We investigate separation properties for neighbourhood spaces in some details within a framework of constructive mathematics, and define corresponding separation properties for quasi-apartness spaces. We also deal with separation properties for spaces with inequality.

Although environmental philosophy and the human exploration of space share common beginnings, scholars from either field have not given adequate attention to the possible connections between them. In this essay, we seek to spur the rapprochement and cross-fertilization of philosophy and space policy by highlighting the philosophic dimensions of space exploration, pulling together issues and authors that have had insufficient contact with one another. We do so by offering an account of three topics: planetary exploration, planetary protection (...) and the search for extraterrestrial life, and terraforming. The resulting synthesis seeks to change our thinking about earthbound environmental ethics as it considers the philosophical dimensions of space exploration, and introduces the possible benefits of a humanities-oriented approach to space policy. (shrink)

Although environmental philosophy and the human exploration of space share common beginnings, scholars from either field have not given adequate attention to the possible connections between them. In this essay, we seek to spur the rapprochement and cross-fertilization of philosophy and space policy by highlighting the philosophic dimensions of space exploration, pulling together issues and authors that have had insufficient contact with one another. We do so by offering an account of three topics: planetary exploration, planetary protection (...) and the search for extraterrestrial life, and terraforming. The resulting synthesis seeks to change our thinking about earthbound environmental ethics as it considers the philosophical dimensions of space exploration, and introduces the possible benefits of a humanities-oriented approach to space policy. (shrink)

The Cartesian concept of nature, which has determined modern thinking until the present time, has become obsolete. It shall be shown that Hegel's objective-idealistic conception of nature discloses, in comparison to that of Descartes, new perspectives for the comprehension of nature and that this, in turn, results in possibilities of actualizing Hegel's philosophy of nature. If the argumentation concerning philosophy of nature is intended to catch up with the concrete Being-of-nature and to meet it in its concretion, then this is (...) impossible for the finite spirit in a strictly a priori sense – this is the thesis supported here which is not at all close to Hegel. As the argumentation rather has to consider the conditions of realization concerning the Being-of-nature, too, it is compelled to take up empirical elements – concerning the organism, for instance, system-theoretical aspects, physical and chemical features of the nervous system, etc. With that, on the one hand, empirical-scientific premises are assumed (e.g. the lawlikeness of nature), which on the other hand become (now close to Hegel) possibly able to be founded in the frame of a Hegelian-idealistic conception. In this sense, a double strategy of empirical-scientific concretization and objective-idealistic foundation is followed up, which represents the methodical basic principle of the developed considerations. In the course of the undertaking, the main aspects of the whole Hegelian design concerning the philosophy of nature are considered – space and time, mass and motion, force and law of nature, the organism, the pro-blem of evolution, psychic being – as well as Hegel's basic thesis concerning the philosophy of nature, that therein a tendency toward coherence and idealization manifests itself in the sense of a (categorically) gradually rising succession of nature: from the separateness of space to the ideality of sensation. In the sense of the double strategy of concretization and foundation it is shown that on the one hand possibilities of philosophical penetration concerning actual empirical-scientific results are opened, and on the other hand – in turn – a re-interpretation of Hegel's theorem on the basis of physical, evolution-theoretical and system-theoretical argumentation also becomes possible. In this mutual crossing-over and elucidation of empirical and Hegelian argumentation not only do perspectives of a new comprehension of nature become visible, but also, at the same time – as an essential consequence of this methodical principle – thoughts on the possibilities of actualizing Hegel's philosophy of nature. (shrink)

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture (...) can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means (...) the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer–Specker group Zω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}^{\omega}}}$$\end{document}. The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and (...) Steprāns :52–59, 2006). (shrink)

Editor’s NoteThanks to the initiative of Alan Norrie, we are pleased to present here a symposium on Nick Wilson’s book The Space that Separates: A Realist Theory of Art. Several authors have contri...

In this paper we will consider two possible definitions of projective subsets of a separable metric space X. A set A subset of or equal to X is Σ11 iff there exists a complete separable metric space Y and Borel set B subset of or equal to X × Y such that A = {x ε X : there existsy ε Y ε B}. Except for the fact that X may not be completely metrizable, this is (...) the classical definition of analytic set and hence has many equivalent definitions, for example, A is Σ11 iff A is relatively analytic in X, i.e., A is the restriction to X of an analytic set in the completion of X. Another definition of projective we denote by ΣX1 or abstract projective subset of X. A set of A subset of or equal to X is ΣX1 iff there exists an n ε ω and a Borel set B subset of or equal to X × Xn such that A = {x ε X:there existsy ε Xn ε B}. These sets ca n be far more pathological. While the family of sets Σ11 is closed under countable intersections and countable unions, there is a consistent example of a separable metric space X where ΣX1 is not closed under countable intersections or countable unions. This takes place in the Cohen real model. Assuming CH, there exists a separable metric space X such that every Σ11 set is Borel in X but there exists a Σ11 set which is not Borel in X2. The space X2 has Borel subsets of arbitrarily large rank while X has bounded Borel rank. This space is a Luzin set and the technique used here is Steel forcing with tagged trees. We give examples of spaces X illustrating the relationship between Σ11 and ΣX1 and give some consistent examples partially answering an abstract projective hierarchy problem of Ulam. (shrink)

We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [2], Mayer-Wolf [5], and Weaver [8]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c to (...) be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

PREFACE: - BY his theory of relativity Albert Einstein has provoked a revolution of thought in physical science. The achievement consists essentially in this Einstein has succeeded in separating far more completely than hitherto the share of the observer and the share of external nature in the things we see happen. The perception of an object by an observer depends on his own situation and circumstances for example, distance will make it appear smaller and dimmer. We make allowance for this (...) almost unconsciously in interpreting what we see. But it now appears that the allowance made for the motion of the observer has hitherto been too crude a fact overlooked because in practice all observers share nearly the same motion, that of the earth. Physical space and time are found to be closely bound up with this motion of the observer and only an amorphous combination of the two is left inherent in the external world. When space and time are relegated to their proper source the observer the world of nature which remains appears strangely unfamiliar but it is in reality simplified, and the underlying unity of the principal phenomena is now clearly revealed. The deductions from this new outlook have, with one doubtful exception, been confirmed when tested by experiment. It is my aim to give an account of this work without intro ducing anything very technical in the way of mathematics, physics, or philosophy. The new view of space and time, so opposed to our habits of thought, must in any case demand unusual mental exercise. The results appear strange and the incongruity is not without a humorous side. For the first nine chapters the task is one of interpreting a clear-cut theory, accepted in all its essentials by a large and growing school of physicists although perhaps not everyone would accept the authors views of its meaning. Chapters x and xi deal with very recent advances, with regard to which opinion is more fluid. As for the last chapter, containing the authors specula tions on the meaning of nature, since it touches on the rudiments of a philosophical system, it is perhaps too sanguine to hope that it can ever be other than controversial. A non-mathematical presentation has necessary limitations and the reader who wishes to learn how certain exact result follow from Einsteins, or even Newtons, law of gravitation m bound to seek the reasons in a mathematical treatise. But thj limitation of range is perhaps less serious than the limitation of intrinsic truth. There is a relativity of truth, as there is a relativity of space. For is and IS-NOT though with Rule and Line And UP-AND-DOWK without, I could define, Alas It is not so simple. We abstract from the phenomena that which is peculiar to the position and motion of the observer but can we abstract that which is peculiar to the limited imagina tion of the human brain We think we can, but only in the symbolism of mathematics. As the language of a poet rings with a truth that eludes the clumsy explanations of his commentators, so the geometry of relativity in its perfect harmony expresses a truth of form and type in nature, which my bowdlerised version misses. But the mind is not content to leave scientific Truth in a dry husk of mathematical symbols, and demands that it shall be alloyed with familiar images. The mathematician, who handles x so lightly, may fairly be asked to state, not indeed the in scrutable meaning of a in nature, but the meaning which x conveys to him. Although primarily designed for readers without technical knowledge of the subject, it is hoped that the book may also appeal to those who have gone into the subject more deeply. A few notes have been added in the Appendix mainly to bridge the gap between this and more mathematical treatises, and to indicate the points of contact between the argument in the text and the parallel analytical investigation. It is impossible adequately to express my debt to con temporary literature and discussion... (shrink)

We define qualia space Q to be the space of all possible conscious experience. For simplicity we restrict ourselves to perceptual experience only, though other kinds of experience could also be considered. Qualia space is a highly idealized concept that unifies the perceptual experience of all possible brains. We argue that Q is a closed pointed cone in an infinite-dimensional separable real topological vector space. This quite technical structure can be explained for the most part (...) in a simple, intuitive way. The structure of qualia space allows us to consider and even answer in a precise way such questions as: Is there a continuous path from the sensation of blue to the sensation of pain? Once we fix a desired accuracy of approximation, do there exist finitely many perceptual experiences such that any possible perceptual experience is approximately equal to one of them? What should be meant by ‘fundamentally different’ perceptual experiences? There is the possibility of additional structure, such as a Hilbert space structure on the vector space in which Q is embedded. (shrink)

Moore attempts to show that privacy, conceived as "control over access to oneself and to information about oneself" is "necessary" for human well-being. Moore grounds his argument in an analysis of the need for physical separation, which Moore suggests is universal among animal species. Moore notes, "One basic finding of animal studies is that virtually all animals seek periods of individual seclusion or small-group intimacy." Citing several studies involving rats and other animals, Moore points out that a lack of such (...) separate space frequently results in threats to survival. Moore goes on to suggest, quite plausibly, that since we evolved from such animals, we share some need for separation. I argue such reasoning involves a conceptual mistake, as a need for physical space and separation is not obviously tantamount to a need for privacy of any kind - much less a need for information privacy. (shrink)

The growing body of research on temporal and spatial experience lacks a comprehensive theoretical approach. Drawing on Giddens’ framework, we present time-space distanciation as a construct for theorizing the relations between culture, time, and space. TSD in a culture may be understood as the extent to which time and space are abstracted as separate dimensions and activities are extended and organized across time and space. After providing a historical account of its development, we outline a multi-level (...) conceptualization of TSD supported by research on cultural differences in the experience of time and space. We impact this conceptualization by examining two ethnographic case studies. We conclude by highlighting future research directions. TSD is an integrative, interdisciplinary, multi level construct with the potential to guide the burgeoning social science of time and space. (shrink)

We often think of normal childhood as a progressive development towards a fixed—and often tacitly individualistic and masculine—model of what it is to be an adult. By contrast, phenomenologists, psychoanalysts, sociology of childhood, and feminist thinkers have set out to offer richer accounts both of childhood development and of mature existence. This paper draws on accounts of childhood development from phenomenologist Maurice Merleau-Ponty and object relations theorist D. W. Winnicott in order to argue that childhood development takes place in “transitional (...) spaces”; explores typical gendered patterns in the formation of selfhood that “split” relationality and separateness into the “feminine” and the “masculine”; and offers a phenomenology of perception, love, and objectivity in order to show the manner in which, contra individualistic and masculine visions of adulthood, maturity requires an embrace rather than eschewal of ambiguity, and the capacity to continue to dwell in the transitional space between relatedness and separateness. (shrink)

Defining privacy is problematic because the condition of privacy appears simultaneously to require separation from others, and the possibility of choosing not to be separate. This latter feature expresses the inherent normative dimension of privacy: the capacity to control the perceptual and informational spaces surrounding one’s person. Clearly the features of separation and control as just described are in tension because one may easily enough choose to give up all barriers between oneself and the public space. How could the (...) capacity for privacy give rise to its absence? Yet both the separation and control features of privacy do seem indispensable to any sensible understanding of it. In this paper I set out an approach to defining privacy that keeps these features and avoids the tension between them. (shrink)

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤rX, differs from the lower ≤s-cone of X. By c osure (...) under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations. (shrink)

2Public space is an old habit. The words public space are deceptive; when I hear the words, when I say the words, I’m forced to have an image of a physical place I can point to and be in. I should be thinking only of a condition; but, instead, I imagine an architectural type, and I think of a piazza, or a town square, or a city commons. Public space, I assume, without thinking about it, is a (...) place where the public gathers. The public gathers in two kinds of spaces. The first is a space that is public, a place where the public gathers because it has a right to the place; the second is a space that is made public, a place where the public gathers precisely because it doesn’t have the right—a place made public by force.3In the space that is public, the public whose space this is has agreed to be a public; these are people “in the form of the city,” they are public when they act “in the name of the city.” They “own” the city only in quotes. The establishment of certain space in the city as “public” is a reminder, a warning, that the rest of the city isn’t public. New York doesn’t belong to us, and neither does Paris, and neither does Des Moines. Setting up a public space means setting aside a public space. Public space is a place in the middle of the city but isolated from the city. Public space is the piazza, an open space separated from the closure of alleys and dead ends; public space is the piazza, a space in the light, away from the plots and conspiracies in dark smokey rooms. Vit Acconci’s latest show, entitled “Public Places,” was held in 1988 at the Museum of Modern Art, New York. He is currently at work on a park in Detroit, a pedestrian mall in Baltimore, and a housing project in Regensburg, Germany. (shrink)

Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group (...) R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

Summary In the visual arts, the constructions of the spatial and chromatic structures of pictures can hardly be separated from each other. The phenomenological approach from the perspective of the arts provides an independent and worthwhile approach to the topic of colour and space. We address some matters of composition in design and, to some extent, in naturalistic painting. The phenomenological approach from the perspective of the arts reveals various topics that invite further investigation by the means of generic (...) vision science. However, although such studies might lead to increased academic insight, a thorough inquiry by the means of experimental phenomenology might well lead to formal descriptions that come somewhat closer to possible applications in the visual arts. (shrink)

The ArgumentThere can be no doubt about the moral and epistemological significance of what Shapin calls the “physical place” of the scientific laboratory. The physical place is defined by the locales, barriers, ports of entry, and lines of sight that bound the laboratory and separate it from other urban and architectural environments. Shapin's discussion of the emergence of the scientific laboratory in seventeenth-century England provides a convincing demonstration that credible knowledge is situated at an intersection between physical locales and social (...) distinctions. In this paper I take up Shapin's theme of the “siting of knowledge production,” but I give it a different treatment – one based on ethnomethodological studies of work. Without denying all that can be witnessed in the spectacle of the scientist at the bench and of the architectural habitat of the bench, I argue that the “place” of scientific work is defined by locally organizedtopical contextures. The paper describes two examples of such spatial orders – “opticism” and “digitality” – associated with distinct complexes of equipment and practice. These topical spaces might initially be viewed as “ideal” or “symbolic” spaces, but I argue that they are no less material than the “physical setting” of the laboratory; indeed, they are the physical setting. (shrink)

This paper is dedicated to Newton da Costa, who, among his many achievements, was the first to aim at dualising intuitionism in order to produce paraconsistent logics, the C-systems. This paper similarly dualises intuitionism to a paraconsistent logic, but the dual is a different logic, namely closed set logic. We study the interaction between the properties of topological spaces, particularly separation properties, and logical theories on those spaces. The paper begins with a brief survey of what is known about the (...) relation between topology and modal logic, intuitionist logic and paraconsistent logic in respect of the incompleteness and inconsistency of theories. Necessary and sufficient conditions which relate the Tⁱ-property to the properties of logical theories, are obtained. The result is then extended to Hausdorff and Normal spaces. In the final section these methods are used to vary the modelling conditions for identity. (shrink)

A Hilbert-space model for quantum logic follows from space-time structure in theories with consistent state collapse descriptions. Lorentz covariance implies a condition on space-like separated propositions that if imposed on generally commuting ones would lead to the covering law, and such a generalization can be argued if state preparation can be conditioned to space-like separated events using EPR-type correlations. The covering law is thus related to space-time structure, though a final understanding of it, through a (...) self-consistency requirement, will probably require quantum space-time. (shrink)

This paper pushes back against the Democritean-Newtonian tradition of assuming a strict conceptual dichotomy between spacetime and matter. Our approach proceeds via the more narrow distinction between modified gravity/spacetime and dark matter. A prequel paper argued that the novel field Φ postulated by Berezhiani and Khoury's 'superfluid dark matter theory' is as much matter as anything could possibly be, but also below the critical temperature for superfluidity as much spacetime as anything could possibly be. Here we introduce and critically evaluate (...) three groups of interpretations that one should consider for such Janus-faced theories. The consubstantiality interpretation holds that Φ is both matter and a modification of spacetime, analogously to the sense in which Jesus is both human and god. The fundamendalist interpretations consider for each of these roles whether they are instantiated fundamentally or emergently. The breakdown interpretations focus on the question of whether Φ signals the breakdown, in some sense to be specified, of the MG-DM dichotomy and perhaps even the broader spacetime–matter distinction. More generally, it is argued that hybrid theories urge a move towards a single space of theories, rather than two separate spaces of spacetime theories and matter theories, respectively. (shrink)

This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal logic, intuitionist logic (...) and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T 1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity. (shrink)

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding (...) in [4], where the authors prove these theorems under stronger assumptions. Our proof is also somewhat simpler. (shrink)

A definition is proposed to give precise meaning to the counterfactual statements that often appear in discussions of the implications of quantum mechanics. Of particular interest are counterfactual statements which involve events occurring at space-like separated points, which do not have an absolute time ordering. Some consequences of this definition are discussed.

A model of reality is called separable if the state of a composite system is equal to the union of the states of its parts, located in different regions of space. Spekkens has argued that it is trivial to reproduce the predictions of quantum mechanics using a separable ontological model, provided one allows for arbitrary violations of ‘dynamical locality’. However, since dynamical locality is strictly weaker than local causality, this leaves open the question of whether an ontological (...) model for quantum mechanics can be both separable and dynamically local. We answer this question in the affirmative, using an ontological model based on previous work by Deutsch and Hayden. Although the original formulation of the model avoids Bell’s theorem by denying that measurements result in single, definite outcomes, we show that the model can alternatively be cast in the framework of ontological models, where Bell’s theorem does apply. We find that the resulting model violates local causality, but satisfies both separability and dynamical locality, making it a candidate for the ‘most local’ ontological model of quantum mechanics. (shrink)

Information space and time in their socio-cultural dimension act as knowledge components of being, and, consequently, the most important regulator of social relations. At the same time, the social and cultural dominants of the information space and time are determined by the creative activity of a person. In principle, a person is always focused on knowledge, where its elements interact in a comprehensive way, and not on information, in which there is a separation of its elements from the (...) spiritually observed cross-section of reality, and thus from reality, which does not lend itself to the construction of connections, and the construction somewhat reduces the elasticity of the structure of information space and time. (shrink)

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding (...) in [4], where the authors prove these theorems under stronger assumptions. Our proof is also somewhat simpler. (shrink)

Personal space is the distance that people tend to maintain from others during daily life in a largely unconscious manner. For humans, personal space-related behaviors represent one form of non-verbal social communication, similar to facial expressions and eye contact. Given that the changes in social behavior and experiences that occurred during the COVID-19 pandemic, including “social distancing” and widespread social isolation, may have altered personal space preferences, we investigated this possibility in two independent samples. First, we compared (...) the size of personal space measured before the onset of the pandemic to its size during the pandemic in separate groups of subjects. Personal space size was significantly larger in those assessed during the onset of the pandemic. Lastly, we found that the practice of social distancing and perceived risk of being infected with COVID-19 were linked to this personal space enlargement during the pandemic. Taken together, these findings suggest that personal space boundaries expanded during the COVID-19 pandemic independent of actual infection risk level. As the day-to-day effects of the pandemic subside, personal space preferences may provide one index of recovery from the psychological effects of this crisis. (shrink)

We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL 0 over RCA 0 . We show that the separation theorem for separably closed convex sets is equivalent to ACA 0 over RCA 0 . Our strategy for proving these geometrical Hahn-Banach theorems is to reduce to the (...) finite-dimensional case by means of a compactness argument. (shrink)

Much has been written about the history of the work of men and women in the premodern past. It is now generally acknowledged that early modern ideological assumptions about a strict division of work and space between men and productive work outside the house on the one hand, and women and reproduction and consumption inside the house, on the other, bore little relation to reality. Household work strategies, out of necessity, were diverse. Yet what this spatial complexity meant in (...) particular households on a day-to-day basis and its consequences for gender relationships is less clear and has received relatively little historical attention. The aim of this paper is to add to our knowledge through a case study of the way that men and women used and organized space for work in the county of Essex during the “long seventeenth century”. Drawing on critiques of the concept of “separate spheres” and the models of economic change to which it relates, together with local/micro historical methods, it places evidence within an appropriate regional context to argue that spatial patterns were enormously varied in early modern England and a number of factors—time, place, occupation, and status, as well as gender—determined them. Understanding of the dynamic, complex, uneven purchase of patriarchy upon the organization, imagination, and experience of space has important implications for approaches to gender relations in early modern England. It raises additional doubts about the utility of the separate spheres analogy, and particularly the use of binary oppositions of male/female and public/private, to describe gender relations and their changes in this period and shows that a deeper understanding demands more research into the local contexts in which the gendered division and meaning of work was negotiated. (shrink)

THE DOMINATING CONCEPT IN GREEK THOUGHT, SAYS TORRANCE, WAS A RECEPTACLE NOTION OF SPACE. THIS HAD NO PLACE IN THE NICENE THEOLOGY. WITH THE ASCENDANCY OF ARISTOTELIAN PHILOSOPHY THE RECEPTACLE NOTION OF SPACE DOMINATED MEDIEVAL THEOLOGY, AND THIS IS WHAT, DESPITE LUTHER’S INSIGHT INTO THE RELATION BETWEEN THE ONTOLOGICAL AND DYNAMIC WAYS OF THINKING OF THE REAL PRESENCE AND THE INCARNATION, PRODUCED THE SEPARATION BETWEEN THEM. THIS PROBLEM INHERITED BY MODERN THEOLOGY CAN ONLY BE SOLVED IF WE USE (...) THE PATRISTIC UNDERSTANDING OF JESUS CHRIST IN SPACE AND TIME AS GOD’S PLACE IN THIS WORLD WHERE HE IS PRESENT IN OUR PLACE. (BP). (shrink)

We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL$_0$ over RCA$_0$. We show that the separation theorem for separably closed convex sets is equivalent to ACA$_0$ over RCA$_0$. Our strategy for proving these geometrical Hahn-Banach theorems is to reduce to the finite-dimensional case by means of a (...) compactness argument. (shrink)

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, (...) and that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

What is special about all our living exchanges with our surroundings is that they occur within the ceaseless, intertwined flow of many unfolding strands of spontaneously responsive, living activity. This requires us to adopt a kind of fluid, process thinking, a shift from thinking of events as occurring between things and beings existing as separate entities prior to their inter-action, to events occurring within a continuously unfolding, holistic but stranded flow of events, with no clear, already existing boundaries to be (...) found anywhere (Mol & Law, 1994)—a flow of events within which we ourselves are also immersed. We thus become involved in activities within which we find things happening to us, as much as we make things happen in our surroundings—in other words, our surroundings are also agentive in that they can exert “calls” upon us to respond within them in appropriate ways. Consequently, what we can learn in such encounters is not just new facts or bits of information, but new ways of relating or orienting ourselves bodily to the others and othernesses in the world around us—although much can “stand in the way” of our doing this. My concern below is to explore events happening on the (inter)-subjective side of the Cartesian subject/object divide which “shape” our spontaneous ways of acting in, and reflecting on, the “worlds” within which we live out our lives. (shrink)

Volume of essays by Rumsey, Rose, Weiner, Bolton, Redmond, Wassman annotated separately; includes introductory review of previous work comparing Australia and New Guinea ethnology; examination of the spatial relations of knowledge; mythology and place; role of secrecy.