The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” (...) to provide evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to (...) mathematics education. By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical (...) confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger (...) than 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (shrink)

We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the four classes (...) of trees mentioned above. (shrink)

On a now orthodox view, humans and many other animals are endowed with a “number sense”, or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques, with critics maintaining either that numerical content is absent altogether, or else that some primitive analog of number (‘numerosity’) is represented as opposed to number itself. We distinguish three arguments for these claims – the arguments from congruency, confounds, and imprecision (...) – and show that none succeed. We then highlight positive reasons for thinking that the ANS genuinely represents numbers. The upshot is that proponents of the orthodox view should not feel troubled by recent critiques of their position. (shrink)

FDE is a logic that captures relevant entailment between implication-free formulae and admits of an intuitive informational interpretation as a 4-valued logic in which “a computer should think”. However, the logic is co-NP complete, and so an idealized model of how an agent can think. We address this issue by shifting to signed formulae where the signs express imprecise values associated with two distinct bipartitions of the set of standard 4 values. Thus, we present a proof system which consists of (...) linear operational rules and only two branching structural rules, the latter expressing a generalized rule of bivalence. This system naturally leads to defining an infinite hierarchy of tractable depth-bounded approximations to FDE. Namely, approximations in which the number of nested applications of the two branching rules is bounded. (shrink)

Strong nativist views about numerical concepts claim that human beings have at least some innate precise numerical representations. Weak nativist views claim only that humans, like other animals, possess an innate system for representing approximate numerical quantity. We present a new strong nativist model of the origins of numerical concepts and defend the strong nativist approach against recent cross-cultural studies that have been interpreted to show that precise numerical concepts are dependent on language and that they are restricted to (...) speakers of languages with the right kind of structure. (shrink)

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated (...) by a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

The history of twentieth-century American psychology is often depicted as a history of the rise and fall of behaviorism. Although historians disagree about the theoretical and social factors that have contributed to the development of experimental psychology, there is widespread consensus about the growing and declining influence of behaviorism between approximately 1920 and 1970. Since such wide-scope claims about the development of American psychology are typically based on small and unrepresentative samples of historical data, however, the question rises to what (...) extent the received view is justified. This paper aims to answer this question in two ways. First, we use advanced scientometric tools to quantitatively analyze the metadata of 119.278 papers published in American journals between 1920 and 1970. We reconstruct the development and structure of American psychology using co-citation and co-occurrence networks and argue that the standard story needs reappraising. Second, we argue that the question whether behaviorism was the ‘dominant’ school of American psychology is historically misleading to begin with. Using the results of our bibliometric analyses, we argue that questions about the development of American psychology deserve more fine-grained answers. (shrink)

This chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than (...) the ANS having noisy or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain. (shrink)

What Stanislas Debaene dubs "the number sense" is a natural ability humans share with other animals, enabling us to "count" to four virtually instantaneously. This so-called "accumulator" provides "a direct intuition of what numbers mean". Beyond four, our ability to perceive numbers becomes approximate, though concepts enable us to move beyond approximation. Because humans typically learn number concepts in early childhood, we easily forget that our brains retain the number sense throughout life. This book examines the (...) biological basis for this intuitive ability, with nine chapters organized into three readily graspable groups of three. Aside from its frustrating lack of a clear referencing system, the book is a pleasure to read. (shrink)

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with (...) a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS. (shrink)

Clarke and Beck assume that approximate number system representations should be assigned referents from our scientific ontology. However, many representations, both in perception and cognition, do not straightforwardly refer to such entities. If we reject Clarke and Beck's assumption, many possible contents for ANS representations besides number are compatible with the evidence Clarke and Beck cite.

I argue that analogue mental representations possess a canonical decomposition into privileged constituents from which they compose. I motivate this suggestion, and rebut arguments to the contrary, through reflection on the approximate number system, whose representations are widely expected to have an analogue format. I then argue that arguments for the compositionality and constituent structure of these analogue representations generalize to other analogue mental representations posited in the human mind, such as those in early vision and visual imagery.

Beck presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey. According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system, which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being general to (...) any account of inductive learning. While the account of Carey and Beck focuses on the OTS, in this paper I want to reconsider the importance of another empirically well-established cognitive core system for treating numerosities, namely the approximate number system. Since the ANS-based account offers a potential alternative for integer concept acquisition, I show that it provides a good reason to revisit the deviant-interpretation challenge. Finally, I will present a hybrid OTS-ANS model as the foundation of integer concept acquisition and the framework of enculturation as a solution to the challenge. (shrink)

This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered. First Part of this book presents some (...) theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes. Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification. Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well. (shrink)

A number of authors have recently put forward arguments pro or contra various rules for scoring probability estimates. In doing so, they have skipped over a potentially important consideration in making such assessments, to wit, that the hypotheses whose probabilities are estimated can approximate the truth to different degrees. Once this is recognized, it becomes apparent that the question of how to assess probability estimates depends heavily on context.

There exists a huge number of numerical methods that iteratively construct approximations to the solution y(x) of an ordinary differential equation (ODE) y′(x) = f(x,y) starting from an initial value y_0=y(x_0) and using a finite approximation step h that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer – the Infinity Computer (it has been patented and its working prototype exists). The new computer (...) is able to work numerically with finite, infinite, and infinitesimal numbers giving so the possibility to use different infinitesimals numerically and, in particular, to take advantage of infinitesimal values of h. To show the potential of the new framework a number of results is established. It is proved that the Infinity Computer is able to calculate derivatives of the solution y(x) and to reconstruct its Taylor expansion of a desired order numerically without finding the respective derivatives analytically (or symbolically) by the successive derivation of the ODE as it is usually done when the Taylor method is applied. Methods using approximations of derivatives obtained thanks to infinitesimals are discussed and a technique for an automatic control of rounding errors is introduced. Numerical examples are given. (shrink)

A number of authors, including me, have argued that the output of our most complex climate models, that is, of global climate models and Earth system models, should be assessed possibilistically. Worries about the viability of doing so have also been expressed. I examine the assessment of the output of relatively simple climate models in the context of discovery and point out that this assessment is of epistemic possibilities. At the same time, I show that the concept of epistemic (...) possibility used in the relevant studies does not fit available analyses of this concept. Moreover, I provide an alternative analysis that does fit the studies and broad climate modelling practices as well as meshes with my existing view that climate model assessment should typically be of real possibilities. On my analysis, to assert that a proposition is epistemically possible is to assert that it is not known to be false and is consistent with at least approximate knowledge of the basic way things are. I, finally, consider some of the implications of my discussion for available possibilistic views of climate model assessment and for worries about such views. I conclude that my view helps to address worries about such assessment and permits using the full range of climate models in it. (shrink)

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...) Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (shrink)

After 50 years of debate, the ethics of aggregation has reached a curious stalemate, with both sides arguing that only their theory treats people as equals. I argue that, on the issue of equality, both sides are wrong. From the premise that “each counts for one,” we cannot derive the conclusion that “more count for more”—or its negation. The familiar arguments from equality to aggregation presuppose more than equality: the Kamm/Scanlon “Balancing Argument” rests on what social choice theorists call “(Positive) (...) Responsiveness,” Kamm’s “Aggregation Argument” assumes that “equal” lives are fungible, and Hsieh et al. have it that spreading goods broadly best approximates equality. In each case, the crucial premise is not equality itself but a further idea that Taurek, I argue, can safely reject. I conclude with a conjecture: there is no theory-neutral argument that settles the question of whether the numbers count. (shrink)

Recently a number of variously motivated epistemologists have argued that knowledge is closely tied to practical matters. On the one hand, radical pragmatic encroachment is the view that facts about whether an agent has knowledge depend on practical factors and this is coupled to the view that there is an important connection between knowledge and action. On the other hand, one can argue for the less radical thesis only that there is an important connection between knowledge and practical reasoning. (...) So, defenders of both of these views endorse the view that knowledge is the norm of practical reasoning. This thesis has recently come under heavy fire and a number of weaker proposals have been defended. In this paper counter-examples to the knowledge norm of reasoning will be presented and it will be argued that this viewand a number of related but weaker viewscannot be sustained in the face of these counter-examples. The paper concludes with a novel proposal concerning the norm of practical reasoning that is immune to the counter-examples introduced here. (shrink)

The paper proposes a new formal approach to vagueness and vague sets taking inspirations from Pawlak’s rough set theory. Following a brief introduction to the problem of vagueness, an approach to conceptualization and representation of vague knowledge is presented from a number of different perspectives: those of logic, set theory, algebra, and computer science. The central notion of the vague set, in relation to the rough set, is defined as a family of sets approximated by the so called lower (...) and upper limits. The family is simultaneously considered as a family of all denotations of sharp terms representing a suitable vague term, from the agent’s point of view. Some algebraic operations on vague sets and their properties are defined. Some important conditions concerning the membership relation for vague sets, in connection to Blizard’s multisets and Zadeh’s fuzzy sets, are established as well. A classical outlook on a logic of vague sentences (vague logic) based on vague sets is also discussed. (shrink)

There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer - the Infinity Computer - able to work numerically with finite, infinite, and infinitesimal number. It is proved that the Infinity Computer (...) is able to calculate values of derivatives of a higher order for a wide class of functions represented by computer procedures. It is shown that the ability to compute derivatives of arbitrary order automatically and accurate to working precision is an intrinsic property of the Infinity Computer related to its way of functioning. Numerical examples illustrating the new concepts and numerical tools are given. (shrink)

The latest draft (posted 05/14/22) of this short, concise work of proof, theory, and metatheory provides summary meta-proofs and verification of the work and results presented in the Theory and Metatheory of Atemporal Primacy and Riemann, Metatheory, and Proof. In this version, several new and revised definitions of terms were added to subsection SS.1; and many corrected equations, theorems, metatheorems, proofs, and explanations are included in the main text. The body of the text is approximately 18 pages, with 3 sections; (...) 1, an Introduction (with a 108 page listing of key terms & definitions), sect. 2, the Results, sect. 3, Discussion (commentary & predictions), and sect. 4, Works Cited. As much as possible, the style is intended for readability and understanding by very bright children (with some interest & knowledge of maths, etc.) and very interested nonprofessionals. The results of this project also enable upgrades of number theory, set theory, proof theory, metamathematics, the foundations of science, and quantum mechanics theory (etc.). (shrink)

The human brain consists of approximately 100 billion electrically active neurones that generate an endogenous electromagnetic field, whose role in neuronal computing has not been fully examined. The source, magnitude and likely influence of the brain's endogenous em field are here considered. An estimate of the strength and magnitude of the brain's em field is gained from theoretical considerations, brain scanning and microelectrode data. An estimate of the likely influence of the brain's em field is gained from theoretical principles and (...) considerations of the experimental effects of external em fields on neurone firing both in vitro and in vivo. Synchronous firing of distributed neurones phase-locks induced em field fluctuations to increase their magnitude and influence. Synchronous firing has previously been demonstrated to correlate with awareness and perception, indicating that perturbations to the brain's em field also correlate with awareness. The brain's em field represents an integrated electromagnetic field representation of distributed neuronal information and has dynamics that closely map to those expected for a correlate of consciousness. I propose that the brain's em information field is the physical substrate of conscious awareness - the cemi field - and make a number of predictions that follow from this proposal. Experimental evidence pertinent to these predictions is examined and shown to be entirely consistent with the cemi field theory. This theory provides solutions to many of the intractable problems of consciousness - such as the binding problem - and provides new insights into the role of consciousness, the meaning of free will and the nature of qualia. It thus places consciousness within a secure physical framework and provides a route towards constructing an artificial consciousness. (shrink)

One of the key responsibilities of public institutions in liberal democracies is to formulate recommendations for decision makers. However, public institutions realize that decision makers will often partly ignore their recommendations. This situation of “partial compliance” with recommendations raises a number of philosophical issues for institutions. Based on an analysis of 570 recommendations drawn from 40 Quebec public-sector documents and reports, we identify two issues surrounding the structure of public-policy recommendations.

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s (...) Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...) truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

Philosophers interested in the theoretical consequences of predictive processing often assume that predictive processing is an inferentialist and representationalist theory of cognition. More specifically, they assume that predictive processing revolves around approximated Bayesian inferences drawn by inverting a generative model. Generative models, in turn, are said to be structural representations: representational vehicles that represent their targets by being structurally similar to them. Here, I challenge this assumption, claiming that, at present, it lacks an adequate justification. I examine the only argument (...) offered to establish that generative models are structural representations, and argue that it does not substantiate the desired conclusion. Having so done, I consider a number of alternative arguments aimed at showing that the relevant structural similarity obtains, and argue that all these arguments are unconvincing for a variety of reasons. I then conclude the paper by briefly highlighting three themes that might be relevant for further investigation on the matter. (shrink)

Today, we have a mixture of young and older individuals, people with special needs, and people who can care for themselves. Over 1 billion people are estimated to be disabled; this figure corresponds to about 15% of the world's population, with 3.8% (approximately 190 million people) accounting for people aged 15 and up (Organization, 2011). The number of people with disabilities is upward due to the increase in chronic health conditions and many other things. These and other factors have (...) made the need for proper care facilities urgent in today's society. Several care facilities are built to help people with disabilities live their everyday lives and not be left out of the community. (shrink)

The prestige of an academic institution may be determined as a function of affiliations with other academic institutions. Using digital tools to data-scrape, data-mine, and perform network analysis on university websites, an approximation of numbers of academic affiliations may be measured. Especially observing the alma mater institutions of the faculty of employed institutions, these numbers show the relative employment of alumni and a proxy metric for the relative prestige of their degree-granting institutions. These affiliations can be charted and graphed to (...) determine the distributions of affiliations throughout an academic ecosystem from which we might draw conclusions about that system’s hierarchies and inequalities. Here we use anglophone PhD-granting philosophy departments as a case study for this methodology with tentative conclusions. (shrink)

It is wholly uncontroversial that measurements-or, more properly, propositions that are measurement reports-are often paradigmatically good cases of propositions that serve the function of evidence. In normal cases it is also obvious that stating such a report is an utterly pedestrian case of successful assertion. So, for example, there is nothing controversial about the following claims: (1) that a proposition to the effect that a particular thermometer reads 104C when properly used to determine the temperature of a particular patient is (...) evidence that the patient in question has a fever and (2) that there is nothing wrong with asserting the proposition that a particular thermometer reads 104C for appropriate reasons of communication, etc. when the thermometer has been properly used to determine the temperature of a particular patient. Here it will be shown that Timothy Williamson’s commitments to a number of principles about knowledge and assertion imply that a whole class of utterly ordinary statements like these that are used as evidence are not really evidence because they are not knowledge and so are (perversely) unassertable according to his principled commitments. This paper deals primarily with the second of these two problems and an alternative account of the norms of assertion is introduced which allows for the assertability of such measurement reports. (shrink)

In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we ask the (...) question, whether at least the variational principle for Haus- dor® dimension holds, which means that there is a sequence of ergodic measures such that their Hausdor® dimension approximates the Hausdor® dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal di®eomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two di®erent rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of de- velopment of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the ¯rst step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. A±ne dynamical systems represent simple examples of non conformal systems. They are easy to de¯ne, but studying their dimensional theoretical properties does never- theless provide challenging mathematical problems and exemplify interesting phe- nomena. We consider here a special class of self-a±ne repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise a±ne maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as gener- alized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of di- mension theory of a±ne dynamical systems and de¯ne the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and ¯nd representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see the- orem 4.1.). Then in chapter ¯ve we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to ques- tions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the iden- tity of box-counting and Hausdor® dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdor® dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdor® dimension for the repellers. On the other hand the vari- ational principle for Hausdor® dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will ¯nd a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdor® dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical excep- tions to our generic results. The starting point of our considerations in section nine are results of ErdÄos [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for in¯nite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdor® and box-counting dimension for all of our systems. In the ¯rst section of chapter ten we ¯nd parameter values such that the variational principle for Hausdor® dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the ¯rst known examples of dynamical systems for which the variational principle for Hausdor® dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdor® dimension; the question if the variational principle for Hausdor® dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdor® and box-counting dimension can drops because there are number theoretical pecu- liarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will ¯nd two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor JÄorg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFÄoG der Freien UniversitÄat Berlin". (shrink)

The study area was a small lowland river valley (the Łęg river) located in the south-east of Poland. The object of investigation was the macrophytes of 10 river lakes with corresponding active river channel stretches of the same length as the cut-offs. The aim was to check the difference in species diversity between cut-off and active river channels. The second aim was to test the following hypothesis: vegetation of river lake has been shaped under the influence of contiguous river stretch (...) which has left a measurable mark in species abundance and composition. To test this, we checked whether a cut-off channel’s flora is more similar to flora of the contiguous river stretch, than to flora of a farther river stretch. During the course of the study it was found that the average species richness was approximately two times higher in the cut-off channels than in the river stretches. The number of the species exclusive for the river lakes was nine times higher in comparison with the river’s exclusives (not found in the cut-offs). The Shannon diversity index definitely spoke in favor of the river lakes. These results clearly show the significance of river lakes in maintaining biodiversity of aquatic ecosystems in a small river valley. We did not confirm our hypothesis concerning floristic relation between an active channel and its cut-offs. The floristic similarity between a given cut-off channel and the contiguous active river channel stretch is not stronger than the similarity between this lake and more distant river stretches. The cause of such a state of affairs may be high natural dynamics of investigated habitats and anthropogenic transformation of the river valley. (shrink)

The paper argues that the depiction of the Mediterranean coast of Africa in Ptolemy’s Geography was based on a source similar to the Stadiasmus of the Great Sea. Ptolemy’s and the Stadiasmus’ toponymy and distances between major points are mostly in good agreement. Ptolemy’s place names overlap with those of the Stadiasmus by 80%, and the total length of the coastline from Alexandria to Utica on Ptolemy’s map deviates from the Stadiasmus data by only 1% or 1.5%. A number (...) of serious disagreements between Ptolemy’s map and the Stadiasmus regarding the length of particular coastal stretches can be explained by assuming that Ptolemy had to tailor the distance data derived from periploi to his other sources, especially, to the longitudes of the key reference points, such as Cape Phyces, Cyrena, Berenica, Aesporis, Thena, and Carthage. A notable stretching and the subsequent contraction of the coast between Alexandria and Cyrenaica, as are exhibited by Ptolemy’s map relative to the Stadiasmus’ data, can be explained by assuming that several points on this coast were tied to the position of Crete, which was moved to the west being pushed by the westward shift of Rhodes. A sharp contraction of the two coastal stretches of the Great Sirte, oriented along the north-south direction, can be explained by Ptolemy’s erroneously underestimated value for the circumference of the Earth. The analysis of this contraction, as well as of the east-west stretching of the coast between Alexandria and Cyrene in angular terms relative to the modern map, makes it possible to assess the magnitude of Ptolemy’s error and to determine the length of his stade. This analysis shows that Ptolemy’s value for the circumference of the Earth must have been underestimated by approximately 20–27%, and Ptolemy’s stade must have been approximately 175–185 m length. Comparison of the Stadiasmus distance data with the modern map shows that the average length of the stade was close to 179 m or to the “common” stade of 185 m for the stretch between Alexandria and Berenice. (shrink)

Bu kitapta, Ebû İshâk es-Saffâr’ın (öl. 534/1139) kelâmî görüşleri, Telḫîṣü’l-edille li-ḳavâʿidi’t-tevḥîd adlı eserinde Allah’ın isimlerinin anlamlarını açıklarken yaptığı yorumlar çerçevesinde ele alınmaktadır. Ebû İshâk es-Saffâr, 6./12. yüzyıl Hanefî-Mâtürîdî âlimlerinden biridir. Kelâma dair Telḫîṣü’l-edille eserinde esmâ-i hüsnâ konusuna ayrıntılı olarak yer vermektedir. İki cilt hâlinde yayımlanan bu eserin yaklaşık üçte birlik bir kısmını esmâ-i hüsnâ konusu oluşturmaktadır. Bu kısım incelendiğinde, Saffâr’ın Allah’ın varlığı, birliği ve sıfatları ile ilgili konular başta olmak üzere pek çok konuyu 175 esmâ-i hüsnâya dayanarak izah ettiği görülmektedir. (...) O, esmâ-i hüsnâ bölümünde yer vermediği bazı isimlere ise müstakil başlıklar altında değinmektedir. Örneğin el-Mütekkelim ismi kelâm sıfatını bağlamında ve halku’l-Kur’ân ile icâz’ul-Kur’ân gibi konularla ilişkili bir şekilde ele almaktadır. Bu isimler de listeye dahil edildiğinde sayı 178’e ulaşmaktadır. Bu durumda eserin yarısını esmâ-i hüsnâ konusu teşkil etmektedir. -/- Saffâr, esmâ-i hüsnâ bölümünde alfabetik bir sıra içerisinde ele aldığı ilâhî isimleri öncelikle lugavî (semantik) yönden izah etmektedir. Sonrasında ise değerlendirdiği ilahî ismi, bir kelâm konusu ile bağlantı kurarak kelâmî perspektifle açıklamaktadır Esmâ-i hüsnâ temelinde ele alınan konuların hilâfet meselesi hariç diğer kelâm bahislerini kapsadığı görülmektedir. Saffâr öncesi Hanefî-Mâtürîdî kelâm literatürü içinde esmâ-i hüsnânın bu kadar kapsamlı ele alındığı başka bir eser bilinmemektedir. -/- Bu kitap; üç ana bölümden oluşmaktadır. “Metodolojik Çerçeve” başlıklı giriş bölümünde çalışmanın konusu, önemi, amacı, yöntemi ve kaynakları hakkında bilgi verilmiştir. Birinci bölümde Saffâr’ın yaşadığı sosyokültürel çevre olan Mâverâünnehir bölgesi ile Buhara ve Merv şehirlerinin siyasî, sosyal ve dinî durumu ortaya konulmaya çalışılmıştır. İkinci bölümde esmâ-i hüsna konusunun anlaşılmasına temel oluşturan isim, tesmiye, müsemmâ, sıfat ve vasf gibi kavramlar ile esmâ-i hüsnânın sayısı ve ihsâsı gibi kelâmî tartışmalara değinilmiştir. Sonrasında Saffâr öncesi dönemde kaleme alınan esmâ-i hüsnâ litaratürü hakkında bilgi verilmiştir. Bölüm sonuna Saffâr’ın rivayet ettiği 178 ilahî isme dair ayrıntılı bir tablo eklenmiştir. Üçüncü bölümde öncelikle, Saffâr’ın esmâ-i hüsnâyı izah ederken dikkate aldığı kelâmî ilkeler tespit edilmeye çalışılmıştır. Sonrasında ise Saffâr’ın Telḫîṣü’l-edille’de ilâhî isimleri açıklarken ortaya koyduğu kelâmî görüş ve değerlendirmeler belirlenerek sistematik bir şekilde kategorize edilmiştir. Bu kapsamda ele alınan her konunun sonuna ilgili ilâhî isimleri ve bağlantılı olduğu tartışmaları içeren tablolar eklenmiştir. Sonuç bölümünde ise Saffâr’ın esmâ-i hüsnâ anlayışına dayanan kelâm yöntemine dair ulaştığımız sonuçlara yer verilmiştir. Bu kitapta onun, esmâ-i hüsnânın %75’inde kelâmî yorumlarda bulunduğu ve bilgi-varlık bahsinden âhiret hayatına kadar bütün kelâm konularını esmâ-i hüsnâ ile bağlantılı yorumladığı tespit edilmiştir. Ulaşılan bu sonuçlar, Saffâr’ın kelâm anlayışının ilâhî isimlerin yorumuna dayandığını ortaya koymaktadır. [his book discusses the theological views of Abū Isḥāq al-Ṣaffār d. 534/1139), within the framework of his comments on the meanings of Allah’s names, provided in his work titled Talkhīṣ al-adilla. Abū Isḥāq al-Ṣaffār is one of the Ḥanafite-Māturīdite scholars in the 6th/12th century. In his work titled Talkhīṣ al-adilla li-qawāʿid al-tawḥīd on kalām, he spared extensive space for al-asmāʾ al-husnā. Approximately one third of this work, published in two volumes, is devoted to al-asmāʾ al-husnā. An examination of the related section reveals that al-Ṣaffār explains many issues, particularly those related to the existence, unity and attributes of Allah, based on 175 al-asmāʾ al-husnā. He mentions some of the names that he does not include in the al-asmāʾ al-husnā section under separate headings. For example, the name al-Mutakallim is addressed within the context of the attribute of kalām and in relation to subjects, such as the khalq al-Qurʾān and i‘jaz al-Qurʾān. Upon the addition of these names to the list, the number names reaches 178. This means that half of the work deals with the subject of al-asmāʾ al-husnā. -/- al-Ṣaffār lists the divine names in alphabetical order and explains them semantically in the chapter of al-asmāʾ al-husnā. Then he goes on to clarify each divine name through a theological lens with a specific reference to the subject of kalām. In the pre-Saffar Ḥanafite-Māturīdite theological literature, there is no other work that addresses al-asmāʾ al-husnā in such an extensive way. -/- This book consists of three main sections. The first section titled “Methodological Framework”, elaborates on the focus, significance, purpose and method of the study, along with the sources used. The first part describes the political, social and religious status of Transoxiana (Mā-warāʾ al-Nahr) region and the cities of Bukhara and Marw, the sociocultural environment in which Saffar lived. The second chapter addresses various concepts, which promote the understanding of al-asmāʾ al-husnā, such as name, tasmiya, musammā, attribute and qualification in addition to the theological debates such as the number and iḥṣāʾ of al-asmāʾ al-husnā. Then, it provides information about the al-asmāʾ al-husnā literature produced in the pre- Ṣaffār period. The end of each chapter comes with a detailed table with the 178 divine names mentioned by al-Ṣaffār. In the third chapter, the author initially discusses the theological principles that al-Ṣaffār considered while explaining the essence of al-asmāʾ al-husnā. This section also determines and systematically categorizes the theological views and evaluations put forward by al-Ṣaffār while explaining the divine names in Talkhīṣ al-adilla. The tables with the divine names and the related discussions can be seen at the end of the discussion for each subject. The last section presents the conclusions reached, regarding the kalām method based on al-Ṣaffār’s understanding of the essence of al-asmāʾ al-husnā. The present study revealed that he made theological interpretations in 75% of the al-asmāʾ al-husnā and interpreted all theological issues ranging from the subjects of knowledge and existence to the Afterlife in connection with the al-asmāʾ al-husnā. These results indicate that al-Ṣaffār's understanding of kalām is based on the interpretation of the divine names.]. (shrink)

Our major application of the so-called Serendipity-Mindsponge-3D (SM3D) knowledge management theory has been published for some 30 weeks (DOI: 10.1057/s41599-022-01034-6). Since January 2022, the article has attracted approximately nine thousand accesses and has received a considerable number of citations from peer-reviewed articles.

Sikhism, the fifth-largest organized religion in the world, was founded in the fifteenth century in Punjab, India. Guru Nanak Dev and his successor Sikh Gurus established this system of religious philosophy. The sacred scripture, Sri Guru Granth Sahib, is the present Guru of the Sikhs. The religious philosophy of Sikhism is traditionally known as Gurmat. Sikhism originated from the word Sikh, having the Sanskrit root śiṣya meaning "disciple" or "learner." With about 27 million followers or 0.39% of the world population (...) [4], approximately 83% of the Sikhs live in India. Islam is the religion articulated by the Holy Quran, a religious book. Its adherents consider it the verbatim word of the one incomparable God (Allah). The Muslims live by following the Prophet of Islam Hazrat Muhammad's demonstrations and real-life examples (Sunnah). The Sunnah has been collected through Prophet Hazrat Muhammad's companions' narration in collections of Hadith. Islam means submission to God. The word Islam is derived from the Arabic word "Salam," which means peace. With about 1.7 billion followers or 23% of the global population, Islam is the second-largest religion by the number of its adherents. (shrink)

New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work (...) with the exact values of the derivatives, instead of their approximations. (shrink)

Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well know that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas (...) and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones. (shrink)

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within (...) mainstream number cognition research, along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (shrink)

One of the most important abilities we have as humans is the ability to think about number. In this chapter, we examine the question of whether there is an essential connection between language and number. We provide a careful examination of two prominent theories according to which concepts of the positive integers are dependent on language. The first of these claims that language creates the positive integers on the basis of an innate capacity to represent real numbers. The (...) second claims that language’s function is to integrate contents from modules that humans share with other animals. We argue that neither model is successful. (shrink)

The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring to numbers as (...) abstract objects. (shrink)

Suppose that scientific realists believe that a successful theory is approximately true, and that constructive empiricists believe that it is empirically adequate. Whose belief is more likely to be false? The problem of underdetermination does not yield an answer to this question one way or the other, but the pessimistic induction does. The pessimistic induction, if correct, indicates that successful theories, both past and current, are empirically inadequate. It is arguable, however, that they are approximately true. Therefore, scientific realists overall (...) take less epistemic risk than constructive empiricists. (shrink)

Hájek (Erkenntnis 70(2):211–235, 2009) argues that probabilities cannot be the limits of relative frequencies in counterfactual infinite sequences. I argue for a different understanding of these limits, drawing on Norton’s (Philos Sci 79(2):207–232, 2012) distinction between approximations (inexact descriptions of a target) and idealizations (separate models that bear analogies to the target). Then, I adapt Hájek’s arguments to this new context. These arguments provide excellent reasons not to use hypothetical frequencies as idealizations, but no reason not to use them as (...) approximations. (shrink)

The main question addressed in this paper is whether some false sentences can constitute evidence for the truth of other propositions. In this paper it is argued that there are good reasons to suspect that at least some false propositions can constitute evidence for the truth of certain other contingent propositions. The paper also introduces a novel condition concerning propositions that constitute evidence that explains a ubiquitous evidential practice and it contains a defense of a particular condition concerning the possession (...) of evidence. The core position adopted here then is that false propositions that are approximately true reports of measurements can constitute evidence for the truth of other propositions. So, it will be argued that evidence is only quasi-factive in this very specific sense. (shrink)