The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, (...) contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...) Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (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.

Laplace wondered about the minimal choice of initial variables and parameters corresponding to a well-posed initial value problem. Discussions of Laplace’s problem in the literature have focused on choosing between spatiotemporal variables relative to absolute space or merely relative to other material bodies and between absolute masses or merely mass ratios. This paper extends these discussions of Laplace’s problem, in the context of Newtonian Gravity, by asking whether mass needs to be included in the initial state at all, or whether (...) a purely spatiotemporal initial state suffices. It is argued that mass indeed needs to be included; removing mass from the initial state drastically reduces the predictive and explanatory power of Newtonian Gravity. (shrink)

In this article, we present a new conception of internal relations between quantity tropes falling under determinates and determinables. We begin by providing a novel characterization of the necessary relations between these tropes as basic internal relations. The core ideas here are that the existence of the relata is sufficient for their being internally related, and that their being related does not require the existence of any specific entities distinct from the relata. We argue that quantity tropes are, as determinate (...) particular natures, internally related by certain relations of proportion and order. By being determined by the nature of tropes, the relations of proportion and order remain invariant in conventional choice of unit for any quantity and give rise to natural divisions among tropes. As a consequence, tropes fall under distinct determinables and determinates. Our conception provides an accurate account of quantitative distances between tropes but avoids commitment to determinable universals. In this important respect, it compares favorably with the standard conception taking exact similarity and quantitative distances as primitive internal relations. Moreover, we argue for the superiority of our approach in comparison with two additional recent accounts of the similarity of quantity tropes. (shrink)

The concept system around 'quantity' and 'quantity value' is fundamental for measurement science, but some very basic issues are still open on such concepts and their relation. This paper argues that quantity values are in fact individual quantities, and that a complementarity exists between measurands and quantity values. This proposal is grounded on the analysis of three basic 'equality' relations: (i) between quantities, (ii) between quantity values and (iii) between quantities and quantity values. A consistent characterization of (...) such concepts is obtained, which is then generalized to 'property' and 'property value'. This analysis also throws some light on the elusive concept of magnitude. (shrink)

The words in children's language learning environments are strongly predictive of cognitive development and school achievement. But how do we measure language environments and do so at the scale of the many words that children hear day in, day out? The quantity and quality of words in a child's input are typically measured in terms of total amount of talk and the lexical diversity in that talk. There are disagreements in the literature whether amount or diversity is the more critical (...) measure of the input. Here we analyze the properties of a large corpus of speech to children and simulate learning environments that differ in amount of talk per unit time, lexical diversity, and the contexts of talk. The central conclusion is that what researchers need to theoretically understand, measure, and change is not the total amount of words, or the diversity of words, but the function that relates total words to the diversity of words, and how that function changes across different contexts of talk. (shrink)

'Explanation, Quantity and Law' is a sustained elaboration and defence of a theory of explanation, called the instance view, that is able to deal with the characteristic aspects of physical science, such as the use of mathematics, the fact that errors of measurement are ubiquitous, and so forth. The book begins with a summary of 'new directions' in the theory of explanation and continues with a systematic account of the view that to explain is to show that something is an (...) instance of a law of nature. This embraces topics such as the following - Salmon’s casual-mechanistic account and the ontic conception of explanation; theories of quantity from Ellis to Bigelow-Pargetter to quantum mechanics; Armstrong on numerical laws and laws of nature as relations between quantities; explanation and the quantum state. (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)

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. (...) The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation. (shrink)

When discrepancies are discovered between the outcomes of different measurement procedures, two sorts of explanation are open to scientists. Either some of the outcomes are inaccurate or the procedures are not measuring the same quantity. I argue that, due to the possibility of systematic error, the choice between and is underdetermined in principle by any possible evidence. Consequently, foundationalist criteria of quantity individuation are either empty or circular. I propose a coherentist, model-based account of measurement that avoids the underdetermination problem, (...) and use this account to explain how scientists individuate quantities in practice. (shrink)

Gricean pragmatics. Saying vs. implicating ; Discourse and cooperation ; Conversational implicatures ; Generalised vs. particularised ; Cancellability ; Gricean reasoning and the pragmatics of what is said -- The standard recipe for Q-implicatures. The standard recipe ; Inference to the best explanation ; Weak implicatures and competence ; Relevance ; Conclusion -- Scalar implicatures. Horn scales and the generative view ; Implicatures and downward entailing environments ; Disjunction : exclusivity and ignorance ; Conclusion -- Psychological plausibility. Charges of psychological (...) inadequacy ; Logical complexity ; Abduction ; Incremental processing ; The intentional stance ; Alternatives ; Conclusion -- Nonce inference or defaults?. True defaults ; Strong defaultism ; Weak defaultism ; Contextualism ; Conclusion -- Intentions, alternatives, and free choice. Free choice ; Problems with the standard recipe ; Intentions first ; Free choice explained ; Comparing alternatives ; Two flavours of Q-implicature ; Conclusion -- Embedded implicatures : the problems. The problems ; Varieties of conventionalism ; Against conventionalism ; Conclusion -- Embedded implicatures : a Gricean approach. Disjunction ; Belief reports ; Factives and other presupposition inducers ; Indefinites ; Contrastive construals and lexical pragmatics ; Conclusion. (shrink)

Intermediate Quantitifiers presents and analyzes the logical and linguistic features of intermediate quantifiers, in a fashion typical of traditional logic. Intermediate quantifiers express logical quantities which fall between Aristotle's two quantities of categorical propositions - the universal and the particular. This book is the first to use traditional methods to integrate the logic and semantics of intermediate quantifiers with the two traditional quantities. Few, many and most express the most commonly referred to intermediate quantities, yet in (...) this book Peterson argues that an infinite number of additional quantities can be understood through a deeper examination of the logical nature of all intermediate quantifiers. Drawing on the work of Robert Carnes and taking a critical look at James McCawley's grammatical analysis, Peterson concludes that a new approach to foundations of mathematics based on the syllogistic logic of quantifiers is possible.Offering invaluable insights for those studying across areas of logic, linguistics, and the philosophy or semantics of natural language, this book presents the completion of an explanation for extending traditional syllogistic logic to handle intermediate quantifiers. (shrink)

This paper compares the development of the notion of continuous quantity in the work of Francisco Suárez and René Descartes. The discussion begins with a consideration of Suárez’s rejection of the view – common to ‘realists’ such as Thomas Aquinas and ‘nominalists’ such as William of Ockham – that quantity is inseparable from the extension of material integral parts. Crucial here is Suárez’s view that quantified extension exhibits a kind of impenetrability that distinguishes it from other kinds of extension. This (...) view sheds considerable light on initially obscure remarks on impenetrability in Descartes’ late correspondence with Henry More. Though Descartes differs from Suárez and other major scholastic figures in his understanding of the relation of quantity to material substance, he nonetheless requires in the end some version of the Suárezian distinction between quantified and unquantified extension. (shrink)

In this essay I propose a new measure of social welfare. It captures the intuitive idea that quantity, quality, and equality of individual welfare all matter for social welfare. More precisely, it satisfies six conditions: Equivalence, Dominance, Quality, Strict Monotonicity, Equality and Asymmetry. These state that i) populations equivalent in individual welfare are equal in social welfare; ii) a population that dominates another in individual welfare is better; iii) a population that has a higher average welfare than another population is (...) better, other things being equal; iv) the addition of a well-faring individual makes a population better, whereas the addition of an ill-faring individual makes a population worse; v) a population that has a higher degree of equality than another population is better, other things being equal; and vi) individual illfare matters more for social welfare than individual welfare. By satisfying the six conditions, the measure improves on previously proposed measures, such as the utilitarian Total and Average measures, as well as different kinds of Prioritarian measures. (shrink)

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.

The concept system around ‘quantity’ and ‘quantity value’ is fundamental for measurement science, but some very basic issues are still open on such concepts and their relations. This paper proposes a duality between quantities and quantity values, a proposal that simplifies their characterization and makes it consistent.

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)

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as \({XH + \mathbb{P}(X|H)H^c}\) ; in particular (the indicator of) a conditional event E|H is looked at as EH + P(E|H)H c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give (...) a meaning to the sum X|H + Y|K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold. (shrink)

We give derivations of two formal models of Gricean Quantity implicature and strong exhaustivity in bidirectional optimality theory and in a signalling games framework. We show that, under a unifying model based on signalling games, these interpretative strategies are game-theoretic equilibria when the speaker is known to be respectively minimally and maximally expert in the matter at hand. That is, in this framework the optimal strategy for communication depends on the degree of knowledge the speaker is known to have concerning (...) the question she is answering. In addition, and most importantly, we give a game-theoretic characterisation of the interpretation rule Grice (formalising Quantity implicature), showing that under natural conditions this interpretation rule occurs in the unique equilibrium play of the signalling game. (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)

Every day, in many situations, we use expressions which seem only vaguely to provide us with information. The weather forecaster tells us that "some showers are likely in Northern regions during the night", a statement which is vague with respect to number of showers, location, and time. Yet such messages are informative, and often it is not possible for the producer of the message to be more precise. A tutor tells his students that "only a few students fail their exams (...) outright". This does not give a precise incidence. Yet it might be equally misleading to do so. For example, to say that 12% failed outright last year says nothing about other years, while to say an average of 8% over the last five years says nothing about variability. We argue that a precise, numerical statement can be sometimes more misleading in reality than a vague statement. Many researchers in psychology have attempted to capture the meaning of quantities by relating them to scales of quantity. The book explores this idea in detail and shows with original studies how these expressions also serve to control attention and to convey information about the expectations held by those involved in the communication. The book works towards a psychological theory of the meaning of quantifiers and similarly vague terms. New links are drawn between formal theories of quantification and psychological experimentation. (shrink)

On which philosophical foundations is the attribution of numerical magnitudes to qualitative phenomena based? That is, what is the philosophical basis for attributing, through measurement operations, numbers to empirical qualities that our senses perceive in the outside world? This question, nowadays rarely addressed in such a way, actually refers to an old debate about the quantification of qualities. A historical analysis reveals that it was a major issue in the “context of discovery” of the first attempts to mathematize new fields (...) of knowledge, whereas the “context of justification” leading our contemporary perspective on such episodes often overlooks such problems. As a consequence, little attention has so far been given to tracing its emergence back to its original context, leaving in the shadows an important debate about measurement. In this respect, the work of William Stanley Jevons is worth analysis. As a preliminary step toward his ambitious project to mathematize economics, he first attempted to understand the mechanism underlying measurement in physics, through a very innovative approach. Its main merit consisted in trying to overcome the Aristotelian distinction between quantities and qualities. In this article, I will strive to highlight the significance of his work for the history of philosophy of science. (shrink)

Quantities are naturally viewed as functions, whose arguments may be construed as situations, events, objects, etc. We explore the question of the range of these functions: should it be construed as the real numbers (or some subset thereof)? This is Carnap's view. It has attractive features, specifically, what Carnap views as ontological economy. Or should the range of a quantity be a set of magnitudes? This may have been Helmholtz's view, and it, too, has attractive features. It reveals the (...) close connection between measurement and natural law, it makes dimensional analysis intelligible, and explains the concern of scientists and engineers with units in equations. It leaves the philosophical problem of the relation between the structure of magnitudes and the structure of the reals. What explains it? And is it always the same? We will argue that on the whole, construing the values of quantities as magnitudes has some advantages, and that (as Helmholtz seems to suggest in "Numbering and Measuring from an Epistemological Viewpoint") the relation between magnitudes and real numbers can be based on foundational similarities of structure. (shrink)

We give derivations of two formal models of Gricean Quantity implicature and strong exhaustivity in bidirectional optimality theory and in a signalling games framework. We show that, under a unifying model based on signalling games, these interpretative strategies are game-theoretic equilibria when the speaker is known to be respectively minimally and maximally expert in the matter at hand. That is, in this framework the optimal strategy for communication depends on the degree of knowledge the speaker is known to have concerning (...) the question she is answering. In addition, and most importantly, we give a game-theoretic characterisation of the interpretation rule Grice (formalising Quantity implicature), showing that under natural conditions this interpretation rule occurs in the unique equilibrium play of the signalling game. (shrink)

Recent research has suggested that the Pirahã, an Amazonian tribe with a number-less language, are able to match quantities > 3 if the matching task does not require recall or spatial transposition. This finding contravenes previous work among the Pirahã. In this study, we re-tested the Pirahãs’ performance in the crucial one-to-one matching task utilized in the two previous studies on their numerical cognition, as well as in control tasks requiring recall and mental transposition. We also conducted a novel (...) quantity recognition task. Speakers were unable to consistently match quantities > 3, even when no recall or transposition was involved. We provide a plausible motivation for the disparate results previously obtained among the Pirahã. Our findings are consistent with the suggestion that the exact recognition of quantities > 3 requires number terminology. (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)

Since summer 2019 there is a new document that defines what in science should be regarded as being one second, one meter, and one kilogram, respectively. It is the ninth edition of the SI Brochure. Compared with older editions, a new definitional approach has been used. The seven base units are now defined by being directly related to a so-called defining constant. The paper discusses the second, the meter, and the kilogram. One odd salient, but nonetheless not discussed, feature of (...) the formulations of the definitions is that, from a purely linguistic point of view, they are circular. That is, symbols for what is to be defined appear also in the defining expressions. However, underneath the circularities there are, appearances notwithstanding, substantially non-circular definitions. The aim of the paper is to uncover these never before presented definitional structures. It is made by means of the two notions absolutely discrete quantities and metrical coordinative definitions. When the true definitional structures become visible, two important substantial metrological facts can be seen. First, the new definition of the kilogram is very much in need of further discussion. Second, contrary to what the Brochure hints at, there is no need to try to exchange the atomic parameter used in the definition of the second. (shrink)

The article shows that strategic quantity competition can be characterized by behavioral heterogeneity, once competing firms are allowed in a pre-market stage to optimally choose the behavioral rule they will follow in their strategic choice of quantities. In particular, partitions of the population of identical firms in which some of them are profit maximizers while others follow an alternative criterion, turn out to be deviation-proof equilibria both in simultaneous and sequential game structures. Our findings that in a strategic framework (...) heterogeneous behavioral rules may be consistent with individual incentives is a first attempt to provide a game-theoretic microfoundation of heterogeneity. (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)

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 essay explores various problematical aspects of Descartes’ conservation principle for the quantity of motion (size 3 speed), particularly its largely neglected “dual role” as a measure of both durational motion and instantaneous “tendencies towards motion.” Overall, an underlying non‐local, or “holistic,” element of quantity of motion (largely derived from his statics) will be revealed as central to a full understanding of the conservation principle’s conceptual development and intended operation; and this insight can be of use in responding to some (...) of the recent and traditional criticisms of Descartes’ physics. (shrink)

John C. Hall; III—Quantity of Pleasure, Proceedings of the Aristotelian Society, Volume 67, Issue 1, 1 June 1967, Pages 35–52, https://doi.org/10.1093/aristotel.

Quantum Mechanics, and apparently its successors, claim that there are minimum quantities by which objects can differ, at least in some situations: electrons can have various “energy levels” in an atom, but to move from one to another they must jump rather than move via continuous variation: and an electron in a hydrogen atom going from -13.6 eV of energy to -3.4 eV does not pass through states of -10eV or -5.1eV, let along -11.1111115637 eV or -4.89712384 eV.

In Shona , bare plurals and bare singulars seem to have different scope possibilities with respect to a class of modifiers which I term “scopeless quantity words” few’, and ose ‘all’). I argue that this is due to two factors. First, the scopeless quantity words are intersective modifiers rather than quantifying determiners, so that DPs containing them denote entities rather than generalised quantifiers. Second, transitive sentences involving plural arguments are usually interpreted using the **-operator, which gives a cumulative reading; the (...) apparent wide and narrow scope readings of bare plurals are both subcases of this reading, but a narrow scope reading for bare singulars cannot be subsumed by the cumulative reading. I also discuss the scope behaviour of a quantifying determiner, oga-oga ‘every’, which shows a subject/object asymmetry: in subject position, it appears to show the expected scopal ambiguity, but in object position, it appears to have the same readings as a scopeless quantity word; I argue that this is because of the location of existential closure. This adds to the evidence suggesting that many words giving quantity are intersective modifiers rather than quantifying determiners or adverbial quantifiers. (shrink)

Despite changeability of the world, the human mind also ponders on those quantities that remain constant over time. This was the case in ancient times, in the middle ages, and the same applies in modern physics. This paper discusses i.a. Zenon paradoxes, the principle of inertia, and the Emma Noether theorem, ending with the modern, so-called Zeno’s quantum effect. The foot-notes concern the ancient “Achilles” paradox, spot speed, as well as some of the facts taken out of the life-history (...) of Emma Noether, as well as the example of applying her theorem. (shrink)

Despite changeability of the world, the human mind also ponders on those quantities that remain constant over time. This was the case in ancient times, in the middle ages, and the same applies in modern physics. This paper discusses i.a. Zenon paradoxes, the principle of inertia, and the Emma Noether theorem, ending with the modern, so-called Zeno’s quantum effect. The foot-notes concern the ancient “Achilles” paradox, spot speed, as well as some of the facts taken out of the life-history (...) of Emma Noether, as well as the example of applying her theorem. (shrink)

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)

Grice's Quantity maxims have been widely misinterpreted as enjoining a speaker to make the strongest claim that she can, while respecting the other conversational maxims. Although many writers on the topic of conversational implicature interpret the Quantity maxims as enjoining such volubility, so construed the Quantity maxims are unreasonable norms for conversation. Appreciating this calls for attending more closely to the notion of what a conversation requires. When we do so, we see that eschewing an injunction to maximal informativeness need (...) not deprive us of any ability to predict or explain genuine cases of implicature. Crucial to this explanation is an appreciation of how what a conversation, or a given stage of a conversation, requires, depends upon what kind of conversation is taking place. I close with an outline of this dependence relation that distinguishes among three importantly distinct types of conversation. (shrink)

Resemblances obtain not only between objects but between properties. Resemblances of the latter sort - in particular resemblances between quantitative properties - prove to be the downfall of a well-known theory of universals, namely the one presented by David Armstrong. This paper examines Armstrong's efforts to account for such resemblances within the framework of his theory and also explores several extensions of that theory. All of them fail.

Kant's obscure essay entitled An Attempt to Introduce the Concept of Negative Quantities into Philosophy has received virtually no attention in the Kant literature. The essay has been in English translation for over twenty years, though not widely available. In his original 1983 translation, Gordon Treash argues that the Negative Quantities essay should be understood as part of an ongoing response to the philosophy of Christian Wolff. Like Hoffmann and Crusius before him, the Kant of 1763 is at (...) odds with the Leibnizian-Wolffian tradition of deductive metaphysics. He joins his predecessors in rejecting the assumption that the law of contradiction alone can provide proof of the principle of sufficient reason: -/- In his rejection of the possibility of deducing all philosophic truth from the law of contradiction, however, and in the clear recognition that this impossibility has immediate consequences for defense of the law of sufficient reason, Kant's work most definitely and positively constitutes a line of succession from Hoffmann and Crusius (Treash, 1983, p. 25). -/- The recognition that Kant's Negative Quantities essay is part of a response to the tradition of deductive metaphysics is, without a doubt, an important contribution to the Kant literature. However, there is still more to be said about this neglected essay. The full significance of the paper becomes known through its ties to a second, empiricist line of succession. Clues to this second line of succession can be found in Kant's prefatory remarks concerning Euler's 1748 Reflections on Space and Time and Crusius' 1749 Guidance in the Orderly and Careful Consideration of Natural Events. As I will show, these prefatory remarks suggest a reading of Kant's Negative Quantities paper that reaches beyond German deductive metaphysics to engage a debate regarding the application of mathematics in philosophy initiated by George Berkeley. (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)

John C. Hall; III—Quantity of Pleasure, Proceedings of the Aristotelian Society, Volume 67, Issue 1, 1 June 1967, Pages 35–52, https://doi.org/10.1093/aristotel.

Thomists do not have an account of how modern mathematics relates to classical mathematics or more generally fits into the Aristotelian hierarchy of sciences. Rather than treat primarily of Aquinas’s theses on mathematical abstraction, I turn to considering what modern mathematics is in itself, seen from a broadly classical perspective. I argue that many modern quantities can be considered to be, not quantities as such or in themselves, but derived quantities, i.e., quantities that can be defined (...) wholly in terms of the principles of number or magnitude. I also interpret the parts of modern mathematics that study quantitative change as being properly-speaking parts of natural philosophy, for example, probability theory, statistics, calculus, etc. In conclusion, I consider the place that quantity as such has in the order of the world and why we should expect the world to be highly mathematical, as we have found it to be. (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)

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)

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)

Any discipline of human knowledge is characterized by three fundamental elements: the complexity of its content, the method used for its elaboration, and the language used for its expression. This article argues that any method for making knowledge is a particular combination of three main components that we can call science, art, and revelation. The right combination depends on the complexity of the slice of reality that we wish to understand in each case. Is there a relationship between the quantity (...) and quality of a particular piece of knowledge and the quantity and quality of its eventual audience? Such a relationship serves, I believe, to avoid certain old misunderstandings. (shrink)

Is Big Data science a whole new way of doing research? And what difference does data quantity make to knowledge production strategies and their outputs? I argue that the novelty of Big Data science does not lie in the sheer quantity of data involved, but rather in the prominence and status acquired by data as commodity and recognised output, both within and outside of the scientific community and the methods, infrastructures, technologies, skills and knowledge developed to handle data. These developments (...) generate the impression that data-intensive research is a new mode of doing science, with its own epistemology and norms. To assess this claim, one needs to consider the ways in which data are actually disseminated and used to generate knowledge. Accordingly, this article reviews the development of sophisticated ways to disseminate, integrate and re-use data acquired on model organisms over the last three decades of work in experimental biology. I focus on online databases as prominent infrastructures set up to organise and interpret such data and examine the wealth and diversity of expertise, resources and conceptual scaffolding that such databases draw upon. This illuminates some of the conditions under which Big Data needs to be curated to support processes of discovery across biological subfields, which in turn highlights the difficulties caused by the lack of adequate curation for the vast majority of data in the life sciences. In closing, I reflect on the difference that data quantity is making to contemporary biology, the methodological and epistemic challenges of identifying and analysing data given these developments, and the opportunities and worries associated with Big Data discourse and methods. (shrink)

A discussion of Suarez's views on continuous quantity in the context of his place in the history of philosophy. The paper raises issues about conceptual change in intellectual history. It advances original interpretations of Aristotle and Suarez on continuous quantity.