Abstract. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. Alexandroff spaces, as they are called today, have many interesting properties that distinguish them from other topological spaces. In particular, they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, a special type (...) of Alexandroff spaces was used by Ian Rumfitt to elucidate the logic of vague concepts in a new way. According to his approach, conceptual spaces such as the color spectrum give rise to classical systems of concepts that have the structure of atomic Boolean algebras. More precisely, concepts are represented as regular open regions of an underlying conceptual space endowed with a topological structure. Something is subsumed under a concept iff it is represented by an element of the conceptual space that is maximally close to the prototypical element p that defines that concept. This topological representation of concepts comes along with a representation of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical operations that characterize regular open interpretations of classical Boolean propositional logic. In the last 20 years, conceptual spaces have become a popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using prototypes and metrics of similarity spaces, one obtains geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces. Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2 order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The crucial role of order theory for Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical stimuli in favor of a more fine-grained gradual distinction between more-orless prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines. Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology. (shrink)

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all (...) inductive formulae are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)

The aim of this paper is to show that topology has a bearing on<br><br>combinatorial theories of possibility. The approach developed in this article is “mapping account” considering combinatorial worlds as mappings from individuals to properties. Topological structures are used to define constraints on the mappings thereby characterizing the “really possible” combinations. The mapping approach avoids the well-known incompatibility problems. Moreover, it is compatible with atomistic as well as with non-atomistic ontologies.It helps to elucidate the positions of logical (...) atomism and monism with theaid of topological separation axioms. (shrink)

The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early forrunner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be (...) useful in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle. (shrink)

(1) An introduction to the principles of conceptual modelling, combinatorial heuristics and epistemological history; (2) the examination of a number of perennial epistemological-methodological schemata: conceptual spaces and blending theory; ars inveniendi and ars demonstrandi; the two modes of analysis and synthesis and their relationship to ars inveniendi; taxonomies and typologies as two fundamental epistemic structures; extended cognition, cognitio symbolica and model-based reasoning; (3) Plato’s notions of conceptual spaces, conceptual blending and hypothetical-analogical models (paradeigmata); (4) Ramon Llull’s concept analysis and (...) combinatoric spaces; (5) Gottfried Leibniz’s development of compositional analysis and synthesis as a general modelling method and a paradigm for ars inveniendi; (6) Fritz Zwicky’s revival of the morphological method of analysis and construction, and its subsequent computerised applications. (shrink)

The Controversy over the Existence of the World (henceforth Controversy) is the magnum opus of Polish philosopher Roman Ingarden. Despite the renewed interest for Ingarden’s pioneering ontological work whithin analytic philosophy, little attention has been dedicated to Controversy's main goal, clearly indicated by the very title of the book: finding a solution to the centuries-old philosophical controversy about the ontological status of the external world. -/- There are at least three reasons for this relative indifference. First, even at the time (...) when the book was published, the Controversy was no longer seen as a serious polemical topic, whether it was disqualified as an archaic metaphysical pseudo-problem, or taken to be the last remnant of an antiscientific approach to philosophy culminating in idealism and relativism. Second, Ingarden’s reasoning on the matter is highly complex, at times misleading, and even occasionally faulty. Finally, his analysis is not only incomplete – Controversy being unachieved – but also arguably aporetic. -/- One may wonder, then, why it is still worth excavating this mammoth treatise to study an issue apparently no longer relevant to contemporary philosophy. Aside from historical and exegetical purposes, which are of course very interesting in their own right, Ingarden’s treatment of the Controversy remains one of the most detailed and ambitious ontological undertakings of the twentieth century. Not only does it lay out an incredibly detailed map of possible solutions to the Controversy, but it also tries to show why the latter is a genuine and fundamental problem that owes its hasty disqualification to various oversimplifications over the course of the history of philosophy. -/- In this chapter, I first give an overview of Ingarden’s method, which relies mainly on a combinatorial analysis. Then, I summarize his examination of possible solutions to the Controversy, and determine which ones can be ruled out on ontological grounds. Finally, I explain why this ambitious project ultimately leads to a theoretical impasse, leaving Ingarden unable to come up with a definitive solution to the Controversy – regardless of the fact that the book is unachieved. I argue that his analysis of the problem yields a more modest but nonetheless valuable result. (shrink)

This essay is an examination of the relationship between phenomenology and analytic method in the philosophy of law. It proceeds by way of a case study, the requirement of compliance in Raz’s theory of mandatory norms. Proceeding in this way provides a degree of specificity that is otherwise neglected in the relevant literature on method. Drawing on insights from the philosophy of art and cognitive neuroscience, it is argued that the requirement of compliance is beset by a range of epistemological (...) difficulties. The implications of these difficulties are then reviewed for method and normativity in practical reason. A topology of normativity emerges nearer the end of the paper, followed by a brief examination of how certain normative categories must satisfy distinct burdens of proof. (shrink)

The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; (...) and (3) the specific forms of representation and notation shared and selected by the practitioners. Moreover, it is claimed that a broadening of the notion of ‘permissible action’ as introduced by Larvor (2012) with respect to mathematical arguments, allows for a consideration of all these three elements simultaneously. Second, a case from topology – the proof of Alexander’s theorem – is presented to illustrate a concrete analysis of a mathematical practice and to exemplify the proposed method. It is discussed that the attention to the three elements of the practice identified above brings to the emergence of philosophically relevant features in the practice of topology: the need for a revision in the definition of criteria of validity, the interest in tracking the operations that are performed on the notation, and the constant and fruitful back-and-forth from one representation to another in dealing with mathematical content. Finally, some suggestions for further research are given in the conclusions. (shrink)

In parts of his Notebooks, Tractatus and in “Lecture on Ethics”, Wittgenstein advanced a new approach to the problems of the meaning of life. It was developed as a reaction to the explorations on this theme by Bertrand Russell. Wittgenstein’s objective was to treat it with a higher degree of exactness. The present paper shows that he reached exactness by treating themes of philosophical anthropology using the formal method of topology.

What is the philosophical significance of Heidegger’s interpretation of the Japanese notion of kotoba (言葉) for Japanese philosophy? Was his conversation with Tezuka Tomio a real dialogue or not? To answer to these correlated questions, I elucidate Heidegger’s 1954 essay “A Dialogue on Language” by following a topological mode of thinking, and I inquire into the way-making of a “thinking conversation”. First, I problematize whether Heidegger engaged in a genuine dialogue with Tezuka. To that end, I distinguish the hermeneutic (...) horizon of the actual encounter between Tezuka and Heidegger from Heidegger’s essay which places Tezuka (the Japanese) and Heidegger (the Inquirer) in a fictional philosophical conversation. Second, I argue that Heidegger’s topological method of interpretating kotoba can be read as a poetic means of thematizing East-West dialogue. Third and finally, exploring the topological sense of kotoba, I engage with third generation Kyoto School thinker Ueda Shizuteru’sideaof“hollow words” of language, situated in a twofold view of the world. I conclude that the true character of Heidegger’s conversation with Tezuka can be identified neither in Heidegger’s “actual” encounter with Tezuka, nor merely in Heidegger’s “hollow” essay. Departing from Ueda’s account of kotoba, it appears that a genuine conversation with language can be located in the dialogue of actuality and hollowness, which finds it expression in poetic language. (shrink)

Karl Christian Friedrich Krause and Georg Wilhelm Friedrich Hegel are two representatives of German Idealism, both of whom developed impressing category systems. At the core of both systems is the question of the relation of the Absolute to its determinations and the determinations of finite beings. Both idealists try to deduce their respective category systems from the immediacy of the Absolute. Both use combinatorial methods to get from known to new categories or constellations in the system, which then (...) unfold in the world. Krause is thereby considered the eponym of so-called panentheism, the doctrine that “Everything is in God.” Hegel is also often referred to as a panentheist. Through a comparison of the two systems of categories, in this essay the thesis will be advocated that Hegel was in no sense a panentheist. Krause is and remains the gold standard of panentheism. (shrink)

The hope of finding new methods of predicting the course of historical processes could be connected with the recent developments of the theory of self-organisation, also called synergetics. It provides us with knowledge of constructive principles of co-evolution of complex social systems, co-evolution of countries and geopolitical regions being at different stages of development, integration of the East and the West, the North and the South. Due to the growth of population on the Earth in blow-up regime, the general (...) and local instability of development is increasing. The social world seems to go towards a unity, a sustainable commonwealth through pulsations rather than monotonously. It goes via alternation of disintegration, even if partial, and getting more powerful integrations of structures. There is a path of multiple reduction of temporal and material expenses, a path of a resonant excitation of desirable and — no less important — feasible in a given social system structure. In the case of the right topology of integration of geopolitical structures an arising whole organisation can accelerate its tempo of evolution. 1999 Elsevier Science Ltd. All rights reserved. (shrink)

In this article, network science is discussed from a methodological perspective, and two central theses are defended. The first is that network science exploits the very properties that make a system complex. Rather than using idealization techniques to strip those properties away, as is standard practice in other areas of science, network science brings them to the fore, and uses them to furnish new forms of explanation. The second thesis is that network representations are particularly helpful in explaining the properties (...) of non-decomposable systems. Where part-whole decomposition is not possible, network science provides a much-needed alternative method of compressing information about the behavior of complex systems, and does so without succumbing to problems associated with combinatorial explosion. The article concludes with a comparison between the uses of network representation analyzed in the main discussion, and an entirely distinct use of network representation that has recently been discussed in connection with mechanistic modeling. (shrink)

General Morphological Analysis (GMA) is a method for structuring a conceptual problem space – called a morphospace – and, through a process of existential combinatorics, synthesizing a solution space. As such, it is a basic modelling method, on a par with other scientific modelling methods including System Dynamics Modelling, Bayesian Networks and various types graph-based “influence diagrams”. The purpose of this article is 1) to present the theoretical and methodological basics of morphological modelling; 2) to situate GMA within a (...) broader modelling theoretical framework by developing a (morphological) model representing different modelling methods, and 3) to demonstrate some of the basic modelling techniques that can be carried out with GMA using dedicated computer support. (shrink)

Combinatorial behavior involves combining different elements into larger aggregates with meaning. It is generally contrasted with compositionality, which involves the combining of meaningful elements into larger constituents whose meaning is derived from its component parts. Combinatoriality is commonly considered a capacity found in primates and other animals, whereas compositionality often is considered uniquely human. Questioning the validity of this claim, this multidisciplinary special issue of the International Journal of Primatology unites papers that each study aspects of combinatoriality and compositionality (...) found in primate and bird communication systems, tool use, skills, and human language. The majority of authors conclude that compositionality is evolutionarily preceded by combinatoriality and that neither are uniquely human. This introduction briefly introduces readers to the major findings and issues raised by the contributors. -/- . (shrink)

Human language, hominin tool production modes, and multimodal communications systems of primates and other animals are currently well-studied for how they display compositionality or combinatoriality. In all cases, the former is defined as a kind of hierarchical nesting and the latter as a lack thereof. In this article, I extend research on combinatoriality and compositionality further to investigations of everyday primate skills. Daily locomotion modes as well as behaviors associated with subsistence practices, hygiene, or body modification rely on the hierarchical (...) nesting of different behavioral and cognitive actions into complex skills. I introduce a scheme which uses hierarchical organization to differentiate combinatorial from compositional skills. Combinatorial skills are defined either as aggregational or linearly hierarchical depending on whether the skill occurs momentarily in space or unfolds sequentially over time. Compositional skills are defined either as nested or interactionally hierarchical depending on whether the skill results in new constructs or in new interactions between existing constructs. The methodology I propose combines epistemological hierarchy theory with data from primatological field research and experimental and comparative psychological research and provides a means to integrate current constructionist and extended views on cognition and action with older research on behavioral logics in psychology and operational chain thinking in anthropology. The approach furthermore synchronizes with ongoing research on teleonomy, intentionality, and creativity. -/- . (shrink)

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

Dialectica: Mathematica and Physica, Truth and Justice, Trick and Life. Mathematica as the Constructive Metaphysica and Ontology. Mathematica as the constructive existential method. Сonsciousness and Mathematica: Dialectica of "eidos" and "logos". Mathematica is the Total Dialectica. The basic maternal Structure - "La Structure mère". Mathematica and Physica: loss of existential certainty. Is effectiveness of Mathematica "unreasonable"? The ontological structure of space. Axiomatization of the ontological basis of knowledge: one axiom, one principle and one mathematical object. The main ideas and concepts (...) of the ontological construction/ "Point with a vector germ" and "heavenly triangle". "Ordo geometricus" and "Ordo onto-topological". Architecture of the onto-topological basis of knowledge: general framework structure, carcass and foundation. The absolute space and the absolute field. The absolute (natural) system of coordinates of Universum. Eidos of "idea of ideas", the symbol and the "formula of Justice". (shrink)

Ontologies of living things are increasingly grounded on the concepts and practices of current life science. Biological development is a process, undergone by living things, which begins with a single cell and (in an important class of cases) ends with formation of a multicellular organism. The process of development is thus prima facie central for ideas about biological individuality and organismality. However, recent accounts of these concepts do not engage developmental biology. This paper aims to fill the gap, proposing the (...) lineage view of stem cells as an ontological framework for conceptualizing organismal development. This account is grounded on experimental practices of stem cell research, with emphasis on new techniques for generating biological organization in vitro. On the lineage view, a stem cell is the starting point of a cell lineage with a specific organismal source, time-interval of existence, and ‘tree topology’ of branch-points linking the stem to developmental termini. The concept of ‘enkapsis’ accommodates the cell-organism relation within the lineage view; this hierarchical notion is further explicated by considering the methods and results of stem cell experiments. Results of this examination include a (partial) characterization of stem cells’ developmental versatility, and the context-dependence of developmental processes involving stem cells. (shrink)

The book is a history of the concept of experience in philosophy. Minakov focuses mainly on Western 19-20th century philosophical movements and their use of the experience concept. Author uses topological method to describe growth of the conseptual content of experience, as well as decline in its use in the end of 20th century.

The purpose of this paper is to review and critique Wessel Reijers and Bert Gordijn’s paper Moving from value sensitive design to virtuous practice design. In doing so, it draws on recent literature on developing value sensitive design (VSD) to show how the authors’ virtuous practice design (VPD), at minimum, is not mutually exclusive to VSD. This paper argues that virtuous practice is not exclusive to the basic methodological underpinnings of VSD. This can therefore strengthen, rather than exclude the VSD (...) approach. Likewise, this paper presents not only a critique of what was offered as a “potentially fruitful alternative to VSD” but further clarifies and contributes to the VSD scholarship in extending its potential methodological practices and scope. It is concluded that VPD does not appear to offer any original contribution that more recent instantiations of VSD have not already proposed and implemented. (shrink)

In the last two decades, philosophy of neuroscience has predominantly focused on explanation. Indeed, it has been argued that mechanistic models are the standards of explanatory success in neuroscience over, among other things, topological models. However, explanatory power is only one virtue of a scientific model. Another is its predictive power. Unfortunately, the notion of prediction has received comparatively little attention in the philosophy of neuroscience, in part because predictions seem disconnected from interventions. In contrast, we argue that (...) class='Hi'>topological predictions can and do guide interventions in science, both inside and outside of neuroscience. Topological models allow researchers to predict many phenomena, including diseases, treatment outcomes, aging, and cognition, among others. Moreover, we argue that these predictions also offer strategies for useful interventions. Topology-based predictions play this role regardless of whether they do or can receive a mechanistic interpretation. We conclude by making a case for philosophers to focus on prediction in neuroscience in addition to explanation alone. (shrink)

Alsted was a foremost encyclopedist of the early seventeenth century. He provided both a complete presentation of all the subjects of philosophy (of which encyclopedia consisted) and a method to learn them. This method was an original synthesis of the dialectic of Petrus Ramus, the combinatorial art of memory of Raimond Lull and Giordano Bruno, and the method of presentation of philosophical disciplines of Bartholomäus Keckermann. Alsted’s encyclopedism was intended as a remedy to the postlapsarian condition of man and (...) was functional to the pedagogical reform pursued at the Academy of Herborn; this was, in turn, an essential part of the Calvinist state reform of the county of Nassau-Dillenburg. In theology, the importance of Alsted consists of having introduced millenarianism in the Reformed Europe, though his early, optimistic views on the imminent end of the world would change to pessimistic as a consequence of the Thirty Years’ War. (shrink)

While mechanistic explanation and, to a lesser extent, nomological explanation are well-explored topics in the philosophy of biology, topological explanation is not. Nor is the role of diagrams in topological explanations. These explanations do not appeal to the operation of mechanisms or laws, and extant accounts of the role of diagrams in biological science explain neither why scientists might prefer diagrammatic representations of topological information to sentential equivalents nor how such representations might facilitate important processes of explanatory (...) reasoning unavailable to scientists who restrict themselves to sentential representations. Accordingly, relying upon a case study about immune system vulnerability to attacks on CD4+ T-cells, I argue that diagrams group together information in a way that avoids repetition in representing topological structure, facilitate identification of specific topological properties of those structures, and make available to controlled processing explanatorily salient counterfactual information about topological structures, all in ways that sentential counterparts of diagrams do not. (shrink)

The dominant approach to analyzing the meaning of natural language sentences that express mathematical knowl- edge relies on a referential, formal semantics. Below, I discuss an argument against this approach and in favour of an internalist, conceptual, intensional alternative. The proposed shift in analytic method offers several benefits, including a novel perspective on what is required to track mathematical content, and hence on the Benacerraf dilemma. The new perspective also promises to facilitate discussion between philosophers of mathematics and cognitive scientists (...) working on topics of common interest. (shrink)

The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can (...) be used to prove a completeness theorem for S4.1. Further, it is shown that the McKinsey algebra MKX of a space X endoewed with an alpha-topologiy satisfies Esakia's GRZ axiom. (shrink)

The being is derived by a difference in Lacanian ontology. This difference is the basic element in Lacanian theory that grounds the unconscious subject. Because according to Lacan, the existence of the subject can not be self-proclaimed and it is represented by a signifier. Lacan gives the name "object a" to this paradoxical being which is distinguished by this difference or lack, and uses some topological transformations in order to be able to explain the structural paradoxes in the psychological (...) theory. The aim of this study is to explore the ontology of these paradoxical situations and try to decipher the function of topology in Lacan's theory. (shrink)

Mixing methods is a well-known innovative meth- odologic proposal for research in the second half of the 20th century social sciences. Reading literature about it, I observed the aspect that justifies this paper: Authors of theoretical contributions on mixing methods recognized that this was known to be a practice already in use many centuries ago. Some of them even have re-examined the whole history of the scientific method to search precedents. They are however individual and theoretical precedents. I (...) add in this paper the practical projection of these and other methodological theories on people’s training from Greek classical times. My hypothesis is that liberal arts was the basic syllabus in Western -and westernized- educaion for more than a millennium in such a way that results of their training precisely involved to mixing methods. In return, to understand the liberal arts in the light of mixing methods shows new aspects of their historical interest. I study the theoretical basis of this syllabus from Cicero to Alcuin. More important for future research, I conclude that Alcuin’s thesis about the correspondence between the gifts of the Holy Spiritu and the seven liberal arts can be extended to the Aristotelian dianoethics habits and to Dilthey’s 20th-century sciences of mind. (shrink)

The formal ontology here presented is what we might call a typed combinatorial Meinongian mereology. Its author seeks to formulate the laws, here called ‘canons’, regulating how entities can combine together in wholes of different sorts. The method, as in Bergmann’s earlier works, involves the construction of an ideal language of such a sort that the analysis of complex wholes can be achieved by transforming our natural-language representations of reality into what we might think of as artificial characteristic maps (...) or diagrams which allow the relevant ontological structures to be read off immediately from the symbolic representations which results. In former works Bergmann had held that the symbolic language of Principia Mathematica could serve as the appropriate diagrammatic device for the standard first-order functional calculus and develops instead a new sort diagrammatic language. (shrink)

Security and privacy are the major concerns in the internet of things (IoT) which are uncertain and unpredictable. Trust aware routing is one of the recent and effective strategies which ensure better resilience for IoT nodes from different security threats. Towards such concern, this paper proposes a new strategy called independent onlooker withstanding trust aware routing (IOWTAR) for IoT. IOWTAR introduced a new compound trust metric by combining three individual metrics namely independent trust, onlooker trust, and withstanding trust (a combination (...) of resilient trust and immovability trust). Independent trust and onlooker trust are assessed based on direct and indirect experiences of nodes about their neighbor nodes. Withstanding trust is assessed based on the stability and resilience of nodes towards dynamic topological changes and network failures respectively. Further, this work adapted the Firefly algorithm (FFA) to optimize the weights of individual trusts and establishes a secure path. Simulation experiments carried out over the proposed method had shown its superiority in terms of packet delivery ratio, delay, and throughput. The proposed method has gained an average improvement in the throughput is of 23.71%, 20.18%, 17.27%, and 2.88% from PSO, GSA, WOA, and CBBMOR-TSM-IOT methods respectively. (shrink)

The aim of the paper is to give a description of the development of understanding scientific method in the history of science and philosophy of science. The thesis is defended that crucial changes in understanding scientific method had fundamental consequences for the development in scientific knowledge, for the position of science in the system of knowledge and for its role in human culture. Basic attitudes to scientific method in contemporary philosophy and philosophy of science are shortly outlined and the rationality (...) of scientific method is defended. (shrink)

Conceptual models as representations of real-world systems are based on diverse techniques in various disciplines but lack a framework that provides multidisciplinary ontological understanding of real-world phenomena. Concurrently, systems’ complexity has intensified, leading to a rise in developing models using different formalisms and diverse representations even within a single domain. Conceptual models have become larger; languages tend to acquire more features, and it is not unusual to use different modeling languages for different components. This diversity has caused problems with consistency (...) between models and incompatibly with designed systems. Two main solutions have been adopted over the last few years: (1) A currently dominant technology-based solution tries “to harmonize or unify” models, e.g., unifies EER and UML. This solution would solidify modeling achievements, reaping benefits from huge investments over the last thirty years. (2) A less prevalent solution is to pursuit deeper roots that reveal unifying modeling principles and apparatuses. An example of the second method is a “category theory”-based approach that utilizes the strengths of the graph and set theory, along with other topological tools. This manuscript is a sequel in a research venture that belongs to the second approach and uses a model called thinging machines (TMs) founded on Stoic ontology and Lupascian logic. TM modeling contests the thesis that there is no universal approach that covers all aspects of an application, and the paper demonstrates that pursuing such universality is anything but a dead-end method. This paper continues in this direction, with emphasis on TM foundation (e.g., existence and subsistence of things) and exemplifies this pursuit by proposing an alternative representation of set theory. (shrink)

Recent research has identified a tension between the Safety principle that knowledge is belief without risk of error, and the Closure principle that knowledge is preserved by competent deduction. Timothy Williamson reconciles Safety and Closure by proposing that when an agent deduces a conclusion from some premises, the agent’s method for believing the conclusion includes their method for believing each premise. We argue that this theory is untenable because it implies problematically easy epistemic access to one’s methods. Several possible (...) solutions are explored and rejected. (shrink)

Research in experimental philosophy has increasingly been turning to corpus methods to produce evidence for empirical claims, as they open up new possibilities for testing linguistic claims or studying concepts across time and cultures. The present article reviews the quasi-experimental studies that have been done using textual data from corpora in philosophy, with an eye for the modeling and experimental design that enable statistical inference. I find that most studies forego comparisons that could control for confounds, and that only (...) a little less than half employ statistical testing methods to control for chance results. Furthermore, at least some researchers make modeling decisions that either do not take into account the nature of corpora and of the word-concept relationship, or undermine the experiment's capacity to answer research questions. I suggest that corpus methods could both provide more powerful evidence and gain more mainstream acceptance by improving their modeling practices. (shrink)

There is widespread excitement in the literature about the method of arbitrary functions: many take it to show that it is from the dynamics of systems that the objectivity of probabilities emerge. In this paper, I differentiate three ways in which a probability function might be objective, and I argue that the method of arbitrary functions cannot help us show that dynamics objectivise probabilities in any of these senses.

Chiastic order is an ancient expression for cross-classification. Cross-classification, in turn, is one of many terms used for the operation of conjoining or cross-mapping one domain, class or set of concepts with another. As such, it is the primordial form of non-quantified modelling and combinatory heuristics. This article presents a brief epistemological history of non-quantified modelling: its prehistory in the form of rhetorical chiasmus; its early (pre-symbolic) use by Plato as a cross-order (paradigmatic) modelling method; and its “modern” (symbolic) use (...) by Leibniz as a calculus of concepts. It will also be shown how classification theory itself is built on a cross-classificatory construct involving two fundamental logical/structural relationships: subordination and conjunction. Finally, examples of modern computer-aided, non-quantified modal modelling are presented in the areas of design theory, operational research and decision science. (shrink)

Here, I examine the epistemic relation between beliefs about the nature of sexual orientation (e.g., beliefs concerning whether orientation is dispositional) and beliefs about the taxonomy of orientation categories (e.g., beliefs concerning whether polyamorous is an orientation category). Current philosophical research gives epistemic priority to the former class of beliefs, such that beliefs about the taxonomy of orientation categories tend to be jettisoned or revised in cases of conflict with beliefs about the nature of sexual orientation. Yet, considering the influence (...) of ideology on beliefs about socially significant phenomena, I argue for an epistemic reversal. (shrink)

In this paper I discuss a set of problems concerning the method of cases as it is used in applied ethics and in the metaphysical debate about personal identity. These problems stem from research in social psychology concerning our access to the data with which the method operates. I argue that the issues facing ethics are more worrying than those facing metaphysics.

This theoretical paper explores the application of Hegelian inferentialism combined with contemporary quantitative methods to enhance decision-making in healthcare leadership. It proposes a novel conceptual framework that integrates Hegel’s inferentialism with Bayesian analysis and epistemic justice indices to offer a new approach for understanding complex decision processes in healthcare settings. The paper develops theoretical constructs such as the Decision Quality Index (DQI) and the Epistemic Justice Quotient (EJQ), which aim to quantitatively assess leadership effectiveness and ethical considerations in decision-making (...) processes. The discussion includes mathematical proofs for these proposed indices and examines the potential practical implications of this integrated approach, emphasizing its theoretical potential to improve healthcare leadership through more inclusive and evidence-based practices. This work provides a conceptual foundation for future empirical research in healthcare leadership and decision-making. (shrink)

The use of "levels of abstraction" in philosophical analysis (levellism) has recently come under attack. In this paper, we argue that a refined version of epistemological levellism should be retained as a fundamental method, which we call the method of abstraction. After a brief introduction, in section two we make clear the nature and applicability of the (epistemological) method of levels of abstraction. In section three, we show the fruitfulness of the new method by applying it to five case studies: (...) the concept of agenthood, the Turing test, the definition of emergence, quantum observation and decidable observation. In section four, we further characterise and support the method by distinguishing it from three other forms of "levellism": (i) levels of organisation; (ii) levels of explanation and (iii) conceptual schemes. In this context, we also briefly address the problems of relativism and antirealism. In the conclusion, we indicate some of the work that lies ahead, two potential limitations of the method and some results that have already been obtained by applying the method to some long-standing philosophical problems. (shrink)

[English] This paper aims to discuss ten traditional and contemporary methodological paradigms in African philosophy, and demonstrates how they may apply to African environmental ethics research. The methods include: Ethno-philosophy, Sage Philosophy, Conversational Philosophy, Conceptual Mandelanization, Eco-Afrocentricism, Indigenous Language Analysis, Eco-Afro-feminism, Conceptual Decolonization, Storytelling Philosophy, and Cultural Adaptationism. The significance and limitations of the methodologies are highlighted. The concept and principles of African environmental ethics are analyzed and discussed to facilitate an understanding of the conceptual frameworks that underpin the (...) methodologies. In conclusion, the discourse demonstrates that environmental ethics research in Africa should be based on African philosophical methodologies so as to facilitate production of research that will be relevant in African contexts. -/- [Annang] N̄wed nduongo ami ayem ibene nwọ́d mme n̄kaan nne mfa usung unam nduongo ke vilọsọvi Afrịke , nne nte amọ ekeme iwam unam nduongo mbanga ido ukpeme nkan-nkuk ke Afrịke. Mme usung unam nduongo ami nsehe ade ami: vilọsọvi mben ufọk, vilọsọvi n̄kan ideen, vilọsọvi nneme, vilọsọvi Mandela, “Eco-Afrocentricism”, nduongore usem utiit ajid, vilọsọvi ibaan Afrịke, vilọsọvi mkpọk-nyak, vilọsọvi n̄ke, nne ukpep ido idung agwo. N̄wed ami abene awọd mme ikek nne nde mem idem usung nduongo ami. N̄wed ami atang mkpọ abanga se ido ukpeme nkan-nkuk ke Afrịke anwongo nne mme itai amọ, nne nte amọ ewam mbon nduongo ediọhọ nnanga ekpekakpa ibọrọ nduongo amọ. Ke n̄suuk ikọ, ukpep ami mkpọ abene awọd ate ke ekpena ekama usung nduongo vilọsọvi Afrịke enam nduongo ebanga ido ukpeme nkan-nkuk ke Afrịke man ibọrọ nduongo asanga akekem nne se adiiwam Afrịke. (shrink)

Ultrafilters are very important mathematical objects in mathematical research [6, 22, 23]. There are a wide variety of classical theorems in various branches of mathematics where ultrafilters are applied in their proof, and other classical theorems that deal directly with ultrafilters. The objective of this article is to contribute (in a divulgative way) to ultrafilter research by describing the demonstrations of some such theorems related (uniquely or in combination) to topology, Measure Theory, algebra, combinatorial infinite, set theory and first-order (...) logic, also formulating some updated open problems of set theory that refer to non-principal ultrafilters on N, the Mathias’s model and the Solovay’s model. -/- Resumen: Los ultrafiltros son objetos matemáticos muy importantes en la investigación matemática [6, 22, 23]. Existen una gran variedad de teoremas clásicos en diversas ramas de la matemática donde se aplican ultrafiltros en su demostración, y otros teoremas clásicos que tratan directamente sobre ultrafiltros. El objetivo de este artículo es contribuir (de una manera divulgativa) con la investigación sobre ultrafiltros describiendo las demostraciones de algunos de tales teoremas relacionados (de manera única o combinada) con topología, teoría de la medida, álgebra, combinatoria infinita, teoría de conjuntos y lógica de primer orden, formulando además algunos problemas abiertos actuales de la teoría de conjuntos que se refieren a ultrafiltros no principales sobre N, al Modelo de Mathias y al Modelo de Solovay. (shrink)

The author asks whether there was a “scientific ‘68”, and focuses on aspects of two specific methodological proposals defined in the 1940s and 50s by the terms “action research” and “mixing methods”, applied particularly to social sciences. In the first, the climate surrounding the events of 1968 contributed to heightening the participative element to be found –by definition– in “action research”; that is: the importance of making the research subjects themselves participants in the design, execution and application of the (...) study of which they are the focus. This approach captured the democratic and anti-authoritarian spirit at the heart of the proposal, which was part of the prevailing climate in those days. The repercussions of 1968 on “mixing methods” focused on studying what had actually occurred, especially between the youth and workers, and therefore, particularly from the point of view of sociology and social psychology, using a “mixed methods” approach. The author explores the proposal of Norman Denzin; but traces the recent origins of both “mixing methods” and “action research” back to the proposals of mainly Kurt Lewin and the Chicago School. (shrink)

This paper is concerned with learners who aim to learn patterns in infinite binary sequences: shown longer and longer initial segments of a binary sequence, they either attempt to predict whether the next bit will be a 0 or will be a 1 or they issue forecast probabilities for these events. Several variants of this problem are considered. In each case, a no-free-lunch result of the following form is established: the problem of learning is a formidably difficult one, in that (...) no matter what method is pursued, failure is incomparably more common that success; and difficult choices must be faced in choosing a method of learning, since no approach dominates all others in its range of success. In the simplest case, the comparison of the set of situations in which a method fails and the set of situations in which it succeeds is a matter of cardinality (countable vs. uncountable); in other cases, it is a topological matter (meagre vs. co-meagre) or a hybrid computational-topological matter (effectively meagre vs. effectively co-meagre). (shrink)

Philosophical methodology is the central focus of pragmatism’s founding documents. The early works of Peirce, James, and Dewey examine methodological questions such as ‘how do we make philosophical ideas clear?’, ‘what is the best method for fixing belief?’ and ‘how do we know whether a philosophical question is answerable?’. Thus, many consider pragmatism inherently methodological – as a metaphilosophy, a view about how philosophy should or must be done (e.g. Talisse 2017). Any summary of pragmatist methods is therefore a (...) summary of pragmatism itself. Given such an impossibly broad remit, this chapter does only three things. First, it provides four broad claims common to pragmatist approaches to philosophical methodology, claims reflecting its underlying theory of inquiry. Second, it briefly examines three core pragmatist methods – for conceptual clarification, for fixing belief, and for settling or dissolving philosophical disputes. Third, it briefly describes differences between the Classical figures regarding each method. This is merely a brief sketch – the reader should consider all entries in this volume relevant to pragmatism qua philosophical method. (shrink)

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the (...) former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions. (shrink)

Progress in the last few decades in what is widely known as “Chaos Theory” has plainly advanced understanding in the several sciences it has been applied to. But the manner in which such progress has been achieved raises important questions about scientific method and, indeed, about the very objectives and character of science. In this presentation, I hope to engage my audience in a discussion of several of these important new topics.

This paper argues that naturalistic philosophy does not meet its own empiricist mandate. It argues from an empiricist perspective. Naturalists either claim that philosophy is like science in significant ways, or they claim that philosophy ought to be like science. This paper, being chiefly focused on the former claim, argues that naturalistic philosophy is nothing like science. Using Papineau’s markers for the similarities between naturalistic philosophy and science, I argue, counter Papineau, that the method employed in naturalistic philosophy is not (...) a posteriori and its claims are certainly not synthetic in the same way as that of science. This methodological distinction between science and philosophy is one made by Carnap. To show how the methods are distinct I compare two papers; I compare the method employed by Andy Clark in his philosophical paper on the brain as a prediction error minimisation machine with that employed by Rees and Haynes in their neuroscientific paper on mental content. (shrink)

The first aim of this paper is to prove a topological completeness theorem for a weak version of Stalnaker’s logic KB of knowledge and belief. The weak version of KB is characterized by the assumption that the axioms and rules of KB have to be satisfied with the exception of the axiom (NI) of negative introspection. The proof of a topological completeness theorem for weak KB is based on the fact that nuclei (as defined in the framework of (...) point-free topology) give rise to a profusion of topological belief operators that are compatible with the familiar topological knowledge operator. Thereby a canonical topological model for weak KB can be constructed. For this canonical model a truth lemma for the K and B holds such that a completeness theorem for KB can be proved in the familiar way. The second aim of this paper is to show that the topological interpretation of knowledge K comes along with a complete Heyting algebra of belief operators B that all fit the knowledge operator K in the sense that the pairs (K, B) satisfy all axioms of weak KB. This amounts to a pluralistic relation between knowledge and belief: Knowledge does not fully determine belief, rather it designs a conceptual space for belief operators where different (competing) belief operators coexist. (shrink)

Entrepreneurship education is among the key drivers of the country’s economy, and teachers are primarily responsible for its integration into teaching and finding the best and most useful method. The purpose of this research is to determine the level of entrepreneurial competence and teaching methods among entrepreneurship education teachers. Focusing on the concept of entrepreneurial competencies combining conceptual, human relations, strategic, commitment, opportunity, organizational and strategic competencies. Using quantitative – descriptive survey, the study tested its significant difference and relationship (...) when grouped with the profile variables. The findings indicate a significant correlation between the variables particularly, with the technical competence which has the strongest correlation. The paper sheds some light concerning the status of the current entrepreneurial competence of the teachers in the division and recommends training curriculum to develop their competencies in handling the subject. (shrink)