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.

The unified treatment of the Dirac monopole, the Schwinger monopole, and the Aharonov-Bohm problem by Barut and Wilson is revisited via a path integral approach. The Kustaanheimo-Stiefel transformation of space and time is utilized to calculate the path integral for a charged particle in the singular vector potential. In the process of dimensional reduction, a topological charge quantization rule is derived, which contains Dirac's quantization condition as a special case. “Everything that is made beautiful and fair and (...) lovely is made for the eye of one who sees.” Jelaluddin Rumi,Mathnawi [I, 2383]. (shrink)

The Dirac operator arises naturally on $\mathbb{S}^1 \times \mathbb{S}^3 $ from the connection on the Lie group U(1)×SU(2) and maps spacetime rays into rays in the Lie algebra. We construct both simple harmonic and pulse solutions to the neutrino equations on $\mathbb{S}^1 \times \mathbb{S}^3 $ , classified by helicity and holonomy, using this map. Helicity is interpreted as the internal part of the Noether charge that arises from translation invariance; it is topologically quantized in integral multiples of a constant g (...) that converts a Lie-algebra phase shift into an action. The fundamental unit of helicity is associated with a full twist in u(1)×su(2) phase per global lightlike cycle. If we pass to the projective space ℝP1xℝP3, we get half-integral helicity. (shrink)

We examine the time discontinuity in rotating space–times for which the topology of time is S1. A kinematic restriction is enforced that requires the discontinuity to be an integral number of the periodicity of time. Quantized radii emerge for which the associated tangential velocities are less than the speed of light. Using the de Broglie relationship, we show that quantum theory may determine the periodicity of time. A rotating Kerr–Newman black hole and a rigidly rotating disk of dust are also (...) considered; we find that the quantized radii do not lie in the regions that possess CTCs. (shrink)

Topology and geometry have played an important role in our theoretical understanding of quantum field theories. One of the most interesting applications of topology has been the quantization of certain coupling constants. In this paper, we present a general framework for understanding the quantization itself in the light of group cohomology. This analysis of the cohomological aspects of physics leads to reconsider the very foundations of mechanics in a new light.

This paper investigates the fixed-time coordinated control problem of six-degree-of-freedom dynamic model for multiple spacecraft formation flying with input quantization, where the communication topology is assumed directed. Firstly, a new multispacecraft nonsingular fixed-time terminal sliding mode vector is derived by using neighborhood state information. Secondly, a hysteretic quantizer is utilized to quantify control force and torque. Utilizing such a quantizer not only can reduce the required communication rate but also can eliminate the control chattering phenomenon induced by the logarithmic (...) quantizer. Thirdly, a 6-DOF fixed-time coordinated control strategy with adaptive tuning laws is proposed, such that the practical fixed-time stability of the controlled system is ensured in the presence of both upper bounds of unknown external disturbances. It theoretically proves that the relative tracking errors of attitude and position can converge into the regions in a fixed time. Finally, a numerical example is exploited to show the usefulness of the theoretical results. (shrink)

We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong (...) interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity. (shrink)

We survey some philosophical aspects of the search for a quantum theory of gravity, emphasising how quantum gravity throws into doubt the treatment of spacetime common to the two `ingredient theories' (quantum theory and general relativity), as a 4-dimensional manifold equipped with a Lorentzian metric. After an introduction (Section 1), we briefly review the conceptual problems of the ingredient theories (Section 2) and introduce the enterprise of quantum gravity (Section 3). We then describe how three main research programmes in quantum (...) gravity treat four topics of particular importance: the scope of standard quantum theory; the nature of spacetime; spacetime diffeomorphisms, and the so-called `problem of time' (Section 4). These programmes are the old particle-physics approach, superstring theory, and canonical quantum gravity. By and large, these programmes accept most of the ingredient theories' treatment of spacetime, albeit with a metric with some type of quantum nature; but they also suggest that the treatment has fundamental limitations. This prompts the idea of going further: either by quantizing structures other than the metric, such as the topology; or by regarding such structures as phenomenological. We discuss this in Section 5. (shrink)

An overview of the following three related papers in this issue presents the Emergence of Highly Complex Systems such as living organisms, man, society and the human mind from the viewpoint of the current Ontological Theory of Levels. The ontology of spacetime structures in the Universe is discussed beginning with the quantum level; then, the striking emergence of the higher levels of reality is examined from a categorical—relational and logical viewpoint. The ontological problems and methodology aspects discussed in the first (...) two papers are followed by a rigorous paper based on Category Theory, Algebraic Topology and Logic that provides a conceptual and mathematical basis for a Categorical Ontology Theory of Levels. The essential links and relationships between the following three papers of this issue are pointed out, and further possible developments are being considered. (shrink)

A generalization of gauge theory in which the gauge potential1-form is replaced by a p-form is studied. Charged particles are then replaced by elementary extended objects of dimension p−1. It is shown that this extension is compatible with space-time locality only if the gauge group is U(1). A source which is a closed p−1 surface has zero total charge and corresponds to a particle-antiparticle pair. Its quantum rate of production in an external uniform field is evaluated semiclassically. The analog of (...) the Dirac magnetic pole is constructed. It is another extended object, of dimension n−p−3, where n is the dimension of space-time. The “electric” and “magnetic” charges obey the Dirac quantization condition. This condition is derived in two different ways. One method makes use of local gauge patches and the other brings in singular gauge transformations. A topological mass term is introduced and it is shown that it can coexist with a magnetic pole when n=2p+1, provided the topological mass is quantized. (shrink)

The issue of the intrinsic nonlocality of quantum mechanics raised by J. S. Bell is examined from the point of view of the recently developed method of geometro-stochastic quantization and its applications to general relativistic quantum theory. This analysis reveals that a distinction should be made between the topological concept of locality used in formulating relativistic causality and a type of geometric locality based on the concept of fiber bundle, which can be used in extending the strong equivalence (...) principle to the quantum domain. Both play an essential role in formulating a notion of geometro-stochastic propagation based on quantum diffusions, which throws new light on the EPR paradox, on the origin of the arrow of time, and on other fundamental issues in quantum cosmology and the theory of measurement. (shrink)

This book attempts to marry truth-conditional semantics with cognitive linguistics in the church of computational neuroscience. To this end, it examines the truth-conditional meanings of coordinators, quantifiers, and collective predicates as neurophysiological phenomena that are amenable to a neurocomputational analysis. Drawing inspiration from work on visual processing, and especially the simple/complex cell distinction in early vision (V1), we claim that a similar two-layer architecture is sufficient to learn the truth-conditional meanings of the logical coordinators and logical quantifiers. As a prerequisite, (...) much discussion is given over to what a neurologically plausible representation of the meanings of these items would look like. We eventually settle on a representation in terms of correlation, so that, for instance, the semantic input to the universal operators (e.g. and, all)is represented as maximally correlated, while the semantic input to the universal negative operators (e.g. nor, no)is represented as maximally anticorrelated. On the basis this representation, the hypothesis can be offered that the function of the logical operators is to extract an invariant feature from natural situations, that of degree of correlation between parts of the situation. This result sets up an elegant formal analogy to recent models of visual processing, which argue that the function of early vision is to reduce the redundancy inherent in natural images. Computational simulations are designed in which the logical operators are learned by associating their phonological form with some degree of correlation in the inputs, so that the overall function of the system is as a simple kind of pattern recognition. Several learning rules are assayed, especially those of the Hebbian sort, which are the ones with the most neurological support. Learning vector quantization (LVQ) is shown to be a perspicuous and efficient means of learning the patterns that are of interest. We draw a formal parallelism between the initial, competitive layer of LVQ and the simple cell layer in V1, and between the final, linear layer of LVQ and the complex cell layer in V1, in that the initial layers are both selective, while the final layers both generalize. It is also shown how the representations argued for can be used to draw the traditionally-recognized inferences arising from coordination and quantification, and why the inference of subalternacy breaks down for collective predicates. Finally, the analogies between early vision and the logical operators allow us to advance the claim of cognitive linguistics that language is not processed by proprietary algorithms, but rather by algorithms that are general to the entire brain. Thus in the debate between objectivist and experiential metaphysics, this book falls squarely into the camp of the latter. Yet it does so by means of a rigorous formal, mathematical, and neurological exposition – in contradiction of the experiential claim that formal analysis has no place in the understanding of cognition. To make our own counter-claim as explicit as possible, we present a sketch of the LVQ structure in terms of mereotopology, in which the initial layer of the network performs topological operations, while the final layer performs mereological operations. The book is meant to be self-contained, in the sense that it does not assume any prior knowledge of any of the many areas that are touched upon. It therefore contains mini-summaries of biological visual processing, especially the retinocortical and ventral /what?/ parvocellular pathways computational models of neural signaling, and in particular the reduction of the Hodgkin-Huxley equations to the connectionist and integrate-and-fire neurons Hebbian learning rules and the elaboration of learning vector quantization the linguistic pathway in the left hemisphere memory and the hippocampus truth-conditional vs. image-schematic semantics objectivist vs. experiential metaphysics and mereotopology. All of the simulations are implemented in MATLAB, and the code is available from the book’s website. • The discovery of several algorithmic similarities between visison and semantics. • The support of all of this by means of simulations, and the packaging of all of this in a coherent theoretical framework. (shrink)

The interest in data anonymization is exponentially growing, motivated by the will of the governments to open their data. The main challenge of data anonymization is to find a balance between data utility and the amount of disclosure risk. One of the most known frameworks of data anonymization is k-anonymity, this method assumes that a dataset is anonymous if and only if for each element of the dataset, there exist at least k − 1 elements identical to it. In this (...) paper, we propose two techniques to achieve k-anonymity through microaggregation: k-CMVM and Constrained-CMVM. Both, use topological collaborative clustering to obtain k-anonymous data. The first one determines the k levels automatically and the second defines it by exploration. We also improved the results of these two approaches by using pLVQ2 as a weighted vector quantization method. The four methods proposed were proven to be efficient using two data utility measures, the separability utility and the structural utility. The experimental results have shown a very promising performance. (shrink)

The nonadditive properties of mass make it desirable to abandon mass as a basis unit in physics and to replace it by a unit of the dimension of the quantum of action [h]. The ensuing four-unit system of action, charge, length, and time [h, q, l, t] interacts in a much more elucidating fashion with experiment and with the fundamental structure of physics. All space-time differential forms expressing fundamental laws of physics are forms of physical dimensions, h, h/q, or q. (...) Their occurrence as periods (residues) of period integrals of forms makes h and q topological invariants in space-time and stresses in general the global nature of quantization. The coordinate dimensions [l] and [t] only come into play for the form-coefficients, the metric tensor, and their tensorial relatives. One thus obtains a very natural guidance for implementing the principle of general covariance, when coordinate-based descriptions are necessary. The consequences for the general theory of relativity are discussed. (shrink)

An extension of the Hurwitz transformation to a canonical transformation between phase spaces allows conversion of the five-dimensional Kepler problem into that of a constrained harmonic oscillator problem in eight dimensions. Thus a new regularization of the Kepler problem is established. Then, following Dirac, we quantize the extended phase space, imposing constraint conditions as superselection rules. In that way the interchangeability of the reduction and the quantization procedures is proved.

I consider exactly what is involved in a solution to the probability problem of the Everett interpretation, in the light of recent work on applying considerations from decision theory to that problem. I suggest an overall framework for understanding probability in a physical theory, and conclude that this framework, when applied to the Everett interpretation, yields the result that that interpretation satisfactorily solves the measurement problem. Introduction What is probability? 2.1 Objective probability and the Principal Principle 2.2 Three ways of (...) satisfying the functional definition 2.3 Cautious functionalism 2.4 Is the functional definition complete? The Everett interpretation and subjective uncertainty 3.1 Interpreting quantum mechanics 3.2 The need for subjective uncertainty 3.3 Saunders' argument for subjective uncertainty 3.4 Objections to Saunders' argument 3.5 Subjective uncertainty again: arguments from interpretative charity 3.6 Quantum weights and the functional definition of probability Rejecting subjective uncertainty 4.1 The fission program 4.2 Against the fission program Justifying the axioms of decision theory 5.1 The primitive status of the decision-theoretic axioms 5.2 Holistic scepticism 5.3 The role of an explanation of decision theory Conclusion. (shrink)

A new logic, quantized intuitionistic linear logic, is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletier's involutive quantales. Some cut-free sequent calculi with a new property quantization principle and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansing's extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.

The quantum gravity program seeks a theory that handles quantum matter fields and gravity consistently. But is such a theory really required and must it involve quantizing the gravitational field? We give reasons for a positive answer to the first question, but dispute a widespread contention that it is inconsistent for the gravitational field to be classical while matter is quantum. In particular, we show how a popular argument falls short of a no-go theorem, and discuss possible counterexamples. Important issues (...) in the foundations of physics are shown to bear crucially on all these considerations. (shrink)

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \ that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi’s theorem (...) generates a microstate-dependent time parametrization \ even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics. (shrink)

The quantization error in a fixed-size Self-Organizing Map (SOM) with unsupervised winner-take-all learning has previously been used successfully to detect, in minimal computation time, highly meaningful changes across images in medical time series and in time series of satellite images. Here, the functional properties of the quantization error in SOM are explored further to show that the metric is capable of reliably discriminating between the finest differences in local contrast intensities and contrast signs. While this capability of the (...) QE is akin to functional characteristics of a specific class of retinal ganglion cells (the so-called Y-cells) in the visual systems of the primate and the cat, the sensitivity of the QE surpasses the capacity limits of human visual detection. Here, the quantization error in the SOM is found to reliably signal changes in contrast or colour when contrast information is removed from or added to the image, but not when the amount and relative weight of contrast information is constant and only the local spatial position of contrast elements in the pattern changes. While the RGB Mean reflects coarser changes in colour or contrast well enough, the SOM-QE is shown to outperform the RGB Mean in the detection of single-pixel changes in images with up to five million pixels. This could have important implications in the context of unsupervised image learning and computational building block approaches to large sets of image data (big data). (shrink)

The basic structure of a second quantized relativistic quantum theory is outlined. The vector space is over the ring of complex quaternions instead of the usual field of complex numbers. This is motivated by the simple quaternion structure of the Dirac equation.

Does the need to find a quantum theory of gravity imply that the gravitational field must be quantized? Physicists working in quantum gravity routinely assume an affirmative answer, often without being aware of the metaphysical commitments that tend to underlie this assumption. The ambition of this article is to probe these commitments and to analyze some recently adduced arguments pertinent to the issue of quantization. While there exist good reasons to quantize gravity, as this analysis will show, alternative approaches (...) to gravity challenge the received wisdom. These renegade approaches do not regard gravity as a fundamental force, but rather as effective, i.e. as merely supervening on fundamental physics. I will urge that these alternative accounts at least prove the tenability of an opposition to quantization. (shrink)

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

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)

The ontic structural realist stance is motivated by a desire to do philosophical justice to the success of science, whilst withstanding the metaphysical undermining generated by the various species of ontological underdetermination. We are, however, as yet in want of general principles to provide a scaffold for the explicit construction of structural ontologies. Here we will attempt to bridge this gap by utilizing the formal procedure of quantization as a guide to ontic structure of modern physical theory. The example (...) of non-relativistic particle mechanics will be considered and, for that case, it will be shown that a viable candidate for an ontic structural realism framework can be constituted in terms of the combination of a state-space with Poisson bracket structure, and a set of observables, with Lie algebra structure. 1 Introduction2 Formulation Underdetermination and Structural Ontologies3 Quantization and Structural Ontologies4 The Case of Non-relativistic Particle Mechanics4.1 Formulation underdetermination4.2 The classical ontology4.3 Quantum theory and the generalized structural ontology4.4 Interpretation of results5 Conclusion and Prospects. (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)

There has been a lot of interest in generalizing orthodox quantum mechanics to include POV measures as observables, namely as unsharp obserrables. Such POV measures are related to symmetric operators. We have argued recently that only maximal symmetric operators should describe observables.1 This generalization to maximal symmetric operators has many physical applications. One application is in the area of quantization. We shall discuss a scheme, to he called quantization by parts,which can systematically deal with what may be called (...) quantum circuits. As a specific application we shall present a novel derivation of the famous Josephson equation for the supercurrent through a Josephson junction in a superconducting circuit. An interesting effect emerges from our quantization scheme when applied to a superconducting Y-shape circuit configuration. We also propose an experimental test for this effect which is expected to shed light on some conceptual problems on the quantum nature of the condensate. (shrink)

This paper considers a generalisation of the sorites paradox, in which only topological notions are employed. We argue that by increasing the level of abstraction in this way, we see the sorites paradox in a new, more revealing light—a light that forces attention on cut-off points of vague predicates. The generalised sorites paradox presented here also gives rise to a new, more tractable definition of vagueness.

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)

The quantum gravity program seeks a theory that handles quantum matter fields and gravity consistently. But is such a theory really required and must it involve quantizing the gravitational field? We give reasons for a positive answer to the first question, but dispute a widespread contention that it is inconsistent for the gravitational field to be classical while matter is quantum. In particular, we show how a popular argument (Eppley and Hannah 1997) falls short of a no-go theorem, and discuss (...) possible counterexamples. Important issues in the foundations of physics are shown to bear crucially on all these considerations. (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)

Karim Thébault has argued that for ontic structural realism to be a viable ontology it should accommodate two principles: physico-mathematical structures it deploys must be firstly consistent and secondly substantial. He then contends that in geometric quantization, a transitional machinery from classical to quantum mechanics, the two principles are followed, showing that it is a guide to ontic structure. In this article, I will argue that geometric quantization violates the consistency principle. To compensate for this shortcoming, the deformation (...) quantization procedure will be offered. (shrink)

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)

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

The 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)

The introduction of an “elementary length”a representing the ultimate limit for the smallest measurable distance leads to a generalization of Einstein's energy-momentum relation and of the usual Lorentz transformation. The value ofa is left unspecified, but is found to be equal tohc/2E u, whereE u is the total energy content of our universe. Particles of zero rest mass can only move at the velocityc of light in vacuum, while material bodies can move slower or faster than light, whena≠0, without violating (...) the principle of causality. The laws of relativistic mechanics are actually generalized so that they include Mach's principle, since it is found that the universe as a whole can only be in a state of rest for any particular inertial observer. (shrink)

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

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)

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)

The gauge compensation fields induced by the differential operators of the Stueckelberg-Schrödinger equation are discussed, as well as the relation between these fields and the standard Maxwell fields; An action is constructed and the second quantization of the fields carried out using a constraint procedure. The properties of the second quantized matter fields are discussed.

I argue that it may well be the case that space and time do not consist of points, indeed that they have no smallest parts. I examine two different approaches to such pointless spaces : a topological approach and a measure theoretic approach. I argue in favor of the measure theoretic approach.

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.