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)

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)

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 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)

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.

'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)

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)

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)

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)

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.

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 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)

In quantity discrimination tasks, adults, infants and animals have been sometimes observed to process number only after all continuous variables, such as area or density, have been controlled for. This has been taken as evidence that processing number may be more cognitively demanding than processing continuous variables. We tested this hypothesis by training mosquitofish to discriminate two items from three in three different conditions. In one condition, continuous variables were controlled while numerical information was available; in another, the number (...) was kept constant and information relating to continuous variables was available; in the third condition, stimuli differed for both number and continuous quantities. Fish learned to discriminate more quickly when both number and continuous information were available compared to when they could use continuous information only or number only; there was no difference in the learning rate between the two latter conditions. Our results do not support the hypothesis that processing numbers imposes a higher cognitive load than processing continuous variables. Rather, they suggest that availability of multiple information sources may facilitate discrimination learning. (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)

The problem of the failure of value definiteness (VD) for the idea of quantity in quantum mechanics is stated, and what VD is and how it fails is explained. An account of quantity, called BP, is outlined and used as a basis for discussing the problem. Several proposals are canvassed in view of, respectively, Forrest's indeterminate particle speculation, the "standard" interpretation of quantum mechanics and Bub's modal interpretation.

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)

Gundel, Hedberg and Zacharski propose a framework whereby different referring forms conventionally signal different cognitive statuses on an implica-tional 'givenness hierarchy'. Interaction of the hierarchy with Grice's Maxim of Quantity gives rise to scalar implicatures which further constrain the choice among forms and their interpretations when necessary conditions for more than one form are met. Wilson and Matsui show that reference assignment for NPs introduced by the definite article is constrained within Relevance Theory by the automatic selection of an (...) interpretation that yields adequate contextual effects for minimal processing effort. This paper proposes that an optimal account of how people understand the intended referent of an NP must appeal to both the Givenness Hierarchy and Relevance Theory. Such an account still requires the first part of the Quantity Maxim, however, either as a socially based assumption or as part of the Principle of Relevance itself. (shrink)

When does more data help and when does it not in the sciences? In the past decade, this question has become central because of the phenomenon of Big Data. While these discussions started as a result of somewhat naive ideas that have been closely analyzed and mostly rejected in the philosophy of data, the question about the epistemic difference that more or less data makes still matters, especially in light of the impressive performance of data science and machine learning tools, (...) which seem to improve their outcomes when they are trained on large volumes of data. In several areas of the sciences, having more data is also connected to methodological and epistemic benefits and more generally something that research should strive toward. More data is often equated to better science: this elicits crucial questions about the epistemic value of the quantity of data. In this chapter, we discuss this problem in light of current discussions in the life and health sciences and the philosophy of data. (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)

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.

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)

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 (...) class='Hi'>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)

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.

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)

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 (a) science, (b) art, and (c) 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)

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)

QUALITY AND QUANTITY are fairly generally regarded as complementary terms, although the profound reason for their comple- mentarism is often far from being understood, this reason lying in the 'polar' correspondence referred to toward ...

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)

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)

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)

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 (...) class='Hi'>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)

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)

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)

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 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)

In a recent paper (1994) Wesley Salmon has replied to criticisms (e.g., Dowe 1992c, Kitcher 1989) of his (1984) theory of causality, and has offered a revised theory which, he argues, is not open to those criticisms. The key change concerns the characterization of causal processes, where Salmon has traded "the capacity for mark transmission" for "the transmission of an invariant quantity." Salmon argues against the view presented in Dowe (1992c), namely that the concept of "possession of a conserved (...) quantity" is sufficient to account for the difference between causal and pseudo processes. Here that view is defended, and important questions are raised about the notion of transmission and about gerrymandered aggregates. (shrink)

Quantities and units of measure provide an important means by which intelligent agents interact with the physical world. Although multiple ontologies for quantities and units of measure have been p...

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)

This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's ``first theorem'', in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. I emphasise that, for both frameworks, the theorem is underpinned by the idea of cyclic coordinates; and that the Hamiltonian theorem is more powerful. The Lagrangian theorem's main ``ingredient'', apart from cyclic coordinates, (...) is the rectification of vector fields afforded by the local existence and uniqueness of solutions to ordinary differential equations. For the Hamiltonian theorem, the main extra ingredients are the asymmetry of the Poisson bracket, and the fact that a vector field generates canonical transformations iff it is Hamiltonian. (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)