of certain canonical length measurements. An account of these measurements’ success, I argue, requires a modally robust connection between quantitative structure and mereology that is not mediated by the dynamics and is stronger than the constraints imposed by “mere additivity.” I outline what it means to say that length is not just extensive but properly so and then briefly sketch an (...) application of proper extensiveness to the project of providing a reductive ground for metric quantitative structure. (shrink)

Building on the seminal work of Kit Fine in the 1980s, Leon Horsten here develops a new theory of arbitrary entities. He connects this theory to issues and debates in metaphysics, logic, and contemporary philosophy of mathematics, investigating the relation between specific and arbitrary objects and between specific and arbitrary systems of objects. His book shows how this innovative theory is highly applicable to problems in the philosophy of arithmetic, and explores in particular how arbitrary objects can engage with (...) the nineteenth-century concept of variable mathematical quantities, how they are relevant for debates around mathematical structuralism, and how they can help our understanding of the concept of random variables in statistics. This fully worked through theory will open up new avenues within philosophy of mathematics, bringing in the work of other philosophers such as Saul Kripke, and providing new insights into the development of the foundations of mathematics from the eighteenth century to the present day. (shrink)

The representational theory of measurement provides a collection of results that specify the conditions under which an attribute admits of numerical representation. The original architects of the theory interpreted the formalism operationally and explicitly acknowledged that some aspects of their representations are conventional. There have been a number of recent efforts to reinterpret the formalism to arrive at a more metaphysically robust account of physical quantities. In this paper we argue that the conventional elements of the representations afforded by (...) the representational theory of measurement require careful scrutiny as one moves toward such an interpretation. To illustrate why, we show that there is a sense in which the very number system in which one represents a physical quantity such as mass or length is conventional. We argue that this result does not undermine the project of reinterpreting the representational theory of measurement for metaphysical purposes in general, but it does undermine a certain class of inferences about the nature of physical quantities that some have been tempted to draw. (shrink)

This book is about matter. It involves our ordinary concept of matter in so far as this deals with enduring continuants that stand in contrast to the occurrents or processes in which they are involved, and concerns the macroscopic realm of middle-sized objects of the kind familiar to us on the surface of the earth and their participation in medium term processes. The emphasis will be on what science rather than philosophical intuition tells us about the world, and on chemistry (...) rather than the physics that is more usually encountered in philosophical discussions. It is the everyday science of matter characterised by differences of chemical substance that constitutes individuated macroscopic bodies of geological and biological complexity—the continuous matter of macroscopic theory and described by mass terms. -/- The discussion might be characterised as descriptive metaphysics, being content, to paraphrase Strawson’s use of the term, to describe the structure of our scientific thought about the actual world. The method in the central chapters dealing with the nature of matter will be to pursue key steps in the historical development of scientific thought about chemical substance and related concepts with a view to tracing the emergence of a systematic ontological interpretation. Aristotle’s and the Stoics’ discussions of elements and mixtures, based on a continuous view of matter, bear some resemblance to modern, macroscopic conceptions and therefore have some interest as precursors to modern views as well as being of conceptual interest in their own right. Some of the issues they raised are still reflected in modern discussions, despite the considerable increase in complexity of the subject. Other ideas, such as the intimate connection between what modern science distinguishes as substance and phase, maintained their grip on the understanding of matter until the end of the eighteenth century. These are some of the important aspects of the properties of matter that have been neglected by linguistic concerns that have dominated the recent study of mass terms and which the present study seeks to redress. It will be of interest to see how the bounds of possibility are delimited by the laws governing the appropriate use of concepts refined for the purpose of describing the behaviour of matter. But the concern with modality will not stretch to venturing into the realms of metaphysical possibility governed by philosophical speculation about individual essences and underlying natures. -/- Like many contemporary discussions of material objects, this one relies heavily on mereology. Unlike several such discussions, this one adheres to the classical principles of mereology governing the usual dyadic relations of part, overlapping, etc. and the operations of sum, product and difference. These principles apply to the mereological structure of regions of space, intervals of time, processes and quantities of matter. Material objects that gain and lose matter are not understood to gain and lose parts and don’t call for a modification of the classic dyadic mereological relations to triadic relations with the introduction of a third term referring to time. Rather, a distinction is drawn between quantities of matter, which don’t gain or lose parts over time, and individuals, which are typically constituted of different quantities of matter at different times. The proper treatment of the temporal aspect of the features of material objects is a central issue in this book. The topics falling under this heading are addressed by investigating the conditions governing the application of predicates relating time and other entities. Of particular interest here are relations between quantities of matter and times expressing substance kind, phase and mixture. (shrink)

Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity.

Review: J. E. Wolff. The metaphysics of quantity. New York: Oxford University Press, 2020. 240 pp, $72.00 HB.

In this paper we explore the interpretation of quantity expressions in Yudja, an indigenous language spoken in the Amazonian basin, showing that while the language allows reference to exact cardinalities, it does not generally allow reference to exact measure values. It does, however, allow non-exact comparison along continuous dimensions. We use this data to argue that the grammar of exact measurement is distinct from a grammar allowing the expression of exact cardinalities, and that the grammar of counting and the grammar (...) of measurement may use numerals with different, though related interpretations. As Yudja shows, the language of measurement is not automatically acquired along with the knowledge of exact numeral expressions. We show that the ‘gap’ between prelinguistic intuitions about quantity in terms of numerosity and counting, which is bridged by the learning of language expressing exact cardinality, is paralleled by a similar gap between prelinguistic intuitions about quantity on a continuous dimension and measuring: this gap too must be bridged by language which expresses exact measure values. Our results suggest that the enculturation process by which we develop skills to perform abstract operations in the domain of measurement is (1) language dependent and (2) distinct from the process by which we learn to perform abstract calculations in the cardinal domain. (shrink)

This chapter examines four special problems involving properties whether universals or tropes. It looks at various accounts of relational facts, facts that involve properties relating two or more particulars. The chapter examines an important special case of relational facts: those that involve nonsymmetric or ordering relations. It focuses on structural properties, those relational properties that enable many things to form a single structure, like a group or a team. Finally, it considers the problem of measurable quantities. Monadism is the (...) view that all relational truths are grounded in the intrinsic character of substances taken individually. The most difficult problem for Leibniz's version of Monadism is giving a full and adequate theory of monadic representation. A monad's representing the world as being a certain way must consist of a set of intrinsic mental acts. The chapter surveys the various metaphysical options for Anti‐Monadism, and considers the most difficult test case: that of non‐symmetric relations. (shrink)

The principles of Kant's pure physics (conservation of quantity of matter, inertia, equality of action and reaction) are a priori in the same sense as are the principles of the understanding. We account for the empirical content of physics by showing that the pure principles operate as rules for generating wellformed empirical descriptions, and as rules for analysis of motion. The relationship between the metaphysics of matter and empirical descriptions is neither deductive, nor as loose as Buchdahl alleges. Belief (...) that a priori principles will apply to empirical cases requires acceptance of methodological presuppositions, chief among which is the principle of affinity. (shrink)

This paper introduces the reader to Meinong's work on the metaphysics of magnitudes and measurement in his Über die Bedeutung des Weber'schen Gesetzes. According to Russell himself, who wrote a review of Meinong's work on Weber's law for Mind, Meinong's theory of magnitudes deeply influenced Russell's theory of quantities in the Principles of Mathematics. The first and longest part of the paper discusses Meinong's analysis of magnitudes. According to Meinong, we must distinguish between divisible and indivisible magnitudes. He (...) argues that relations of distance, or dissimilarity, are indivisible magnitudes that coincide with divisible magnitudes called "stretches". The second part of the paper is concerned with Meinong's account of measurement as a comparison of parts. According to Meinong, since measuring consists in comparing parts only divisible magnitudes are directly measurable. Indivisible magnitudes can only be measured indirectly, by measuring the divisible stretches that coincide with them. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume's Metaphysical Musicians E. W. Van Steenburgh Having argued to the possible existence ofmathematical points, Hume concedes that checking for their actual existence is difficult because of the minuteness. He writes: the points, which enter into the composition of any line or surface, whether perceiv'd by the sight or touch, are so minute and soconfoundedwith eachother, that'tis utterly impossible for the mind to compute their number.1 He concludesthat theirminutenessrenderspoints (...) uselessforpurposes ofmeasurement. "We are sensible, that the addition or removal ofone of these minute parts, is not discernible either in the appearance or measuring" (T 48). This objection to his points leads Hume to view measuring, not in terms ofpoints, but asjuxtaposition ofobjects one to another or to a common measure, correcting apparent equality by art and instruments. And, he argues, correcting comes to an end when one runs out of art and instruments, although correcting, once set in motion, may go on at the level offancying after art and instruments have announced an end to it. This holds true not only for equality of quantities but of qualities as well. A musician finding his ear become every day more delicate, and correcting himself by reflection and attention, proceeds with the same act ofthe mind, even when the subject failshim, and entertains a notion ofa compleat tierce or octave, without being able to tell whence he derives his standard. (T 48-49) The metaphysical musician, according to Hume, is unable "to tell whence he derives his standard." This objection by Hume looks like it ismissingthe mark. The metaphysical musician could tell from whence he derives his standard. It is derived from sensible appearances. It is derived from the undifferentiated case which, to ears subsequently more discriminating, turns out to be two different notes, not one. The source ofthe standard is specified. What, then, does Hume mean when he states that metaphysical musicians cannot tell us from whence they derive their standard? An apparent standard, if it functions as a standard, has objects forits terms. For the standard is neither equality Volume XVIII Number 2 151 E. W. VAN STEENBURGH itself, whatever that might mean, nor an object; rather, the standard is "being equal to this object," whatever the object might be—in this case a musical note. In other words, an unspecified standard—a standard that is neither this nor that—is not a standard. It becomes a standardifand only ifit is specified for some object. The question now shifts to the object without which no standard is specified. It could be actual. It could be imaginary. Either way, it is an object. Yet, the metaphysical musicians rules that no object satisfies the standard. The outcome is contradiction. If you don't specify the standard, you don't have a standard; and ifyou do, you contradict the ruling that no object satisfies the standard. Hume does not explicitly allege contradiction. However, the other sideofthecontradictioncoin is an unspecifiedstandard. AndthisHume does allege. Hume sometimes does speak as ifan imagined musical note—the object in this case—is itselfa note. He speaks this way out ofdeference to his general theory of ideas as copies of impressions. However, the claim to contradiction, to incomprehensibility, does not depend on Hume's special doctrine of ideas copying impressions. For when we imagine objects, the objects we imagine, if we are consistent, are possible objects. So what we are saying, ifwe say that we can imagine equal objects is this: equal objects are impossible. Either, then, the metaphysical musician's imaginary correction, the imaginary trek toward absolute equahty, is incomprehensible because there is no standard—no objects—by which "correction"makes sense; orelse there is such a standard—object equality is possible—and the general rule that object equality is impossible is contradicted. Suppose we downgrade the a priori necessary rule to a contingent generalization about ordinary objects. Now it is not impossible, only unlikely, that one will find equal actual objects. Given a candidate for prototypicality, one imagines that one is mistaken, that the objects one thinks are equal are not, after all, equal, and soon. Here the only equal objects are always possible objects, never actual ones. One simply refuses to use... (shrink)

In the Critique of Pure Reason, Kant develops a metametaphysical view concerning the domain and source of a distinctively metaphysical cognition of objects of experience, which is given in terms of an analysis of our representational capacity for thought, namely, the understanding, regarding its sub-capacities and their constitutive abilities and acts. In the Analytic of Concepts, more precisely, in the Metaphysical and Subjective Deductions of the Categories, Kant develops an elaborate account of the content and formation of those metaphysical concepts (...) of an object in general through which there is genuine philosophical cognition of objects of experience, namely, the categories. Relating the Analytic of Concepts to the Doctrine of Method, I give a detailed reconstruction of this account in contrast to both empirical and mathematical concepts. I do so by establishing, for the first time (in English) that and how the representational contents of the categories, as general representations of objects, derive from fundamental abilities and acts of a synthesis of an empirical intuition that the understanding contributes to experiences of objects, which thus yield the original contents of the categories. In particular, I argue that the threefold synthesis of the fundamental act-types of the apprehension, reproduction, and recognition of a manifold of sense-impressions, presented in the A-Deduction, holds the key to understanding the origin of the categories, or concepts of an object in general, investigated in their Metaphysical Deduction. It turns out that the threefold synthesis, as it is exercised regarding a manifold of sense-impressions in an empirical intuition of an object, can indeed account for the representational contents of the categories of Quality (apprehension), Quantity (reproduction), Relation (recognition), and Modality (relating to the object through a manifold of sense-impressions). In this sense, the categories are “concepts of synthesis” (A80/B106) or “concepts of the synthesis of possible sensations” (A723/B751). (shrink)

Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately oriented, constructed (...) geometrical spaces is a proper subset of metaphysical space, thus, metaphysical space is infinite. Kant's paradoxical doctrine of metaphysical space is necessary to reconcile his empiricism with his transcendental idealism. (shrink)

This paper explores a specific problem within an important philosophical genre of the fourteenth century: the debates over the perfection of species. It investigates how the problem of defining limits for continuous magnitudes – a problem typical of Aristotelian physics – was integrated into these debates at the levels of genera, species, and individuals as these entities began to be conceptualized in quantitative terms. After explaining the emergence of this problem within fourteenth-century metaphysics, the paper examines the contributions of (...) three philosophers – Hugolinus of Orvieto, John of Ripa, and Paul of Venice – who offered varying solutions to the challenge of defining limits between species. It demonstrates that two primary solutions arose, inspired by continuous and discrete mathematical objects. It is shown that whereas Hugolinus of Orvieto advocates for a continuist model, John of Ripa proposes a discrete one. The last part of the paper examines Paul of Venice's hybrid approach, which combines elements from both models, facilitating a more comprehensive treatment of species particularly difficult to analyse, namely geometric figures. The conclusions of this comparative study underscore how profoundly the metaphysical reflections of the Middle Ages contributed to the analysis of the structure of the continuum and the extension of the notion of quantity to various objects. (shrink)

I defend the conserved quantity theory of causation against two objections: firstly, that to tie the notion of “cause” to conservation laws is impossible, circular or metaphysically counterintuitive; and secondly, that the conserved quantity theory entails an undesired notion of identity through time. My defence makes use of an important meta-philosophical distinction between empirical analysis and conceptual analysis. My claim is that the conserved quantity theory of causation must be understood primarily as an empirical, not a conceptual, analysis of causation.