This paper argues that besides mechanistic explanations, there is a kind of explanation that relies upon “topological” properties of systems in order to derive the explanandum as a consequence, and which does not consider mechanisms or causal processes. I first investigate topological explanations in the case of ecological research on the stability of ecosystems. Then I contrast them with mechanistic explanations, thereby distinguishing the kind of realization they involve from the realization relations entailed by mechanistic explanations, and explain how both (...) kinds of explanations may be articulated in practice. The second section, expanding on the case of ecological stability, considers the phenomenon of robustness at all levels of the biological hierarchy in order to show that topological explanations are indeed pervasive there. Reasons are suggested for this, in which “neutral network” explanations are singled out as a form of topological explanation that spans across many levels. Finally, I appeal to the distinction of explanatory regimes to cast light on a controversy in philosophy of biology, the issue of contingence in evolution, which is shown to essentially involve issues about realization. (shrink)

We reformulate a key definition given by Wáng and Ågotnes to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ???? to be the “local truth” of ????, semantically rendered via an interior operator. This is closely related to the notion of ???? being “knowable”. We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial (...) role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness. (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)

In this paper, I argue that the newly developed network approach in neuroscience and biology provides a basis for formulating a unique type of realization, which I call topological realization. Some of its features and its relation to one of the dominant paradigms of realization and explanation in sciences, i.e. the mechanistic one, are already being discussed in the literature. But the detailed features of topological realization, its explanatory power and its relation to another prominent view of realization, namely the (...) semantic one, have not yet been discussed. I argue that topological realization is distinct from mechanistic and semantic ones because the realization base in this framework is not based on local realisers, regardless of the scale but on global realizers. In mechanistic approach, the realization base is always at the local level, in both ontic and epistemic accounts. The explanatory power of realization relation in mechanistic approach comes directly from the realization relation-either by showing how a model is mapped onto a mechanism, or by describing some ontic relations that are explanatory in themselves. Similarly, the semantic approach requires that concepts at different scales logically satisfy microphysical descriptions, which are at the local level. In topological framework the realization base can be found at different scales, but whatever the scale the realization base is global, within that scale, and not local. Furthermore, topological realization enables us to answer the “why” questions, which according to Polger 2010 make it explanatory. The explanatoriness of topological realization stems from understanding mathematical consequences of different topologies, not from the mere fact that a system realizes them. (shrink)

Stephen Hawking, among others, has proposed that the topological stability of a property of space-time is a necessary condition for it to be physically significant. What counts as stable, however, depends crucially on the choice of topology. Some physicists have thus suggested that one should find a canonical topology, a single ‘right’ topology for every inquiry. While certain such choices might be initially motivated, some little-discussed examples of Robert Geroch and some propositions of my own show that (...) the main candidates—and each possible choice, to some extent—faces the horns of a no-go result. I suggest that instead of trying to decide what the ‘right’ topology is for all problems, one should let the details of particular types of problems guide the choice of an appropriate topology. (shrink)

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an (...) interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group. (shrink)

A topological inevitability of early developmental events through the use of classical topological concepts is discussed. Topological dynamics of forms and maps in embryo development are presented. Forms of a developing organism such as cell sets and closed surfaces are topological objects. Maps (or mathematical functions) are additional topological constructions in these objects and include polarization, singularities and curvature. Topological visualization allows us to analyze relationships that link local morphogenetic processes and integral developmental structures and also to find stable spatio-temporal (...) topological characteristics that are invariant for a taxonomic group. The application of topological principles reveals a topological imperative: certain topological rules define and direct embryogenesis. A topological stability of embryonic morphogenesis is proposed and a topological scenario of developmental and evolutionary transformations is presented. (shrink)

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects living in infinite-dimensional spaces, working through many examples from cosmology. I focus on the relation of topological to measure-theoretic notions of and relating to probability, how they diverge in unpleasant ways in the infinite-dimensional case, and are even difficult to work with on their own. Even in cases where an appropriate family of spacetimes (...) is finite-dimensional, and so admits a measure of the relevant sort, however, it is always the case that the family is not a compact topological space, and so does not admit a physically significant, well behaved probability measure. Problems of a different but still deeply troubling sort plague arguments about likelihood in that context, which I also discuss. I conclude that most standard forms of argument used in cosmology to estimate the likelihood of the occurrence of various properties or behaviors of spacetimes have serious mathematical, physical and conceptual problems. (shrink)

Homologous operational localization processes are effectuated in terms of generalized topological covering systems on structures of physical events. We study localization systems of quantum events' structures by means of Gtothendieck topologies on the base category of Boolean events' algebras. We show that a quantum events algebra is represented by means of a Grothendieck sheaf-theoretic fibred structure, with respect to the global partial order of quantum events' fibres over the base category of local Boolean frames.

Fundamental concrete particulars are needed to explain facts about non-fundamental concrete particulars. However, the former can only play this explanatory role if they are properly discernible from the latter. Extant theories of how to discern fundamental concreta primarily concern mereological structure. Those according to which fundamental concreta can bear, but not be, proper parts are motivated by the possibilities that all concreta bear proper parts and that some properties of wholes are not fixed by the properties of their proper parts. (...) In response, theorists who hold that the fundamental concrete particulars can be proper parts may appeal to the possibility that every concrete particular is a proper part—that there is no mereologically maximal whole world, as well as to the intuition that fundamental concreta are qualitatively homogeneous “blocks” from which non-fundamental concreta are built. After motivating the plausibility of gunk and junk, the present essay proposes a constraint on fundamental concrete particulars based on topology instead of mereology: the fundamental concrete particulars must be appropriately connected. This constraint has the unique advantage of consistency with each of gunk, emergence, junk, and the building block intuition. (shrink)

Dynamic Topological Logic ( ) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems , which are pairs consisting of a topological space X and a continuous function f : X → X . The function f is seen as a change in one unit of time; within one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is (...) particularly interesting is that of minimal systems ; these are dynamic topological systems which admit no proper, closed, f -invariant subsystems. In such systems the orbit of every point is dense, which within translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. (shrink)

A topological constraint language is a formal language whose variables range over certain subsets of topological spaces, and whose nonlogical primitives are interpreted as topological relations and functions taking these subsets as arguments. Thus, topological constraint languages typically allow us to make assertions such as “region V1 touches the boundary of region V2”, “region V3 is connected” or “region V4 is a proper part of the closure of region V5”. A formula f in a topological constraint language is said to (...) be satisfiable if there exists an assignment to its variables of regions from some topological space under which φ is made true. This paper introduces a topological constraint language which, in addition to the usual mechanisms for expressing Boolean combinations of regions and their topological closures, includes primitives for bounding the number of components of a region. We call this language T CC, a rough acronym for “topological constraint language with component counting”. Our main result is that the problem of determining the satisfiability of a T CC-formula is NEXPTIME-complete. (shrink)

We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.

Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, □ is interpreted as topological interior. Thus S4 can be understood as the logic of topological spaces, and □ can be understood as a topological modality. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system (...) be a topological space X together with a continuous function f. f can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces. Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □, and two temporal modalities, ○ and *, both interpreted using the continuous function f. In particular, ○ expresses f’s action on X from one moment to the next, and * expresses the asymptotic behaviour of f. (shrink)

We prove a topological completeness theorem for the modal logic $$\textsf{GLP}$$ GLP containing operators $$\{\langle \xi \rangle :\xi \in \textsf{Ord}\}$$ { ⟨ ξ ⟩ : ξ ∈ Ord } intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence $$\phi $$ ϕ consistent with $$\textsf{GLP}$$ GLP can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the (...) finitely many modalities that occur in $$\phi $$ ϕ. (shrink)

갈등은 일차적으로 개인(주관)의 심리문제일 것이지만, 결국은 사회(주체)의 정치적 문제가 된다. 그리고 이 정치적 문제는 갈등의 제거가 아니라 갈등의 승인에서 가능하다. 이는 갈등에 대한 정치적 논의가 사회적 현실에 대한 “탈구와 재구성”의 역학, 즉 ‘해방의 변증법’을 수반하는 ‘갈등의 위상학’을 통해 이루어져야 한다는 것을 의미한다. 주체에 억압된 욕망, 부연하자면 ‘존재론적 욕망’으로 회귀하며 “내재성”의 논리에 입각해 갈등의 ‘위상학’(topology)을 선명히 가리킨 것은 현대의 철학, 특히 후설 이후 메를로퐁티와 하이데거, 그리고 라캉과 들뢰즈의 철학이었다. 이 글은, 이들에 함축된 ‘시(詩)적 인식’의 위상학적 양상을 메를로퐁티의 관점으로 수렴하는 자리에서, (...) 갈등이 갖는 해방의 변증법적 의의를 짚은 후, 우리사회에서 폭발하는 갈등이 위기가 아니라 사실은 〈외재적 초월의 유토피아적 허구를 넘는 ‘내재적 초월’의 현실에 입각해 정치적 상상력을 재정립할 기회〉의 한 계기임을 제시할 것이다. (shrink)

We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition, we give a description of minimal subflows and the Ellis group of its universal definable flow in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to. This provides a range of counterexamples to a question by Newelski whether the Ellis group (...) is isomorphic to. We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort and a sort for an abelian group. (shrink)

Indeterminism, understood as a notion that an event may be continued in a few alternative ways, invokes the question what a region of chanciness looks like. We concern ourselves with its topological and spatiotemporal aspects, abstracting from the nature or mechanism of chancy processes. We first argue that the question arises in Montague-Lewis-Earman conceptualization of indeterminism as well as in the branching tradition of Prior, Thomason and Belnap. As the resources of the former school are not rich enough to study (...) topological issues, we investigate the question in the framework of branching space-times of Belnap (Synthese 92:385–434, 1992). We introduce a topology on a branching model as well as a topology on a history in a branching model. We define light-cones and assume four conditions that guarantee the light-cones so defined behave like light-cones of physical space-times. From among various topological separation properties that are relevant to our question, we investigate the Hausdorff property. We prove that each history in a branching model satisfies the Hausdorff property. As for the satisfaction of the Hausdorff property in the entire branching model, we prove that it is related to the phenomenon of passive indeterminism, which we describe in detail. (shrink)

In this highly original text, Christopher Steinsvold explores an alternative semantics for logics of rational belief. Topologies, as mathematical objects, are typically interpreted in terms of space; here topologies are re-interpreted in terms of an agent with rational beliefs. The topological semantics tells us that the agent can never, in principle, know everything; that the agent's beliefs can never be complete. -/- A number of completeness proofs are given for a variety of logics of rational belief. Beyond this, the author (...) explores the philosophical question of why our beliefs can never be complete, and considers the possibility that a totality of truths is a dialethia. -/- This work will be of interest to all philosophers interested in epistemology, and modal logicians as well. (shrink)

In this work, we investigate the relationship between paraconsistent semantics and some well-known topological spaces such as connected and continuous spaces. We also discuss homotopies as truth preserving operations in paraconsistent topological models.

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)

Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a ‘syntax-semantics’ duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantic topological groupoid of models and isomorphisms of a theory. It is then shown how to extract a theory from equivariant sheaves on a topological groupoid in such a way that the result is a contravariant adjunction between theories and groupoids, the restriction of which is a (...) duality between theories with enough models and semantic groupoids. Technically a variant of the syntax-semantics duality constructed in 1 for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids—reflecting the use of ‘indexed’ models in the representation theorem—which in several respects simplifies the construction and the characterization of semantic groupoids. (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 (...) class='Hi'>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)

A topological class logic is an infinitary logic formed by combining a first-order logic with the quantifier symbols O and C. The meaning of a formula closed by quantifier O is that the set defined by the formula is open. Similarly, a formula closed by quantifier C means that the set is closed. The corresponding models are a topological class spaces introduced by Ćirić and Mijajlović (Math Bakanica 1990). The completeness theorem is proved.

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the (...) Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets (...) of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y, then for a large set of tuples $\langle \overline{a}_{1},\ldots,\overline{a}_{2^{k}}\rangle \in X^{2^{k}}$ , where k = dim(Y), the union of fibers $S_{\overline{a}_{1}}\cup \cdots \cup S_{\overline{a}_{2^{k}}}$ is large in Y. Finally, given a weakly o-minimal structure ������, we find various conditions equivalent to the fact that the topological dimension in ������ enjoys the addition property. (shrink)

Topology is integral to a shift in socio-cultural theory from a linguistic to a mathematical paradigm. This has enabled in Badiou and Žižek a critique of the symbolic register, understood in terms of pure conceptual abstraction. Drawing on topology, this article understands it instead in terms of the figure. The break with the symbolic and language necessitates a break with form, but topologically still preserves a logic of the figure. This becomes a process of figuration, indeed a process (...) of `deformation'. Badiou/Žižek will then presume a break with the symbolic for `the real'. But topology entails the centrality of not the real but the imaginary. With Castoriadis, this imaginary is understood as productive and social, and with Sloterdijk as spatial. In our times this is a self-organizing socio-technical imaginary. For Niklas Luhmann these socio-technical systems engage in coupling. They structurally couple with other social imaginaries. Their self-organization, going beyond pure functionality, operates through semantic excess: an excess that organizes the system and is their structure. Such structural coupling entails semantic exchange, consisting of not just information but also of images. (shrink)

The aim of this paper is to present a new perspective under which branching-time semantics can be viewed. The set of histories (maximal linearly ordered sets) in a tree structure can be endowed in a natural way with a topological structure. Properties of trees and of bundled trees can be expressed in topological terms. In particular, we can consider the new notion of topological validity for Ockhamist temporal formulae. It will be proved that this notion of validity is equivalent to (...) validity with respect to bundled trees. (shrink)

The standard presentation of topological spaces relies heavily on (naïve) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are required to achieve just the basic results about topological spaces. Concentrating on the mathematical structures, category theory offers the possibility to look synthetically at the structure of continuous transformations between topological spaces addressing specifically how the fundamental notions of point and open come about. As (...) a byproduct of this, one may look at the different approaches to topology from an external perspective and compare them in a unified way. Technically, the category of sober topological spaces can be seen as consisting of (co)algebraic structures in the exact completion of the elementary category of sets and relations. Moreover, the same abstract construction of taking the exact completion, when applied to the category of topological spaces and continuous functions produces an extension of it which is cartesian closed. In other words, there is one general mathematical construction that, when applied to a very elementary category, generates the category of topological spaces and continuous functions, and when applied to that category produces a very suitable category where to deal with all sorts functions spaces. Yet, via such free constructions it is possible to give a new meaning to Marshall Stone’s dictum: “always topologize” as the category of sets and relations is the most natural way to give structure to logic and the category of topological spaces and continuous functions is obtained from it by a good mix of free – i.e. syntactic – constructions. (shrink)

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker’s core insights, we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and what is knowable; we argue that the foundational axioms of Stalnaker’s system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the (...) principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an “evidence-in-hand” conception of knowledge and a weaker “evidence-out-there” notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker’s postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics. (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 (...) 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)

Male sexual arousal (SA) has been known as a multidimensional experience involving closely interrelated and coordinated neurobehavioral components that rely on widespread brain regions. Recent functional neuroimaging studies have shown relation between abnormal/altered dynamics in these circuits and male sexual dysfunction. However, alterations in the topological1 organization of structural brain networks in male sexual dysfunction are still unclear. Here, we used graph theory2 to investigate the topological properties of large-scale structural brain networks, which were constructed using inter-regional correlations of cortical (...) thickness between 78 cortical regions (subcortical regions were not included due to the used cortical surface model) in 40 patients with psychogenic erectile dysfunction (pED) and 39 normal controls. Compared with normal controls, pED patients exhibited a less optimal global topological organization with reduced global and increased local efficiencies. Our results suggest disrupted neural integration among distant brain regions in pED patients, consistent with previous reports of impaired white matter structure and abnormal functional integrity in pED. Additionally, disrupted global network topology in pED was observed to be primarily relevant to altered subnetwork and nodal properties within the networks mediating the cognitive, motivational and inhibitory processes of male SA, possibly indicating disrupted integration of these networks in the whole brain networks and might account for pED patients’ abnormal cognitive, motivational and inhibitory processes for male SA. In total, our findings provide evidence for disrupted integrity in large-scale brain networks underlying the neurobehavioral processes of male SA in pED and provide new insights into the understanding of the pathophysiological mechanisms of pED. (shrink)

Social epistemology studies knowledge and justified belief acquisition through organized group cooperation. To do this, the way such group cooperation is structured has to be modeled. The obvious way of modeling a group structure is with a directed graph; unfortunately, most types of social cooperation directed at epistemological aims are variably implementable, including in their structural expression. Furthermore, the frequency with which a practice is implemented in a certain way can vary with topology. This entails that the topology (...) of social practices directed toward epistemological ends has to be modeled by a set of directed graphs, or their equivalent, together with a probability distribution over that set. In theory, this is eminently possible; however, there are considerable practical obstacles to the specification of a practice’s topology in this way. This paper examines these practical difficulties and concludes that todays sampling protocols are either far too slow to handle pratices with 10 or more participants, or else prone to produce misleading evaluations. (shrink)

We show that the Axiom of Choice is equivalent to each of the following statements: A product of closures of subsets of topological spaces is equal to the closure of their product ; A product of complete uniform spaces is complete.

In social and cultural theory, topology has been used to articulate changes in structures and spaces of power. In this introduction, we argue that culture itself is becoming topological. In particular, this ‘becoming topological’ can be identified in the significance of a new order of spatio-temporal continuity for forms of economic, political and cultural life today. This ordering emerges, sometimes without explicit coordination, in practices of sorting, naming, numbering, comparing, listing, and calculating. We show that the effect of these (...) practices is both to introduce new continuities into a discontinuous world by establishing equivalences or similitudes, and to make and mark discontinuities through repeated contrasts. In this multiplication of relations, topological change is established as being constant, normal and immanent, rather than being an exceptional form, which is externally produced; that is, forms of economic, political and cultural life are identified and made legible in terms of their capacities for continuous change. Outlining the contributions to this Special Issue, the introduction discusses the meaning of topological culture and provides an analytic framework through which to understand its implications. (shrink)

Must space be a unity? This question, which exercised Aristotle, Descartes and Kant, is a specific instance of a more general one; namely, can the topology of physical space change with time? In this paper we show how the discussion of the unity of space has been altered but survives in contemporary research in theoretical physics. With a pedagogical review of the role played by the Euler characteristic in the mathematics of relativistic spacetimes, we explain how classical general relativity (...) (modulo considerations about energy conditions) allows virtually unrestrained spatial topology change in four dimensions. We also survey the situation in many other dimensions of interest. However, topology change comes with a cost: a famous theorem by Robert Geroch shows that, for many interesting types of such change, transitions of spatial topology imply the existence of closed timelike curves or temporal non-orientability. Ways of living with this theorem and of evading it are discussed. (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)

Territorial borders just like other boundaries are involved in a politics of belonging, a politics of “us” and “them”. Border management regimes are thus part of processes of othering. In this article, we use the management of borders and populations in Europe as an empirical example to make a theoretical claim about race. We introduce the notion of the phenotypic other to argue that race is a topological object, an object that is spatially and temporally folded in distributed technologies of (...) governance. To elaborate on these notions, we first examine a number of border management technologies through which both race and Europe are brought into being. More specifically we focus on how various such technologies aimed at monitoring the movement of individuals together with the management of populations have come to play crucial roles in Europe. Different border management regimes, we argue, do not only enact different versions of Europe but also different phenotypic others. We then shift the focus from border regimes to internal practices of governance, examining forensic DNA databanks to unravel articulations of race in the traffic between databases and societies. (shrink)

The concept of "the flesh" (la chair) derives from the writings of Maurice Merleau-Ponty. This was the word he used to name the concrete realm of sentient bodies and life processes that has been eclipsed by the abstractions of science, technology, and modern culture. Topology, to conventional understanding, is the branch of mathematics that concerns itself with the properties of geometric figures that stay the same when the figures are stretched or deformed. Topologies of the Flesh blends continental thought (...) and mathematical imagination, opening up a new area of philosophical inquiry: topological phenomenology. Through its unique application of qualitative mathematics, this work extends the approaches of Merleau-Ponty and Heidegger so as to offer a detailed exploration of previously uncharted dimensions of human experience and the natural world. (shrink)

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also étale, this association is compatible with but independent from the theory of localic étale groupoids and their quantales [9] of P. Resende. As a motivating example, we describe the connection between the quantale and the C*-algebra that both classify Penrose tilings, which was left as an open problem in [5].

Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, is interpreted as topological interior. Thus S4 can be understood as the logic of topological spaces, and can be understood as a topological modality. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system be a (...) topological space X together wError: Corrupted memory profileError: read ICCBased color space profile errorith a continuous function f. f can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces. Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality, and two temporal modalities, and ∗, both interpreted using the continuous function. (shrink)

The quantization of the magnetic flux in superconducting rings is studied in the frame of a topological model of electromagnetism that gives a topological formulation of electric charge quantization. It turns out that the model also embodies a topological mechanism for the quantization of the magnetic flux with the same relation between the fundamental units of magnetic charge and flux as there is between the Dirac monopole and the fluxoid.

(1997). The topological inventions of life: From the specialization of multicellular colonies to the functioning of the vertebrate brain. World Futures: Vol. 50, No. 1-4, pp. 617-631.

Stalnaker, 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of the closure of the interior operator on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge with a new topological semantics for belief. We prove that the belief (...) logic KD45 is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. We also study belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our setting based on extremally disconnected spaces, however, encounters problems when extended with dynamic updates. We then propose a solution consisting in interpreting belief in a similar way based on hereditarily extremally disconnected spaces, and axiomatize the belief logic of hereditarily extremally disconnected spaces. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic. (shrink)

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

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)

We consider semi‐algebraic sets and properties of these sets that are expressible by sentences in first‐order logic over the reals. We are interested in first‐order properties that are invariant under topological transformations of the ambient space. Two semi‐algebraic sets are called topologically elementarily equivalent if they cannot be distinguished by such topological first‐order sentences. So far, only semi‐algebraic sets in one and two‐dimensional space have been considered in this context. Our contribution is a natural characterisation of topological elementary equivalence of (...) regular closed semi‐algebraic sets in three‐dimensional space, extending a known characterisation for the two‐dimensional case. Our characterisation is based on the local topological behaviour of semi‐algebraic sets and the key observation that topologically elementarily equivalent sets can be transformed into each other by means of geometric transformations, each of them mapping a set to a first‐order indistinguishable one. (shrink)