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)

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)

The next generation of Geographic Information Systems is anticipated to automate some of the reasoning required for spatial analysis. An important step in the development of such systems is to gain a better understanding and corresponding modeling practice of when to apply arithmetic operations to quantities. The concept of extensivity plays an essential role in determining when quantities can be aggregated by summing them, and when this is not possible. This is of particular importance to geographic information systems, (...) which serve to quantify phenomena across space and time. However, currently, multiple contrasting definitions of extensivity exist, and none of these suffice for handling the different practical cases occurring in geographic information. As a result, analysts predominantly rely on intuition and ad hoc reasoning to determine whether two quantities are additive. In this paper, we present a novel approach to formalizing the concept of extensivity. Though our notion as such is not restricted to quantifications occurring within geographic information, it is particularly useful for this purpose. Following the idea of spatio-temporal controls by Sinton, we define extensivity as a property of measurements of quantities with respect to a controlling quantity, such that a sum of the latter implies a sum of the former. In our algebraic definition of amounts and other quantities, we do away with some of the constraints that limit the usability of older approaches. By treating extensivity as a relation between amounts and other types of quantities, our definition offers the flexibility to relate a quantity to many domains of interest. We show how this new notion of extensivity can be used to classify the kinds of amounts in various examples of geographic information. (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)

Truth Be Told explains how truth and falsity result from relations that sentences and their constituents have to the circumstances at which they are evaluated. It offers a precise analysis of truth and a diagnosis of the Liar paradox. Current semantic theory employs generalized quantifiers as the extensions of noun phrases. The book provides simpler extensions for noun phrases. These permit intuitive compositions of truth-values and a diagnosis of the Liar and Grelling paradoxes.

In The Quantity of the Soul, Augustine puts forward the view that the soul is immaterial and that its quantity (quantitas) must be understood in terms of power rather than spatial extension. Against this view, his friend and interlocutor Evodius raises an important objection, The Objection from Touch, which argues that the soul’s exercise of tactile sensation requires that it be extended through the parts of the body. This paper examines Evodius’s objection and Augustine’s response to it. Particular attention is (...) given to certain features of Augustine’s theory of sensation that this exchange reveals, especially his view that the eyes undergo passion-at-a-distance or are acted on at a place where they are not present. (shrink)

This paper concerns the ways in which one can/cannot combine extensive quantities. Given a particular theory of extensive measurement, there can be no alternative ways of combining extensive quantities, where 'alternative' means that one combining operation can be used instead of another causing only a change in the number assigned to the quantity. As a consequence, rectangular concatenation cannot be an alternative combining operation for length as was suggested by Ellis and agreed by Krantz, Luce, (...) Suppes, and Tversky. I argue as well that a theory which imposes such restrictions on the combining operation is more desirable than less stringent rival theories. (shrink)

This thesis concerns alternative concatenating operations in extensive measurements and the degree to which concatenating operations are matter of convention. My arguments are directed against Ellis' claim that what prevents us from choosing alternative ways of combining extensive quantities is only convenience and simplicity and that the choice is not based on empirical reasons. ;My first argument is that, given certain relational theories of measurement, there can be no more than one concatenating operation per quantity; because combining (...) operations are the only conclusive distinguishing characteristic among distinct extensive measurements and thus the only thing that can serve as an indicator for the identity of a quantity. Rectangular and collinear concatenation, for example, cannot both be used as a way of combining lengths. However, rectangular concatenation could be used to measure some other extensive quantity since it fulfills the necessary conditions for adequate numerical assignment. ;For those that adopt theories of measurement which can support the claim that dinches measure length, I demonstrate, in the second argument, that rectangular concatenation would be a bad alternative and thus no alternative at all. Implementation of rectangular concatenation requires a new science; a science compatible with dinches violates present invariance principles and a new set must be provided, if possible. It is not clear that there is a set of invariance principles compatible with dinches which also gives basis for a science that explains the phenomena as well as the old one. Moreover, new invariance principles would be unintuitive and would require that certain fundamental concepts change their meaning. For these reasons the choice between them is guided by a number of empirical reasons and is not only a matter of convention. (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)

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.

Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. (...) This is interpreted as involving arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement. (shrink)

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)

0. The ancient and honorable role of philosophy as a servant to the learning, development and use of scientific knowledge, though sadly underdeveloped since Grassmann, has been re-emerging from within the particular science of mathematics due to the latter's internal need; making this relationship more explicit (as well as further investigating the reasons for the decline) will, it is hoped, help to germinate the seeds of a brighter future for philosophy as well as help to guide the much wider learning (...) of mathematics and hence of all the sciences. 1. The unity of interacting opposites "space vs. quantity", with the accompanying "general vs. particular" and the resulting division of variable quantity into the interacting opposites "extensive vs. intensive", is susceptible, with the aid of categories, functors, and natural transformations, of a formulation which is on the one hand precise enough to admit proved theorems and considerable technical development and yet is on the other hand general enough to admit incorporation of almost any specialized hypothesis. Readers armed with the mathematical definitions of basic category theory should be able to translate the discussion in this section into symbols and diagrams for calculations. 2. The role of space as an arena for quantitative "becoming" underlies the qualitative transformation of a spatial category into a homotopy category, on which extensive and intensive quantities reappear as homology and cohomology. 3. The understanding of an object in a spatial category can be approached through definite Moore-Postnikov levels; each of these levels constitutes a mathematically precise "unity and identity of opposites", and their ensemble bears features strongly reminiscent of Hegel's Science of Logic. This resemblance suggests many mathematical and philosophical problems which now seem susceptible of exact solution. (shrink)

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ℏ→0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. (...) Moreover, we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ℏ→0 case) on the fiber C*-algebras. (shrink)

According to Kant, the division of the categories “is not the result of a search after pure concepts undertaken at haphazard,” but is derived from the “complete” classification of judgments developed by traditional logic. However, the sorts of judgments that he enumerates in his table of judgments are not all ones that traditional logic has dealt with; consequently, we must say that he chose the sorts of judgments in question with a certain intention. Besides, we know that his choice of (...) judgments and categories is strongly influenced by certain views of natural science that he fully accepts. For this reason, his argumentations are sometimes seriously inconsistent. As to Kant’s argumentations of categories, many problems have already been pointed out, but in this paper, I take up the categories of quantity and quality once more, and make clear his argumentations’ hidden logic and its distortion from the point of view of the history of logic and natural science. First, I confirm that there are non-negligible problems in his explanation to the effect that his derivation of the categories of quantity and quality is based on the quantity and quality of judgments. Next, I reconsider the meaning of his treating the categories of quantity and quality as pure concepts of the understanding. Finally, I conclude that by having recourse to the categories of quantity and quality Kant tried unjustly to apriorize the distinction between the “extensive magnitude” and “intensive magnitude” that has a long formational history since Aristotle. (shrink)

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

The use of the axial vector representing a three-dimensional rotation makes the rotation representation much more compact by extending the trigonometric functions to vectorial arguments. Similarly, the pure Lorentz transformations are compactly treated by generalizing a scalar rapidity to a vector quantity in spatial three-dimensional cases and extending hyperbolic functions to vectorial arguments. A calculation of the Wigner rotation simplified by using the extended functions illustrates the fact that the rapidity vector space obeys hyperbolic geometry. New representations bring a Lorentz-invariant (...) fundamental equation of motion corresponding to the Galilei-invariant equation of Newtonian mechanics. (shrink)

In this paper, I analyze William Crathorn’s view on extension and compare it to William Ockham’s reductionist view, according to which extension is not really distinct from substance or quality. In my view, Crathorn elaborates a metaphysical machinery based on mereological and topological relationships in order to solve what he considers to be problems in Ockham’s account of quantity. In order to make my point, I reconstruct Crathorn’s main arguments in favor of his finitist atomism. Crathorn claims that certain fundamental (...) spatial structures are of a mereological nature and that the mereological properties of located entities perfectly match those of their locations. This idea lies at the core of his solution to the paradox of touching and the metrical paradox. It allows him to redefine contiguity in such a way that extensionless magnitudes can touch and succeed each other. Crathorn goes one step further and claims that indivisibles are extended in the sense that they occupy an extended incorporeal space, which is their dimension. In this sense, they can come together to form a continuous magnitude. Thus, Crathorn succeeds in defining a concept of continuity that is compatible with his finitist atomism and that addresses what he takes to be a major weakness in Ockham’s view on extension, namely a murky concept of impenetrability. (shrink)

One approach to understanding how the human cognitive system stores and operates with quantifiers such as “some,” “many,” and “all” is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link between quantifier processing and nonsymbolic quantity processing has been considered in the past, it has never been discussed extensively. Simultaneously, there is a long line of research within the field of numerical cognition on the (...) relationship between processing exact number symbols (such as “3” or “three”) and nonsymbolic quantity. This accumulated knowledge can potentially be harvested for research on quantifiers since quantifiers and number symbols are two different ways of referring to quantity information symbolically. The goal of the present review is to survey the research on the relationship between quantifiers and nonsymbolic quantity processing mechanisms and provide a set of research directions and specific questions for the investigation of quantifier processing. (shrink)

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

Plants from a handful of species provide the primary source of food for all people, yet this source is vulnerable to multiple stressors, such as disease, drought, and nutrient deficiency. With rapid population growth and climate uncertainty, the need to produce crops that can tolerate or resist plant stressors is more crucial than ever. Traditional plant breeding methods may not be sufficient to overcome this challenge, and methods such as highOthroughput sequencing and automated scoring of phenotypes can provide significant new (...) insights. Ontologies are essential tools for accessing and analysing the large quantities of data that come with these newer methods. As part of a larger project to develop ontologies that describe plant phenotypes and stresses, we are developing a plant disease extension of the Infectious Disease Ontology (IDOPlant). The IDOPlant is envisioned as a reference ontology designed to cover any plant infectious disease. In addition to novel terms for infectious diseases, IDOPlant includes terms imported from other ontologies that describe plants, pathogens, and vectors, the geographic location and ecology of diseases and hosts, and molecular functions and interactions of hosts and pathogens. To encompass this range of data, we are suggesting inOhouse ontology development complemented with reuse of terms from orthogonal ontologies developed as part of the Open Biomedical Ontologies (OBO) Foundry. The study of plant diseases provides an example of how an ontological framework can be used to model complex biological phenomena such as plant disease, and how plant infectious diseases differ from, and are similar to, infectious diseases in other organism. (shrink)

John Locke’s treatment of quantity in the Essay Concerning Human Understanding is not nearly as extensive or as well-known as his treatment of quality and his distinction between primary and secondary qualities. Yet I contend that a close examination of Locke’s comments on quantity in the Essay reveals that he endorses a general theory of quantity that not only distinguishes quantities from qualities, but also plays several other important roles in his overall philosophy—particularly in his treatments of infinity (...) and demonstrative knowledge. Moreover, I argue that this theory does not appear out of thin air, but is in certain respects one and the same as the approach to quantity found in Aristotle and developed by medieval Aristotelian thinkers. Identifying the core tenets of this approach in the texts of Aristotle and medieval Aristotelians, and formulating them clearly enough to shed light on Locke’s theory of quantity in the Essay, is the principal task of this paper. (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)

It is a received view that Kant’s formal logic (or what he calls “pure general logic”) is thoroughly intensional. On this view, even the notion of logical extension must be understood solely in terms of the concepts that are subordinate to a given concept. I grant that the subordination relation among concepts is an important theme in Kant’s logical doctrine of concepts. But I argue that it is both possible and important to ascribe to Kant an objectual notion of logical (...) extension according to which the extension of a concept is the multitude of objects falling under it. I begin by defending this ascription in response to three reasons that are commonly invoked against it. First, I explain that this ascription is compatible with Kant’s philosophical reflections on the nature and boundary of a formal logic. Second, I show that the objectual notion of extension I ascribe to Kant can be traced back to many of the early modern works of logic with which he was more or less familiar. Third, I argue that such a notion of extension makes perfect sense of a pivotal principle in Kant’s logic, namely the principle that the quantity of a concept’s extension is inversely proportional to that of its intension. In the process, I tease out two important features of the Kantian objectual notion of logical extension in terms of which it markedly differs from the modern one. First, on the modern notion the extension of a concept is the sum of the objects actually falling under it; on the Kantian notion, by contrast, the extension of a concept consists of the multitude of possible objects—not in the metaphysical sense of possibility, though—to which a concept applies in virtue of being a general representation. While the quantity of the former extension is finite, that of the latter is infinite—as is reflected in Kant’s use of a plane-geometrical figure (e.g., circle, square), which is continuum as opposed to discretum, to represent the extension in question. Second, on the modern notion of extension, a concept that signifies exactly one object has a one-member extension; on the Kantian notion, however, such a concept has no extension at all—for a concept is taken to have extension only if it signifies a multitude of things. This feature of logical extension is manifested in Kant’s claim that a singular concept (or a concept in its singular use) can, for lack of extension, be figuratively represented only by a point—as opposed to an extended figure like circle, which is reserved for a general concept (or a concept in its general use). Precisely on account of these two features, the Kantian objectual extension proves vital to Kant’s theory of logical quantification (in universal, particular and singular judgments, respectively) and to his view regarding the formal truth of analytic judgments. (shrink)

This paper analyzes certain technical details of Floridi’s Theory of Strongly Semantic Information. It provides a clarification regarding desirable properties of degrees of informativeness functions by rejecting three of Floridi’s original constraints and proposing a replacement constraint. Finally, the paper briefly explores the notion of quantities of inaccuracy and shows an analysis that mimics Floridi’s analysis of quantities of vacuity.

This paper analyzes certain technical details of Floridi’s Theory of Strongly Semantic Information. It provides a clarification regarding desirable properties of degrees of informativeness functions by rejecting three of Floridi’s original constraints and proposing a replacement constraint. Finally, the paper briefly explores the notion of quantities of inaccuracy and shows an analysis that mimics Floridi’s analysis of quantities of vacuity.

The 16th and 17th centuries witnessed a philosophical shift that would help pave the way for modern science, a shift from metaphysical theories of material objects to other views embracing only the empirically-accessible parts of material things. One much-debated topic in the course of this shift was regarding prime matter. The late scholastic Jacobus Zabarella (1533-1589) arrived upon his views about prime matter via his version of the regressus method, a program for a sort of scientific reasoning. In his De (...) rebus naturalibus, Zabarella defends the position that prime matter is extended. However, it is less clear how he accounts for its extension. There is an important text where he apparently suggests that prime matter is extended in and of itself. However, there are two other texts apparently stating that matter is extended in virtue of a distinct accident in the category of quantity. I argue that a decisive reading based solely upon any particular texts is not available. Nevertheless, examining Zabarella’s writing on the topic as a whole, the trajectory of his overall argument is remarkably sympathetic to Averroes, who famously took the accidentally quantified position. Thus, from this systematic feature, the accidentally quantified reading is more reasonable. (shrink)

This rich book differs from much contemporary philosophy of mathematics in the author’s witty, down to earth style, and his extensive experience as a working mathematician. It accords with the field in focusing on whether mathematical entities are real. Franklin holds that recent discussion of this has oscillated between various forms of Platonism, and various forms of nominalism. He denies nominalism by holding that universals exist and denies Platonism by holding that they are concrete, not abstract - looking to (...) Aristotle for inspiration. (shrink)

This paper examines the Leibnizian influence in Deleuze's theory of the spatium. Leibniz's critique of Cartesian extension and Newtonian space leads him to a conception of space in terms of internal determination and internal difference. Space is thus understood as a structure of individual relations internal to substances. Making some Nietzschean corrections to Leibniz, Deleuze understands the spatium in terms of individuating differences instead of individual relations. Leibnizian space is thus transformed into a genetic space producing both extension (quantity) and (...) quality. (shrink)

This article motivates and develops a reductive account of the structure of certain physical quantities in terms of their mereology. That is, I argue that quantitative relations like "longer than" or "3.6-times the volume of" can be analyzed in terms of necessary constraints those quantities put on the mereological structure of their instances. The resulting account, I argue, is able to capture the intuition that these quantitative relations are intrinsic to the physical systems they’re called upon to describe (...) and explain. (shrink)

According to the realist interpretation, measurement commits us not just to the logically independent existence of things in space and time, but also to the existence of quantitatively structured properties and relations, and to the existence of real numbers, understood as relations of ratio between specific levels of such attributes. Measurement is defined as the estimation of numerical relations (or ratios) between magnitudes of a quantitative attribute and a unit. The history of scientific measurement, from antiquity to the present may (...) be interpreted as revealing a progressive deepening in the understanding of this position. First, the concept of ratio was broadened to include ratios between incommensurable magnitudes; second, the concept of a quantitative attribute was broadened to include non-extensive quantities; third, quantitative structure and its relations to ratios and real numbers were elaborated; and finally, the issue of empirically distinguishing between quantitative and non-quantitative structures was addressed. This interpretation of measurement understands it in a way that is continuous with scientific investigation in general, i.e., as an attempt to discover independently existing facts. (shrink)

Commonplace syntactic constructions in natural language seem to generate ontological commitments to a dazzling array of metaphysical categories - aggregations, sets, ordered n-tuples, possible worlds, intensional entities, ideal objects, species, intensive and extensive quantities, stuffs, situations, states, courses of events, nonexistent objects, intentional and discourse objects, general objects, plural objects, variable objects, arbitrary objects, vague kinds and concepts, fuzzy sets, and so forth. But just because a syntactic construction in some natural language appears to invoke a new category (...) of entity, are we theoreticians epistemically justified in holding that there are such entities? This would hardly seem sufficient. To be epistemically justified, the ontology to which we theoreticians are committed must pass strict standards: the entities must be of the sort required by our best comprehensive theory of the world. The thesis of this paper is that fine-grained type-free intensional entities are like this. If the thesis is right, these entities have a special objective status perhaps not possessed by some of the other ontological categories associated with special syntactic constructions in natural language. In fact, it is plausible to hold that fine-grained type-free intensional entities provide the proper minimal framework for constructing logical and linguistic theories. In this paper my strategy will be to survey the competing conceptions of fine-grained type-free intensionality and to present arguments in support of one of them. Following this narrowing down process, I will go on to the indicated epistemological considerations. (shrink)

In everyday life, people typically observe fragments of causal networks. From this knowledge, people infer how novel combinations of causes they may never have observed together might behave. I report on 4 experiments that address the question of how people intuitively integrate multiple causes to predict a continuously varying effect. Most theories of causal induction in psychology and statistics assume a bias toward linearity and additivity. In contrast, these experiments show that people are sensitive to cues biasing various integration rules. (...) Causes that refer to intensive quantities (e.g., taste) or to preferences (e.g., liking) bias people toward averaging the causal influences, whereas extensive quantities (e.g., strength of a drug) lead to a tendency to add. However, the knowledge underlying these processes is fallible and unstable. Therefore, people are easily influenced by additional task‐related context factors. These additional factors include the way data are presented, the difficulty of the inference task, and transfer from previous tasks. The results of the experiments provide evidence for causal model and related theories, which postulate that domain‐general representations of causal knowledge are influenced by abstract domain knowledge, data‐driven task factors, and processing difficulty. (shrink)

Entropy in thermodynamics is an extensive quantity, whereas standard methods in statistical mechanics give rise to a non-extensive expression for the entropy. This discrepancy is often seen as a sign that basic formulas of statistical mechanics should be revised, either on the basis of quantum mechanics or on the basis of general and fundamental considerations about the distinguishability of particles. In this article we argue against this response. We show that both the extensive thermodynamic and the non- (...) class='Hi'>extensive statistical entropy are perfectly alright within their own fields of application. Changes in the statistical formulas that remove the discrepancy must be seen as motivated by pragmatic reasons rather than as justified by basic arguments about particle statistics. (shrink)

The aim of this paper is to reconstruct the historical evolution of the so-called Measurement Theory. MT has two clearly different periods, the formation period and the mature theory, whose borderline coincides with the publication in 1951 of Suppes' foundational work, ‘A set of independent axioms for extensive quantities’. In this paper two previous research traditions on the foundations of measurement, developed during the formation period, come together in the appropriate way. These traditions correspond, on the one hand, (...) to Helmholtz's, Campbell's and Hölder's studies on axiomatics and real morphisms and, on the other, to the work undertaken by Stevens and his school on scale types and transformations. These two lines of research are complementary in the sense that neither of them is enough taken alone, but together they contain all that is necessary to develop the theory, and it is in Suppes that these complementary approaches converge and all the elements of the theory are appropriately integrated for the first time. With Suppes' work, then, begins what may be called the ‘mature’ theory, which was to develop rapidly later on, especially during the 1960s. Our historical reconstruction is divided into two parts, each part devoted to one of the periods mentioned. Part I also contains a conceptual introduction which aims to establish the use of some notions, specifically those of measurement and metrization. Although the reconstruction is not exhaustive, it intends to be quite complete and up to date compared to what is available in measurement literature; in this sense the aim of this paper is mainly historical but, although secondarily, it also attempts to make some conceptual and metascientific clarifications on the subject of the theory. (shrink)

The relation between extension and impenetrability is a major issue in the Descartes-More correspondence, which implies an analysis of the concept of extension. The mereological structure partes extra partes is a crucial element here. Both philosophers hold two opposed views of this mereological structure. I try to show that these two views can be traced back to scholastic discussions on quantity’s relation to extension. This background provides a vantage point, which enables to propose a new construal of the argumentative exchange (...) around impenetrability, and which casts some light too on our understanding of two properties – spissitude and indiscerpibility – which More attributes to spiritual extension in his later writings. (shrink)

This collection contains twenty-three papers published by Suppes over the last eighteen years. For the most part they are foundational studies ranging over a wide variety of topics in the philosophy of science. The first two of four parts contain papers on methodological issues like models, measurement, probability and utility. There are two papers on models, an axiomatic treatment of extensive quantity and two papers on measurement. The six papers in Part II deal with probability theory and decision theory (...) with reference to theories of behavior, economics, and other topics. Part III contains studies in the axiomatic foundations of physics, one on relativistic kinematics and three on probability in quantum mechanics. The final and longest section contains eight papers on the foundations of psychology. Several of these deal with the psychological or behavioral bases of mathematics. Others deal with unpredictability in human behavior, finite automata, cognition and other topics. The book is nicely printed and those who have learned from Suppes work in the past will be grateful for this collection of his most important papers.--R. H. K. (shrink)

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan’s Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in (...) particular we examine the Fréchet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction Cn+1 of n + 1 conditional events to the family {Cn,En+1|Hn+1}. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid. (shrink)

The extensive philosophical literature devoted to "reduction to basic quantities" and "supervenience" generally trades upon a tacit looseness with respect to basic logical issues. Specifically, working physics commonly traffics in quantities pegged to different length scales whose interrelationships conceal a good deal of logical and structural complexity. So let us begin with a somewhat pedantic warning about the subtleties of scale dependent location attributions. Let us presume — and this is an assumption we shall want to revisit (...) later — that we are operating within a standard form of classical physics whose "basic entities" are point masses that act upon one another through action-at-a-distance central forces. (shrink)

The John of Saint Thomas' critique of the suarecian concept of the prime matter The figure of Juan Poinsot is inserted in the movement of return to a Thomism free of nominalist adherences promoted in the Hispanic Thomism by relevant figures of the order of preachers. In this sense must be understood the Poinsot’s critique of some ideas of Scotistic and Ockhamistic origin, present in the metaphysics of Suarez, referred to the ontological status of the material substance, such as: the (...) actuality of matter, the substantial mode of union, the immediate inhesion of the quantity in matter and the idea of entitative extension. (shrink)

This is a preprinted excerpt from: Kochiras, “By ye Divine Arm: God and Substance in De gravitatione”, Religious Studies (Sept. 2013), 49(3): 327-356 (available at: https://www.cambridge.org/core/journals/religious-studies/article/by-ye-divine-arm-god-and-substance -in-de-gravitatione/08D21B2C2611624FA11A0D6B115849AD ). In this preprinted excerpt, I explicate the concepts of matter and space that Newton develops in De gravitatione. As I interpret Newton’s account of created substances, bodies are constructed from qualities alone, as configured by God. Although regions of space and then “determined quantities of extension” appear to replace the Aristotelian substrate by (...) functioning as property-bearers, they actually serve only as logical subjects. An implication of the interpretation I develop is that only space is extended by having parts outside parts; material bodies are spatially extended only in a derivative sense, via the presence of their constitutive qualities or powers in space. (shrink)

This paper aims to show that Deleuze’s ideas on space and extension are heavily in debt to Leibniz. The focus is on chapter five, ‘the Asymmetrical Synthesis of the Sensible’, of Difference and Repetition. Concepts such as ‘intensive magnitude’, ‘distance’, ‘order’ and most importantly ‘spatium’ are shown to have their origin in Leibniz’s philosophy. In order to do so, the article starts with Leibniz’s critique on Cartesian mechanics and how this leads Leibniz to a conception of space that goes beyond (...) the Cartesian concept of extension. The latter lacks difference, potentiality and individuality. Leibniz argues that the Cartesian homogeneous plenum should therefore be grounded in a deeper more substantial space: the spatium. Understood as such, space is not an absolute substance but a structure of relations between substances. Although Leibniz’s texts offer enough to understand this spatium, they do not explain how the genesis of extension out of space should be understood. It is on this point that Deleuze offers a fruitful interpretation of Leibniz by conceptualizing such a genesis in terms of difference. Extension is understood as a homogenization and externalization of the internal and pure differences in space. In the end, Leibniz helps us to understand Deleuze’s theory, while Deleuze offers us an enriching interpretation of Leibniz that succeeds where authors such as Yvon Belaval and Martial Gueroult have failed. (shrink)

To represent extension of objects in particle physics, a modified Weyl theory is used by gauging the curvature radius of the local fibers in a soldered bundle over space-time possessing a homogeneous space G/H of the (4, 1)-de Sitter group G as fiber. Objects with extension determined by a fundamental length parameter R0 appear as islands D(i) in space-time characterized by a geometry of the Cartan-Weyl type (i.e., involving torsion and modified Weyl degrees of freedom). Farther away from the domains (...) D(i), space-time is identified with the pseudo-Riemannian space of general relativity. Extension and symmetry breaking are described by a set of additional fields ( $(\tilde \xi ^a (x),R(x)) \in {G \mathord{\left/ {\vphantom {G H}} \right. \kern-0em} H}x R^ +$ , given as a section on an associated bundle $\tilde E(B,{G \mathord{\left/ {\vphantom {G H}} \right. \kern-0em} H}x R^ + ,\tilde G)$ over space-time B with structural group $\tilde G$ = G ⊗ D(1), where D(1) is the dilation group. Field equations for the quantities defining the underlying bundle geometry and for the fields $\tilde \xi ^a (x)$ are established involving matter source currents derived from a generalized spinor wave function. Einstein's equations for the metric are regarded as the part of the $\tilde G$ -gauge theory related to the Lorentz subgroup H of G exhibiting thereby the broken nature of the $\tilde G$ -symmetry for regions outside the domains D(i). (shrink)

We introduce stellar photometry, its purpose and relationship to other astronomical quantities, presenting it within the context of astronomical research at the Vatican Observatory. We demonstrate the usefulness of the Vilnius Photometric System previously shown at Vilnius Astronomical Obs., Edinburgh Royal Obs., and Università di Roma “La Sapienza”, and then adopted by the Vatican Observatory for use with its Vatican Advanced Technology Telescope on Mount Graham, AZ, USA. The development of astronomical observations has led from photographic plates, through photoelectric (...) detectors to the use of sensitive CCD cameras. Expanding the photoelectric databank of “Vilnius Stars”, we describe the development of new techniques of data-acquisition, calibration to photometric standards, stellar classification and interstellar-matter characterization. Looking ahead, we briefly describe the extensive data set, and our plans for further detailed analyses. (shrink)

The quality and quantity of training data are crucial to the performance of a deep-learning-based brain-computer interface system. However, it is not practical to record EEG data over several long calibration sessions. A promising time- and cost-efficient solution is artificial data generation or data augmentation. Here, we proposed a DA method for the motor imagery EEG signal called brain-area-recombination. For the BAR, each sample was first separated into two ones by left/right brain channels, and the artificial samples were generated by (...) recombining the half-samples. We then designed two schemas corresponding to the single- and multi-subject scenarios. Extensive experiments using the classifier of EEGnet were conducted on two public datasets under various training set sizes. In both schemas, the BAR method can make the EEGnet have a better performance of classification. To make a comparative investigation, we selected two common DA methods, and the BAR method beat them. Further, using the proposed BAR for augmentation, EEGnet achieved up to 8.3% improvement than a typical decoding algorithm CSP-SVM, note that both the models were trained on the augmented dataset. This study shows that BAR usage can significantly improve the classification ability of deep learning to MI-EEG signals. To a certain extent, it may promote the development of deep learning technology in the field of BCI. (shrink)

I provide a fairly systematic analysis of when quantities that are variant under a dynamical symmetry transformation should be regarded as unobservable, or redundant, or unreal; of when models related by a dynamical symmetry transformation represent the same state of affairs; and of when mathematical structure that is variant under a dynamical symmetry transformation should be regarded as surplus. In most of these cases the answer is `it depends': depends, that is, on the details of the symmetry in question. (...) A central feature of the analysis is that in order to draw any of these conclusions for a dynamical symmetry it needs to be understood in terms of its possible extensions to other physical systems, in particular to measurement devices. (shrink)

At A71/B96–7 Kant explains that singular judgements are ‘special’ because they stand to the general ones as Einheit to Unendlichkeit. The reference to Einheit brings to mind the category of unity and hence raises a spectre of circularity in Kant’s explanation. I aim to remove this spectre by interpreting the Einheit-Unendlichkeit contrast in light of the logical distinctions among universal, particular and singular judgments shared by Kant and his logician predecessors. This interpretation has a further implication for resolving a controversy (...) over the correlation between the logical moments of quantity (universal, particular, singular) and the categorial ones (unity, plurality, totality). (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)