The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations (...) used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Newton's Metaphysics: Essays by Eric SchliesserMarius StanEric Schliesser. Newton's Metaphysics: Essays. Oxford: Oxford University Press, 2021. Pp. 328. Hardback, $99.90.Newton owes his high regard to the quantitative science he left us, but his overall picture of the world had some robustly metaphysical threads woven in as well. Posthumous judgment about the value of these threads has varied wildly. Christian Wolff thought him a metaphysical rustic, as did Hans (...) Reichenbach some two centuries later ("Die Bewegungslehre bei Newton, Leibniz und Huygens," Kant-Studien 29 [1924]: 416–38). In the 1960s, the tide would turn, as Howard Stein and James Edward McGuire separately began to show that Newton's metaphysics was not just sophisticated, but often more compelling than its early modern alternatives (see respectively "Newtonian Spacetime," Texas Quarterly 10 [1967]: 174–200; and Tradition and Innovation: Newton's Metaphysics of Nature [Dordrecht: Kluwer Academic Publishers, 1995]). Eric Schliesser's book unfolds in that same register of appreciative, high scholarship, and it attests to the enduring attraction of its subject.The book grew out of previous discrete papers, to which he added a few more for this occasion. Accordingly, it does not come with a master argument. Rather, it is an in-depth exploration of some key metaphysical themes in Newton. As such, it befits its subject figure, who reflected on themes and concepts while stopping short of working out systems.The first major theme is Newton and Spinozism. The latter does not denote Spinoza's metaphysics. In fact, Schliesser explains, Newton shared with Spinoza some weighty commitments, for example, to space being actually infinite, and to substance monism: "for Newton, there is strictly speaking only one genuine substance," namely, God (33). Rather, "Spinozism" is an interpreter's category for a bundle of three theses: identifying God and nature; denying final causation in the physical world; and the assumption of "blind" metaphysical necessity. Some counted Hobbes and John Toland as Spinozist in this sense, and even Epicurus, avant la lettre. Newton and his followers argued vehemently against this package, as Schliesser shows in chapters 4, 5, and 8. Their chief complaint was that Spinozism is unable to account for the "origin of motion," for a certain type of order, and for the stability of cosmological structure—and to do so in a way that "meets the standards of Newtonian mechanics" (122). This interpretive lens allows Schliesser to draw Kant in, by reading his youthful Theory of Heaven as a possible Spinozist reply to their objections (chapter 3). [End Page 157]A second theme is the metaphysics of mechanics, where Schliesser weaves together several strands. One is the ontology of gravity, for which he offers a novel construal: for Newton, gravity counts as real, but in a qualified sense. Namely, it is not essential and not intrinsic: "even after creation, a lonely partless particle of matter in the universe would not be said to gravitate." And gravity is relational: a "shared quality" of two or more bits of matter, which obtains in virtue of their sharing a nature (19). Another strand is Newton's ontology of time—really, a subtle, difficult topic long neglected. Famously, Newton in Principia defends absolute, true, and mathematical time. The question is what these three qualifiers denote, and whether Newton thought they were synonymous. Schliesser in chapter 7 argues for the provocative view that "absolute" and "true" time are not identical concepts; rather, they pick out different entities. These entities share a common structure, namely, metric and topological. That makes them species of "mathematical" time (179). But, Schliesser contends, Newton has unequal warrant for his two concepts of time.Yet another theme is formal causation. It comes to the fore in chapter 5, which grapples with Newton's opaque idea that space is an "emanative effect." Schliesser explains that here, "emanation picks out a species of formal causation" (147), but in a novel, early modern sense that comes from Bacon, not Aristotle. If space has a formal cause, then so has its twin brother, time, the topic of chapter 7. God is their common formal cause. Then in chapter 6 (coauthored with Zvi Biener) Schliesser adds that, for Newton, laws... (shrink)

In 1982, MIT physicist Frank Wilczek predicted and named ANYONS, quasiparticles (particle-like formations) that are confined to 2 dimensions. The name might come from Prof. Wilczek's lighthearted comment "anything goes". This article's main goal is to show that anyons could be another name for 1) virtual particles, 2) Mobius strips, and 3) figure-8 Klein bottles. Along the way, we'll see the picture painted by the article confirm that Einstein's dream of gravitational-electromagnetic unity fits in with anyons being Mobius strips. We'll (...) also have encounters with intergalactic travel and imaginary computers. They really could exist but are imaginary in the sense that they use imaginary time (as well as space-time warping). (shrink)

Computational modeling should play a central role in philosophy. In this introduction to our topical collection, we propose a small topology of computational modeling in philosophy in general, and show how the various contributions to our topical collection fit into this overall picture. On this basis, we describe some of the ways in which computational models from other disciplines have found their way into philosophy, and how the principles one found here still underlie current trends in the field. Moreover, we (...) argue that philosophers contribute to computational modeling not only by building their own models, but also by thinking about the various applications of the method in philosophy and the sciences. In this context, we note that models in philosophy are usually simple, while models in the sciences are often more complex and empirically grounded. Bridging certain methodological gaps that arise from this discrepancy may prove to be challenging and fruitful for the further development of computational modeling in philosophy and beyond. (shrink)

The author attempts to show how mereology, taken together with certain topological notions, can yield the foundations for future investigations in formal ontology. He also attempts to show how the mereological framework allows for the direct and natural formulation of a series of theses - for example pertaining to the concept of a boundary - which can be only indirectly formulated (if at all) in set-theoretic terms. The far-reaching ain of the present framework is to serve as a basis (...) for a formal ontology of the common-sense world. The author is interested in producing formally precise theories of structures of certain sorts in such a way that it is the structures themselves that intrest him and not formal machinery that has been set up to describe them. Hence the axioms are chosen primarily for the sake of the light they throw on the intended subject-matters (and not for their logical independence). The world itself - or its picture given in ordinary experience - is the only model the paper concentrates upon. Thus for example the presented system allows to prove that every boundary is the boundary of something, and that in particular no point exists in isolation from a large extended whole that is its boundary. (shrink)

Undoubtedly the Penrose-Hameroff Orch OR model may be considered as a good theory for describing information processing mechanisms and holistic phenomena in the human brain, but it doesn’t give us satisfactory explanation of human perception. In this work a new approach explaining our perception is introduced, which is in good agreement with Orch OR model and other mainstream science theories such as string theory, loop quantum gravity and holographic principle. It is shown that human perception cannot be explained in the (...) terms of elementary particles and we should introduce new indivisible holistic objects with geometry based on smooth infinitesimal analysis - elastic membranes. The example of such a membrane is our Universe which is an indivisible whole. It is shown that our perception may be considered as the result of elastic oscillations of two dimensional (2D) elastic membranes with closed topology embedded in our bodies. Only one elastic membrane responsible for its perceptions will correspond to the selected organism, but there may be other membranes, even at the cell level. In other words, reality may be considered as the process of time evolution of holistic energetically very weak macro objects - elastic membranes with the geometry based on smooth infinitesimal analysis. An embedded membrane in this multidimensional world will look different for the external and internal observers: from the outside it will look like a material object with smooth infinitesimal geometry, while from the inside our Universe-like space-time fabric. When interacting with elementary particles and other membranes, a membrane will transform their energy into its elastic energy (a new form of energy) - the energy of stretching of the infinitesimal segments. The theory postulates that these elastic deformations will not be observable from the point of view of the internal observer. Heisenberg’s uncertainty principle will work in this physics only from the point of view of the internal observer. For the external observer each embedded elastic membrane may be stretched and even a very small region will become observable. For example, living organisms play the role of internal observers of the Universe, and at the same time they serve as external observers for 2D membranes embedded into our Universe. We can observe our 2D self-membranes through our perceptions, which are encoded in elastic oscillations of the elastic membrane. According to the theory, elastic membranes occupy energetically favorable positions around microtubules involved into Orch OR. The theory not only gives us a really multidimensional holistic picture of reality, but it also provides us with a new method for understanding such phenomena as perception, self-awareness and will. (shrink)

The file on this site provides the slides for a lecture given in Hangzhou in May 2018, and the lecture itself is available at the URL beginning 'sms' in the set of links provided in connection with this item. -/- It is commonly assumed that regular physics underpins biology. Here it is proposed, in a synthesis of ideas by various authors, that in reality structures and mechanisms of a biological character underpin the world studied by physicists, in principle supplying detail (...) in the domain that according to regular physics is of an indeterminate character. In regular physics mathematical equations are primary, but this constraint leads to problems with reconciling theory and reality. Biology on the other hand typically does not characterise nature in quantitative terms, instead investigating in detail important complex interrelationships between parts, leading to an understanding of the systems concerned that is in some respects beyond that which prevails in regular physics. It makes contact with quantum physics in various ways, for example in that both involve interactions between observer and observed, an insight that explains what is special about processes involving observation, justifying in the quantum physics context the replacement of the unphysical many-worlds picture by one involving collapse. The link with biology furthermore clarifies Wheeler’s suggestion that a multiplicity of observations can lead to the ‘fabrication of form’, including the insight that this process depends on very specific ‘structures with power’ related to the 'semiotic scaffolding' of the application of sign theory to biology known as biosemiotics. -/- The observer-observed 'circle' of Wheeler and Yardley is a special case of a more general phenomenon, oppositional dynamics, related to the 'intra-action' of Barad's Agential Realism, involving cooperating systems such as mind and matter, abstract and concrete, observer and observed, that preserve their identities while interacting with one another in such a way as to act as a unit. A third system may also be involved, the mediating system of Peirce linking the two together. Such a situation of changing connections and separations may plausibly lead in the future to an understanding of how complex systems are able to evolve to produce 'life, the universe and everything'. -/- (Added 1 July 2018) The general structure proposed here as an alternative to a mathematics-based physics can be usefully characterised by relating it to different disciplines and the specialised concepts utilised therein. In theoretical physics, the test for the correctness of a theory typically involves numerical predictions, corresponding to which theories are expressed in terms of equations, that is to say assertions that two quantities have identical values. Equations have a lesser significance in biology which typically talks in terms of functional mechanisms, dependent for example on details of chemistry and concepts such as genes, natural selection, signals and geometrical or topologically motivated concepts such as the interconnections between systems and the unfolding of DNA. Biosemiotics adds to this the concept of signs and their interpretation, implying novel concepts such as semiotic scaffolding and the semiosphere, code duality, and appreciation of the different types of signs, including symbols and their capacity for abstraction and use in language systems. Circular Theory adds to this picture, as do the ideas of Barad, considerations such as the idea of oppositional dynamics. The proposals in this lecture can be regarded as the idea that concepts such as those deriving from biosemiotics have more general applicability than just conventional biology and may apply, in some circumstances, to nonlinear systems generally, including the domain new to science hypothesised to underlie the phenomena of present-day physics. -/- The task then has to be to restore the mathematical aspect presumed, in this picture, not to be fundamental as it is in conventional theory. Deacon has invoked a complex sequence of evolutionary steps to account for the emergence over time of human language systems, and correspondingly mathematical behaviour can be subsumed under the general evolutionary mechanisms of biosemiotics (cf. also the proposals of Davis and Hersh regarding the nature of mathematics), so that the mathematical behaviour of physical systems is consistent with the proposed scheme. In conclusion, it is suggested that theoretical physicists should cease expecting to find some universal mathematical ‘theory of everything’, and focus instead on understanding in more detail complex systems exhibiting behaviour of a biological character, extending existing understanding. This may in time provide a more fruitful understanding of the natural world than does the regular approach. The essential concepts have an observational basis from both biology and the little-known discipline of cymatics (a discipline concerned with the remarkable patterns that specific waveforms can give rise to), while again computer simulations also offer promise in providing insight into the complex behaviours involved in the above proposals. -/- References -/- Jesper Hoffmeyer, Semiotic Scaffolding of Living Systems. Commens, a Digital Companion to C. S. Peirce (on Commens web site). Terrence Deacon, The Symbolic Species, W.W. Norton & Co. Karen Barad, Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning, Duke University Press. Philip Davis and Reuben Hersh, The Mathematical Experience, Penguin. Ilexa Yardley, Circular Theory. (shrink)

This chapter provides a systematic overview of topological explanations in the philosophy of science literature. It does so by presenting an account of topological explanation that I (Kostić and Khalifa 2021; Kostić 2020a; 2020b; 2018) have developed in other publications and then comparing this account to other accounts of topological explanation. Finally, this appraisal is opinionated because it highlights some problems in alternative accounts of topological explanations, and also it outlines responses to some of the main (...) criticisms raised by the so-called new mechanists. (shrink)

Hobbes emphasized that the state of nature is a state of war because it is characterized by fundamental and generalized distrust. Exiting the state of nature and the conflicts it inevitably fosters is therefore a matter of establishing trust. Extant discussions of trust in the philosophical literature, however, focus either on isolated dyads of trusting individuals or trust in large, faceless institutions. In this paper, I begin to fill the gap between these extremes by analyzing what I call the topology (...) of communities of trust. Such communities are best understood in terms of interlocking dyadic relationships that approximate the ideal of being symmetric, Euclidean, reflexive, and transitive. Few communities of trust live up to this demanding ideal, and those that do tend to be small (between three and fifteen individuals). Nevertheless, such communities of trust serve as the conditions for the possibility of various important prudential epistemic, cultural, and mental health goods. However, communities of trust also make possible various problematic phenomena. They can become insular and walled-off from the surrounding community, leading to distrust of out-groups. And they can lead their members to abandon public goods for tribal or parochial goods. These drawbacks of communities of trust arise from some of the same mecha-nisms that give them positive prudential, epistemic, cultural, and mental health value – and so can at most be mitigated, not eliminated. (shrink)

We provide three innovations to recent debates about whether topological or “network” explanations are a species of mechanistic explanation. First, we more precisely characterize the requirement that all topological explanations are mechanistic explanations and show scientific practice to belie such a requirement. Second, we provide an account that unifies mechanistic and non-mechanistic topological explanations, thereby enriching both the mechanist and autonomist programs by highlighting when and where topological explanations are mechanistic. Third, we defend this view against (...) some powerful mechanist objections. We conclude from this that topological explanations are autonomous from their mechanistic counterparts. (shrink)

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent (...) class='Hi'>topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness. (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)

Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, from the central position it (...) used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)

The relationship between topological explanation and mechanistic explanation is unclear. Most philosophers agree that at least some topological explanations are mechanistic explanations. The crucial question is how to make sense of this claim. Zednik (Philos Psychol 32(1):23–51, 2019) argues that topological explanations are mechanistic if they (i) describe mechanism sketches that (ii) pick out organizational properties of mechanisms. While we agree with Zednik’s conclusion, we critically discuss Zednik’s account and show that it fails as a general account (...) of how and when topological explanations are mechanistic. First, if topological explanations were just mechanism sketches, this implies that they could be enriched by replacing topological terms with mechanistic detail. This, however, conflicts how topological explanations are used in scientific practice. Second, Zednik’s account fails to show how topological properties can be organizational properties of mechanisms that have a place in mechanistic explanation. The core issue is that Zednik’s account ignores that topological properties often are _global_ properties while mechanistic explanantia refer to _local_ properties. We demonstrate how these problems can be solved by a recent account of mechanistic completeness (Craver and Kaplan in Br J Philos Sci 71(1):287–319, 2020; Kohár and Krickel in Calzavarini and Viola (eds) Neural mechanisms—new challenges in the philosophy of neuroscience, Springer, New York, 2021) and use a multilayer network model of Alzheimer’s Disease to illustrate this. (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)

It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it. But it is questionable whether mathematical concepts could ever play such a role in psychology or philosophy. In this paper, we set out to examine a rather unobvious example of the application of topology, in the form of the theory of persons proposed by Kurt Lewin in his Principles of Topological Psychology. Our aim is (...) to show that this branch of mathematics can furnish a natural conceptual system for Gestalt psychology, in that it provides effective tools for describing global qualitative aspects of the latter’s object of investigation. We distinguish three possible ways in which mathematics can contribute to this: explanation, explication and metaphor. We hold that all three of these can be usefully characterized as throwing light on their subject matter, and argue that in each case this contrasts with the role of explanations in physics. Mathematics itself, we argue, provides something different from such explanations when applied in the field of psychology, and this is nevertheless still cognitively fruitful. (shrink)

It is well-known that the global structure of every space-time model for relativistic cosmology is observationally underdetermined. In order to alleviate the severity of this underdetermination, it has been proposed that we adopt the Cosmological Principle because the Principle restricts our attention to a distinguished class of space-time models (spatially homogeneous and isotropic models). I argue that, even assuming the Cosmological Principle, the topology of space remains observationally underdetermined. Nonetheless, I argue that we can muster reasons to prefer various (...) class='Hi'>topological properties over others. In particular, I favor the adoption of multiply connected universe models on grounds of (i) simplicity, (ii) Machian considerations, and (iii) explanatory power. We are able to appeal to such grounds because multiply connected topologies open up the possibility of finite universe models (consistent with our best data), which in turn avoid thorny issues concerning the postulation of an actually infinite universe. (shrink)

Considering topology as an extension of mereology, this paper analyses topological variants of mereological essentialism (the thesis that an object could not have different parts than the ones it has). In particular, we examine de dicto and de re versions of two theses: (i) that an object cannot change its external connections (e.g., adjacent objects cannot be separated), and (ii) that an object cannot change its topological genus (e.g., a doughnut cannot turn into a sphere). Stronger forms of (...) structural essentialism, such as morphological essentialism (an object cannot change shape) and locative essentialism (an object cannot change position) are also examined. (shrink)

In this paper I want to show that topology has a bearing on the theory of tropes. More precisely, I propose a topological ontology of tropes. This is to be understood as follows: trope ontology is a „one-category”-ontology countenancing only one kind of basic entities, to wit, tropes. 1 Hence, individuals, properties, relations, etc. are to be constructed from tropes.

List of Illustrations Introduction 1 The Didactic and the Elegant: Some Thoughts on Scientific and Technological Illustrations in the Middle Ages and Renaissance 3 2 Temples of the Body and Temples of the Cosmos: Vision and Visualization in the Vesalian and Copernican Revolutions 40 3 Descartes’s Scientific Illustrations and ’la grande mecanique de la nature’ 86 4 Illustrating Chemistry 135 5 Representations of the Natural System in the Nineteenth Century 164 6 Visual Representation in Archaeology: Depicting the Missing-Link in Human (...) Origins 184 7 Towards an Epistemology of Scientific Illustration 215 8 Illustration and Inference 250 9 Visual Models and Scientific Judgment 269 10 Are Pictures Really Necessary? The Case of Sewall Wright’s ’Adaptive Landscapes’ 303 Bibliography 339 Notes on Contributors 373 Index 377. (shrink)

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)

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)

Semantics traditionally focuses on linguistic meaning. In recent years, the Super Linguistics movement has tried to broaden the scope of inquiry in various directions, including an extension of semantics to talk about the meaning of pictures. There are close similarities between the interpretation of language and of pictures. Most fundamentally, pictures, like utterances, can be either true or false of a given state of affairs, and hence both express propositions (Zimmermann, 2016; Greenberg, 2013; Abusch, 2015). Moreover, sequences (...) of pictures, like sequences of utterances, can be used to tell stories. Wordless picture books, comics, and film are cases in point. In this paper I pick up the project of providing a dynamic semantic account of pictorial story-telling, started by Abusch (2012) and continued by Abusch & Rooth (2017); Maier & Bimpikou (2019); Fernando (2020). More specifically, I propose here a semantics of speech and thought bubbles by adding event reference to PicDRT. To get there I first review the projection-based semantics for pictures (section 1), noting the fundamental distinction between symbolic and iconic meaning that makes speech bubbles especially interesting (section 2). I then review the dynamic PicDRT framework for pictorial narratives (section 3), add events (section 4), and propose an account of speech bubbles as quotational event modification (section 5). I end with a brief look at other conventional, symbolic enrichments in comics (section 6). (shrink)

In “Topology of Balasaguni’s Kutadgu Bilig: Thinking the Between,” Onur Karamercan focuses on the philosophical dimension of Kutadgu Bilig, a poetic work of Yūsuf Balasaguni, an 11th century Central Asian thinker, poet, and statesman. Karamercan pays special attention to the meaning of betweenness and, in the first step of his argument, discusses the hermeneutic and topological implications of the between, distingushing the dynamic sense of betweenness from a static sense of in-betweenness. He then moves on to analyze Balasaguni’s notion (...) of language, which he interprets as an early critique of the instrumental account of language and, by examining several selected fragments from Kutadgu Bilig, illustrates Balasaguni’s designation of language as an inexhaustible phenomenon. In the process, he also points to the possible parallels between Balasaguni’s and Heidegger’s ideas of language. In the final section of the article, building on his argument, Karamercan thematizes the margins of Turkic languages and of Islamic philosophy, suggesting that they need to be reexamined. He problematizes the very meaning of Asia itself by decentering what he calls “its internal East–West antagonism” and puts forth instead a framework based on in-betweenness reinterpreted from a topological perspective, proposing it as an alternative view which might help us make sense of the hermeneutic neighborhoods of Asian philosophies. (shrink)

I aim to give a new account of picture perception: of the way our visual system functions when we see something in a picture. My argument relies on the functional distinction between the ventral and dorsal visual subsystems. I propose that it is constitutive of picture perception that our ventral subsystem attributes properties to the depicted scene, whereas our dorsal subsystem attributes properties to the picture surface. This duality elucidates Richard Wollheim’s concept of the “twofoldness” of our experience of (...) class='Hi'>pictures: the “visual awareness not only of what is represented but also of the surface qualities of the representation.” I argue for the following four claims: (a) the depicted scene is represented by ventral perception, (b) the depicted scene is not represented by dorsal perception, (c) the picture surface is represented by dorsal perception, and (d) the picture surface is not necessarily represented by ventral perception. (shrink)

This is a revised version of the introductory essay in C. Eschenbach, C. Habel and B. Smith (eds.), Topological Foundations of Cognitive Science, Hamburg: Graduiertenkolleg Kognitionswissenschaft, 1994, the text of a talk delivered at the First International Summer Institute in Cognitive Science in Buffalo in July 1994.

This paper argues for a possible worlds theory of the content of pictures, with three complications: depictive content is centred, two-dimensional and structured. The paper argues that this theory supports a strong analogy between depictive and other kinds of representation and the platitude that depiction is mediated by resemblance.

Irigaray’s engagement with Aristotelian physics provides a specific diagnosis of women’s ontological and ethical situation under Western metaphysics: Women provide place and containership to men, but have no place of their own, rendering them uncontained and abyssal. She calls for a reconfiguration of this topological imaginary as a precondition for an ethics of sexual difference. This paper returns to Aristotelian cosmological texts to further investigate the topologies of sexual difference suggested there. In an analysis both psychoanalytic and phenomenological, the (...) paper rigorously traces a teleological and oedipal narrative implicit in the structure of the Aristotelian cosmos, in which desire for the mother is superseded by love for the father. Further, the paper argues that this narrative is complicated by certain other elements in the Aristotelian text—aporias involving the notion of boundary and the relationship between space and time, the fallenness of the feminine, and the ineliminably aleatory qualities of matter. The paper concludes that such elements may provide material for disrupting this teleology of gender, opening onto not merely an ethics of sexual difference, but providing space and place for a proliferation of non-normative, queer, transgender and intersex modes of sexed and gendered subjectivity. (shrink)

In this paper, I present a general theory of topological explanations, and illustrate its fruitfulness by showing how it accounts for explanatory asymmetry. My argument is developed in three steps. In the first step, I show what it is for some topological property A to explain some physical or dynamical property B. Based on that, I derive three key criteria of successful topological explanations: a criterion concerning the facticity of topological explanations, i.e. what makes it true (...) of a particular system; a criterion for describing counterfactual dependencies in two explanatory modes, i.e. the vertical and the horizontal; and, finally, a third perspectival one that tells us when to use the vertical and when to use the horizontal mode. In the second step, I show how this general theory of topological explanations accounts for explanatory asymmetry in both the vertical and horizontal explanatory modes. Finally, in the third step, I argue that this theory is universally applicable across biological sciences, which helps to unify essential concepts of biological networks. (shrink)

A central concept for information retrieval is that of similarity. Although an information retrieval system is expected to return a set of documents most relevant to the query word(s), it is often described as returning a set of documents most similar to the query. The authors argue that in order to reason with similarity we need to model the concept of discriminating power. They offer a simple topological notion called resolution space that provides a rich mathematical framework for reasoning (...) with limited discriminating power, avoiding the vagueness paradox. (shrink)

Although philosophers of art and aesthetics regularly appeal to a notion of ‘pictorial content’, there is little agreement over its nature. The present paper argues that pictures have propositional contents. This conclusion is reached by considering a style of argument having to do with the phenomenon of negation intended to show that pictures must have some kind of non-propositional content. I first offer reasons for thinking that arguments of that type fail. Second, I show that when properly understood, (...) such arguments can in fact be turned on their heads and shown to support the propositionalist position. (shrink)

This review investigates the potential of non-orientable topology as a fundamental framework for understanding the Poincaré conjecture and its implications across various scientific disciplines. Integrating insights from Dokuchaev (2020), Rapoport, Christianto, Chandra, Smarandache (under review), and other pioneering works, this article explores the theoretical foundations linking non-orientable spaces to resolving the Poincaré conjecture and its broader implications in theoretical physics, geology, cosmology, and biology.

Although cognitivism has lost some ground recently in the philosophical circles, it is still the favorite view of many scholars of emotions. Even though I agree with cognitivism's insight that emotions typically involve some type of evaluative intentional state, I shall argue that in some cases, less epistemically committed, non-propositional evaluative states such as mental pictures can do a better job in identifying the emotion and providing its intentional object. Mental pictures have different logical features from propositions: they (...) are representational, and some may or may not portray actual objects aptly. Yet, unlike propositional attitudes, mental pictures do not allow for objective criteria by which one can judge that a certain picture is an apt portrait of someone or something. (shrink)

Winning essay of the American Society for Aesthetics' inaugural Peter Kivy Prize. Extends Kivy's notion of sonic picturing through engagement with recent work in philosophy of perception. Argues that sonic pictures are more widespread and more aesthetically and artistically important than even Kivy envisioned. Topics discussed include: the nature of sonic pictures; the nature of sounds; what we can (and more importantly, cannot) conclude from musical listening; sonic pictures in film; beatboxing as an art of sonic picturing; (...) and cover songs as sonic pictures. To be published in the Journal of Aesthetics and Art Criticism. (shrink)

The aim of this paper is to show that topology has a bearing on Leibniz’s Principle of the Identity of Indiscernibles (PII). According to (PII), if, for all properties F, an object a has property F iff object b has property F, then a and b are identical. If any property F whatsoever is permitted in PII, then Leibniz’s principle is trivial, as is shown by “identity properties”. The aim of this paper is to show that topology can make a (...) contribution to the problem of giving criteria of how to restrict the domain of properties to render (PII) non-trivial. In topology a wealth of different Leibnizian principles of identity can be defined - PII turns out to be just the weakest topological separation axiom T0 in disguise, stronger principles of can be defined with the aid of higher separation axioms Ti, i > 0. Topologically defined properties have a variety of nice features, in particular they are stable in a natural sense. Topologically defined properties do not have a monopoly on defining “good” properties. In the final section of the paper it is show that the topological approach is closely related to Gärdenfors’s approach of conceptual spaces based on the concept of convexity. (shrink)

Proponents of ontic conceptions of explanation require all explanations to be backed by causal, constitutive, or similar relations. Among their justifications is that only ontic conceptions can do justice to the ‘directionality’ of explanation, i.e., the requirement that if X explains Y , then not-Y does not explain not-X . Using topological explanations as an illustration, we argue that non-ontic conceptions of explanation have ample resources for securing the directionality of explanations. The different ways in which neuroscientists rely on (...) multiplexes involving both functional and anatomical connectivity in their topological explanations vividly illustrate why ontic considerations are frequently (if not always) irrelevant to explanatory directionality. Therefore, directionality poses no problem to non-ontic conceptions of explanation. (shrink)

We now found nine new topologies, such as: NonStandard Topology, Largest Extended NonStandard Real Topology, Neutrosophic Triplet Weak/Strong Topologies, Neutrosophic Extended Triplet Weak/Strong Topologies, Neutrosophic Duplet Topology, Neutrosophic Extended Duplet Topology, Neutrosophic MultiSet Topology, and recall and improve the seven previously founded topologies in the years (2019-2023), namely: NonStandard Neutrosophic Topology, NeutroTopology, AntiTopology, Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, SuperHyperTopology, and Neutrosophic SuperHyperTopology. They are called avantgarde topologies because of their innovative forms.

The general aim of this paper is to introduce some ideas of the theory of infinite topological games into the philosophical debate on supertasks. First, we discuss the elementary aspects of some infinite topological games, among them the Banach-Mazur game.Then it is shown that the Banach-Mazur game may be conceived as a Newtonian supertask.In section 4 we propose to conceive physical experiments as infinite games. This leads to the distinction between determined and undetermined experiments and the problem of (...) how it is related to that between determinism and indeter-minism. Finally the role of the Axiom of Choice as a source of indetermi-nacy of supertasks is discussed. (shrink)

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo (...) E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Kim. (shrink)

The issue of whether or not there are visual arguments has been an issue in informal logic and argumentation theory at least since 1996. In recent years, books, sections of prominent conferences and special journals issues have been devoted to it, thus significantly raising the profile of the debate. In this paper I will attempt to show how the views of the later Wittgenstein, particularly his views on images and the no- tion of “picturing”, can be brought to bear on (...) the question of whether there are such things as “purely visual” arguments. I shall draw on Wittgenstein’s remarks in the Blue and Brown Books and in Philosophical Investigations in order to argue that al- though visual images may occur as elements of argumentation, broadly conceived, it is a mistake to think that there are purely visual arguments, in the sense of illative moves from premises to conclusions that are conveyed by images alone, without the support or framing of words. (shrink)

We develop a simple framework called ‘natural topology’, which can serve as a theoretical and applicable basis for dealing with real-world phenomena.Natural topology is tailored to make pointwise and pointfree notions go together naturally. As a constructive theory in BISH, it gives a classical mathematician a faithful idea of important concepts and results in intuitionism. -/- Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye, and various (...) real-number representations. -/- We compare classical mathematics (CLASS), intuitionistic mathematics (INT), recursive mathematics (RUSS), Bishop-style mathematics (BISH) and formal topology, aiming to reduce the mutual differences to their essence. To do so, our mathematical foundation must be precise and simple. There are links with physics, regarding the topological character of our physical universe. -/- Any natural space is isomorphic to a quotient space of Baire space, which therefore is universal. We develop an elegant and concise ‘genetic induction’ scheme, and prove its equivalence on natural spaces to a formal-topological induction style. The inductive Heine-Borel property holds for ‘compact’ or ‘fanlike’ natural subspaces, including the real interval [g, h]. Inductive morphisms respect this Heine-Borel property, inversely. This partly solves the continuous-function problem for BISH, yet pointwise problems persist in the absence of Brouwer’s Thesis. -/- By inductivizing the definitions, a direct correspondence with INT is obtained which allows for a translation of many intuitionistic results into BISH. We thus prove a constructive star-finitary metrization theorem which parallels the classical metrization theorem for strongly paracompact spaces. We also obtain non-metrizable Silva spaces, in infinite-dimensional topology. Natural topology gives a solid basis, we think, for further constructive study of topological lattice theory, algebraic topology and infinite-dimensional topology. The final section reconsiders the question of which mathematics to choose for physics. Compactness issues also play a role here, since the question ‘can Nature produce a non-recursive sequence?’ finds a negative answer in CTphys . CTphys , if true, would seem at first glance to point to RUSS as the mathematics of choice for physics. To discuss this issue, we wax more philosophical. We also present a simple model of INT in RUSS, in the two-players game LIfE. (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)

The leading models of disability struggle to fully encompass all aspects of “disability.” This difficulty arises, the author argues, because the models fundamentally misunderstand the nature of disability. Current theoretical approaches to disability can be understood as “nounal,” in that they understand disability as a thing that is caused or embodied. In contrast, this paper presents an adverbial perspective on disability, which shows that disability is experienced as a personally irremediable impediment to daily-living tasks or goals-like-ours. The picture theory of (...) disability technically constitutes a species of relational approach because its analysis references the interplay between an individual and their environment; it differs from other relational accounts, however, by interpreting disability as a certain kind of negative experience—rather than a function of that relationship. This purely descriptive theory makes no normative claims about disability and operates as both a mechanism for the evaluation of the experience of disability and a heuristic device for the proper interpretation of disability. When disability is reframed in this way, the theory offers a particularist perspective which shows if, when, where, and how disability is experienced. (shrink)

Do representational pictures have propositional contents? The current paper argues that the characteristic contents of pictures are predicative rather than propositional: pictures characterise things as looking certain ways, and they thereby express properties of visual perspectives. The paper argues that the characteristic predicative contents of pictures are nonetheless able to feature in fully-fledged propositional contents once they are combined with contents of other suitable sorts. Various facts about communicative uses of pictures are then explained. The (...) paper concludes by considering the bearing of its conclusions upon questions about the relationships between linguistic representation and pictorial representation. (shrink)

The concept of neutrosophic locally compact and neutrosophic compact open topology are introduced and some interesting propositions are discussed.

The Japanese scholar Miura Baien (1723-1789) worked throughout his life to produce a philosophical analysis of the natural world. Misinterpretations of his intentions arise from drawing diagrams on his behalf that are inconsistent with his text, or by applying to his text Western academic terms that are quite foreign to his thought. When Baien's text is examined in his own terms we can understand its significant role in the scientific thought of the Edo period.

In this paper, we introduce the concept of ""neutrosophic crisp neighborhoods system for the neutrosophic crisp point ". Added to, we introduce and study the concept of neutrosophic crisp local function, and construct a new type of neutrosophic crisp topological space via neutrosophic crisp ideals. Possible application to GIS topology rules are touched upon.