Topology

Review of Slavoj Žižek, "Sex and the Failed Absolute".

This paper takes as its point of departure recent research into the possibility that human beings can perceive single photons. In order to appreciate what quantum perception may entail, we first explore several of the leading interpretations of quantum mechanics, then consider an alternative view based on the ontological phenomenology of Maurice Merleau-Ponty and Martin Heidegger. Next, the philosophical analysis is brought into sharper focus by employing a perceptual model, the Necker cube, augmented by the topology of the Klein bottle. (...) This paves the way for addressing in greater depth the paper’s central question: Just what would it take to observe the quantum reality of the photon? In formulating an answer, we examine the nature of scientific objectivity itself, along with the paradoxical properties of light. The conclusion reached is that quantum perception requires a new kind of observation, one in which the observer of the photon adopts a concretely self-reflexive observational posture that brings her into close ontological relationship with the observed. (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)

The subject of my dissertation is the structure of continua and, in particular, of physical space and time. Consider the region of space you occupy: is it composed of indivisible parts? Are the indivisible parts, if any, extended? Are there infinitesimal parts? The standard view that space is composed of unextended points faces both \textit{a priori} and empirical difficulties. In my dissertation, I develop and evaluate several novel approaches to these questions based on metaphysical, mathematical and physical considerations. In particular, (...) I develop and evaluate two infinitesimal theories of space based on Robinson's nonstandard analysis. I argue that \textit{Infinitesimal Gunk}, according to which every region is further divisible and some regions have infinitesimal sizes, has distinct advantages over alternative gunky views. I also advance a new account of distance for atomistic space, \textit{the mixed account}, in response to Weyl's tile argument, which is an influential argument against the view that space is composed of indivisible regions. Having these theories in stock, we make progress in discovering the best theory of continua. (shrink)

According to Brentano's theory of boundaries, no boundary can exist without being connected with a continuum. But there is no specifiable part of the continuum, and no point, which is such that we may say that it is the existence of that part or of that point which conditions the boundary. - An adequate theory of the continuum must now recognize that boundaries be boundaries only in certain directions and not in others. This leads to consequences in other areas, too.

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...) algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. (shrink)

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev’s logic (...) of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic. (shrink)

In “On Drawing Lines on a Map” (1995), I suggested that the different ways we have of drawing lines on maps open up a new perspective on ontology, resting on a distinction between two sorts of boundaries: fiat and bona fide. “Fiat” means, roughly: human-demarcation-induced. “Bona fide” means, again roughly: a boundary constituted by some real physical discontinuity. I presented a general typology of boundaries based on this opposition and showed how it generates a corresponding typology of the different sorts (...) of objects which boundaries determine or demarcate. In this paper, I describe how the theory of fiat boundaries has evolved since 1995, how it has been applied in areas such as property law and political geography, and how it is being used in contemporary work in formal and applied ontology, especially within the framework of Basic Formal Ontology. (shrink)

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

Let be a topological space and *X a nonstandard extension of X. Sets of the form *G, where G $\in$ T. form a base for the "standard" topology $^ST$ on *X. The topological space will be used to study compactifications of in a systematic way.

Kant maintains that in face of the failure of the traditional arguments for the existence of God it is necessary to provide an entirely fresh centre of gravity for the notion of religious consciousness. To explicate Kant's critique this paper develops, as a special hermeneutic device, the idea of a kind of Husserlian reduction and realization', in terms of which the various uses of Kant's concept of thing' or object' are given a new interpretation,using this to provide a novel approach (...) to the whole structure of the Kantian world. (shrink)

Physical reality is all the reality we have, and so physical theory in the standard sense is all the ontology we need. This, at least, was an assumption taken almost universally for granted by the advocates of exact philosophy for much of the present century. Every event, it was held, is a physical event, and all structure in reality is physical structure. The grip of this assumption has perhaps been gradually weakened in recent years as far as the sciences of (...) mind are concerned. When it comes to the sciences of external reality, however, it continues to hold sway, so that contemporary philosophers B even while devoting vast amounts of attention to the language we use in describing the world of everyday experience B still refuse to see this world as being itself a proper object of theoretical concern. Here, however, we shall argue that the usual conception of physical reality as constituting a unique bedrock of objectivity reflects a rather archaic view as to the nature of physics itself and is in fact incompatible with the development of the discipline since Newton. More specifically, we shall seek to show that the world of qualitative structures, for example of colour and sound, or the commonsense world of coloured and sounding things, can be treated scientifically (ontologically) on its own terms, and that such a treatment can help us better to understand the structures both of physical reality and of cognition. (shrink)

The fact that boundaries are ontologically dependent entities is agreed by Franz Brentano and Roderick Chisholm. This article studies both authors as a single metaphysical account about boundaries. The Brentano-Chisholm theory understands that boundaries and the objects to which they belong hold a mutual relationship of ontological dependence: the existence of a boundary depends upon a continuum of higher spatial dimensionality, but also is a conditio sine qua non for the existence of a continuum. Although the view that ordinary material (...) objects and their boundaries (or surfaces) ontologically depend on each other is correct, it does not grasp their asymmetric relationship: while the existence of a surface rigidly depends upon the existence of the very object it belongs to, the existence of a physical object generically depends upon having some surface. In modal terms, both are two kinds of de re ontological dependence that this article tries to distinguish. (shrink)

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

Rosen's modelling relations constitute a conceptual schema for the understanding of the bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems used in this study refers to information structures constructed as algebraic rings of observable attributes of natural systems, in which the notion of observable signifies a physical attribute that, in principle, can be measured. Due to the fact that modelling relations are bidirectional by construction, they admit a precise categorical formulation in terms (...) of the category-theoretic syntactic language of adjoint functors, representing the inverse processes of information encoding/decoding via adjunctions. As an application, we construct a topological modelling schema of complex systems. The crucial distinguishing requirement between simple and complex systems in this schema is reflected with respect to their rings of observables by the property of global commutativity. The global information structure representing the behaviour of a complex system is modelled functorially in terms of its spectrum functor. An exact modelling relation is obtained by means of a complex encoding/decoding adjunction restricted to an equivalence between the category of complex information structures and the category of sheaves over a base category of partial or local information carriers equipped with an appropriate topology. (shrink)

We develop a general covariant categorical modeling theory of natural systems' behavior based on the fundamental functorial processes of representation and localization-globalization. In the second part of this study we analyze the semantic bidirectional process of localization-globalization. The notion of a localization system of a complex information structure bears a dual role: Firstly, it determines the appropriate categorical environment of base reference contexts for considering the operational modeling of a complex system's behavior, and secondly, it specifies the global compatibility conditions (...) of local contextual information. A localization system acts on the global information structure of a complex system, partitions it into sorts, and eventually, forces the consistent sheaf-theoretic fibering of the latter over the base category of commutative reference contexts. In this manner, the sheafification of the Spectrum functor of a complex information structure takes place by imposing on the uniform and homologous fibered structure of elements of the Spectrum presheaf the following two requirements of coherence in relation to the localization-globalization process: [i]. Compatibility of information under restriction from the global to the local level, and [ii]. Compatibility of information under extension from the local to the global level. Correspondingly, the options of local and global receive a concrete mathematical meaning with respect to a suitable notion of topology (categorical Grothendieck topology) defined on the base category of commutative reference contexts. Finally, the accurate functorial process modeling of a complex system's behavior, respecting the processes of representation and localization-globalization, is being effectuated by means of establishing a categorical dual equivalence between the category of complex information structures, and the topes of sheaves over the base category of partial or local information carriers, equipped with the categorical topology of epimorphosis families. (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.

We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algebras and the category of quantum measure algebras. The sheaf-theoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the level of (...) events, to the sheaf-theoretic gluing conditions at the level of Boolean localization systems. (shrink)

We develop a relativistic perspective on structures of quantum observables, in terms of localization systems of Boolean coordinatizing charts. This perspective implies that the quantum world is comprehended via Boolean reference frames for measurement of observables, pasted together along their overlaps. The scheme is formalized categorically, as an instance of the adjunction concept. The latter is used as a framework for the specification of a categorical equivalence signifying an invariance in the translational code of communication between Boolean localizing contexts and (...) quantum systems. Aspects of the scheme semantics are discussed in relation to logic. The interpretation of coordinatizing localization systems, as structure sheaves, provides the basis for the development of an algebraic differential geometric machinery suited to the quantum regime. (shrink)

This volume develops a fundamentally different categorical framework for conceptualizing time and reality. The actual taking place of reality is conceived as a “constellatory self-unfolding” characterized by strong self-referentiality and occurring in the primordial form of time, the not yet sequentially structured “time-space of the present.” Concomitantly, both the sequentially ordered aspect of time and the factual aspect of reality appear as emergent phenomena that come into being only after reality has actually taken place. In this new framework, time functions (...) as an ontophainetic [H1] platform, i.e., as the stage on which reality can first occur. Events are merely the “tracks” that the actual taking place of reality leaves behind on the co-emergent “canvas’’ of local spacetime. -/- The view of time proposed here is particularly relevant to the recent debate over the “ER=EPR” conjecture targeting the relation between quantum physics and general relativity theory. The novelty of this radically different framework is that it allows quantum reduction and singularities to be addressed as inverse transitions into and out of the factual layer of reality: In quantum physical state reduction, reality “gains” the chrono-ontological format of facticity, and the sequential aspect of time becomes applicable. In singularities, by contrast, the opposite happens: Reality loses its local spacetime formation and reverts back to its primordial, pre-local shape – making the use of causality relations, Boolean logic and the dichotomization of subject and object obsolete in the process. -/- For our understanding of the relation between quantum and relativistic physics, this new view opens up fundamentally new perspectives: Both are legitimate views of time and reality; they simply address very different chrono-ontological portraits, and thus should not lead us to erroneously prefer one view over the other. -/- The task of the book is to provide a formal framework in which this radically different view of time and reality can be suitably addressed. The mathematical approach is based on the logical and topological features of the Borromean Rings, and draws upon concepts and methods from algebraic and geometric topology – especially the theory of sheaves and links, group theory, logic and information theory in relation to the standard constructions employed in quantum mechanics and general relativity, shedding new light on the problems of their compatibility. The intended audience includes physicists, mathematicians and philosophers with an interest in the conceptual and mathematical foundations of modern physics. (shrink)

All the typical global quantum mechanical observables are complex relative phases obtained by interference phenomena. They are described by means of some global geometric phase factor, which is thought of as the “memory” of a quantum system undergoing a “cyclic evolution” after coming back to its original physical state. The origin of a geometric phase factor can be traced to the local phase invariance of the transition probability assignment in quantum mechanics. Beyond this invariance, transition probabilities also remain invariant under (...) the operation of complex conjugation. Most important, geometric phase factors distinguish between unitary and antiunitary transformations in terms of complex conjugation. These two types of invariance functions as an anchor point to investigate the role of loops and based loops in the state space of a quantum system as well as their links and interrelations. We show that arbitrary transition probabilities can be calculated using projective invariants of loops in the space of rays. The case of the double slit experiment serves as a model for this purpose. We also represent the action of one-parameter unitary groups in terms of oppositely oriented-based loops at a fixed ray. In this context, we explain the relation among observables, local Boolean frames of projectors, and one-parameter unitary groups. Next, we exploit the non-commutative group structure of oriented-based loops in 3-d space and demonstrate that it carries the topological semantics of a Borromean link. Finally, we prove that there exists a representation of this group structure in terms of one-parameter unitary groups that realizes the topological linking properties of the Borromean link. (shrink)

Human cognitive acts are directed towards entities of a wide range of different types. What follows is a new proposal for bringing order into this typological clutter. A categorial scheme for the objects of human cognition should be (1) critical and realistic. Cognitive subjects are liable to error, even to systematic error of the sort that is manifested by believers in the Pantheon of Olympian gods. Thus not all putative object-directed acts should be recognized as having objects of their own. (...) Broadly, the objects towards which human cognition is directed should be parts of reality in a sense that is at least consistent with the truths of natural science. But such a scheme should also be (2) comprehensive: it should do justice to each sort of object on its own terms, and not attempt to eliminate objects of one sort in favour of objects of other, more favoured sorts. Linguistic and other forms of idealism, as well as Meinongian theories, which assign to each and every referring expression or intentional act an object tailored to fit, yield categorial schemes which fail to satisfy (1). Physicalistic and other forms of reductionism yield categorial schemes which fail to satisfy (2). What follows is a categorial scheme that is both critically realistic and comprehensive. Thus it enjoys some of the benefits of linguistic idealism and physicalism, without (or so it is hoped) the corresponding disadvantages of each. (shrink)

This paper attempts to explore a possibility to visualize the structure of time-consciousness in a knot shape. By applying Louis Kauffman’s knot-logic, the consistency of subjective consciousness, the plurality of now’s, and the necessary relationship between subjective and intersubjective consciousness will be represented in topological space.

Brouwer's views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke (...) semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the semantic hierarchy. (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)

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...) of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

The discrepancy between syntax and semantics is a painstaking issue that hinders a better comprehension of the underlying neuronal processes in the human brain. In order to tackle the issue, we at first describe a striking correlation between Wittgenstein's Tractatus, that assesses the syntactic relationships between language and world, and Perlovsky's joint language-cognitive computational model, that assesses the semantic relationships between emotions and “knowledge instinct”. Once established a correlation between a purely logical approach to the language and computable psychological activities, (...) we aim to find the neural correlates of syntax and semantics in the human brain. Starting from topological arguments, we suggest that the semantic properties of a proposition are processed in higher brain's functional dimensions than the syntactic ones. In a fully reversible process, the syntactic elements embedded in Broca's area project into multiple scattered semantic cortical zones. The presence of higher functional dimensions gives rise to the increase in informational content that takes place in semantic expressions. Therefore, diverse features of human language and cognitive world can be assessed in terms of both the logic armor described by the Tractatus, and the neurocomputational techniques at hand. One of our motivations is to build a neuro-computational framework able to provide a feasible explanation for brain's semantic processing, in preparation for novel computers with nodes built into higher dimensions. (shrink)

In the author’s previous contribution to this journal (Rosen 2015), a phenomenological string theory was proposed based on qualitative topology and hypercomplex numbers. The current paper takes this further by delving into the ancient Chinese origin of phenomenological string theory. First, we discover a connection between the Klein bottle, which is crucial to the theory, and the Ho-t’u, a Chinese number archetype central to Taoist cosmology. The two structures are seen to mirror each other in expressing the psychophysical (phenomenological) action (...) pattern at the heart of microphysics. But tackling the question of quantum gravity requires that a whole family of topological dimensions be brought into play. What we find in engaging with these structures is a closely related family of Taoist forebears that, in concert with their successors, provide a blueprint for cosmic evolution. Whereas conventional string theory accounts for the generation of nature’s fundamental forces via a notion of symmetry breaking that is essentially static and thus unable to explain cosmogony successfully, phenomenological/Taoist string theory entails the dialectical interplay of symmetry and asymmetry inherent in the principle of synsymmetry. This dynamic concept of cosmic change is elaborated on in the three concluding sections of the paper. Here, a detailed analysis of cosmogony is offered, first in terms of the theory of dimensional development and its Taoist (yin-yang) counterpart, then in terms of the evolution of the elemental force particles through cycles of expansion and contraction in a spiraling universe. The paper closes by considering the role of the analyst per se in the further evolution of the cosmos. (shrink)

Causal set theory and the theory of linear structures share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin’s more general framework and I characterise what Maudlin’s topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of (...) causal sets that can be described in Maudlin’s theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin’s theory. The value of this theory depends crucially on whether it is true that its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On the one hand, my theorems support : the theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine : standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin’s framework. This fact poses a challenge for the proponents of Maudlin’s theory to prove it fruitful. (shrink)

The general theme of this article is the actual practice of how definitions are justified and formulated in mathematics. The theoretical insights of this article are based on a case study of topological definitions of chaos. After introducing this case study, I identify the three kinds of justification which are important for topological definitions of chaos: natural-world-justification, condition-justification and redundancy-justification. To my knowledge, the latter two have not been identified before. I argue that these three kinds of justification are widespread (...) in mathematics. After that, I first discuss the state of the art in the literature about the justification of definitions in the light of actual mathematical practice. I then go on to criticize Lakatos’s account of proof-generated definitions—the main account in the literature on this issue—as being limited and also misguided: as for topological definitions of chaos, in nearly all mathematical fields various kinds of justification are found and are also reasonable. (shrink)

To what in reality do the logically simple sentences with empirical content correspond? Two extreme positions can be distinguished in this regard: 'Great Fact' theories, such as are defended by Davidson; and trope-theories, which see such sentences being made the simply by those events or states to which the relevant main verbs correspond. A position midway between these two extremes is defended, one according to which sentences of the given sort are made tme by what are called 'dependence structures', or (...) in other words by certain complex concrete portions of reality between the parts of which relations of dependence are defined. Principles governing such dependence-structures are laid down, principles of an ontologically motivated sort which serve as basis for a topological semantics conceived as an altemative to standard set-theoretic approaches to semantics of the Tarskian sort. These principles are then used to resolve certain puzzles generated by the (semantically motivated) theory of events put forward by Davidson. (shrink)

The Clifford tori in $S^3$ constitute a one-parameter family of flat, two-dimensional, constant mean curvature submanifolds. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE methods. That is, one can create a submanifold that has almost everywhere constant mean curvature by gluing a re-scaled catenoid into the neighbourhood of each point of a sub-lattice (...) of the Clifford torus; and then one can show that a constant mean curvature perturbation of this submanifold does exist. (shrink)

The paper introduces the concepts at the heart of point-set-topology and of mereotopology (topology founded in the non-atomistic theory of parts and wholes) in an informal and intuitive fashion. It will then seek to demonstrate how mereotopological ideas can be of particular utility in cognitive science applications. The prehistory of such applications (in the work of Husserl, the Gestaltists, of Kurt Lewin and of J. J. Gibson) will be sketched, together with an indication of the field of possibilities in linguistics, (...) perceptual psychology, cate¬gorization and geographic information systems. Topological structures will be shown to play a central role in studies of naive physics not least in virtue of the fact that even well-attested departures from true physics on the part of common sense leave the topology and vectorial orientation of the underlying physical phenomena invariant: our common sense would thus seem to have a veridical grasp of the topology and broad general orientation of physical phenomena, both static and dynamic, even where it illegitimately modifies the relevant shape and metric properties. The implications of this and related insights for the methodology of psychology will be explored. (shrink)

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

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 this paper I will try to argue for a new version in philosophy entitled as Philosophical Topology. It is inspired by the thought of Peter Strawson as well as ones of some of so-called Continental philosophers like Heidgger. Unlike any of metaphilosophy in general, the philosophical topology focuses rather on analyses of processes of make-up in philosophers’ thinking, especially by revealing the internal logic of philosophical ideas in making and processing in order to explain the intrinsic continuation of philosophical (...) ideas in particular philosophers’ thinking. In this sense the philosophical topology is not one of philosophical methods but a new branch in philosophy that is characterized as being concerned with continuation of ideas as its basic task. So, in this way, it is available to analyses of various doctrines in philosophy. Main issues in the philosophical topology are as follows: 1) to analyze the internal law-likes in the development of philosophy with the perspective of topology; 2) to interpret the general routes of Western philosophy in terms of the philosophical topology; 3) to view the philosophical topology as a way to inquire into metaphysics; 4) to deal properly with the relation of the philosophical topology to other branches in contemporary philosophy such as the philosophy of language, of logic, of science and ethics. (shrink)

Understanding of elementary topological equivalencies is impaired by preconceptions about the topological structure of ordinary objects, so that the equivalencies turn out to be counterintuitive. Here I will discuss some of these preconceptions, namely the dominance of gestalt properties of the visual display of the configuration, the neglect of holistic properties, the dominance of transformations the preserve metric properties over those that preserve topological properties only, the assumption that holes are objects of their own. These factors delineate an empirical research (...) field, intuitive topology. An explanatory hypothesis for the preconceptions suggests that they are hard-wired. (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)

The Aharonov-Bohm effect is typically called ``topological.'' But it seems no more topological than magnetostatics, electrostatics or Newton-Poisson gravity. I distinguish between two senses of ``topological.''.

In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of rest (...) and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises. (shrink)

In this paper we propose an approach to vagueness characterised by two features. The first one is philosophical: we move along a Kantian path emphasizing the knowing subject’s conceptual apparatus. The second one is formal: to face vagueness, and our philosophical view on it, we propose to use topology and formal topology. We show that the Kantian and the topological features joined together allow us an atypical, but promising, way of considering vagueness.

This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of Łukasiewicz n -valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n -valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal Łukasiewicz n -valued logic with truth constants, which generalizes Jónsson-Tarski duality for modal algebras to the (...) n -valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics. (shrink)