We present two discrete dualities for double Stone algebras. Each of these dualities involves a different class of frames and a different definition of a complex algebra. We discuss relationships between these classes of frames and show that one of them is a weakening of the other. We propose a logic based on double Stone algebras.

Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and we prove a discrete duality for them.

In this paper, we continue with the study of tense operators on Nelson algebras (Figallo et al. in Studia Logica 109(2):285–312, 2021, Studia Logica 110(1):241–263, 2022). We define the variety of algebras, which we call tense Nelson D-algebras, as a natural extension of tense De Morgan algebras (Figallo and Pelaitay in Logic J IGPL 22(2):255–267, 2014). In particular, we give a discrete duality for these algebras. To do this, we will extend the representation theorems for Nelson algebras given (...) in Sendlewski (Studia Logica 43(3):257–280, 1984) to the tense case. (shrink)

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety $${\mathcal {A}}$$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to (...) show that topological and discrete dualities for $${\mathcal {A}}$$ exist in partnership. (shrink)

This paper starts from a nonlinear fermion field equation of motion with a strongly coupled self-interaction. Nonperturbative quark solutions of the equation of motion are constructed in terms of a Reggeized infinite component free spinor field. Such a field carries a family of strongly interacting unstable compounds lying on a Regge locus in the analytically continued quark spin. Such a quark field is naturally confined and also possesses the property of asymptotic freedom. Furthermore, the particular field self-regularizes the interactions and (...) naturally breaks the chiral invariance of the equation of motion. We show why and how the existence of such a strongly coupled solution and its particle-family, wave duality forces a change in the field equation of motion such that it conserves C, P, T, although its individual interaction terms are of V-A and thus C, P nonconserving type. (shrink)

We present dualities for implicative and residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orłowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated (...) modal logic, as well as full Stone duality for residuated lattices. Our results have immediate applications to the semantics of related substructural logical calculi. (shrink)

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, (...) a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages. (shrink)

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We give (...) the bi-tense logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)

Recently, two apparently quite different duality-based approaches to automata minimisation have appeared. One is based on ideas that originated from the controllability-observability duality from systems theory, and the other is based on ideas derived from Stone-type dualities specifically linking coalgebras with algebraic structures derived from modal logics. In the present paper, we develop a more abstract view and unify the two approaches. We show that dualities, or more generally dual adjunctions, between categories can be lifted to dual adjunctions (...) between categories of coalgebras and algebras, and from there to automata with initial as well as final states. As in the Stone-duality approach, algebras are essentially logics for reasoning about the automata. By exploiting the ability to pass between these categories, we show that one can minimize the corresponding automata. We give an abstract minimisation algorithm that has several instances, including the celebrated Brzozowski minimisation algorithm. We further develop three examples that have been treated in previous works: deterministic Kripke frames based on a Stone-type duality, weighted automata based on the self-duality of semimodules, and topological automata based on Gelfand duality. As a new example, we develop alternating automata based on the discrete duality between sets and complete atomic Boolean algebras. (shrink)

Based on Alasdair Urquhart’s representation of not necessarily distributive bounded lattices we exhibit several discrete dualities in the spirit of the “duality via truth” concept by Orłowska and Rewitzky. We also exhibit a discrete duality for Urquhart’s relevant algebras and their frames.

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = f for the graph Laplacian Δθ, potential functions q, p built from the (...) probabilities, and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ıψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations. (shrink)

The paper addresses the problem, which quantum mechanics resolves in fact. Its viewpoint suggests that the crucial link of time and its course is omitted in understanding the problem. The common interpretation underlain by the history of quantum mechanics sees discreteness only on the Plank scale, which is transformed into continuity and even smoothness on the macroscopic scale. That approach is fraught with a series of seeming paradoxes. It suggests that the present mathematical formalism of quantum mechanics is only partly (...) relevant to its problem, which is ostensibly known. The paper accepts just the opposite: The mathematical solution is absolute relevant and serves as an axiomatic base, from which the real and yet hidden problem is deduced. Wave-particle duality, Hilbert space, both probabilistic and many-worlds interpretations of quantum mechanics, quantum information, and the Schrödinger equation are included in that base. The Schrödinger equation is understood as a generalization of the law of energy conservation to past, present, and future moments of time. The deduced real problem of quantum mechanics is: “What is the universal law describing the course of time in any physical change therefore including any mechanical motion?”. (shrink)

For many among the scientifically informed public, and even among physicists, Heisenberg's uncertainty principle epitomizes quantum mechanics. Nevertheless, more than 86 years after its inception, there is no consensus over the interpretation, scope, and validity of this principle. The aim of this chapter is to offer one such interpretation, the traces of which may be found already in Heisenberg's letters to Pauli from 1926, and in Dirac's anticipation of Heisenberg's uncertainty relations from 1927, that stems form the hypothesis of finite (...) nature. Instead of a mere mathematical theorem of quantum theory, or a manifestation of "wave-particle duality", the uncertainty relations turn out to be a result of a more fundamental premise, namely, the inherent limitation on spatial resolution that follows from the bound on physical resources. The implication of this view are far reaching: it depicts the Hilbert space formalism as a phenomenological, "effective", formalism that approximates an underlying discrete structure; it supports a novel interpretation of probability in statistical physics that sees probabilities as deterministic, dynamical transition probabilities which arise from objective and inherent measurement errors; and it helps to clarify several puzzles in the foundations of statistical physics, such as the status of the "disturbance" view of measurement in quantum theory, or the tension between ontology and epistemology in the attempts to describe nature with physical theories whose formalisms include subjective probabilities. Finally, this view also renders obsolete the entire class of interpretations of quantum theory that adhere to "the reality of the wave function". (shrink)

An elementary review of the origin of quantum theory, with focus on the nature of the quantum dynamic variables, reveals the essential wave-likeness of quantum dynamics. The introduction of the concept of point-particle entities resulted from over-use of classical perspectives, and an issue of language: conflation of the concepts of point-particle localization, and discreteness of quantum detections. Keeping in mind the distinction between point-localization and discreteness of quantum exchange, it is clear that there is no experimental evidence for point-localization. A (...) simple review of the origin of quantum theory, and review of several experiments designed to explore "wave-particle duality" and "complementarity" support this perspective. Normal 0 false false false EN-AU X-NONE X-NONE MicrosoftInternetExplorer4. (shrink)

Did protolanguage users use discrete words that referred to objects, actions, locations, etc., and then, at some point, combine them; or on the contrary did they have words that globally indexed whole semantic complexes, and then come to divide them? Our answer is: early humans were forming language units consisting of global and discrete dimensions of semiosis in dynamic opposition. These units of thinking-for-speaking, or ‘growth points’ (GPs) were, jointly, analog imagery (visuo-spatio-motoric) and categorically-contrastive (-emic) linguistic encodings. This (...) discrete-global duality was a new mode of embodied cognition that enabled thinking and acting in new ways: the dawn of protolanguage. Where did this mode of cognition come from? We have some suggestions based on the hypothesis that gestures gained the power to orchestrate actions, manual and vocal, with significances other than those of the actions themselves, giving rise to cognition framed in the proposed dual terms. Note, however, our proposal is not one of the ‘gesture-first’ theories of language origins. Such theories predict what did not evolve: a language of pantomime; rather than what did evolve: an integrated system of synchronized gestures and spoken forms. GP theory is an account of the cognition underlying such an integrated system. A scenario for the evolutionary selection of this cognitive mode is ‘Mead’s Loop’, a model in which one’s cognition is enriched by one’s own gestures, insofar as they are objects in social interactions. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Realm of Mimesis in Plato: Orality, Writing, and the Ontology of the Image by Mariangela EspositoDoug Al-MainiESPOSITO, Mariangela. The Realm of Mimesis in Plato: Orality, Writing, and the Ontology of the Image. Boston: Brill, 2023. xiv + 173 pp. Cloth, $143.00This manuscript grew out of the author’s original interest in Platonic aesthetics, itself developing into a more particularized examination of Plato’s account of beauty. Plato’s interest in (...) the arts is famously bound up in a broad criticism of their imitative nature; the relation between the original and the imitation, or the eidos and the eidolon, is the core topic of this book. When this binary nature of mimesis is emphasized, it is difficult not to see the two poles as opposed to each other, and indeed there are several passages in the Platonic corpus that tell us directly to value the eidos over the eidolon. “The critique of the arts, in general terms, is based on the assumption that they produce multiple, false, sensorial images (eidola ) that are far removed from the truth, while the truth is a process through which multiplicities are gathered together into an intellectual unity (eidos).” While this duality shows up in the dialogues in numerous ways, Esposito seeks to show a method that transcends it, a method that depends on the experience of [End Page 347] beauty. Beauty is the entity that allows for a transition from the sensual to the intellectual, a point Plato makes in both the Symposium and the Phaedrus, and thereby supports the view that the binary pair of image and original are not so discrete as we might originally believe.Esposito’s analysis of the relation between image and original is made concrete through its application to writing and speech. The book is divided into an introduction, three chapters, and a conclusion that track the progress of our ascent to the eidos, with the first chapter devoted to Plato’s critique of writing, the second to the critique of speech, and the third to an analysis of how beauty facilitates going beyond their opposition. The main sources for the critique of writing are the Phaedrus and the Seventh Letter; Socrates in the Phaedrus explicitly declares writing to be an imitation of speech and the Seventh Letter supports this position. Esposito follows Derrida’s famous essay, Plato’s Pharmacy, stating, “In Derrida’s account of the Phaedrus, this is the main charge against the written word: first, it imitates the original, and then—its second crime—substitutes itself for the original.” The ability of the copy to substitute itself for the original shows a sensitivity on Derrida’s part to the kind of project that Esposito wants to engage in, and indeed a major theme of Derrida’s work was his use of the concept of the pharmakon as something that transgressed a boundary between seemingly discrete opposites; the term pharmakon could refer to either a poison or a cure. But Esposito claims that focusing on the opposition between speech and writing ensures that it will never be overcome, and so this criticism of writing (as an imitation) must be demoted in importance. Esposito therefore ultimately distances herself from Derrida and also spends time taking on the importance of the so-called Unwritten Doctrines, especially as outlined by the Tübingen School. The Tübingen School straightforwardly accepts the criticism of writing in comparison to speech and so develops a theory of unwritten doctrines that Plato would only transfer orally. Given her project of synthesizing the tension between writing and speech, it is unsurprising that Esposito would be hostile to the project of the Tübingen School, “which this work does not consider sustainable.”In the second chapter Esposito moves on to the critique of speech. Here the core of her analysis is given to the Sophist, which provides us with “the best ontological expression of the criticism of image making.” The wider criticism of the Sophist is centered on the observation that appearances are bound up with nonbeing itself; “the biggest challenge posed by the sophist and the main danger of the mimetic mechanism is revealed: both allow the pseudos... (shrink)

There inevitably comes a time when even the best informed minds are tempted to yield to the lure of binarism: when we are no longer concerned with the details of a system, but rather with our vision of the relationship between man and the world. Without going any further it can already be said that the problem of existence is presented to us in terms of a man/world duality, as though we were unable to exceed our subjective vision of (...) things so as to assess their intrinsic reality.When linguists discovered that they could isolate discrete units called phonemes, they were tempted to oppose the discontinuity of these phonemes – reflecting the underlying discontinuity of the signifiers – to a continuum of prelinguistic experience, a continuum that is not arranged as discrete elements except with reference to any significant units of the language selected for the purpose of communicating this experience to others. At the outset we have nothing but a nebulous mass where only the application of a linguistic grid allows us to discern identifiable units. Even if, in the final analysis, it will prove necessary to doubt the universal validity of this vision of things, we must recognize its usefulness in concretizing a concept of linguistic facts – a concept that has for some time played a beneficent role in research. At least for some of us, this vision should reflect the manner in which the world is gradually organized in our minds each time a perceptible reality is isolated from its context at the instant it receives a name. Allow me to introduce an illustration from personal experience. It was the last day of July in 1914. In our little village in the Savoy the international tension – which would eventually end in general mobilization – was very perceptible; our classes were idle, and teachers and pupils had settled down in a field near the school. I was six. Sitting on the grass, my attention was attracted by a plant with large flat leaves. At that moment someone said the word plantain “plantain.” The plant had not been heretofore exactly unknown to me; it didn't strike me, at that moment, as a discovery; but it acquired a distinct existence only at that instant in the green carpet of the field when a signifier had definitively isolated it. It was most assuredly the gravity of the hour that engraved this incident in my memory. (shrink)

It is argued that a realistic interpretation of quantum mechanics is possible and useful. Current interpretations, from “Copenhagen” to “many worlds” are critically revisited. The difficulties for intuitive models of quantum physics are pointed out and possible solutions proposed. In particular the existence of discrete states, the quantum jumps, the alleged lack of objective properties, measurement theory, the probabilistic character of quantum physics, the wave–particle duality and the Bell inequalities are analyzed. The sketch of a realistic picture of (...) the quantum world is presented. It rests upon the assumption that quantum mechanics is a stochastic theory whose randomness derives from the existence of vacuum fields. They correspond to the vacuum fluctuations of quantum field theory, but taken as real rather than virtual. (shrink)

Alfred North Whitehead’s metaphysics provides a means for overcoming the dualism embedded in J. Baird Callicott’s “postmodern” axiology. Indeed, the lessons Callicott draws from the new physics and ecology imply Whitehead’s position. While Callicott holds that subjectivity and valuing require consciousness, Whitehead argues that subjectivity and valuing characterize all metaphysically basic entities, conscious and non-conscious. Removing the constraint that valuing requires consciousness is a slight shift, but it makes all the difference. By jettisoning this constraint, we can develop a robust (...) account of intrinsic value that overcomes Callicott’s duality, while retaining his insights that valuing requires a valuer and fluent energy is more fundamental than discrete entities. (shrink)

Alfred North Whitehead’s metaphysics provides a means for overcoming the dualism embedded in J. Baird Callicott’s “postmodern” axiology. Indeed, the lessons Callicott draws from the new physics and ecology imply Whitehead’s position. While Callicott holds that subjectivity and valuing require consciousness, Whitehead argues that subjectivity and valuing characterize all metaphysically basic entities, conscious and non-conscious. Removing the constraint that valuing requires consciousness is a slight shift, but it makes all the difference. By jettisoning this constraint, we can develop a robust (...) account of intrinsic value that overcomes Callicott’s duality, while retaining his insights that valuing requires a valuer and fluent energy is more fundamental than discrete entities. (shrink)

Using the mainstreaming of socially responsible investing as our empirical context, we show that as the divestment movement in the late twentieth century got institutionalized by being incorporated as a business strategy into more mainstream financial instruments like mutual funds, the prior meanings and categorical definition of ethical investing became ambiguous due to fuzzy boundaries, duality of virtue inherent in the portfolio targets, and exercise of discretion by portfolio managers. We find that increased heterogeneity in standards led to greater (...) ambiguity about who belongs to a category, and fund managers adopted distinct measures-based, values-based and expertise-based approaches to resolve this ambiguity. One consequence of such ambiguity is that individual self-expression, self-consciousness and agency that are central in such movements become appropriated by discretionary and meaning-making work at the institutional level. (shrink)

A fundamental problem in spoken language is the duality between the continuous aspects of phonetic performance and the discrete aspects of phonological competence. We study 2 instances of this problem from the phenomenon of voicing neutralization and vowel harmony. In each case, we present a model where the experimentally observed continuous distinctions are linked to the discreteness of phonological form using the mathematics of nonlinear dynamics.

Cyclic mechanic is intended as a suitable generalization both of quantum mechanics and general relativity apt to unify them. It is founded on a few principles, which can be enumerated approximately as follows: 1. Actual infinity or the universe can be considered as a physical and experimentally verifiable entity. It allows of mechanical motion to exist. 2. A new law of conservation has to be involved to generalize and comprise the separate laws of conservation of classical and relativistic mechanics, and (...) especially that of conservation of energy: This is the conservation of action or information. 3. Time is not a uniformly flowing time in general. It can have some speed, acceleration, more than one dimension, to be discrete. 4. The following principle of cyclicity: The universe returns in any point of it. The return can be only kinematic, i.e. per a unit of energy (or mass), and thermodynamic, i.e. considering the universe as a thermodynamic whole. 5. The kinematic return, which is per a unit of energy (or mass), is the counterpart of conservation of energy, which can be interpreted as the particular case of conservation of action “per a unit of time”. The kinematic return per a unit of energy (or mass) can be interpreted in turn as another particular law of conservation in the framework of conservation of action (or information), namely conservation of wave period (or time). These two counterpart laws of conservation correspond exactly to the particle “half” and to the wave “half” of wave-particle duality. 6. The principle of quantum invariance is introduced. It means that all physical laws have to be invariant to discrete and continuous (smooth) morphisms (motions) or mathematically, to the axiom of choice. The list is not intended to be exhausted or disjunctive, but only to give an introductory idea. (shrink)

It has been argued within some philosophy of quantum gravity circles that endorsing Lewisian modal metaphysics is incompatible with endorsing the fundamental physical ontology of any quantum gravity theory. Speaking concisely, the unsolvable tension would be between Lewis' metaphysical commitment to the fundamentality of space and time, and the physical lesson of quantum gravity about the disappearance of space and time from the fundamental structure of the world. In this essay I argue against the idea that the tension is unsolvable. (...) The analysis does not apply to quantum gravity in general, but only to the specific perspective delivered by quantum string theory. In the first two sections I describe what I think is the general formal payoff of Lewisian possible worlds semantics, and more importantly the general metaphysical payoff generated by some Lewisian metaphysical applications of that formal framework. Lewisian possible worlds semantics is not among the topics of this essay, still a concise presentation of its basic features is required for a more accurate description of the type of modal metaphysics Lewis delivers by applying that formal machinery. The moral of the story in the first two sections is that an attempt of preserving crucial features of Lewisian modal metaphysics, although endorsing the quantum string theory lesson on spacetime non fundamentality, would be not only possible, but also philosophically worthy in a general sense. Then, section three is divided in three parts. In the first part I summarize some formal tools of the methodology used elsewhere - (Vistarini, Routledge, 2019, chapter 6)- to argue for string theory background independence. More precisely, I briefly re-describe some crucial features of a topological-vector bundle structure there defined on a "space" associated to the theory. Such structure is there shown to carry physical and dynamical information about quantum strings systems supporting the thesis of background independence of quantum strings laws (string theory background independence is not a topic of this essay, and the construction of the full formal structure as well as the full argumentative line can be found in the work cited above). The rationale behind the choice of including these formal tools is explained in the second and third parts of section three: that very same topological-fiber bundle structure associated to string theory, if read differently, is shown to also carry structural properties of the Lewisian metaphysical scheme. More precisely, in the second part of section three I attempt to show that such topological-fiber bundle structure owns a topology which qualifies as a plausible accessibility structure extending the Lewisian similarity accessibility. To avoid confusion, I will denote the extended similarity order with S*, or S*-similarity. The latter has some interesting properties. It is quantitatively describable, and even more interestingly its spectrum of variation is a countable one. Then, insofar Lewisian similarity order can be regained from the S*-similarity, one might say that the latter is a metaphysical accessibility relation underlying the former because "more fundamental". As we will see, the explanation of what "more fundamental" means in this metaphysical context mainly involves the idea that the formal nature of S*-similarity, in particular its property of having a countable spectrum of variation, calibrates a metaphysical similarity that reflects the discreteness of the fundamental quantum nature of reality. The extended modal metaphysics with accessibility structure S*-similarity is informed by quantum string physics. Within the extended scheme, the Lewisian claim "things could have been different in countless ways" (Lewis 1973) is an approximate one because only permitted by ordinary language. As soon as one refines the accessibility structure by using the formal language of quantum string theory, it appears things could have been different in countable ways. But the Lewisian variety of alternative states of affairs can be regained from the underlying countable variety via a specific interpretation of the Humean supervenience thesis, one that changes the metaphysical status of the thesis' content and restricts the thesis' domain of application. Finally, in the third part of section three I attempt to show that the same topological-fiber bundle structure used to extend Lewisian similarity has a bundle component producing the formal scheme for a tentative extension of Lewisian nomological accessibility through the lens of string dualities. This will prove to be a non trivial task, and the Lewisian accessibility structure will be regained from the extended one only partially. Finally, the essay contains two short appendices summarizing two argumentative lines outside the main topic of the essay. They briefly describe different ways in which the non fundamentality of spatiotemporal structures can be found in quantum string theory. These sections can be skipped. Nevertheless, they might contribute to understand more robustly the general view endorsed by this essay, i.e. that there isn't any unsolvable tension between Lewis' commitment to the fundamentality of spatiotemporal relations and quantum gravity's lesson about their non fundamentality. -/- . (shrink)

This paper uses the locale theory approach to topology. Two descriptions are given of all locale limits, the first description using suplattice constructions and the second preframe constructions. The symmetries between these two approaches to locale theory are explored. Given an informal assumption that open locale maps are parallel to proper maps we argue that various pairs of locale theory results are ‘parallel’, that is, identical in structure but prove facts about proper maps on one side of the pair and (...) about open maps on the other. The pairs of results are: pullback stability of proper/open maps, regularity of the category of compact Hausdorff/discrete locales, and theorems on information systems. Some remarks are included on a possible formalization of this parallel as a duality. (shrink)

continent. 1.1 (2011): 60-67. At the beginning of Martin Heidegger’s lecture “Time and Being,” presented to the University of Freiburg in 1962, he cautions against, it would seem, the requirement that philosophy make sense, or be necessarily responsible (Stambaugh, 1972). At that time Heidegger's project focused on thinking as thinking and in order to elucidate his ideas he drew comparisons between his project and two paintings by Paul Klee as well with a poem by Georg Trakl. In front of Klee's (...) Saints from the Window and Death of Fire —though we wouldn’t absolutely understand what we were seeing—he writes, “we should want to stand…a long while.” In a similar manner, of Trakl’s poem “Septet of Death”—although it is likely we are unsure in what we hear—Heidegger states that, “we should want to hear…[it] often.” Heidegger further states that in appreciating these, “we “should abandon any claim that [they] be immediately intelligible” (1). So also we must we approach, Heidegger continues, the realm of theoretical physics, in which the difficult work of Werner Heisenberg, be listened to “without protest” and without “any claim that he be immediately understood.” These works, like his own project, merit the time they take to be originally (mis)understood. But this is not necessarily true for philosophy, Heidegger advises, because, “That thinking is supposed to offer ‘worldly wisdom’ and perhaps even a ‘way to a blessed life’” (1). Philosophy is in the unique position of being both abstract (what do we talk about when we talk?) and at the same moment, useful. There are demands that it make sense, that it be, grounded, immediate, and most importantly, rational. Heidegger draws these comparisons between the works of Klee and Trakl and Heisenberg not to claim that philosophy is totally irresponsible. He does not claim that the poet, the painter, nor the physicist have acted irresponsibly. Rather, he would say, they are rising to the highest levels of intelligibility, though it is perhaps an intelligibility that is commonly unavailable to us. In Trakl’s case, as we shall see, mastery is the end result of difficult words. Instead, Heidegger seems be making pleas: for a period of uninterrupted unintelligibility (pure unintelligibility); and that there is a time (that time is now) when, “thinking is…placed in a position which demands of it reflections that are far removed from any useful, practical wisdom” (1-2). Heidegger argues that philosophy requires a period of time in which, instead of focusing on the practical—or even on the worldly—the discipline draw its “determination” instead from the place of painting and poetry and physics. In doing so, “we should have to abandon any claims to immediate intelligibility” (2). But Heidegger doesn’t offer this as a way out. We still have to turn up, we “still have to listen, because we must think what is inevitable, but preliminary” (2). The point for Heidegger, is not to listen to a series of propositions, but instead to “follow the movement of showing” (2). I. … Maybe we're here only to say: house, bridge, well, gate, jug, olive tree, window— at most: pillar, tower … but to say them, remember, oh, to say them in a way that the things themselves never dreamed of existing so intensely.… Rainer Maria Rilke, “The Ninth Elegy” There is little more fundamental, preliminary, in the world than language. We use language, in the form of speech, constantly, whether we are saying anything or not. “Man speaks,” Heidegger writes, and goes on to describe in his essay “Language” (2001) the constant speaking that we do. “We are always speaking, even when we do not utter a single word aloud, but merely listen or read, and even when we are not particularly speaking or listening but are attending to some work or taking a rest” (187). Speaking is expression, an utterance of something internal. It is a recognition of a world. It is a way of communicating thought and it is an activity that we all do, inevitably. Speaking separates the human from the animal world, and despite advances in primatology that seek to give ‘voice’ to primates and other non-human animals, it can safely be said that our form of communication—what we call speech and know as language—is the most advanced, the most complicated. We use it to present and represent the world around us; through actual utterances (vocalization) or through the written word or through ‘unvoiced thoughts’ and dreams. We use language, but more specifically speech—naming—to transmit moods, thoughts, desires, aversions, and feelings. These expressions, mere utterances, nearly always find a source in words, whether spoken or not. Heidegger writes that this common view of language means “that only speech enables man to be the living being he is as man. It is as the one who speaks that man is—man” (187). Speaking then—utterance—surrounds us constantly, whether in the form of careful thought—in the form of an academic paper, say—or in the half mutterings and forgotten thoughts of a nearly remembered dream. Like scaffolding, the apparatus of speech sustains and explains the world, making, in a sense, the world rational, making it apparent. When we speak, we describe, and in doing so, name the world. We use words, through this process of naming, to interpret and sustain the world we see, and the world we imagine. Like Rilke’s naming of jug, and bridge, and window, and stream, we describe—and inscribe upon—the world through an activity of naming. Are we here, perhaps, to name? Is it possible not to name? Is it possible to regard and to view and to look around without naming what we see? Is it possible to feel—to experience—world without giving utterance to that feeling, that process? Sadness, grief, joy, ecstasy, hunger, thirst. Desk, light, room, pen, book, world. White, black, tomorrow, today. Each of these words is a name given to a thing I see in front of me, or a concept that I imagine (in the case of tomorrow or world). What arises in my mind has already existed. If I imagine it, it is named. The word for ‘tomorrow’ and ‘desk’ and ‘light’ precedes me, and precedes my concept of it. The idea of it is already pre-informed, and I must, in a certain sense, bend my ideas to it. When I imagine a table, I imagine my own table (or I imagine an idea of table) but that imagining must follow certain general rules; while, perhaps, it might not always require four legs and a horizontal surface, it must, at least not be, say, a pool of water, or a pile of excrement. It must have some tableness to it to be a table. It must, with Heidegger, table. Otherwise, speech is reduced to gibberish. For Heidegger however, the discussion of language points to something deeper than its “scientific and analytic” study as a communicative device. Indeed, Heidegger seeks to understand language not in reference to man or woman, not as an utterance of humanity, but in reference to itself. In doing so he abandons the conversation—that is, he casts away the traditional arguments surrounding philosophy of language; that it is a means of expression; that it is a human activity; and that it is somehow a representation of something that is—in order to seek to understand language as language, on its own terms. Heidegger is not a philosopher of language, but a philosopher of world, of being. Despite these “correct ideas” holding sway “over the whole field of the varied scientific perspectives on language…they ignore completely the oldest natural cast of language” (191). What is this “oldest natural cast of language? It is the act of language itself. Language is language and language speaks. “Most often,” Heidegger writes, “and too often, we encounter what is spoken only as the residue of a speaking long past” (195). Speech, as we normally encounter it, is like Echo herself calling out to Narcissus, doomed to repeat what has already been said, a mere remnant of what was once language, a trace left behind in the gathering silence of becoming. In both the essay “Language” and in his three lecture course “The Nature of Language” (1982), Heidegger attempts to unpack the seeming tautology of language as language. In each, he focuses on poetry as a way out, or into, a true discussion of the oldest natural cast of language. Poetry is pure speech. In poetry, language is brought to language and language speaks ( Die Sprach spricht ). In the act of poetry, the act of pure speech, the poet names (on the surface not different from how I name this table, this computer) but in the poet naming, the naming does not “hand out titles,” “apply terms,” but rather the naming is a call, a calling forth of entities that bring them into their own, allow them to take their place purely, without compromise. This calling, Heidegger (2001) states, “here calls into a nearness. But even so the call does not wrest what it calls away from the remoteness, in which it is kept by the calling there. The calling calls into itself and therefor always here and there—here into presence, there into absence” (196). II. Window with falling snow is arrayed, Long tolls the vesper bell, The house is provided well, The table is for many laid. Wandering ones, more than a few, Come to the door on darksome courses. Golden blooms the tree of graces Drawing up the earth’s cool dew. Wanderer quietly steps within; Pain has turned the threshold to stone. There lie, in limpid brightness shown, Upon the table bread and wine. Georg Trakl “A Winter Evening” Heidegger is often given short shrift as an abysmally difficult writer, as one that makes no sense and is needlessly difficult. In doing this though, we forget sometimes his eloquence, his simple beauty in writing. Of the above opening stanza, “Window with falling snow is arrayed/ Long tolls the vesper bell.” he writes (2001), This speaking names the snow that soundlessly strikes the window late in the waning day, while the vesper bell rings. In such a snowfall, everything lasts longer. Therefore the vesper bell, which daily rings for a strictly fixed time, tolls long. The speaking names the winter evening time (197). The speaking names the winter evening time. It is almost impossible to comment on that one line by Heidegger. It is as though it must exist on its own completely, without elucidation—as though in front of it we must stand as we do in front of a painting by Klee, that is, we must abandon any claim . Silent and devoted. The speaking of the poem here is not clearly different from common speech ( rede ) yet there is, via Heidegger, an invitation to experience that auracular quality of light and stillness that a gentle, dusk tinged snowfall gives; words evoking a quality so clear, so poignant that, in a sense, Heidegger’s work, at this moment, has been done. But what does this naming accomplish? What does the call call? Remember, it is not the poet that calls, but the naming which calls. The poet has only brought the words forward; it is now the words— snow, vesper bell, window— that take new life in the pure language of the poem. Entities are called forth into presence, like the speaking that names the winter evening time. Not to be present amongst us now, however; naming ‘table’ in the poem does not place it in front of us in this room. Rather, the naming places entities into a gathering which is also a sheltering. The naming brings them to be. In an act of appropriation, things come to be purely as their own, unimpeded by a predetermined expectation. They are called to themselves into an arrival. In Being and Time (1962), Heidegger famously describes the movement between present-at-hand and ready-to-hand in his analysis of the hammer, and of equipment in general. This analysis is already known, if not always clear, to most readers of Heidegger. Briefly, a hammer is ordinarily zuhanden or ready-to-hand; it is part of the background of the world, equipment used and never thought about, like this desk, this sheet of paper, this room. Our interaction with it is temporary and it is historically different for each entity and each relation. We need a hammer, we use a hammer. If all goes according to plan, the hammer remains ready-to-hand; it remains, in a sense, undisclosed and certainly unobtrusive. Our world is undisturbed by the hammer. Only when the hammer or the car or the computer is broken (or sometimes unused or unrecognized or missing) does it intrude into our world, become conspicuous as an object present-at-hand, or vorhanden . In this case, we reach for the hammer, it is broken and we suddenly notice it’s being, broken though it is. The thing which was always ready to hand—handy—is suddenly abstract, something to be examined, if only in its absence or disfunction. This can be exhibited for all entities, and it is important to note that in this analysis the zuhanden / vorhanden split is not restricted to a specific form of constructed material; zuhanden does not refer to the car, and vorhanden the sunset. No entity is ever exclusively ready-to-hand or present-at-hand; they are instead, interpenetrated with each others mode of being, one informing the other in a way that both brings things forth into the world—discloses is the word Heidegger chooses—and at the same time conceals them. As one mode is coming to be so another mode is withdrawing. This flux and movement between modes is precisely what brings the world forward, and what makes it manifest. It exists beyond where we tread and before man took dubious control of the world. We go to find a book or turn on the computer and it is missing or broken and we become aware of the object, as though for the first time. We walk to the water to glimpse the sunset and miss it, or it is less than stellar; in the sunset’s grayness, we become aware of it’s being sunset. The thing is disclosed in its withdrawal. Echoing Heraclitus, we can say that as something is coming to be, it is already becoming something other. Of similar importance to the fact that things are never all of one, or all of the other, they are also never alone. The hammer is never a single object, but is in relation always to the whole. Heidegger (1962) states, Equipment—in accordance with its equipmentality—always is in terms of its belonging to other equipment: ink-stand, pen, ink, paper, blotting pad, table, lamp, furniture, windows, doors, room. These ‘Things’ never show themselves proximally as they are for themselves…What we encounter as closest to us…is the room; and we encounter it not as something ‘between four walls’ in a geometrical, spatial sense, but as equipment for residing…it is in this that any ‘individual’ item of equipment shows itself. Before its does so, a totality of equipment has already been discovered (97). Equipment resides—dwells—in its relations, in its proximal being to other beings. Within the structure of totality, a series of relations is always occurring. The hammer is on the workbench in the carpenter’s hand in the workshop in the village under the sky and under the sun. It doesn’t stop there. Equipment surrounds us and the focus is not on what it does, or what one does with it—the carpenter with the hammer, the writer with laptop—but with the fact that it is. Equipment occurs in relation and is always occurring around us overhead, underfoot, by the stream and in the city. There is a constant exchange of relations happening, and as this occurs, so the world occurs, so the world both discloses and withdraws, into and out of itself. In a very real sense, language is also equipment. It is the thing that we use most often without thought, it is ready-to-hand (except when it’s not.) As we re- cognize the world around us, as we offer names for things and make lists, we are using language much like we might use a hammer, that is, bluntly. Most of the time it is invisible and we draw on it without wondering how we are going to say something. When language does intrude, it does so in an awkward moment, that moment when we can’t remember the right word, when we forget a phrase. At that second, it “speaks itself as language,” it reveals itself as malfunctioning equipment, and we “undergo” an experience with language, in Heidegger’s famous formulation. Language is not a tool anymore that we use to bludgeon an object, but something instead that we submit to, that we experience. In “The Nature of Language,” Heidegger (1982) writes, But when does language speak to us as language? Curiously enough, when we cannot find the right word for something that concerns us, carries us away, oppresses or encourages us. Then we leave unspoken what we have in mind and, without rightly giving it a thought, undergo moments in which language itself has distantly and fleetingly touched us with its essential being (59). The poet’s words become themselves, become appropriate, only when they no longer function in the prosaic world; they instead intrude as they come to be. Language itself brings itself to language. Names, like the entities they indicate, are always becoming something else. As noted above, language, through the poet, has brought forth entities from words previously “known.” I thought that I knew snow, but through Trakl’s re- presentation of the word, language calls forth a new image of snow, indeed calls forth snow itself. The vesper bell tolls longer, the table is for many laid. In bringing forth things , language has brought the world to presence. In everyday naming, word occludes world, preventing, in its everydayness, its coming forth, its disclosing. But what is this world that word has been brought forth into? In the same way that language speaks, world worlds. World, left alone, un-interfered with, comes into itself. It worlds. Again, this sounds like a tautology, but it is essential to Heidegger’s thought (and in my mind is more of a “god killer” than Nietzsche.) One of the most overlooked (and under-appreciated) aspects of Heidegger is his later examination and enthusiasm for the “fourfold,” or the system through which things come to be, through which things thing in a world worlding. The fourfold is the interaction of earth and sky, mortals and gods. Things come to be in the interstices and gathering of the fourfold. The fourfold provides both a place of being, and a sheltering, a place to dwell. Heidegger writes that “the things that were named, thus called, gather to themselves sky and earth, mortals and divinities. The four are united primally in being toward one another, a fourfold.” The poet has called, through the act of pure naming, things to come forth. In the purity of the fourfold—that is, when that is all that there is, when there are no other distractions, definitions, things—entities themselves can come to be. It is important to note that Heidegger is not saying that there are four formal things in the world, autonomous entities unto themselves. He is not evoking a pre-Socratic formula as to what makes up the world; instead, the four mirror each other constantly. (Heidegger calls this the ‘mirror-play.’) They interpenetrate in the same way that the modes of being of things interpenetrate themselves. There is no discrete exclusivity in being or the fourfold. What makes up things is not a precise recipe of the four main components; what makes up things is the action of the fourfold coming together, the movement of the fourfold which is a becoming. Heidegger (2001) writes that “this gathering, assembling, letting stay is the thinging of things.” And he later adds that “thinging, things are things. Thinging, they gesture—gestate—world” (197). Language brings the world to be. It works not again as a recipe added to things, but instead it is a bridge, or more precisely, a relation. Language relates world to thing, brings world to thing. In a sense, it does not say anything; rather it allows, or calls in its movement. Heidegger (2001) writes that, The intimacy of world and thing is not a fusion. Intimacy obtains only where the intimate—world and thing—divides itself cleanly and remains separated. In the midst of the two, in the between of world and thing, in their inter, division prevails: a dif-ference (199). It is in this inter that language prevails. Language is difference, it is the differential aspect between world and thing that brings world to thing. In the final stanza of Trakl’s poem “The Winter Evening,” Trakl evokes this difference in the second line when he writes, “Pain has turned the threshold to stone.” Christopher Fynsk, in his essay “Noise at the Threshold,” draws attention to this point when he writes “it is the figure of the threshold that is language itself, inasmuch as language is defined as the articulation of difference by which difference comes about” (25). Language, as used in the pure language of the poem, draws together world and thing, bridging relation between entity and world. The calling of language calls world to thing, world to being. Heidegger (2001) describes this difference as unique; “of itself, it holds apart the middle in and through which world and things are at one with each other” (200). III. Neither reading nor writing, nor speaking—and yet it is by those paths that we escape what has been said already, and knowledge, and reciprocity, and enter the unknown space, the space of distress where what is given is perhaps not received by anyone (99). Maurice Blanchot The Writing of Disaster So far, we have allowed Heidegger to put forward what language does, how it functions as a relation and how it operates as a threshold, as a bridge. What interests me is what happens beyond language, beyond the relation. What happens to the thing without the naming, without the poet, or even without the everyday chatter—Fynsk calls this ‘noise’—intruding on being? If language allows things to become by bringing thing to world, what happens when we remove this bringing, this threshold turned to pain? In “The Nature of Language,” Heidegger examines the work of the poet Stefan George, specifically “The Word”: Wonder or dream from distant land I carried to my county’s strand And waited till the twilight norn Had found the name within her bourn— Then I could grasp it close and strong It blooms and shines now the front along… Once I returned from happy sail, I had a prize so rich and frail, She sought for long and tidings told: “No like of these depths unfold.” And straight it vanished from my hand, The treasure never graced my land… So I renounced and sadly see: Where word breaks off no thing may be. That final line, “Where word breaks off no thing may be” is evoked on nearly every page of Heidegger’s essay. It is the line to which he returns over and over and bears repeating. Where word breaks off no thing may be. Where naming ends, no thing. We can interpret this in two ways (at least.) Where word—naming—breaks off, then there is nothing . Or, as I choose to read it, where word breaks off no thing may be. In this I see a hint forward, a marker left behind by Heidegger. What could this look like? What does no thing look like? Like a zen koan ( there is no mirror ) it is as thrilling and horrifying as contemplating what preceded the Big Bang. Because language as naming wasn’t always here; we weren’t always here. One or maybe two aspects of the fourfold (depending on your view of gods) were not always here, and there is no guarantee that we will always be around. What then? Heidegger (1982) describes the landscape that the poet finds, “It names the realm into which the renunciation must enter: it names the call to enter into that relation between thing and word which has now been experienced” (65). The poet, in renouncing, allows for the “may be” of “where word breaks off no thing may be.” This “may be” becomes “a kind of imperative, a command which the poet follows, to keep it from then on.” No thing is lacking. Where word breaks off, there may be, a totality, a completion. If we place the emphasis on may be , we make it affirmative, make it positive. One allows it. No thing is where the word, that is the name, is lacking. If we remove then (if we can remove) the word, the name, than that is where no thing is, that is where no thing may blossom, enrich, belong, become. What is no thing? Heidegger writes that “thing is anything that in any way is.” And just after this he writes that the “world alone gives being to the thing.” But what happens if there is no word? Word, in this formulation can be seen as an enframing, a challenging of language. By naming, by drawing a perimeter around an object, we hold it, by its definition (that is a brick) to an ordered future. If we borrow from Heidegger’s essay “The Question Concerning Technology” (1993) the idea of this standing-reserve of an ordered future, we see that “everywhere everything is ordered to stand by, to be immediately on hand, indeed to stand there just so that it may be on call for a further ordering” (322). What happens if there is no naming of the thing? Silence perhaps. Stillness. We can name—and do name—that which we know. We equate knowledge with knowing the name of something. A brick is a brick, a hammer is a hammer, the universe is the universe. By naming a thing, we create, and draw its parameters, the parameters of the thing, not as thing thinging in the fourfold, but as blunt object apparent. In the four dimensions relatively available to us, we observe (and name) that the brick takes up possibly six by four by two inches and is, in the sense that it currently occupies this time slot. It fulfills its destiny, its being, its brickness, it bricks . But what happens when we remove the name for this brick. We no longer know what to call this no-thing (if indeed we can even arrive at the point of uncalled calling.) In fact the it (this brick) is no longer a thing in the sense that by not naming—by removing the name—it still occupies the same dimension but is indiscernible from the world. It simply is , un-reliant, un- needed by me. By removing the subject (me) from it (the brick) do I not then also remove the object—or at least the duality objectifying it? Why is this important? Why does this matter? I have not really removed anything. I have not changed any thing , per se . The brick still occupies the same space in geographic and temporal dimensions. I have literally not even touched the brick sitting on my desk. But what I have done is removed the name, removed the word (the bridge, according to Heidegger) and in this (again, if this is even possible) there is something vertiginously liberating, not only for me (and my way of thinking) but also for the brick itself. Like the poet who calls the thing forward, by refusing to name, by avoiding any thing that demands me to name it, I release the thing into the fourfold. I am no longer challenging the thing to be there for me ; I do not en frame it through language. Rather I, in an act of extreme responsibility, am refusing the challenge. By refusing the name (refusing to name) one allows, (or no one allows no thing) the brick to be all things , to manifest its manifold being, to incorporate all things into its thinging . It becomes, quite literally, every thing . Because, in its infinite manifestness, it incorporates everything; the mud that gave it its current being, the water that formed the mud, the sun, the stars, the universe and it also allows it to become mud again, to become landfill, to become again, water and sun and stars and universe in an endless, infinite cycle of coming to be some thing (else). Perhaps this is what Heidegger is suggesting when he talks about the stillness at the end of his essay, “Language” (2001): The dif-ference stills particularly in two ways: it stills the things in thinging and the world in worlding. Thus stilled, thing and world never escape from the dif-ference. Rather, they rescue it in the stilling, where the dif-ference is itself the stillness (206). It is in this stillness that I can imagine a gellasenheit (here I mean both “releasement in the Heideggarean sense, as well as Meister Eckhart’s use of the term meaning “letting the world go and giving oneself to God”) of thing and world, a releasing into the stillness and silence of no thing beyond where word breaks off. It is here where I may no longer be, and yet no thing is, but may be. (shrink)

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this (...) perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises. (shrink)

[A dilettante admires a constellation from a hot tub, and considers it the next day. Is it a prose poem, creative nonfiction, or "light" verse; phenomenology, astrophysics, metaphysics, or half-lit paronomasia?]A photon hit my retina. Again, again; others also hit. The sensation seemed continuous, albeit twinkling, rather than as discrete and separated points. It was like dust, but I didn't blink.It came from a thing I would call bright, in front of me, over my head, on a dark night; (...) it is said to be a star from the cluster of a pattern I learned to call Cassiopeia, and just in a similar way, I learned about my retina, and I learned about photons, and I learned that some consider them a duality of either wavering... (shrink)

A geometric model previously developed to analyze the quantum structure of Hardy’s paradox and explore the generic structure of universal states is applied to the Moebius band. The Moebius band’s conformity to the model strengthens its hypothesis. The claim made is that universal states cannot be represented as singular structures without reference to a partition of entangled parts that do not share property with each other. In other words, universal states categorically incorporate a systemic feature of inconsistency within their frameworks. (...) In logic and mathematics, this is the prohibition to forming theoretical principles that are singularly fundamental as absolute truths. The Moebius band contains a dualism of two categorically separate formats. It has, firstly, two discrete and observable sides and secondly, a unitary path, in which the sides are entangled and indistinguishable. Quantum structures display a similar attribute of duality. However, unlike quantum structure, the Moebius band displays the full duality of its structure at the classical level. The geometric model demonstrates the common structural basis shared in the Moebius band and quantum structure. (shrink)

Nonlocality is a puzzling issue in modern physics. I propose that, aside from the experimental determination of nonlocality, the concept of atomistic lightmdash;discrete, self-bounded photonsmdash;breaks down toward something like nonlocality when subjected to philosophical scrutiny. Louis de Broglie made a similar argument regarding the material atom: the concept of the classical atom, when interrogated, collapses upon itself to offer a glimpse of wave-particle duality. Light atoms or photons, I argue, similarly collapse toward the contradictory possibility of nonlocality.

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 25.1 (2001) 155-165 [Access article in PDF] Critical Discussions Big Guys, Babies, and Beauty Nancy Easterlin Art and Intimacy: How the Arts Began, by Ellen Dissanayake; xvii & 265 pp. Seattle: University of Washington Press, 2000, $29.95. The intellectual climate of postmodernism has not been particularly encouraging for the development of an evolutionary theory of the arts. Concentrated in constructionist modes of analysis and interpretation for (...) the past thirty years, the social sciences and humanities have asserted that aesthetic objects and practices are produced and regulated by social and cultural structures, and serve overwhelmingly to reinforce those structures. In its most extreme form, postmodern constructionism claims that all meaning or significance derives from social and cultural factors, and that perceived truth is a matter of agreement or a function of power. If, for instance, we like Edward Weston's photographs and Georgia O'Keefe's paintings because we observe striking similarities between the organic forms represented in them and features of the human form, postmodernism counsels us that our preferences are dictated by social and cultural regulators, such as current discourses about nature and art, and not in any way by predisposed attractions to types of natural objects or to the relationships between them.Certainly, it would be foolhardy to argue that social and cultural context have no influence on either the type of art that is produced or its reception in any given culture, but a thoroughgoing or strong [End Page 155] constructionism is another matter. 1 Strong constructionism cannot explain--or, to be more accurate, refuses to explain--why we have art in the first place. Recent intellectual fashion has dictated, in sum, that the most interesting questions about art, the truly thought-provoking and hard questions, should not be asked. Yet even in an era inimical to the study of complexity, some scholars will continue to grapple with the interesting questions. So it is that within the past twelve years Ellen Dissanayake has published three books that together stand as the major contribution to an evolutionary understanding of the arts.Dissanayake's three books--What Is Art For? (1988); Homo Aestheticus: Where the Arts Come From and Why (1992); and, most recently, Art and Intimacy: How the Arts Began (2000)--build on one another, elaborating a functional account of the genesis and adaptive value of the arts. 2 It is fitting that each volume amplifies Dissanayake's central theory, representing, as it were, a new stage in that theory's evolution, for elaboration is a key concept in Dissanayake's work.First and foremost, Dissanayake establishes the need for a functional account of art in What Is Art For? before developing her central thesis. Pointing out that what we know of pre-industrial cultures is inconsistent with the traditional approach to aesthetics, Dissanayake fruitfully suggests we readjust our intellectual lens. Whereas aestheticians have been preoccupied for centuries with discovering a transcendent essence for art, a functional and evolutionary account reveals the myopia of such an approach. Premodern cultures, because they do not separate life into discrete categories including the real and everyday, the imaginative and aesthetic, and the religious, have no concept of art apart from everyday activities or utilitarian considerations and, therefore, the study of these cultures calls the existence of a transcendent art essence into question. As constructionist colleagues would be happy to point out, the quest for essences does indeed attest to a cultural bias, that particular bias being an overwhelming tendency to conceptualize in rigid dualities within modern western culture; simultaneously, the quest for essences conflicts with an evolutionary understanding of human beings and their practices. In thus preparing the ground for her central argument, Dissanayake recommends persuasively the need for a major shift in focus among aestheticians, who are encouraged to look at how art behaviors work in adaptively significant ways within human social life.In Dissanayake's account, all of those activities, adornments, and the like in pre-industrial culture that we today see as the ancestors of [End Page 156] modern art forms exemplify a predisposition toward elaboration or, in her chosen phrase... (shrink)

The main focus of the paper is to reflect analytically on the likely place/role of international solidarity in global refugee protection context in the coming years. Following a short introduction, the paper begins with brief discussions of certain preliminary questions related to the nature of the concept of international solidarity. These discussions are followed by a consideration of some discrete issues related to the “norm/practice chasm” in the operation of international solidarity in global refugee protection. Thereafter, the future of (...) international solidarity in the global refugee protection context is pondered in its duality and complexity, especially the claim that the future of international solidarity is likely to be shaped by a harsher trend that is styled “de-solidarity” in the paper. (shrink)

A ‘duality’ is a formal mapping between the spaces of solutions of two empirically equivalent theories. In recent times, dualities have been found to be pervasive in string theory and quantum field theory. Naïvely interpreted, duality-related theories appear to make very different ontological claims about the world—differing in e.g. space-time structure, fundamental ontology, and mereological structure. In light of this, duality-related theories raise questions familiar from discussions of underdetermination in the philosophy of science: in the presence of (...) dual theories, what is one to say about the ontology of the world? In this paper, we undertake a comprehensive and non-technical survey of the landscape of possible ontological interpretations of duality-related theories. We provide a significantly enriched and clarified taxonomy of options—several of which are novel to the literature. (shrink)

The aim of this article is to discuss the extent to which certain substructural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collect...

The main aim of this paper is to make a remark about the relation between dualities between theories, as `duality' is understood in physics and equivalence of theories, as `equivalence' is understood in logic and philosophy. The remark is that in physics, two theories can be dual, and accordingly get called `the same theory', though we interpret them as disagreeing---so that they are certainly not equivalent, as `equivalent' is normally understood. So the remark is simple: but, I shall argue, (...) worth stressing---since often neglected. My argument for this is based on the account of duality developed by De Haro: which is illustrated here with several examples, from both elementary physics and string theory. Thus I argue that in some examples, including in string theory, two dual theories disagree in their claims about the world. I also spell out how this remark implies a limitation of proposals to understand theoretical equivalence as either logical equivalence or a weakening of it. (shrink)

There exists a common view that for theories related by a ‘duality’, dual models typically may be taken ab initio to represent the same physical state of affairs, i.e. to correspond to the same possible world. We question this view, by drawing a parallel with the distinction between ‘interpretational’ and ‘motivational’ approaches to symmetries.

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.

In this paper I develop a framework for relating dualities and emergence: two notions that are close to each other but also exclude one another. I adopt the conception of duality as 'isomorphism', from the physics literature, cashing it out in terms of three conditions. These three conditions prompt two conceptually different ways in which a duality can be modified to make room for emergence; and I argue that this exhausts the possibilities for combining dualities and emergence. I (...) apply this framework to gauge/gravity dualities, considering in detail three examples: AdS/CFT, Verlinde's scheme, and black holes. My main point about gauge/gravity dualities is that the theories involved, qua theories of gravity, must be background-independent. I distinguish two senses of background-independence: minimalistic and extended. I argue that the former is sufficiently strong to allow for a consistent theory of quantum gravity; and that AdS/CFT is background-independent on this account; while Verlinde's scheme best fits the extended sense of background-independence. I argue that this extended sense should be applied with some caution: on pain of throwing the baby out with the bath-water. Nevertheless, it is an interesting and potentially fruitful heuristic principle for quantum gravity theory construction. It suggests some directions for possible generalisations of gauge/gravity dualities. The interpretation of dualities is discussed; and the so-called 'internal' vs. 'external' viewpoints are articulated in terms of: epistemic and metaphysical commitments; parts vs. wholes. I then analyse the emergence of gravity in gauge/gravity dualities in terms of the two available conceptualisations of emergence; and I show how emergence in AdS/CFT and in Verlinde's scenario differ from each other. Finally, I give a novel derivation of the Bekenstein-Hawking black hole entropy formula based on Verlinde's scheme; the derivation sheds light on several aspects of Verlinde's scheme and how it compares to Bekenstein's original calculation. (shrink)

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the ‘flavour’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a duality as an ‘isomorphism’ between theories: more precisely, as appropriate bijections between the two theories’ sets of (...) states and sets of quantities. The first point is that this conception of duality meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge’. The second point is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory. (shrink)

We argue that dualities offer new possibilities for relating fundamentality, levels, and emergence. Namely, dualities often relate two theories whose hierarchies of levels are inverted relative to each other, and so allow for new fundamentality relations, as well as for epistemic emergence. We find that the direction of emergence typically found in these cases is opposite to the direction of emergence followed in the standard accounts. Namely, the standard emergence direction is that of decreasing fundamentality: there is emergence of less (...) fundamental, high-level entities, out of more fundamental, low-level entities. But in cases of duality, a more fundamental entity can emerge out of a less fundamental one. This possibility can be traced back to the existence of different classical limits in quantum field theories and string theories. (shrink)

This is the first of two papers concerned with the philosophical significance of dualities as applied in recent fundamental physics. The general idea is that, for its peculiarity, this ‘new’ ingredient in theory construction can open unexpected perspectives in the current philosophical reflection on contemporary physics. In particular, today’s physical dualities represent an unusual type of intertheory relation, the meaning of which deserves to be investigated. The aim is to show how discussing this point brings into play, at the same (...) time, what is intended by a 'theory’ and in which sense dualities are to be considered 'symmetries'. This paper is introductory and focusses on the first form of duality explicitly applied in twentieth century physics, that is electromagnetic duality as discussed in Dirac’s theory of magnetic poles. The extension of electromagnetic duality in the context of quantum field theory and string theory is explored in a forthcoming companion paper. (shrink)

String dualities establish empirical equivalence between theories that often look entirely different with respect to their basic ontology and physical structure. Therefore, they represent a particularly interesting example of empirical equivalence in physics. However, the status of duality relations in string physics differs substantially from the traditional understanding of the role played by empirical equivalence. The paper specifies three important differences and argues that they are related to a substantially altered view on the underdetermination of theory building.

We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.

Weak/strong duality is usually accompanied by what seems a puzzling ontological feature: the fact that under this kind of duality what is viewed as 'elementary' in one description gets mapped to what is viewed as 'composite' in the dual description. This paper investigates the meaning of this apparent 'particle democracy', as it has been called, by adopting an historical approach. The aim is to clarify the nature of the correspondence between 'dual particles' in the light of an historical (...) analysis of the developments of the idea of weak/strong duality, starting with Dirac's electric-magnetic duality and its successive generalizations in the context of field theory, to arrive at its first extension to string theory. This analysis is then used as evidential basis for discussing the 'elementary/composite' divide and, after taking another historical detour by analysing an instructive analogy case, drawing some conclusions on the particle-democracy issue. (shrink)

Advocates of dynamic systems have suggested that higher mental processes are based on continuous representations. In order to evaluate this claim, we first define the concept of representation, and rigorously distinguish between discrete representations and continuous representations. We also explore two important bases of representational content. Then, we present seven arguments that discrete representations are necessary for any system that must discriminate between two or more states. It follows that higher mental processes require discrete representations. We also (...) argue that discrete representations are more influenced by conceptual role than continuous representations. We end by arguing that the presence of discrete representations in cognitive systems entails that computationalism (i.e., the view that the mind is a computational device) is true, and that cognitive science should embrace representational pluralism. (shrink)

Causal connectionists need to provide an account of causal linkage and of causal direction. This chapter distinguishes between two kinds of causal connection, namely, discrete and continuous. Causal connectionists have a number of options for explaining the linkage between causes and effects in the case of discrete causation. The chapter provides some popular options. If some causation is discrete, and the exercise of causal powers provides a direction to discrete causation, then the causal direction of processes (...) can be derived from the causal direction of discrete interaction. David Hume famously made the claim about the knowledge of causation in his Enquiry Concerning Human Understanding. Hume offers, in defense of his principle, a set of thought‐experiments that seem to show that we can always imagine any event's having a different result from the one we in fact observe. Causal processes like locomotion may provide counterexamples to Hume's Epistemic Principle. (shrink)