Non-Duality Questions, Non-Duality Answers is an organic, interactive exploration of non-duality--the understanding that in life there is no separation, only seamless Oneness. Designed to answer questions as they arise, this book invites readers to participate in the argument for non-separation, inquire about the nature of the self and the experience of liberation, and understand how non-dual awareness can impact our lives. With a unique blend of patience, compassion, respect, provocation, and humor, author Richard Sylvester helps frustrated spiritual (...) seekers gain clarity on what non-duality is--and what it isn't. (shrink)

This chapter focuses on non-dualism as an affirmation of the exclusive unity of Reality, and as an apophatic approach of the Ultimate through negations. The two points of view are esoteric in that they appear to contradict the ordinary dualistic experience of phenomenal reality on the one hand, and the predicative focus of exoteric perspectives on the other hand, since the latter rest upon dogmatic affirmations. Thus, for instance, the Advaitin doctrine of ajāti, or non-creation, denies the reality of creation. (...) As such, it is counter-intuitive from a conventional point of view and seems to undermine the very basis of religious consciousness which posits, by definition, a relationship between the Divine and the terrestrial. Thus, this chapter explores the challenges and paradoxes of esoteric knowledge as it delves into the implications of non-duality. How is such a knowledge possible without annulling the very conditions of knowledge that is the duality of the knower and the known, and is not its exclusive identification of knowledge with Unity incompatible with the relativity (esoteric/exoteric) that the esoteric entails? (shrink)

Self – organizing is closely connected to the phenomenon of life. The creation of complexity, which is necessary for the creation of life, is connected to the process of storing information in molecules of which the living cell is constituted. All living systems, either unicellular or multicellular organisms, are extremely complex systems compared to all the other species of the non living matter that exist in the Universe. Complexity is the result of effect processes, leading to systems with great organization, (...) containing large information stocks. This organization of the molecules of a living organism, a result of accumulation of information, is what makes them able to produce useful work. The useful work involves both the fulfillment of basic biological processes, such as metabolism and reproduction, and the further increase in the information content that builds up in living systems. This last process is subject to the great chain of evolution of biological systems, this development is governed by the law of natural selection. The capability of reproduction, mutation and metabolism are necessary conditions for the latter. (shrink)

Resumo: Devido à tendência contemporânea do “Budismo Humanista”, na China, a difusão da cultura budista tem exibido tendências peculiares de modernização e secularização. Desde a década de 1990, as animações budistas de Fujian e Taiwan têm tido um amplo impacto em Hong Kong, Macau, Taiwan, Sudeste Asiático e outras comunidades da diáspora chinesa. No contexto atual, os ideais de “Não dualidade do princípio e dos fenômenos” e “Integração do ensino doutrinário e da prática Chan”, na tradição budista chinesa, refletem-se vividamente (...) nesses efeitos. Há um espectro de transição do “sagrado” para o “secular”: não só está sujeito à iconografia tradicional e aos enredos padronizados, mas também a ajustes formais às especificidades da mídia audiovisual. Isso significa que as diferentes tradições da mídia audiovisual apresentam variações, especialmente aquelas que preparam animações. Além disso, na mediação prolongada, por um lado, do “Princípio” em comparação com os “Fenômenos” e, por outro, da “fala” em comparação com o “Chan”, surgiram gradualmente três formas principais de animações budistas, incluindo Animação das Escrituras, Animação Doutrinária e Animação da Natureza Búdica. (shrink)

Inquiry into the felt dimension of experience can be deeply transformative, but more than promoting transcendence it can help us make contact with the intricacy of our open and embodied self-….

When comparing diverse philosophical traditions, it becomes necessary to establish a common point of departure. This paper offers a comparative analysis of Advaita Vedānta Hinduism and esoteric Christianity, as represented by the two highly celebrated figures of Śaṅkara and Nicholas Cusanus, respectively. The common point of departure on which I base this comparison is the concept of “non-duality”—a concept that is fitting for at least two reasons. First, it is general enough to encompass both traditions, pervading the work of (...) each figure, and thus allowing for a kind of “shared language.” Second, it is specific enough to identify a set of core and well-defined principles amenable to systematic study, chief among which are the notions of the “Absolute” as an infinite unity that transcends all determinations and exceeds all oppositions, and the world as an ontologically ambiguous “reflection” that simultaneously hides and manifests its meta-ontological Principle. In drawing these connections, I hope to show how the concept of non-duality provides the possibility for a mutual understanding among diverse traditions at the philosophical level. (shrink)

This paper takes inspiration from the Silk Road crossroads of Dunhuang and proposes an “opportunistic trade”of concepts between distant intellectual traditions, in particular Mahāyāna Buddhism and contemporary dark matter cosmology. Dark matter presents philosophy of science with a tension: it is fundamental to the standard model of cosmology, yet directly unobservable, known only through its gravitational manifestations accessed by an extensive infrastructure of telescopes and computational models. The paper argues that the Mahāyāna Buddhist doctrines of emptiness (空) and dependent origination (...) (緣起), which entered sinified Buddhism through Nāgārjuna’s Madhyamaka (Middle Way, 中觀宗) and Heart Sūtra (Prajñāpāramitāhrdaya, 般若波羅蜜多心經), allow us to characterize dark matter as a scientific postulate that gathers a stabilized pattern of manifestations without reifying any substrate beneath. After that, the paper presents the thesis of New Confucian philosopher Xiong Shili (熊十力) on non-duality of reality and function (體用不二), which emerges from his critical and syncretic re-reading of Mahāyāna Buddhism, in order to mobilise the Buddhist concept of 不二 to dissolve the residual philosophical anxiety about dark matter’s ontological standing. The paper advertises a philosophical stance towards dark matter that is (a) comfortable with ambiguity, (b) resistant to premature ontological commitment, and (c) attuned to the mediated character of contemporary cosmological knowledge. The motivation behind this operation is to demonstrate that philosophical positions originating in or inspired by Mahāyāna Buddhism constitute a conceptual toolkit of equal theoretical importance for the philosophy of cosmology alongside the pathways grounded in the Western canon. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Duality and Non-Duality in Christian Practice:Reflections on the Benefits of Buddhist-Christian Dialogue for Constructive TheologyWendy FarleyThe question before us is the desirability of Buddhist-Christian dialogue in the work of (what Christians call) constructive theology. As a feminist theologian whose work is ever more deeply shaped by such a dialogue, my immediate answer is an unequivocal yes.1 This dialogue fits a general pattern over two thousand years in (...) which theologians have drawn from the wisdom of other traditions or cultures to better understand the mysteries of human experience. Without neo-Platonism and Aristotle there could hardly have been anything we recognize as Christian theology. Without pre-Christian Irish religion, there would be no Saint Brigit or Celtic Christianity. Without phenomenology, process, and Marxist philosophy the greatest works of twentieth-century Christian thought would not exist. Christianity, like all religions, is a moving river changed by time and culture and nearly as interiorly diverse as its members. The dialogue with Buddhism might be seen as a contemporary example of Christianity's openness to exchange with other ways of thinking that broadens and deepens its best insights. It is perhaps analogous to Origen's reliance on neo-Platonism or Thomas Aquinas's inspiration by the rediscovery of Aristotle.2The porous boundaries between Christianity and its culture open it to wisdom beyond itself but they also open it to cruelty and ignorance. Nothing protects religions from participating in the human condition and thus the prejudices and power structures that mutilate human society. Consumer culture and oppressions by class, ethnicity, and gender are seamlessly incorporated into Christian identity. The tacit synthesis of a culture's values with those of a religion neutralizes possible critique and can elevate a society's oppressions to a sacred duty. For this reason, too, dialogue with Buddhism is important to call Christianity back to its ethical moorings in the ideal of universal love and compassion.Religions are porous to the world around them, but they tend to conceal this from themselves and often imagine themselves to be not only untainted but also uniquely [End Page 135] qualified to mediate salvation. Religions provide human beings with language and practice through which dimensions of reality that cannot simply be read off of nature or experience are (in some sense) encountered. This is their great gift. The difficulty of religions is that tradition can become an end in itself. Because a tradition really does offer meaning and tools for transformation it is easy to slip into the existential posture of so idealizing a tradition that it seems in some direct and exclusive way to correlate to ultimate truth. The curvature of the ego onto itself, so attached to its own pain and pleasure that everything else seems a mere shadow, moves from the microcosm of the self to the macrocosm of religious tradition. The particular religion—or that element within it with which I identify—becomes identical with sacred reality. In the hostility to other religions characteristic of many Christians we see the illusion of egocentrism in its religious form. Paul Tillich described this tendency to absolutize itself as the demonic distortion of religion.3 Religious dialogue is a potent medicine against this overattachment to one's own tradition.These different senses in which a religion integrates into itself the wisdom and illusions of its time and place create enormous diversity within a religious tradition. Root concepts within Christianity concerning soteriology, divinity, and suffering do not have any agreed upon meaning. That is, there is no "Christian" view of salvation. Gregory of Nyssa's ideal of theosis, Luther's emphasis on salvation by grace through faith, an evangelical's experience of being born again, a liberal Protestant's commitment to social justice are so diverse that they seem to represent almost different religions.In light of the malleability and interior diversity of traditions, dialogue may not be between two self-sufficient religions but between participants who inhabit their tradition in a variety of ways.4 With this in mind, I would like to experiment with practice as a basis for some kinds of dialogue. Practice is concretely located in a religious tradition and yet reflects... (shrink)

In its classical formulation as found in Svātmārāma’s Haṭhapradīpikā, haṭhayoga is a Śaiva appropriation of an older extra-Vedic soteriological method. But this appropriation was not accompanied by an imposition of Śaiva philosophy. In general, the texts of haṭhayoga reveal, if not a disdain for, at least an insouciance towards metaphysics. Yoga is a soteriology that works regardless of the yogin’s philosophy. But the various texts that were used to compile the Haṭhapradīpikā (a table identifying these borrowings is given at the (...) end of the article) were not composed in metaphysical vacua. Analysis of their allusions to doctrine shows that the texts from which Svātmārāma borrowed most were products of a Vedantic milieu—bearing testament to Vedānta’s newfound interest in yoga as a complement to jñāna—but that many others were Śaiva non-dual works. Because of the lack of importance given to the niceties of philosophy in haṭhayogic works, these two non-dualities were able to combine happily and thus the Śaiva tenets incorporated within haṭhayoga survived the demise of Śaivism as part of what was to become in the medieval period the dominant soteriological method in scholarly religious discourse in India. (shrink)

NON-DUALISM, DUALISM AND MONISM Q.1 What do the following terms, often used by the Vedanta: dualism, monism, monotheism and non-dualism, mean? A. Every philosophical or cosmological vision which affirms two opposing and irreducible...

A fascinating compilation of the life stories of perfectly ordinary people, all of whom have been through extraordinary experiences leading to amazing new perceptions. Some have been seekers from a very young age, whilst others had previously never even thought in terms of seeking. Nevertheless, all shared a common sense of dissatisfaction with their lives. This collection of stories explores the remarkable means by which each individual went about achieving an end to their suffering.

We agree with Josipovic that a fundamental differentiating feature of meditation techniques is whether they remain within the dualistic subject–object cognitive structure, or they transcend this structure to reveal an underlying level of non-dual awareness. Further discussion is needed to delineate the basic non-dual experience in meditation, where all phenomenal content is absent, from the more advanced experience of non-duality in daily life, where phenomenal content is obviously present as well. In this discussion, it is important to recognize that (...) the experiencer–object relation makes the experience dual or non-dual, rather than the nature of the object experienced. (shrink)

The discussion on the Buddhist roots of contemporary mindfulness practices is dominated by a narrative which considers the Theravāda tradition and Theravāda-based ‘neo-vipassanā movement’ as the principal source of Buddhist influences in mindfulness-based stress reduction (MBSR) and related mindfulness-based programmes (MBPs). This Theravāda bias fails to acknowledge the significant Mahāyāna Buddhist influences that have informed the pioneering work of Jon Kabat-Zinn in the formation of the MBSR programme. In Kabat-Zinn’s texts, the ‘universal dharma foundation’ of mindfulness practice is grounded in (...) pan-Buddhist teachings on the origins and cessation of suffering. While MBSR methods derive from both Theravāda-based vipassanā and non-dual Mahāyāna approaches, the philosophical foundation of MBSR differs significantly from Theravāda views. Instead, the characteristic principles and insights of MBSR practice indicate significant similarities and historical continuities with contemporary Zen/sŏn/thiền and Tibetan Dzogchen teachings based on doctrinal developments within Indian and East Asian Mahāyāna Buddhism. (shrink)

This paper describes the contemporary phenomenological understanding of subjectivity as non dual awareness.

This essay considers some meta-ethical questions that emerge from a consideration of the phenomena of terrorism in the context of Buddhist metaphysics: what, in the Buddhist view, ultimately causes terrorism (and its subsidiary effects)? What resources do the Buddhist metaphysical claims of no-self, karma, emptiness and related concepts bring to a meta-ethical understanding of terrorism and its effects?

Inquiry into object happiness -- What is enlightenment? -- The means of knowledge -- Qualifications -- The self -- Obstructions -- Inquiry into karma and dharma -- Inquiry into practice -- Love -- The assimilation of experience -- Lifestyle -- Knowledge yoga -- Meditation -- After enlightenment -- The teachings of Ramana Maharshi -- Neo-Advaita.

This article problematizes the way of thinking schooling in discourse of sign system, which involves opposition, and double gesture of inclusion/exclusion. Drawing on two fundamental texts of Taoism, the Tao Te Ching and Chuang Tzu, this article puts forward the seemingly passive Non-Word and Non-Self to resist the hierarchy ordering of conceptions and man, and to undo duality of binary opposition. It links the history of assessment and PISA to the rethinking of evidence and sign in contemporary movements. The (...) second section proposes teaching as wandering (you, 游), the transcendence into Non-Self with response to men’s natural inquiry, but not artificially manipulating and shaping man by privileging certain knowledge, value, and thoughts over other. Instead of making categories of race, gender, and normal/abnormal, Taoist provides a way of reconsider the self operating on assessing, standardizing, normalizing, correcting and predicting others in school accountability. It is built on the ontology of considering individual as belonging to an ultimate sameness, the oneness. The article concludes with underlying assumption that students are subjected to change, and their difference must be eliminated through standard test. It further questions tactics of reason that educational reform face in the current regime of school accountability. (shrink)

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that (...) are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. (shrink)

The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topological and non-topological Kripke-style models for these types of logics. Moreover, we show that, under certain additional assumptions on the variety (...) of the algerabic models of the given logics, soundness and completeness with respect to these classes of Kripke-style models follows by using entirely algebraical arguments from the soundness and completeness of the logic with respect to its algebraic models. (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...

Recently, a number of epistemologists have argued that there are no non-conceptual elements in representational content. On their view, the only sort of non-conceptual elements are components of sub-personal organic hardware that, because they enjoy no veridical role, must be construed epistemologically irrelevant. By reviewing a 35-year-old debate initiated by Dagfinn F.

Many results in logic and mathematics rely on techniques that allow for concrete, often visual, representations of abstract concepts. A primary example of this phenomenon in logic is the distinction between syntax and semantics, itself an example of the more general duality in mathematics between algebra and geometry. Such representations, however, often rely on the existence of certain maximal objects having particular properties such as points, possible worlds or Tarskian first-order structures. -/- This dissertation explores an alternative to such (...) representations known as possibility semantics. Its core idea is to replace maximal objects with ordered systems of partial approximations. Although it originates in the semantics of modal logic and the representation of abstract ordered structures, I argue that it has far-reaching mathematical, foundational and philosophical significance, especially in the context of semiconstructive mathematics, a foundational framework that does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. -/- The dissertation is divided in two main parts. The first part explores various applications of the mathematical framework underlying possibility semantics to lattice theory and non-classical propositional logics. A major theme is the development of constructive dualities for various categories of lattices, which are related to standard non-constructive dualities via Vietoris constructions. -/- The second part of the dissertation explores the alternative foundational setting of semi-constructive mathematics, focusing on three applications of possibility semantics for classical first-order logic to the philosophy of the mathematical infinite. In particular, we introduce generic powers, a semi-constructive analogue of ultrapowers in classical model theory, and we explore the merits of these structures from a foundational, conceptual and historical viewpoint. (shrink)

Kumar and Banerjee (Some algebras and logics from quasiorder-generated covering-based approximation spaces. J Appl Non-classical Log 2024;34:248-68) characterized a subclass of quasiorder-generated covering-based approximation spaces for which the algebra of definable sets forms a Stone algebra. In this paper, we characterize those subclasses for which the definable sets form dual Stone, regular double Stone, linear Heyting, and well-connected Heyting algebras. As a consequence, discrete dualities of the aforementioned algebras are also obtained. Furthermore, we provide representation theorems of the aforementioned algebras (...) in terms of rough sets determined by a quasiorder. (shrink)

The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to the variety generated by the smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) for the category of non-trivial (0,1)-lattices belonging to the same variety with (0,1)-lattice homomorphisms as morphisms. Although the two categories coincide on their finite objects, the presented dualities essentially differ mostly but not only by the fact that the duality for the second category uses topology. Using the presented (...) dualities and some known in the literature results we prove that the Q-lattice of any non-trivial variety of (0,1)-lattices is either a 2-element chain or is uncountable and non-distributive. (shrink)

In this paper, we will analyse the superloop space formalism for a four dimensional supersymmetric Yang–Mills theory in deformed superspace. We will deform the \ superspace by imposing imposing non-anticommutativity. This non-anticommutative deformation of the superspace will break half the supersymmetry of the original theory. So, this theory will have \ supersymmetry. We will analyse the superloop space duality for this deformed supersymmetric Yang–Mills theory using the \ superspace formalism. We will demonstrate that the sources in the original theory (...) will become monopoles in the dual theory, and the monopoles in the original theory will become sources in the dual theory. (shrink)

In many philosophical traditions, it is held that reality is non-dual. Of course, to be non-dual, as opposed to dual, is itself to partake of a certain duality. If reality really is non-dual, it must transcend this duality too. But what could this mean? Can one make coherent sense of it? To keep the discussion focussed, I will locate it in one specific tradition: the Mahāyāna Buddhist tradition. The idea that ultimate reality is non-dual goes back to the (...) earliest Mahāyāna sūtras at the turn of the Common Era. Thereafter, the question of what it means to transcend duality plays a central role in Buddhist philosophy. The point that reality must transcend even the duality between duality and non-duality plays a significant role in the Chinese Sanlun philosopher Jizang (, 549–623). His discussion points the way to an answer to our problem which may be articulated with the techniques of contemporary paraconsistent logic, as we will see. (shrink)

A key working hypothesis in neuroscience is ‘materialistic reductionism’, i.e., the assumption whereby all physiological, behavioral or cognitive phenomena is produced by localized neurochemical brain activation (but not vice versa). However, analysis of sub-threshold Weber’s psychophysical stimulation indicates its computational irreducibility to the direct interaction between psychophysical stimulation and any neuron/s. This is because the materialistic-reductionistic working hypothesis assumes that the determination of the existence or non-existence of any psychophysical stimulation [s] may only be determined through its direct interaction [di1] (...) with a given neuron/s [N] that together forms the ‘neural registry’ computational level [NR/di1]. But, this implies that in cases of (initial) sub-threshold (sensory-specific) psychophysical stimulation which is increased above the sensory-specific threshold but below Weber’s psychophysical ‘dv’—the psychophysical computational processing [PCP] produces an apparently ‘computationally indeterminate’ output. This is because materialistic reductionism asserts the contingency of PCP upon the existence of a direct interaction between ‘s’ and ‘N’ within the NR/di1 level, but in the special case of Weber’s sub-threshold psychophysical stimulation the same PCP/di1 also asserts the non-existence of ‘s’ (as demanded by Weber’s psychophysical law). However, given robust empirical evidence indicating the capability of PCP to determine whether (or not) ‘s’ exists, we must conclude that PCP may not be carried out from within NR’s direct interaction between a particular psychophysical stimulation and any set of neuron/s in the brain. Hence, the Duality Principle asserts the conceptual irreducibility of sub-threshold psychophysical stimulation to any direct NR/di1: s-N interaction, thereby challenging the current materialistic-reductionistic assumption. (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)

: Frege’s remarks about the first-person pronoun in Der Gedanke have elicited numerous commentaries, but his insight has not been fully appreciated or developed. Commentators have overlooked Frege’s reasons for claiming that there are two distinct first-person senses, and failed to realize that his remarks easily generalize to all indexicals. I present a perspectival theory of indexicals inspired by Frege’s claim that all indexical types have a dual meaning which, in turn, leads to a duality of senses expressed by (...) indexical tokens. Keywords : Indexicals; First Person; Perspective; Senses La doppia natura degli indessicali: una posizione fregeana Riassunto : Le osservazioni di Frege sul pronome di prima persona contenute in Der Gedanke hanno sollevato numerosi commenti, ma le sue intuizioni non sono state pienamente comprese o sviluppate. I commentatori di Frege hanno trascurato le ragioni per le quali egli sosteneva che ci sono due distinti sensi della prima persona e non hanno colto come queste sue osservazioni possono essere facilmente estese a tutti gli indessicali. Intendo presentare qui una posizione prospettivista sugli indessicali, ispirata dall’affermazione di Frege per cui tutti i tipi di indessicali hanno un doppio significato che, a sua volta, porta a una doppia natura dei sensi espressi dalle occorrenze indessicali. Parole chiave : Indessicali; Prima persona; Prospettiva; Sensi. (shrink)

A method of defining semantics of logics based on not necessarily distributive lattices is presented. The key elements of the method are representation theorems for lattices and duality between classes of lattices and classes of some relational systems . We suggest a type of duality referred to as a duality via truth which leads to Kripke-style semantics and three-valued semantics in the style of Allwein-Dunn. We develop two new representation theorems for lattices which, together with the existing (...) theorems by Urquhart and Bimbo-Dunn, constitute a complete, in a sense, representation theory for lattices. As observed by Dunn and Hardegree, variations of Urquhart's duality arise by varying his disjointness assumption on the canonical frame. Four possible assumptions – disjoint, exhaustive, non-disjoint and non-exhaustive – are discussed in the paper. Each of the four corresponding representation theorems is expanded to a duality via truth. Based on these dualities we suggest four corresponding types of semantics for lattice-based logics. We also discuss a new topological representation of lattices. (shrink)

Haar measure has two spectral regimes determined entirely by the compactness of the underlying group. On a non-compact group the regular representation decomposes by Plancherel measure over a continuous unitary dual; on a compact group it decomposes by Peter–Weyl summation over finite-dimensional irreducibles. We prove this dichotomy in the forms needed for two applications in the shadow framework. For the non-compact case we show that the half-density Mellin transform on L²(ℝ⁺, dx) is unitary exactly on the line Re(s) = 1/2, (...) and that the functional equation ξ(s) = ξ(1−s) of the completed zeta function is the restriction of Haar inversion self-duality to the arithmetic spectrum via Tate's thesis. Under Δ = 2s, the same inversion is the shadow symmetry transform ∆ ←→2 - ∆ of celestial conformal field theory, which identified as the boundary theory's version of the fundamental CPT-symmetry of regular four-dimensional quantum field theory in the bulk provides a means of resolving the googly problem regarding the perceived asymmetry problem in twistor-theoretic quantum gravity. For the compact case we prove that Haar projection onto SU(N)-singlets is the orthogonal projection onto the physical gauge sector, and that the Sugawara conformal weight of the first adjoint-current excitation gives a mass gap M = 2N/(k+N)·Λ_QCD > 0 in the continuum celestial Yang–Mills construction. The Riemann and Yang–Mills mechanisms are the two complementary spectral faces of one mathematical fact: non-compact Haar measure gives a Plancherel axis, and compact Haar measure gives a Peter–Weyl gap. (shrink)

Constructive dualities have recently been proposed for some lattice-based algebras and a related project has been outlined by Holliday and Bezhanishvili, aiming at obtaining “choice-free spatial dualities for other classes of algebras [ $$\ldots $$ ], giving rise to choice-free completeness proofs for non-classical logics”. We present in this article a way to complete the Holliday–Bezhanishvili project (uniformly, for any normal lattice expansion). This is done by recasting in a choice-free manner recent relational representation and duality results by the (...) author. These results addressed the general representation and duality problem for lattices with quasi-operators, extending the Jónsson–Tarski approach for BAOs, and Dunn’s follow-up approach for distributive generalized Galois logics, to contexts where distributivity may not be assumed. To illustrate, we apply the framework to lattices (and their logics) with some form or other of a (quasi)complementation operator, obtaining correspondence results and canonical extensions in relational frames and choice-free dualities for lattices with a minimal, or a Galois quasi-complement, or involutive lattices, including De Morgan algebras, as well as Ortholattices and Boolean algebras, as special cases. (shrink)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover (...) the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices. (shrink)

I review at a non-technical level the use of the gauge-string duality to study aspects of heavy ion collisions, with special emphasis on the trailing string calculation of heavy quark energy loss. I include some brief speculations on how variants of the trailing string construction could provide a toy model of black hole formation and evaporation. This essay is an invited contribution to “Forty Years of String Theory” and is aimed at philosophers and historians of science as well as (...) physicists. (shrink)

The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the (...) development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations. (shrink)

In this paper we present a schema for describing dualities between physical theories, and illustrate it in detail with the example of bosonization: a boson-fermion duality in two-dimensional quantum field theory. The schema develops proposals in De Haro : these proposals include construals of notions related to duality, like representation, model, symmetry and interpretation. The aim of the schema is to give a more precise criterion for duality than has so far been considered. The bosonization example, or (...) boson-fermion duality, has the feature of being simple, yet rich enough, to illustrate the most relevant aspects of our schema, which also apply to more sophisticated dualities. The richness of the example consists, mainly, in its concern with two non-trivial quantum field theories: including massive Thirring-sine-Gordon duality, and non-abelian bosonization. This prompts two comparisons with the recent philosophical literature on dualities:--- Unlike the standard cases of duality in quantum field theory and string theory, where only specific simplifying limits of the theories are explicitly known, the boson-fermion duality is known to hold {\it exactly}. This exactness can be exhibited explicitly. The bosonization example illustrates both the cases of isomorphic and {\it non-isomorphic} models: which we believe the literature on dualities has not so far discussed. (shrink)

Karl Popper developed a theory of deductive logic in the late 1940s. In his approach, logic is a metalinguistic theory of deducibility relations that are based on certain purely structural rules. Logical constants are then characterized in terms of deducibility relations. Characterizations of this kind are also called inferential definitions by Popper. In this paper, we expound his theory and elaborate some of his ideas and results that in some cases were only sketched by him. Our focus is on Popper's (...) notion of duality, his theory of modalities, and his treatment of different kinds of negation. This allows us to show how his works on logic anticipate some later developments and discussions in philosophical logic, pertaining to trivializing connectives, the duality of logical constants, dual-intuitionistic logic, the conservativeness of language extensions, the existence of a bi-intuitionistic logic, the non-logicality of minimal negation, and to the problem of logicality in general. (shrink)

This paper examines Theodor Adorno’s critical reappropriation of the Kantian notion of Spontaneity. By tracing two distinct senses of spontaneity in Kant’s Critique of Pure Reason – the relative spontaneity of the understanding and the absolute spontaneity of pure reason – the paper highlights how each addresses the problem of the subject-object relation. Adorno’s account of subjectivity within his critique of idealism and materialist doctrine of the “preponderance of the object” draws upon Kant’s relative spontaneity of the mind precisely because (...) it establishes contact between subject and object. Adorno’s rereading of spontaneity, which seeks to critically appropriate it from Kant’s idealist conception of the subject, is shown to be central to his critique of Kantian moral philosophy in Negative Dialectics. For Adorno, the impulsive, bodily dimension of free action, which Kant’s conception of the will glosses over, corresponds to the relative spontaneity of the mind used in his materialist conception of subjectivity. This paper concludes by demonstrating how Adorno’s reinterpretation of Kantian spontaneity underpins two of his late projects, namely, the development of an ethics after Auschwitz and a messianic, micrological approach to metaphysical thought that would be attentive to the non-identical in experience. (shrink)

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.

Humans can consolidate and carry out habits other animals cannot. This statement is mainly sustained by the fact that humans have a unique cognitive control of their actions: we can let our attention fade away to perform automatic tasks more efficiently, we can detect if there has been an unexpected problem in their implementation, and we can regain conscious control of the action if necessary. We tend to dichotomize this cognitive process into two “systems”, namely goal-directed versus habitual (Dickinson, 1985), (...) conscious versus unconscious (Crick and Koch, 1998), or slow versus fast (Kahneman, 2011). If we just put it in those terms, these two ways of tackling the challenges of a changing environment seem to be present in non-human animals. However, all dichotomies imply a difficulty to deal with: the regulation of the transition between the two systems. Is this carried out by a third element, or regulated by one of the systems? Could it be more convenient to view it as a continuum, rather than a dichotomy? In any case, we believe this transition has a level of complexity in humans that makes it qualitatively different from its analogue in animals. In fact, this ‘cognitive bridge’ might be a major feature to characterize a reliable behavior, since a particular task or problem is more efficiently tackled when the transition between the two systems is more adequate. Moreover, the integrity of this link could be an indicator to detect prodromal psychiatric conditions, as it has been suggested for slips-of-action (Gillan et al., 2011). In order to justify these ideas, we will focus first on Kahneman’s distinction between systems 1 and 2 (Kahneman, 2011). On the one hand, System 1 is responsible for making decisions rapidly. The purpose of this system is to give us an assessment of the environment around us as quickly as possible so that we are able to respond as fast as possible. To perform this task, System 1 follows general rules or guidelines (heuristics). In all, System 1 is intended to help us make decisions more quickly, and is very useful (let’s say ‘just fine’) in most cases. However, one of the characteristics of these decisions is the lack of voluntary control, what is a problem considering this system is responsible of many of the decisions and judgments we make. Given to its “automatic” nature, System 1 also has biases and systematic errors that are likely to happen in some situations.On the other hand, System 2 acts when a problem which System 1 has no solution for arises. System 2, apparently, can take control of the whole process at any time. It is somewhat triggered by some external or internal alarm that draws its attention and makes it take “conscious” control of the situation. One of the problems of this system is that it is lazy and can be easily exhausted. Therefore, it usually accepts the decisions of System 1 without monitoring them. One proof of System 2’s negligence is what Kahneman calls WYSIATI ("What You See Is All There Is"), a general rule that “facilitates the achievement of coherence and of the cognitive ease that causes us to accept a statement as true”. System 1 easily gets that coherence, and System 2 usually allows it to jump to conclusions and act. In different sets of experiments, Kahneman demonstrates that humans are not good at all with statistics or handling mathematics; in his opinion, this is because humans simplify judgments to make them more understandable and deal with them just through heuristics that System 1 can handle. This general view of humans as poor rational decision-makers is also supported by other authors (see, for example, Ariely, 2008).In our opinion, this division of human cognition into two systems fits well with the usual opposition between goal-directed versus habitual systems (Dickinson, 1985). In general, goal-directed actions are viewed as conscious, flexible, and sensitive to outcome devaluation, whereas habits are mainly unconscious, rigid and insensitive to changes in the value of the outcome. The features of goal-directed and habit systems were mainly drawn from studies in animals. The typical experiment about this subject consists on teaching the contingency between an instrumental action (for example, a lever press) and a reward to the animal (Adams and Dickinson, 1981). At the beginning, the animal’s behavior is goal directed, and it performs the action to obtain the reward. However, this behavior becomes “habitual” (in this context, a motor routine) after many repetitions. When that happens, the value of the reward is transferred to the lever press itself: even though the outcome is devalued (gets the animal sick) or the animal is sated, it keeps pressing the lever. This is why habits have been opposed to goal-directed behavior.A quick look suggests that habits and goal-directed actions are intimately related to Systems 1 and 2, respectively. This is also supported by the identification of the goal-directed system with a model-based reinforcement learning scheme, since it can be viewed “in terms of sophisticated, computationally demanding, prospective planning, in which a decision tree of possible future states and actions is built using a learned internal model of the environment” (Dolan and Dayan, 2013). The habitual system, on the other hand, follows a model-free scheme, which “is computationally efficient, since it replaces computation (i.e., the burdensome simulation of future states) with memory (i.e., stored discounted values of expected future reward); however, the forward-looking nature of the prediction error makes it statistically inefficient” (Dolan and Dayan, 2013). Following these analogies between systems, we can assume that some actions that at the beginning fall under the domain of System 2 might be transferred to System 1 through learning, like goal-directed actions become habits through experience.Concerning the neural bases of these systems, the striatum and its cortical afferents and –indirect– target areas in the cortex play a major role. It is widely accepted that the cognitive part of the striatum –caudate nucleus and anterior putamen– are involved in the planning and execution of goal-directed actions, together with the prefrontal cortex (Balleine et al., 2007). On the other hand, the sensorimotor striatal aspects –mainly the posterior putamen– and the supplementary motor area of the cortex are particularly active when the agent is performing a well-learned action (Ashby et al., 2010; Miyachi et al., 2002). Furthermore, the activity of the neurons in these areas follows a “chunked” pattern: they are mainly active at certain stages of the motor routine (for example at the beginning and the end of the sequence, when a particular switch or turn is needed, etc), and this activity is reduced in the rest of the motor sequence (Graybiel, 1998). Although some authors question a sharp neuroanatomical basis of Kahneman’s Systems 1 and 2 (Gold and Shadlen, 2007), our train of thought in this manuscript suggests that the more reflective System 2 should be based in the prefrontal cortex –both dorsal and ventral–, and the cognitive regions of the basal ganglia. Likewise, the more automatic System 1 would lie on motor and premotor cortical regions, as well as on the sensorimotor aspects of those subcortical nuclei.This neuroscientific framework identifies the habit system with automaticity, rigidity and unconsciousness; however, we are intending to challenge this view in past and forthcoming contributions (Bernacer et al., 2014; Bernacer and Gimenez-Amaya, 2013). In a nutshell, we propose to view the phenomenon of action from the point of view of the agent as a whole, and not from an isolated movement. Hence, it could be more convenient to understand System 1 –or habits– as a resource of System 2, rather than as opposed systems in competition. Whereas a motor routine (i.e., what is commonly called “habit” in neuroscience) implies the sequential and unconscious performance of movements, they usually pursue the goal set by the agent. In fact, the more engrained the routine is, the easier for the agent to achieve that goal. Furthermore, the agent can consciously stop or correct the movement at any point, since the habit releases the higher cognitive regions of the brain to improve the performance of the action. A very simple example of this is a tennis service, which should be “goal directed” to place the ball wherever the player wants. It involves a set of movements such as throwing the ball upwards, moving the feet, putting the arm back, etc. Only when these motor routines are learned correctly, the player is able to concentrate on other aspects of the service such as the speed, spin, or exploiting the weaknesses of the receiver. This can be also exemplified with other kinds of habits such as driving, playing an instrument, tackling a mathematical problem, and so on. They all suggest that “automatic” routines are governed by higher cognitive functions to better achieve a particular goal.Kahneman’s Systems 1 and 2 allow as well this release of consciousness from everyday decisions to focus on more complicated situations. As Kahneman himself and other authors defend (Ariely, 2008), the problem arises when System 2 is rarely used or either system is applied to inadequate situations. However, we believe that the most effective agent does not exclusively rely on System 2, but efficiently uses all resources of ‘each system’ in the right situation and, more importantly, carries out an appropriate transition between them. That is, in our opinion, a ‘rational’ agent. This could be also said about goal-directed and habitual systems. Moreover, we believe that this transition between systems is subject to learning, and it can be performed more effectively through experience.If we understand these fragmentations of cognition as independent systems in competition, we encounter an important problem: is there an additional mechanism in charge of the transition between systems, or is this regulated by one of the systems itself? If the first option were true, we would find the difficulty of defining the nature –both conceptually and anatomically– of a “third system” qualitatively different than the other two. This would lead us to an ad infinitum process –the need of a fourth element to regulate the activity of the third, and so on–, and therefore we believe this hypothesis should be rejected. Considering the second option, it seems that only the highly cognitive System 2 could be in charge of leading the transition between systems, which in turn dissolves a rigorous separation in two systems. The role of System 2 in leading the transition is clear when the agent decides to regain conscious control of a task generally performed in an unconscious manner. In this sense, the interaction of the orbitofrontal cortex with either the cognitive or sensorimotor aspects of the striatum plays a central role in shifting between goal-directed actions and motor routines (Gremel and Costa, 2013). In other situations, an external cue such as an error may set the alarm for System 2 to retake control of the action. Regarding this, the anterior cingulate cortex has been reported to be active in highly-conflictive decision making situations (Goñi et al., 2011); for that reason, some authors relate this cortical area with error monitoring (Carter, 1998; Botvinick et al., 2004). A recent report suggests a new model of reinforcement learning and conflict monitoring, which involves a wide network including different areas of the cortex (posterior parietal, precentral, anterior cingulate and prefrontal) and the basal ganglia (Zendehrouh et al., 2013).To sum up, this opinion article suggests viewing Kahneman’s systems as analogous to the goal-directed/habits dichotomy in order to improve the understanding of some aspects of human cognition. Further, we believe a strict separation between systems in competition is problematic, since System 2 is always in charge of governing the interplay between systems: therefore, System 1 –or habits– should be understood as a resource of System 2. This view could shed some light on the understanding of habits as a source of learning, plasticity and freedom for the agent. Finally, an inappropriate cognitive control of habits could be an indicator of certain psychiatric conditions. (shrink)