We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics (...) class='Hi'>duality. Whereas in classical algebraic geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding domains. (shrink)

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics.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 →.

Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I consider possible arguments for this view. Major elements of this argument are out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They also require a special form of consistency requirement.

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)

Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I attempt to construct a serious argument for this view. Certain major elements of this argument are then considered at length. They’re found to be out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They’re also found to require a special form of consistency requirement.

The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during (...) the nineteenth century. Specifically, we want to see in which ways geometers’ preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been influenced by reflections on duality. (shrink)

The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry 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 (...) class='Hi'>geometry, 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)

The AdS/CFT duality has been a source of several strong conceptual claims in the physics literature that have yet to be explored by philosophers. In this paper I focus on one of these: the extent to which spacetime geometry and locality can be said to emerge from this duality, so that neither is fundamental. I argue: that the kind of emergence in question is relatively weak, involving one kind of spacetime emerging from another kind of spacetime; inasmuch (...) as there is something conceptually interesting to say about the emergence of spacetime and locality , it is no different from that already well known to those within canonical quantum gravity; that at the core of AdS/CFT is an issue of representation and redundancy in representation. (shrink)

This essay addresses Cartesian duality and how its implicit dialectic might be repaired using physics and information theory. Our agenda is to describe a key distinction in the physical sciences that may provide a foundation for the distinction between mind and matter, and between sentient and intentional systems. From this perspective, it becomes tenable to talk about the physics of sentience and ‘forces’ that underwrite our beliefs (in the sense of probability distributions represented by our internal states), which may (...) ground our mental states and consciousness. We will refer to this view as Markovian monism, which entails two claims: (1) fundamentally, there is only one type of thing and only one type of irreducible property (hence monism). (2) All systems possessing a Markov blanket have properties that are relevant for understanding the mind and consciousness: if such systems have mental properties, then they have them partly by virtue of possessing a Markov blanket (hence Markovian). Markovian monism rests upon the information geometry of random dynamic systems. In brief, the information geometry induced in any system—whose internal states can be distinguished from external states—must acquire a dual aspect. This dual aspect concerns the (intrinsic) information geometry of the probabilistic evolution of internal states and a separate (extrinsic) information geometry of probabilistic beliefs about external states that are parameterised by internal states. We call these intrinsic (i.e., mechanical, or state-based) and extrinsic (i.e., Markovian, or belief-based) information geometries, respectively. Although these mathematical notions may sound complicated, they are fairly straightforward to handle, and may offer a means through which to frame the origins of consciousness. (shrink)

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, (...) in this case 2.In the present work, we generalize the entire arrangement from propositional to first-order logic, using a representation result of Butz and Moerdijk. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed primarily in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology.The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we give an elementary proof of Butz and Moerdijkʼs result in logical terms. (shrink)

The fundamental duality theories relating algebra and geometry that were discovered in the mid-twentieth century can also be applied to logic via its algebraization under categorical logic. They thereby result in known and new completeness theorems. This idea can be taken even further via what is sometimes called “categorification” to establish a new connection between logic and geometry, a glimpse of which can also be had in topos theory.

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of (...) correct independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

Shahn Majids philosophy of physics is critically presented. In his view the postulate that the universe should be self-explaining implies that no fundamental theory of physics is complete unless it is self-dual. Majid shows that bicrossproduct Hopf algebras have this property. His philosophy is compared with other approaches to the ultimate explanation and briefly analyzed.

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

The aim of this article is to contribute to a better understanding of Frege’s views on semantics and metatheory by looking at his take on several themes in nineteenth century geometry that were significant for the development of modern model-theoretic semantics. I will focus on three issues in which a central semantic idea, the idea of reinterpreting non-logical terms, gradually came to play a substantial role: the introduction of elements at infinity in projective geometry; the study of transfer (...) principles, especially the principle of duality; and the use of counterexamples in independence arguments. Based on a discussion of these issues and how nineteenth century geometers reflected about them, I will then look into Frege’s take on these matters. I conclude with a discussion of Frege’s views and what they entail for the debate about his stance towards semantics and metatheory more generally. (shrink)

ArgumentA plagiarism charge in 1827 sparked a public controversy centered between Jean-Victor Poncelet (1788–1867) and Joseph-Diez Gergonne (1771–1859) over the origin and applications of the principle of duality in geometry. Over the next three years and through the pages of various journals, monographs, letters, reviews, reports, and footnotes, vitriol between the antagonists increased as their potential publicity grew. While the historical literature offers valuable resources toward understanding the development, content, and applications of geometric duality, the hostile nature (...) of the exchange seems to have deterred an in-depth textual study of the explicitly polemical writings. We argue that the necessary collective endeavor of beginning and ending this controversy constitutes a case study in the circulation of geometry. In particular, we consider how the duality controversy functioned as a medium of communicating new fundamental principles to a wider audience of practitioners. (shrink)

Let be n events in a probability space, and suppose that we have only partial information about the distribution: The probabilites of the events themselves, and their pair intersections. With this partial information we cannot, usually, deternine the probability of an event B in the algebra generated by the 's, but we can obtain lower and upper bounds. This is done by a linear program related to the correlation polytope c(n), a structure introduced in [3], [4]. In the first part (...) of the paper I demonstrate how laws of large numbers (for sequences of events which are not necessarily independent) can be proved, using only the duality theorem of linear programming. These include the weak law of large numbers (necessary and sufficient condition) and various sufficient conditions for strong laws. The connection between these laws and the facet structure of the correlation polytope is established. In the second part of the paper I consider a more general case. Assume that our information consists of the values of the probabilities of all intersections of the 's up to size k, k < n. The techniques of linear programming lead naturally to an application of the theory of polynomial approximation in estimating the size of various events. In particular, I prove an approximate version of the central limit theorem. (shrink)

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff. The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. (...) The second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice. (shrink)

In this paper, I propose the idea that the French mathematician Michel Chasles developed a foundational programme for geometry in the period 1827–1837. The basic concept behind the programme was to show that projective geometry is the foundation of the whole of geometry. In particular, the metric properties can be reduced to specific graphic properties. In the attempt to prove the validity of his conception, Chasles made fundamental contributions to the theory of polarity and also understood that (...) a satisfactory development of projective geometry has to overcome the specificity of that theory itself. In this perspective, he developed his ideas on duality and homography, showing their dependence on the concept that he was going to pose at the basis of geometry: the anharmonic ratio, today called cross-ratio. The conceptual itinerary that starts from Chasles’ studies on polarity and ends with his results based on the cross-ratio represents the itinerary of his foundational programme. I will follow this complex and interesting line of thought, place it in the context of what was then called descriptive geometry, and outline a connection with the later great results of Von Staudt, Cayley, and Klein. (shrink)

The physical singularity of life phenomena is analyzed by means of comparison with the driving concepts of theories of the inert. We outline conceptual analogies, transferals of methodologies and theoretical instruments between physics and biology, in addition to indicating significant differences and sometimes logical dualities. In order to make biological phenomenalities intelligible, we introduce theoretical extensions to certain physical theories. In this synthetic paper, we summarize and propose a unified conceptual framework for the main conclusions drawn from work spanning a (...) book and several articles, quoted throughout. (shrink)

We argue that the projective geometrical component of the Projective Consciousness Model can account for key aspects of pre-reflective self-consciousness and can relate PRSC intelligibly to another signal feature of subjectivity: perspectival character or point of view. We illustrate how the projective geometrical versions of the concepts of duality, reciprocity, polarity, closedness, closure, and unboundedness answer to salient aspects of the phenomenology of PRSC. We thus show that the same mathematics that accounts for the statics and dynamics of perspectival (...) character also accounts for PRSC. More generally, we argue that introducing higher-level geometrical concepts into the theory of PRSC, and into the theory of consciousness broadly, as the PCM does, promises to break longstanding theoretical impasses and dialectical stalemates. (shrink)

Formulating quantum gravity is the greatest challenge that 21st century physics must address. If quantum physics refuses to blend with general relativity, it may be that relativity does not represent a good description of the universe in line with gravity and all its effects. This opens a path for us: first understanding quantum physics and what it reveals about nature and then analyzing the boundaries of relativistic cosmology and reconsidering the whole matter. Physicists consider entanglement as a fundamental property derived (...) from quantum mechanics, which is a physics of communication. The involvement of quantum computing shows that the entanglement/geometry duality works by employing a "quantum algorithm" that corrects the information. Most physical theories make use of gauges. Gauges give us evidence about two things: how the cosmos is organized; and why and how nature is "captured" by mathematical theories, opening a line of inquiry about the relationship between man and the universe. (shrink)

The idea of objectivity is primary in Gottlob Frege’s thought, not only for his conception of logic and mathematics, but also for his philosophy as a whole. He deals with the topic for the first time in 1884, in Grundlagen der Arithmetik, precisely in paragraph 26. He distinguishes the objective from the subjective, of course, but also from the real, what he calls the actual (wirklich in German). In order to be perfectly understood, he gives as example the colors but, (...) above all, to explain what is objective in geometry and space, he proceeds to a «thought-experiment», founded on the principle of duality. It is this aspect that I emphasize here, in the general framework ofparagraph 26 and in conjunction with Frege’s other works. (shrink)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, (...) neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change. (shrink)

The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.

The essay brings a summation of human efforts seeking to understand our existence. Plato and Kant & cognitive science complete reduction of philosophy to a neural mechanism, evolved along elementary Darwinian principles. Plato in his famous Cave Allegory explains that between reality and our experience of it there exists a great chasm, a metaphysical gap, fully confirmed through particle-wave duality of quantum physics. Kant found that we have two kinds of perception, two senses: By the spatial outer sense we (...) perceive phenomena, objects in space. Our temporal inner sense lets us perceive our inner state, noumena . Kant's two senses are fully confirmed through bilateral brain duality of cognitive science. The bilateral brain serves inner, temporal sense, logic & noumena in the left hemisphere. The right brain half is for the outer, spatial sense, geometry & phenomena. We form a whole system, a bilateral interior cosmos, a kind of world model, expressing what may lie beyond the metaphysical gap. In the right brain we build a phenomenal cosmos, to resemble the outer environment. In the brain's left hemisphere a noumenal cosmos, a model of our private world including Saint Teresa of Avila's castillo interior. The cosmos is regularly updated with novel sense data integrated into our memory banks. This was recognized early by Helmholtz (1860s). Our lives, health & well-being depend on us keeping our interior cosmos in good order. This used to be seen as saving one's precious soul from perdition. While we endeavour to keep our phenomenal cosmos neat & orderly by protecting the environment from harm, our noumenal cosmos to be livable requires us to engage in ethical conduct. This bilateral cosmos is responsible for our common sense judgment power, that we depend on for being able to lead a good & benign life. The 200 million neurons of BA10 in the prefrontal lobe have global access to the interior cosmos, & apply massive feedback to identify environmental conditions rapidly. The global access also gives conscious presence of the individual to itself, all that is present in the interior cosmos, for total freedom of action. In the left brain private ego, we are guarded by faith in divine mercy, & guided to choose the best course of action, able to survive under the most adverse conditions. For this, we receive directions from the Holy Ghost in Saint Teresa's castillo interior. (shrink)

The hexagonal structure for ‘the geometry of logical opposition’, as coming from Aristoteles–Apuleius square and Sesmat–Blanché hexagon, is presented here in connection with, on the one hand, geometrical ideas on duality on triangles (construction of ‘companion’), and on the other hand, constructions of tripartitions, emphasizing that these are exactly cases of borromean objects. Then a new case of a logical interest introduced here is the double magic tripartition determining the semi-ring ${\mathcal{B}_3}$ and this is a borromean object again, (...) in the heart of the semi-ring ${{\rm Mat}_{3}(\mathbb{B}_{\rm Alg})}$ . With this example we understand better in which sense the borromean object is a deepening of the hexagon, in a logical vein. Then, and this is our main objective here, the Post-Mal’cev full iterative algebra ${\mathbb{P}_4 = \mathbb{P}(\mathbb{F}_4)}$ of functions of all arities on ${\mathbb{F}_4}$ , is proved to be a borromean object, generated by three copies of ${\mathbb{P}_2}$ in it. This fact is induced by a hexagonal structure of the field ${\mathbb{F}_4}$ . This hexagonal structure is seen as precisely a geometrical addition to standard boolean logic, exhibiting ${\mathbb{F}_4}$ as a ‘boolean manifold’. This structure allows to analyze also ${\mathbb{P}_4}$ as generated by adding to a boolean set of logical functions a very special modality, namely the Frobenius squaring map in ${\mathbb{F}_4}$ . It is related to the splitting of paradoxes, to modified logic, to specular logic. It is a setting for a theory of paradoxical sentences, seen as computations of movements on the bi-hexagonal link among the 12 classical logics on a set of 4 values. (shrink)

ZusammenfassungDas Axiomensystem der ebenen projektiven Geometrie ist dual in dem Sinne, dass es bei Vertauschung der Begriffe « Punkt » und « Gerade » in sich übergeht. Daraus folgt, dass mit jedem Satz auch der duale Satz aus den Axiomen beweisbar ist. Dagegen kann aus der Dualität des Axiomensystems nicht geschlossen werden, dass in einem Modell mit jedem Satz auch der duale Satz gilt; noch weniger folgt, dass ein Modell eine eineindeutige Abbildung zulässt, welche Punkte and Geraden unter Erhaltung der (...) Inzidenz vertauscht. Im Falle der projektiven Geometrie sind solche Modelle bekannt; für die einfache Typentheorie wird gezeigt, dassa die Existenz solcher Modelle äquivalent ist mit der Widerspruchsfreiheit des Systems « New Foundations ».Zusatz. Der folgende Satz beantwortet die beiden in der Arbeit aufgeworfenen Fragen: Eine vollständige Theorie mit einem Automorphismus hat ein Modell, welches einen entsprechenden Automorphismus zulässt. Insbesondere ist somit NF widerspruchsfrei, wenn die einfache Typentheorie mit den zusätzlichen Axiomen S ≡ S* widerspruchsfrei ist.RésuméLe système d'axiomes de la géométrie projective plane est dualistique dans le sens qu'il se transforme en lui‐même si I'on y échange les notions « points » et « droite ». Cette dualité entraîne un parallélisme des théorémes et de leurs démonstrations. Par contre, la dualité d'un système d'axiomes ne nous permet pas de conclure au parallélisme des énoncés qui soient vrais dans un modéle donné du systéme; moins encore peut‐on insérer I'existence d'une transformation biunivoque échangeant points et droites et conservant la relation d'incidence. Pour la géométric projective, de tels modèles sont bien connus; pour la théorie simple des types, I'existence d'un tel modèle est équivalent a la cohérence du système « New Foundations ».Remarque. Le théoréme suivant résout les deux problèmes proposés dans le travail: Toute théorie complète admettant un automorphisme possède un modèle admettant un automorphisme correspondant. NF est par conséquent cohérent si la théorie simple des types avec les axiomes supplémentaires S ≡ S * est cohérente.The axiom system of plane projective geometry is dual in the sense that it is transformed into itself by exchange of the notions « point » and « line.» It follows that for every theorem the dual sentence is also a theorem. However, from the duality of the axiom system one cannot conclude that in a model the truth of a sentence implies that of the dual sentence; even less can one conclude that each model admits a 1‐1‐transformation interchanging points and lines and preserving the incidence relation. For projective geometry, models of this kind are well known. For the simple theory of types it is shown that the existence of such models is equivalent to the consistency of « New Foundations. »Additional remark. The following theorem answers both of the questions proposed in the paper: If it is complete, then a theory with an automorphism has a model with a corresponding automorphism. NF is therefore consistent if simple theory of type with the additional axioms S ≡ S* is consistent. (shrink)

This paper explores the meaning of time from three points of view: (1) David Bohm's concepts of "vertical implicate order" and "holomovement"; (2) Alfred North Whitehead's idea of the "actual occasion"; and (3) the author's notion of "nondual duality." The author argues that Bohm and Whitehead alike implicitly divide time into dual and nondual aspects and that, in failing to adequately reconcile these, time, in effect, is denied. The alternative offered seeks to thoroughly integrate dual and nondual (holistic) modalities (...) in the understanding that time as becoming entails a dynamic interpenetration of its opposing aspects. Visual geometry and topology are employed to flesh out the "nondual duality" of temporal structure. (shrink)

I examine the relationship between \\)-dimensional Poincaré metrics and d-dimensional conformal manifolds, from both mathematical and physical perspectives. The results have a bearing on several conceptual issues relating to asymptotic symmetries in general relativity and in gauge–gravity duality, as follows: I draw from the remarkable work by Fefferman and Graham on conformal geometry, in order to prove two propositions and a theorem that characterise which classes of diffeomorphisms qualify as gravity-invisible. I define natural notions of gravity-invisibility that apply (...) to the diffeomorphisms of Poincaré metrics in any dimension. I apply the notions of invisibility, developed in, to gauge–gravity dualities: which, roughly, relate Poincaré metrics in \ dimensions to QFTs in d dimensions. I contrast QFT-visible versus QFT-invisible diffeomorphisms: those gravity diffeomorphisms that can, respectively cannot, be seen from the QFT. The QFT-invisible diffeomorphisms are the ones which are relevant to the hole argument in Einstein spaces. The results on dualities are surprising, because the class of QFT-visible diffeomorphisms is larger than expected, and the class of QFT-invisible ones is smaller than expected, or usually believed, i.e. larger than the PBH diffeomorphisms in Imbimbo et al. :1129, 2000, Eq. 2.6). I also give a general derivation of the asymptotic conformal Killing equation, which has not appeared in the literature before. (shrink)

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of (...) a polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\) . We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry. (shrink)

Za svjetlo možemo reći da je istinski razbuđivač mišljenja , kako u metafizici tako u prirodnoj filozofiji i znanosti. Želimo ukazati na posebnost Petrića kao onoga mislioca s kojim je dokončana epoha od antike nasljeđene metafizike svjetla i koji otvara put za novovjekovnu fiziku svjetla, naročito u djelu Panaugia, ali ne samo u njemu. Nakon Petrića uslijedila su brojna uzbudljiva znanstvena otkrića koja su rezultirala dubokim uvidima u narav svjetla, u geometriji, a posebice u fizikalnoj optici: svjetlo se širi konačnom (...) brzinom, složeno je, dio je elektromagnetskog spektra zračenja, djeluje na materiju , dovedeno je u svezu s masom i energijom. Ono nas začuđuje dualnošću svoje prirode čestice i vala. Istraživanjem zračenja tamnog tijela, suprotnosti vidljivog svjetla, otvorena je epoha kvantne fizike. Pitamo: Možemo li na osnovu mnoštva znanja o svjetlu do kojih su došli fizičari govoriti o bîti svjetla ? Fotonika je nova epoha u komunikacijskoj tehnologiji. Što tek treba očekivati od istraživanja fotosinteze?Light has been defined as a true instigator of thought both in metaphysicsand in philosophy of natureand science. This article focuses on the exceptional position of Frane Petrić as the philosopher with whom the epoch of metaphysics of light inherited fromantiquityended and the path for the new epoch of the physics of light started – especially inhiswork Panaugia,though not only in it. AfterPetrić, numerousexciting scientific discoveries occurred that resulted in profoundinsights into the nature of light – in geometry,and especiallyinphysical optics: light travels with finite speed, it is complex, it is part of electromagneticspecterof radiation, it has effect on matter , it has been relatedto mass and energy.Lightamazes us with the duality of its natureas both a particle and a wave. The researchon theradiationof dark objects and the contrast with visible light has opened the epoch of quantumphysics.The question this lectureposes is: Based on the body of knowledge about light accumulatedby the physicists, could we talk about the essence of light ? Photonics isanew epoch in communication technology.What then could we expect from the research on photosynthesis? (shrink)

Following a bi-cylindrical model of geometrical dynamics, our study shows that a 6D-gravitational equation leads to geodesic description in an extended symmetrical time–space, which fits Hubble-like expansion on a microscopic scale. As a duality, the geodesic solution is mathematically equivalent to the basic Klein–Gordon–Fock equations of free massive elementary particles, in particular, the squared Dirac equations of leptons. The quantum indeterminism is proved to have originated from space–time curvatures. Interpretation of some important issues of quantum mechanical reality is carried (...) out in comparison with the 5D space–time–matter theory. A solution of lepton mass hierarchy is proposed by extending to higher dimensional curvatures of time-like hyper-spherical surfaces than one of the cylindrical dynamical geometry. In a result, the reasonable charged lepton mass ratios have been calculated, which would be tested experimentally. (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)

We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John (...) ellipsoid we relate these geometric quantum states to the notion of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by Hilgevoord and Uffink. (shrink)

This complex and subtle book is difficult to summarize. The author intends it as a supplement to existing commentaries on Plato’s Meno, rather than as a straightforward commentary of his own. His approach to Plato builds upon that of Leo Strauss, Jacob Klein, and the Tübingen School, but is not reducible to any of these and contains other influences as well, such as Heidegger. In addition to taking with minute seriousness the dramatic composition of the dialogue, Brague combines precise and (...) sensitive philological tools with a marked ontological interest. He reads the Platonic dialogue in a bipolar perspective. Dialogues are tests of friendship but at the same time, as "works" or literary texts rather than objects, they may have a sense in themselves. Brague cites Proust in this context, but one thinks also of Derrida. Consequently the dialogue has both an intrinsic order, corresponding to the rhythmic development of its theme, and also an allusive dimension or a lacuna suggested by the text, to be filled in by the reader. In a similar sense one may say that the theme of the Meno is bipolar. The first pole is the theme of the nature of the soul, hence of virtue, and more specifically, of Meno’s soul. Meno’s defective nature is the basis for a continuous "degradation" in the stages of the dialogue. Alternatively, it is the basis for Meno’s "anthropological" approach to virtue. The key to the Meno is that the Good is missing from the dialogue. This observation provides a transition to the second thematic pole. The structure of the dialogue reflects the stages of Plato’s esoteric teaching as presented by K. Gaiser : dialectic, geometry, politics. This raises a question for Brague, which he does not seem to me to answer : if this is the order of Plato’s secret system, how is the degraded presentation of the system, relative to Meno’s defective soul, isomorphic to the hierarchical structure of the system? Granting that politics is lower than dialectic, there must be a better and a worse presentation of politics. Brague’s approach entails that the degraded image of the structure nevertheless reveals the structure itself. But if the Good is absent from the image, how are we to infer that it is present in the structure? A similar problem suggests itself with respect to what Brague calls the key to the movement of the dialogue: the distinction between the limiting and the limited and correspondingly, "the powerlessness of the limit" in Meno’s case. In the last and longest section of the book, there are many fine analyses and illuminating insights, too many to do justice to here. One may call special attention to the connection between the episode of recollection and geometry on the one hand, and the duality of principles in Plato’s system. I also found unusually helpful Brague’s exposition of the "feminine" nature of Plato’s teaching of genesis, of the link between Plato’s use of anakineisthai and the dramatic device of a prologue, and the significance of katasëmainesthai in conjunction with the theme of "growing" and preserving virtuous youths. In sum: this is a book to be read carefully and slowly by every serious student of Plato. It is almost as difficult, and hence as controversial, as a Platonic dialogue. It is marked throughout, even when influenced by contemporary schools of thought, by a genuine interest in Plato’s manner of thinking and writing. This is not exactly a common property of contemporary books on Plato.—S.R. (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 final version of the paper is published pp. 117-166 in: Myrdene Anderson and Floyd Merrell (eds.): On Semiotic Modeling . Mouton de Gruyter, Berlin and New York, 1991.

We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.

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.

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)

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)