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)

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.

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)

Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller, have thought that the answer to this question is No - that the status of individual continuous quantities is very different in quantum mechanics than in (...) classical mechanics. On the contrary, I shall show that the same subtle issues arise with respect to continuous quantities in classical and quantum mechanics; and that it is, after all, possible to describe a particle as possessing a sharp position value without altering the standard formalism of quantum mechanics. (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.

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 development of the notion of continuous physical quantity is traced from Aristotle to Aquinas to Suarez. It is concluded that Aristotle’s divisibility definition fails to excavate the ontological core of material quantification. Although the basic germ of the solution to the problem is discovered in Aquinas, it is Suarez who fully articulates the essence of continuous physical quantity with his explicit concept of aptitudinal extension — which has crucial theological implications. Résumé Nous considérons ici le (...) développement de la notion de quantité physique continue, d’Aristote à Thomas d’Aquin à Suarez. Nous concluons que la définition d’Aristote en termes de divisibilité ne parvient pas à dégager le noyau ontologique de la quantification matérielle. Encore que le germe fondamental de la solution au problème soit découvert par Thomas d’Aquin, c’est Suarez qui articule pleinement l’essence de la quantité physique continue, grâce à son concept explicite d’extension « aptitudinale » — dont les implications théologiques sont cruciales. (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)

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 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)

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)

The paper is a continuation of three previous papers , which have discussed the issues of measure and measurement, as well as the views of K. Berka and B. Ellis on this issue. This paper gives a restatement of those views from the point of view of the unity of qualitative and quantitative determinations of measure. Further it deals with Ellis’ conventionalism in measurement theory. Finally it provides a differentiated typology of measurement.

Second Scholasticism greatly developed the medieval theory of continuous quantity as the Aristotelian notion for thematizing spatial extension, paving the way for the idea of space as extension in early modern natural philosophy. The article analyzes the section related to the category of continuous quantity in the Coimbra commentary on the Dialectics (1606), showing that it is indebted to the novel theory of Francisco Suárez on quantity as bestowing extension to a body in a particular (...) sense, something which had been overlooked by previous research. The scholarly debate on quantity was brought to China, and here the Chinese translation is examined of the section on quantity in the fourth volume of the Mingli Tan, published in China in 1636-­1639. (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)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular (constant) quantity, and the properties of such quantities are then akin to, and derived from, the..

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the field of real numbers in many ways, but instead form a ring. Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets (...) . In this expository paper, such applications are explained to the non-specialist. Some recent results are mentioned, including a new completeness theorem for higher-order logic. (shrink)

A wide array of species throughout the animal kingdom has shown the ability to distinguish between quantities. Aside from being important for optimal foraging decisions, this ability seems to also be of great relevance in group-living animals as it allows them to inform their decisions regarding engagement in between-group conflicts based on the size of competing groups. However, it is often unclear whether these animals rely on numerical information alone to make these decisions or whether they employ other cues that (...) may covary with the differences in quantity. In this study we used a touch screen paradigm to investigate the quantity discrimination abilities of two closely related group-living species, wolves and dogs, using a simultaneous visual presentation paradigm. Both species were able to successfully distinguish between stimuli of different quantities up to 32 items and ratios up to 0.80, and their results were in accordance with Weber's law (which predicts worse performances at higher ratios). However, our controls showed that both wolves and dogs may have used continuous, non-numerical cues, such as size and shape of the stimuli, in conjunction with the numerical information to solve this task. In line with this possibility, dogs' performance greatly exceeded that which they had shown in other numerical competence paradigms. We discuss the implications these results may have on these species' underlying biases and numerical capabilities, as well as how our paradigm may have affected the animals' ability to solve the task. (shrink)

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that (...) quantity, in the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut. (shrink)

This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...) does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

: Descartes provides an original and puzzling argument for the traditional theological doctrine that the world is continuously created by God. His key premise is that the parts of the duration of anything are "completely independent" of one another. I argue that Descartes derives this temporal independence thesis simply from the principle that causes are necessarily simultaneous with their effects. I argue further that it follows from Descartes's version of the continuous creation doctrine that God is the instantaneous and (...) total cause of everything that happens, and that this is just what his physics demands. But although God is the total cause of everything, he is not the only cause, since Descartes considers it obvious that finite minds have the power to move bodies. In the face of this apparent paradox, several recent commentators have suggested that Descartes accepted the late scholastic view that God and finite causes somehow collaborate or concur in the production of numerically identical effects. But close examination reveals that the case for Cartesian concurrentism is weak. Fortunately, there is a simpler and more fruitful solution to the problem of reconciling divine and human action. I argue that that in Descartes's world certain motions, such as voluntary movements of our bodies, are causally overdetermined This account allows Descartes to avoid occasionalism without embracing an elaborate metaphysics of secondary causality. Finally, I argue that the overdeterminist interpretation sheds new light on two longstanding problems of Cartesian metaphysics. First, it resolves a serious difficulty with a standard reading of Descartes's conception of human freedom. Second, it offers a novel approach to the old problem of reconciling genuine human action with the principle of the conservation of total quantity of motion. (shrink)

This article defends a continuity position. Infants can abstract numerosity and young preschool children do respond appropriately to tasks that tap their ability to use a count and cardinal value and/or arithmetic principles. Active use of a nonverbal domain of arithmetic serves to enable the child to find relevant data to build knowledge about the language and use rules of numerosity and quantity.

Pour Aristote, sous le rapport de sa composition en parties, le continu est divisible mais sous le rapport de ses limites (point, ligne, surface et profondeur), le continu est indivisible. Walter Burley, comme ses contemporains, a commenté la coexistence problématique de la divisibilité et de l’indivisibilité dans la structure du continu. Bien plus, aux prises avec sa célèbre polémique contre son adversaire Guillaume d’Ockham à propos de l’ontologie de la catégorie de quantité, il admet une structure du continu originale qui (...) semble contenir à la fois des intervalles ou parties divisibles et des points ou indivisibles. (shrink)

Pour Aristote, sous le rapport de sa composition en parties, le continu est divisible mais sous le rapport de ses limites (point, ligne, surface et profondeur), le continu est indivisible. Walter Burley, comme ses contemporains, a commenté la coexistence problématique de la divisibilité et de l’indivisibilité dans la structure du continu. Bien plus, aux prises avec sa célèbre polémique contre son adversaire Guillaume d’Ockham à propos de l’ontologie de la catégorie de quantité, il admet une structure du continu originale qui (...) semble contenir à la fois des intervalles ou parties divisibles et des points ou indivisibles. (shrink)

In order to find positive solutions for third-degree equations, which he did not know how to solve for roots, m proceeds by the intersections of conic sections. The representation of an algebraic equation by a geometrical curve is made possible by the choices of units of measure for lengths, surfaces, and volumes. These units allow a numerical quantity to be associated with a geometrical magnitude. Is there a trace of this unit in the mathematicians to whom al-Khayyām refers directly (...) in his Algebra? How does this unit enable the measurement of quantities and rational and irrational relations? We find answers to these questions in the commentaries to Books V and X of the Elements. (shrink)

A formalism developed in previous papers for the description of continual observations of some quantities in the framework of quantum mechanics is reobtained and generalized, starting from a more axiomatic point of view. The statistics of the observations of continuous state trajectories is treated from the beginning as a generalized stochastic process in the sense of Gel'fand. An effect-valued measure and an operation-valued measure on the σ-algebra generated by the cylinder sets in the space of trajectories are introduced. The (...) properties of the characteristic functional for the “operation-valued stochastic process” are discussed and, through a suitableansatz, a significant class of such processes is explicitly constructed, which contains the examples of the preceding papers as particular cases. (shrink)

In an old paper of our group in Milano a formalism was introduced for the continuous monitoring of a system during a certain interval of time in the framework of a somewhat generalized approach to quantum mechanics. The outcome was a distribution of probability on the space of all the possible continuous histories of a set of quantities to be considered as a kind of coarse grained approximation to some ordinary quantum observables commuting or not. In fact the (...) main aim was the introduction of a classical level in the context of QM, treating formally a set of basic quantities, to be considered as beables in the sense of Bell, as continuously taken under observation. However the effect of such assumption was a permanent modification of the Liouville-von Neumann equation for the statistical operator by the introduction of a dissipative term which is in conflict with basic conservation rules in all reasonable models we had considered. Difficulties were even encountered for a relativistic extension of the formalism. In this paper I propose a modified version of the original formalism which seems to overcome both difficulties. First I study the simple models of an harmonic oscillator and a free scalar field in which a coarse grain position and a coarse grained field respectively are treated as beables. Then I consider the more realistic case of spinor electrodynamics in which only certain coarse grained electric and magnetic fields are introduced as classical variables and no matter related quantities. (shrink)

This book collects six unpublished and published academic studies on the thought of Francisco Suárez, which is addressed through accurate textual analyses and meticulous contextualization of his doctrines in the Scholastic debate. The present essays aim to portray two complementary aspects coexisting in the work of the Uncommon Doctor: his innovative approach and his adherence to the tradition. To this scope, they focus on some pivotal, but often neglected, topics in Suárez’s metaphysics and psychology – such as his theories of (...) cognition and truth, angelology, continuous quantity – thereby developing an original inquiry into a crucial moment in the development of Western philosophy. (shrink)

Le problème abordé par cette étude concerne l’interprétation du statut de l’imagination dans les Méditations métaphysiques, dans son rapport avec l’intellection et la conception de l’espace, de ses objets et des propriétés mathématiques. Il s’agit en particulier de savoir comment l’imagination distincte paraît assurer la réalité de son objet lorsqu’il s’agit de l’espace en général et lorsqu’il s’agit des figures particulières.

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...) this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

DMet 10: Prime matter is the origin of all quantities. Hence it is the origin of every dimension of continuous quantity whatever. ...

The base units of the SI include six units of continuous quantities and the mole, which is defined as proportional to the number of specified elementary entities in a sample. The existence of the mole as a unit has prompted comment in Metrologia that units of all enumerable entities should be defined though not listed as base units. In a similar vein, the BIPM defines numbers of entities as quantities of dimension one, although without admitting these entities as base (...) units. However, there is a basic ontological distinction between continuous quantities and enumerable aggregates. The distinction is the basis of the difference between real and natural numbers. This paper clarifies the nature of the distinction: (i) in terms of a set of measurement axioms stated by Hölder; and (ii) using the formalism known in metrology as quantity calculus. We argue that a clear and unambiguous scientific distinction should be made between measurement and enumeration. We examine confusion in metrological definitions and nomenclature concerning this distinction, and discuss the implications of this distinction for ontology and epistemology in all scientific disciplines. (shrink)

Translator's introduction -- Mir Damad's introduction -- 1st qabas. On the kinds of creation (huduth) and the divisions of existence according to them, establishing the bases of judgment, and defining the area of dispute -- 2nd qabas. On the three kinds of essential antecedence and constructing the demonstration by way of essential priority (taqaddum bi'l-dhat) -- 3rd qabas. The two kinds of separate posteriority and constituting the demonstration by way of eternal priority -- 4th qabas. Quotations from the book of (...) god and the traditions of his messenger, and from the traditions of the lofty and pure imams -- 5th qabas. Concerning the mode of existence of the unqualified natures and the path of the demonstration by the mode of the existence of the nature -- 6th qabas. On the continuity of time and motion; setting up the course of the demonstration with respect to the continuity of the magnitude of time according to the natural system in two ways; establishing the finitude of extended continuous quantity; and invalidating a numerical infinity -- With respect to successive temporal creatures -- 7th qabas. A series of abridged arguments, unsound dialectical arguments, and criticism of certain syllogisms and controversial sophistical doubts according to the two extremes of the two groups -- 8th qabas. Inquiry into the power of God and his will after completing what remains in the care of the intellect, through the use of decisive utterance, to solve some of the difficulties and dilemmas caused by doubt and the confusions arising from idle fancies -- 9th qabas. On establishing the intelligible substances and the stages of the system of existence in the two chains of beginning and return -- 10th qabas. The decisive doctrine on the secret of predetermination (qada') and fate (qadar), how evil is related to predetermination, the colocynth of truth on prayer and its granting, and the return of the command to god in the beginning and the end -- Glossary-- Bibliography. (shrink)

A mathematical model in science can be formulated as a counterfactual conditional, with the model’s assumptions in the antecedent and its predictions in the consequent. Interestingly, some of these models appear to have assumptions that are metaphysically impossible. Consider models in ecology that use differential equations to track the dynamics of some population of organisms. For the math to work, the model must assume that population size is a continuous quantity, despite that many organisms are necessarily discrete. This (...) means our counterfactual representation of the model can have an impossible antecedent, giving us a counterpossible. Analogous counterpossibles arise in other sciences, as we’ll see. According to a prominent view in counterfactual semantics, the vacuity thesis, all counterpossibles are vacuously true, that is, true merely because their antecedents are necessarily false. But some counterpossible formulations of differential equation models in science are not all vacuously true—some are non-vacuously true, and some are false. I go on to show how an alternative semantics, one that employs impossible worlds, can deliver this judgment. (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)

Most of our best scientific descriptions of the world employ rates of change of some continuous quantity with respect to some other continuous quantity. For instance, in classical physics we arrive at a particle’s velocity by taking the time-derivative of its position, and we arrive at a particle’s acceleration by taking the time-derivative of its velocity. Because rates of change are defined in terms of other continuous quantities, most think that facts about some rate of (...) change obtain in virtue of facts about those other continuous quantities. For example, on this view facts about a particle’s velocity at a time obtain in virtue of facts about how that particle’s position is changing at that time. In this paper we raise a puzzle for this orthodox reductionist account of rate of change quantities and evaluate some possible replies. We don’t decisively come down in favour of one reply over the others, though we say some things to support taking our puzzle to cast doubt on the standard view that spacetime is continuous. (shrink)

Jeffrey conditionalization is a rule for updating degrees of belief in light of uncertain evidence. It is usually assumed that the partitions involved in Jeffrey conditionalization are finite and only contain positive-credence elements. But there are interesting examples, involving continuous quantities, in which this is not the case. Q1 Can Jeffrey conditionalization be generalized to accommodate continuous cases? Meanwhile, several authors, such as Kenny Easwaran and Michael Rescorla, have been interested in Kolmogorov’s theory of regular conditional distributions as (...) a possible framework for conditional probability which handles probability-zero events. However the theory faces a major shortcoming: it seems messy and ad hoc. Q2 Is there some axiomatic theory which would justify and constrain the use of rcds, thus serving as a possible foundation for conditional probability? These two questions appear unrelated, but they are not, and this paper answers both. We show that when one appropriately generalizes Jeffrey conditionalization as in Q1, one obtains a framework which necessitates the use of rcds. It is then a short step to develop a general theory which addresses Q2, which we call the theory of extensions. The theory is a formal model of conditioning which recovers Bayesian conditionalization, Jeffrey conditionalization, and conditionalization via rcds as special cases. (shrink)

According to the Spatial Quantity Association of Response Codes (SQUARC), people hold a mental association between horizontal position and quantity (lower quantities left, higher quantities right). While a large body of research has explored this effect for response speed and judgment accuracy, the affective downstream consequences of the SQUARC remain unexplored. Aiming to address this gap, the present two experiments (pre-registered, total N = 521) investigated whether stimulus arrangements that are compatible with the SQUARC for luminance are affectively (...) preferred to stimulus arrangements that are incompatible. SQUARC-compatible square arrangements (dark-left, bright-right) were preferred over SQUARC-incompatible square arrangements (dark-right, bright-left). The preference for SQUARC compatibility was not moderated by the horizontal orientation of the response scale. Our results confirm the direction of the spatial-luminance association and provide initial support that the cognitive processing of SQUARC compatibility is hedonically marked and appears sufficient to impact affective evaluations. (shrink)

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, (...) is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity. (shrink)

The thesis that numbers are ratios of quantities has recently been advanced by a number of philosophers. While adequate as a definition of the natural numbers, it is not clear that this view suffices for our understanding of the reals. These require continuous quantity and relative to any such quantity an infinite number of additive relations exist. Hence, for any two magnitudes of a continuous quantity there exists no unique ratio. This problem is overcome by (...) defining ratios, and hence real numbers, as binary relations between infinite standard sequences. This definition leads smoothly into a new definition of measurement consonant with the traditional view of measurement as the discovery or estimation of numerical relations. The traditional view is further strengthened by allowing that the additive relations internal to a quantity are distinct from relations observed in the behaviour of objects manifesting quantities. In this way the traditional theory can accommodate the theory of conjoint measurement. This is worth doing because the traditional theory has one great strength lacked by its rivals: measurement statements and quantitative laws are able to be understood literally. 1 This paper was completed in 1990-1. while the author was a visiting scholar at the Irvine Research Unit in Mathematical Behavioral Sciences. University of California. Irvine. The author wishes to thank the Director. Professor R. D. Luce, for the generous provision of space and facilities within the Unit and for his critical comments upon some of the ideas expressed herein: Professor L. Narens. for his trenchant criticisms: and the University of Sydney, for granting Special Study Leave and financial assistance to make the visit possible. (shrink)

SOME MODERN THOMISTS claiming to follow the lead of Thomas Aquinas, hold that the objects of the types of mathematics known in the thirteenth century, such as the arithmetic of whole numbers and Euclidean geometry, are real entities. In scholastic terms they are not beings of reason but real beings. In his once-popular scholastic manual, Elementa Philosophiae Aristotelico-Thomisticae, Joseph Gredt maintains that, according to Aristotle and Thomas Aquinas, the object of mathematics is real quantity, either discrete quantity in (...) arithmetic or continuous quantity in geometry. The mathematician considers the essence of quantity in abstraction from its relation to real existence in bodily substance. "When quantity is considered in this way," he writes, "it is not a being of reason but a real being. Nevertheless it is so abstractly considered that it leaves out of account both real and conceptual existence." Recent mathematicians, Gredt continues, extend their speculation to fictitious quantity, which has conceptual but not real being; for example, the fourth dimension, which by its essence positively excludes a relation to real existence. According to Gredt this is a special, transcendental mathematics essentially distinct from "real mathematics," and belonging to it only by reduction. (shrink)

The International System of Units tries to find or construct something that does not change with time and place, since such constancy is the best possible ground for definitions of fundamental measurement units. This problem of constancy has received scant attention within the philosophy of science, but is the topic of the paper. The paper first highlights inevitable kinds of circularities, semantic and epistemic, that belongs to the search for constancy, and then discusses contingent dependencies between unit definitions. The New (...) SI proposal is criticized for not paying due attention to the fact that it defines units for one kind of quantity by means of a constancy that belongs to another. This inattention flaws the kilogram definition. The New SI definitions of the mole and the second neglect the distinction between discrete and continuous quantities; which make the definitions refer to constancies that are not invariants of nature. (shrink)

The concept of information tempts us as a theoretical primitive, partly because of the respectability lent to it by highly successful applications of Shannon’s information theory, partly because of its broad range of applicability in various domains, partly because of its neutrality with respect to what basic sorts of things there are. This versatility, however, is the very reason why information cannot be the theoretical primitive we might like it to be. “Information,” as it is variously used, is systematically ambiguous (...) between whether it involves continuous or discrete quantities, causal or noncausal relationships, and intrinsic or relational properties. Many uses can be firmly grounded in existing theory, however. Continuous quantities of information involving probabilities can be related to information theory proper. Information defined relative to systems of rules or conventions can be understood relative to the theory of meaning . A number of causal notions may possibly be located relative to standard notions in physics. Precise specification of these distinct properties involved in the common notion of information can allow us to map the relationships between them. Consequently, while information is not in itself the kind of single thing that can play a significant unifying role, analyzing its ambiguities may facilitate headway toward that goal. (shrink)

Numbers are symbols manipulated in accord with the axioms of arithmetic. They sometimes represent discrete and continuous quantities, but they are often simply names. Brains, including insect brains, represent the rational numbers with a fixed-point data type, consisting of a significand and an exponent, thereby conveying both magnitude and precision.

In his writings about hypergeometric functions Gauss succeeded in moving beyond the restricted domain of eighteenth-century functions by changing several basic notions of analysis. He rejected formal methodology and the traditional notions of functions, complex numbers, infinite numbers, integration, and the sum of a series. Indeed, he thought that analysis derived from a few, intuitively given notions by means of other well-defined concepts which were reducible to intuitive ones. Gauss considered functions to be relations between continuous variable quantities while (...) he regarded integration and summation as appropriate operations with limits. He also regarded infinite and infinitesimal numbers as a façon de parler and used inequalities in order to prove the existence of certain limits. He took complex numbers to have the same legitimacy as real quantities. However, Gauss’s continuum was linked to a revised form of the eighteenth-century notion of continuous quantity: it was not reducible to a set of numbers but was immediately given. (shrink)