We consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.

We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincaré transformations of a plane. In particular, a 1-parametric family of natural deformations of (...) the Poincaré group is described. We also visualized the underlying groups of Galilean, Euclidean, and pseudo-Euclidean rotations within the special linear group. (shrink)

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates . This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted (...) to those known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry. (shrink)

We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to analyse certain family of affine logics with inclusion and convexity as primitives interpreted over real spaces of increasing dimensionality. In this article we show that logics of different dimensionalities must have different theories, thus justifying further work (...) on different dimensions. We then focus on the three-dimensional case, exploring the expressiveness of this logic and consequently showing that it is possible to construct formulas describing a three-dimensional coordinate frame. The final result, making use of the high expressive power of this logic, is that every region satisfies an affine complete formula, meaning that all regions satisfying it are affine equivalent. (shrink)

A complete set of mutually unbiased bases for a Hilbert space of dimension N is analogous in some respects to a certain finite geometric structure, namely, an affine plane. Another kind of quantum measurement, known as a symmetric informationally complete positive-operator-valued measure, is, remarkably, also analogous to an affine plane, but with the roles of points and lines interchanged. In this paper I present these analogies and ask whether they shed any light on the existence or (...) non-existence of such symmetric quantum measurements for a general quantum system with a finite-dimensional state space. (shrink)

We introduce and study a variety of modal logics of parallelism, orthogonality, and affine geometries, for which we establish several completeness, decidability and complexity results and state a number of related open, and apparently difficult problems. We also demonstrate that lack of the finite model property of modal logics for sufficiently rich affine or projective geometries (incl. the real affine and projective planes) is a rather common phenomenon.

The uniqueness of the parallel lines is independent from the analogous statement on parallel planes and the usual further axioms of three-dimensional affine geometry. MSC: 51A15, 03F65.

This essay examines Deleuze’s relation to new materialism through an engagement with new materialist claims about the human and nonhuman relation and about agency. It first considers the work of Elisabeth Grosz and then moves on to a consideration of Deleuze’s own conception of a new materialism/new naturalism. I seek to show that Deleuze is an ethically motivated naturalist concerned with an ethical pedagogy of the human, which he derives from his reading of Spinoza. I seek to illuminate some of (...) the principal features of this ethically guided materialism/naturalism and show that even in his later work with Felix Guattari, which situates all life, human and nonhuman, on a plane of immanence, there remains a recognition that the human animal is ethically distinguished as the inventive species par excellence. My main claim, then, is that Deleuze’s project cannot be aligned with a new materialism that supposes a flat ontology and that does away with an ethical distinction between the human and the nonhuman. Although Deleuze bequeaths a complex legacy to post-modern thought in his thinking about the human, it should not be supposed that he has no affinities with aspects of a humanist position and pedagogy. (shrink)

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows (...) solutions to geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

The essay focuses on Gloria Anzaldúa’s narrative of overcoming shame in Borderlands/La Frontera. It addresses the question of coming to terms with multiple conditions of oppression obstructing the creative agency of radical Latina subjects. The discussion occurs along two intersecting planes: (1) the existential question of self-knowledge as the self undergoes the difficult process of identifying and releasing the weight of past oppressions and (2) the situated character of Anzaldúa’s Chicana/Latina condition in light of the heteronormative and epistemic constraints she (...) faces as a Latina writer whose work is intimately interlinked with her sense of self. Anzaldúa’s message of overcoming shame is shown to have some affinities with Friedrich Nietzsche’s reflections on freedom from shame as the ultimate seal of liberation. Considered as Anzaldúa’s companion in a journey in self-overcoming at the crossroads of indigenous and Chicana/Tejana cultures, her new mestiza takes on a new significance. (shrink)

An experience of helplessness during the production of a collective autobiographical narrative offers an opportunity to explore points of convergence between Adriana Cavarero's postural philosophy and Bracha Lichtenberg Ettinger's matrixial borderlinking. The narrative is treated as a live scene that enfolds the movement of the drive conceptualized in such a way as to avoid the prioritization of death over life. Five successive moves, inspired by Ettinger's rotation of the phallic prism, illuminate affinities between the two thinkers. The first rotation explores (...) the matrixial and relational qualities that compose the scene. The second rotation approaches death from a notional beginning, prioritizing the presence of the mother and her assymmetrical relationship to the infant. The third rotation emphasizes the transformational character of Ettinger's matrixial movement and border-crossing, especially when examined in conjunction with Cavarero's maternal inclination. In the fourth rotation we show how inclination in Lacanian psychoanalysis is linked to the disappearance of the mother from the scene. The introduction of the mother disrupts the Lacanian logic with the possibility of a beginning just as immanent as desire and lack. Movement, qua embrace and unity-disunity, becomes an indispensable component of feminist ontology that captures the immanence of sharing in the matrixial-maternal plane. (shrink)

The aim of this paper is to use the approaches of the Korean philosopher Byung-Chul Han, on the one hand, to show the challenges that formulate and define our times and, on the other, to focus this view on education. In this sense, it is necessary to define the post-pandemic 21st century society marked by psychopolitics. Education is subject to a framework of unprecedented complexity, due to digitalisation and the acceleration of a world that is perceived as progressing, and yet (...) is becoming increasingly ill due to an excess of positivity. The pedagogy that operates through pleasure and the new relational plane of social networks, which formulate affine and disembodied encounters, eliminates negativity. Education lends itself to a demanding examination as the starting point of this society and is seen as the place where the possibility of transformation arises. In the latter sense it is argued that philosophy in our classrooms has a crucial role to play. (shrink)

In the chapter on multiplicity and unity, the affective or anthropological motifs are both more complex and more interesting. Wölfflin’s initial distinction is between “the articulated system of forms of classic art and the flow of the baroque” . Imagery of fluidity pervades the chapter, for water, according to Wölfflin, “was the period’s favourite element” . “Now, and now only,” he says, “the greatness of the sea could find its representation”, and as if to inculcate this affinity he places the (...) reproduction of a baroque seascape by Jan van Goyen at the head of the introduction to the book and a riverscape by Peter Brueghel at the head of this chapter, even though neither painting is discussed where it is reproduced. In fact it is worth observing that Wölfflin does not discuss any water paintings in this chapter, though of course he does so elsewhere. Where fluidity becomes the meaning of his category, it is absent from the contents of the paintings. Wölfflin’s procedure, as I have argued, is both objectively analytical and subjectively interpretive, and in this chapter he seems careful to preserve the distance between the forms he describes and the significances he reveals. Were he to treat water paintings here, he would obscure the fact that his analyses are always the prelude to translations.Though he conceals the fact, Wölfflin has here effected a translation of the baroque into itself, of water painting into fluidity. Baroque art has declared its true meaning, which is to be an art of flux—of time and, throughout this chapter, of momentariness. Suddenly here the baroque comes into its own, with a surprising reversal in Wölfflin’s categories. Until now he has associated the baroque with lawlessness and confusion, and classicism with the unifying force of symmetrical organization around a center. Unity is repose—the equation had been made explicitly in the discussion in Classic Art of Michelangelo’s Medici Madonna —and clearly in Principles of Art History Wölfflin seems to say that the unification achieved in Leonardo’s Last Supper was later lost Tiepolo’s version.21 As Wölfflin says in the first sentence of chapter 4, “The principle of closed form of itself presumes the conception of the picture as a unity.” But as the baroque now comes into its own, it appears that the unity of classicism is an illusory, “multiple unity,” whereas the true or “unified unity” actually pertains to the baroque. It is the usurping baroque, rather than the deposed classic, that now has “a dominating central motive.” And So Wölfflin returns in this chapter to the two Last Suppers in order to rescind his earlier position. He still claims that Leonardo’s painting is unified, but he offers Tiepolo’s version to illustrate the “possibility of surpassing this unity” . In becoming itself, baroque art has overthrown classicism.21. “Tiepolo composed a Last Supper which, while it cannot be compared with Leonardo as a work of art, stylistically presents the absolute opposite. The figures do not unite in the plane, and that decides”. (shrink)

This article discusses epistemological and methodological problems brought forth during the study of management practices in companies and organisations based on an ethnomethodological approach. Ethnomethodological issues and knowledge in organaisation management and the complexity of the involvement of the researcher will be discussed by way of analysis of six controversial reports on the involvement of the researcher. Our aim is to clarify the nature of the work carried out by the researcher. Therefore the questions of the neutrality of the researchers (...) and the dialectical immersion-distancing with regards to the objects being studied will be discussed. The aim of this article is to show the theoretical and epistemological interest of the organisation ethnomethodology in a constructivist perspective with regards to management sciences research. (shrink)

The plane of the present is a concept that is useful for discussing the various paradigms of time. Here by ‘plane of the present’ we mean the temporal interface that represents the present instant and that forms the boundary between the past and the future. We use the geometrical term ‘plane’ to indicate an extended surface in the space-time continuum, as opposed to a ‘point’ on some time axis. This point/plane dichotomy is intended to raise issues (...) of extension and simultaneity and to examine the degree to which these are meaningful concepts from various physical viewpoints. We will show by example in the present work that the plane of the present is a pivotal concept that offers considerable power in differentiating between various views of the nature of time. The concept of time within the main stream of physics thinking has followed a rather convoluted path over the past three millennia. Anticipating the modern motion picture, Zeno of Elea (c.490-c.430 B.C) questioned whether time should appropriately be viewed as a continuously flowing river, or should more properly be considered as a rapid sequence of stop-motion ‘freeze-frames’, in effect rendering geometrical each instant as a separate infinitesimal point on the line of time. Adopting this view, he asked how physical motion could occur. He argued paradoxically that motion is not possible, since it appears to happen only between the frozen frames of time instants.1 From the viewpoint of Zeno, the plane of the present would be simply the last and most recent in this sequence of freeze-frames. It would be that frozen instant, spanning the universe, which changes progressively as the instant we call ‘now’ becomes the frozen past and future possibility freezes into the ‘now’ of present reality. We note that the plane of the present as a concept does not resolve the arrow paradox that Zeno raised. It only provides a way of thinking about it. (shrink)

With the recent report of a live birth after use of Mitochondrial replacement therapy, sometimes called ‘Three-parent IVF’, the clinical application of the technique is fast becoming a reality. While the United Kingdom allows the procedure under regulatory scrutiny, it remains effectively outlawed in many other countries. We argue that such prohibitions may violate individuals’ procreative rights, grounded in individuals’ interest in genetic affinity. The interest in genetic affinity was recently endorsed by Singapore’s highest court, reflecting an emphasis on the (...) importance of biological ties found across the globe. We apply that reasoning to make the case for a right to ‘Three-parent IVF’. (shrink)

We show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically “constructivize” classical definitions, handling the resulting bookkeeping (...) automatically. (shrink)

Elective Affinities: Musical Essays on the History of Aesthetic Theory collects a selection of Lydia Goehr's recent essays. In them she traces “a history of attraction and reaction … of music to philosophy, drama, birdsong, crime, film, and nationhood” . Goehr examines the ways that philosophers, the ideas that they present, and works of art display “elective affinities”. Her procedure is like that of an art historian who presents parallel slides to reveal visual affinities, even between artists who themselves were (...) unaware of each other. Her analyses are erudite, lucid, and always suggestive, but what I found most admirable in Elective Affinities is Goehr's extraordinarily brave experimentation with a novel form of philosophy-writing, the adumbration of which is the focus of this review. Her book is strange enough to be genuinely magnificent and lastingly influential. (shrink)

In Black Panther, T'Challa and Erik "Killmonger" Stevens experience audiences with their departed fathers. Careful analysis of the second and especially the third Ancestral Plane visits downplays the metaphysical, leading toward a metaphor for better understanding one's place in the world. M'Baku fights gallantly, but T'Challa prevails, and bolsters his worthiness for the throne by sparing M'Baku's life. T'Challa rises from the ground, not in the entombed Hall of Kings, but in what appears to be an African savanna where (...) the sky is rippled in majestic blues and purples, giving off a twilight ambiance. Ingesting it increases one's strength, speed, agility, and stamina. It is akin to Captain America's super soldier serum, which reminds us of that awesome footrace between T'Challa and Steve in Avengers: Infinity War. The Heart Shaped Herb potion, then, helps T'Challa and Killmonger achieve greater focus on each man's past and allows a better perspective on who each one. (shrink)

Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP q ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem (...) forCP 2-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE +, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem. (shrink)

The medicalization of mental health remains a point of contention for bioethicists, especially as it concerns the epistemic capabilities of those diagnosed with an illness or disorder. Gosselin (2019) argues that biomedicalization commits epistemic injustices against these persons and consequently entraps them in a “cycle of vulnerability”; in response, she proposes principles of justice to defend them from such affronts. This paper builds off of her work and responds particularly to the demand for a “sociocentric view of the self as (...) essentially relational.” I present a theory of interdependent agency and affiliation that I contend conceptually bolsters her normative principles. I explain how an expanded use of relational epistemologies, united with a non-ideal theory of mental health, can enrich our hermeneutical resources with respect to those with mental illnesses/disorders. My account introduces normative considerations premised on interdependency, most notably from care theory. Concepts such as vulnerability, relational autonomy, attentiveness, and responsiveness ground a relationally-situated approach that (1) improves the epistemic positionalities of patients, (2) informs more suitable dynamics of care/treatment, and (3) unites groups of mutually interested actors against harm and injustice. I thus use a framework of care to promote affiliations of similarly disadvantaged persons under shared causes and initiatives. I refer to these assemblages as “caring affinity networks.”. (shrink)

A new dimension is given to modulo theory by defining MOD planes. In this book, the authors consolidate the entire four quadrant plane into a single quadrant plane defined as the MOD planes. MOD planes can be transformed to infinite plane and vice versa. Several innovative results in this direction are obtained. This paradigm shift will certainly lead to new discoveries.

This article considers the affinities in the socio-political thought of Emmanuel Levinas and Richard Rorty. The writings of both display considerable concern for the suffering of others. Both authors note the importance of a self-critical subject becoming more aware of its own injustice as very important for recognizing our responsibilities to others. Furthermore, both stress the importance of recognizing the other outside of the usual, objectifying categories, since it is the uniqueness of the other that reminds us of our responsibility (...) for the other. Both writers view the liberal state as the best political forum in which to realize a fuller recognition of and responsibility towards the other, a form of state in which the ethical constantly interrupts the political. Rorty and Levinas disagree, however, on the legitimacy of not responding to the other. Key Words: Critchley • irony • justice • Levinas • liberal state • other • responsibility • Rorty • sentimental education. (shrink)

A personalized recommender system is broadly accepted as a helpful tool to handle the information overload issue while recommending a related piece of information. This work proposes a hybrid personalized recommender system based on affinity propagation, namely, APHPRS. Affinity propagation is a semisupervised machine learning algorithm used to cluster items based on similarities among them. In our approach, we first calculate the cluster quality and density and then combine their outputs to generate a new ranking score among clusters for the (...) personalized recommendation. In the first phase, user preferences are collected and normalized as items rating matrix. This generated matrix is then clustered offline using affinity propagation and kept in a database for future recommendations. In the second phase, online recommendations are generated by applying the offline model. Negative Euclidian similarity and the quality of clusters are used together to select the best clusters for recommendations. The proposed APHPRS system alleviates problems such as sparsity and cold-start problems. The use of affinity propagation and the hybrid recommendation technique used in the proposed approach helps in improving results against sparsity. Experiments reveal that the proposed APHPRS performs better than most of the existing recommender systems. (shrink)

Uno de los puntos más recurrentes en los últimos años en las políticas sociales en la Argentina es la presencia cada vez más importante de las mujeres como "sujetos" de dichas políticas. En contraste con la difusión oficial de los beneficios de la intervención en los contextos de pobreza y expulsión social, los relatos de las mujeres dibujan un conjunto de experiencias y sensibilidades de segregación y dependencia cada vez mayor. El presente artículo tiene por objetivo hacer evidente desde la (...) voz de las mujeres pobres del conurbano bonaerense, las políticas de los cuerpos y de las emociones que los planes que ellas reciben construyen e implican. La estrategia argumentativa es la siguiente: a) se sintetiza el punto de partida teórico y metodológico del trabajo, b) se presenta un bosquejo de la problemática desde los relatos de las mujeres, y c) se resumen algunas de las conclusiones del estudio realizado. (shrink)

The history of the Nicaraguan Revolution has received considerable analytical attention. Typically, the successful overthrow of the Somoza regime in the late 1970s is associated with the Frente Sandinista de Liberación Nacional, a Marxist/socialist inspired vanguard group. While the role Christians played in the revolution is often acknowledged as a significant one, in part because many Sandinista cadres were Christian revolutionaries, little attention has been paid to the degree to which Sandinismo, as a unique perspective on socialism, shares elective affinities (...) with liberation theology, a prophetic expression of Christianity. This manuscript sets out to explore the relationship between liberation theology and Sandinismo-as-socialism. It starts by considering the perspective of Christian revolutionaries on this relationship. It then identifies the electives affinities between the aforementioned cultural frameworks, and it evaluates the Nicaraguan Revolution in light of these elective affinities. The latter evaluation includes exploring the connection between Saint Paul and what it means to be a Sandinista revolutionary. (shrink)

We show that K-generic projective planes have Morley rank either two or infinity. We also show give a direct argument that such planes are not Desarguesian.

In his book, Earth and Other Ethics, Christopher Stone attempts to account for the moral dimension of our lives insofar as it extends to nonhuman animals, plants, species, ecosystems, and even inanimate objects. In his effort to do this, he introduces a technical notion, the moral plane. Moral planes are defined both by the ontological commitments they make and by the governance mIes (moral maxims) that pertain to the sorts of entities included in the plane. By introducing these (...) planes, Stone is left with a set of problems. (1) Do the planes provide anything more objective than a set of alternative ways of looking at moral problems? (2) How can one resolve apparent conflicts between the recommendations forthcoming from distinct planes? (3) Why do certain entities constitute moral planes; and how do we decide which planes to “buy into?” Stone’s answers to these questions endorse aseries of concessions to moral relativism. In this paper I outline an alternative to Stone’s moral planes which, while sympathetic to his ethical concerns, comes down squarelyon the side of moral realism. (shrink)

The paper is the first of two to be published in Cognitio which explore the hypothesis that the thought of Ralph Waldo Emerson (1803- 1882), brilliantly expounded in the generation before Charles Sanders Peirce (1839-1914), anticipated, if not provided the direct provenance of, Peirce’s mature metaphysical ideas. The papers provide running commentaries on Emerson’s later-phase essays, “Poetry and Imagination” (1854, published in 1876) and “The Natural History of Intellect” (1870). “Poetry and Imagination” is shown to contain the seeds of Peirce’s (...) objective idealism — namely, that “matter is effete mind,” or “mind hide-bound with habits,” — set within an evolutionary cosmology that grounds the human mind’s connaturality or affinity with the laws of nature, a doctrine subtending his abductory logic of scientific discovery. Mutatis mutandis, all these components were already present in Emerson’s essay. Peirce spoke of his “buddhisto-christian religion,” which was another name for Emersonian cosmosemiosis. His originality consisted in applying Emerson’s emphasis on the poetic to the scientific imagination, though the seeds of the latter are present in Emerson’s “The Natural History of Intellect” and other essays as well. (shrink)

El presente artículo aborda un tema novedoso dentro del Derecho, siendo objeto de estudio del Derecho Urbanístico en especial, tanto para la enseñanza pregraduada como postgraduada, que analiza el alcance jurídico de los planes de ordenamiento territorial y urbanismo como auténticas normas jurídicas, su contenido, elaboración y vía de aprobación, constatando la importancia de otorgarle a dichos planes autenticidad jurídica, no como simple instrumento jurídico, sino como norma que integra el ordenamiento normativo de nuestra sociedad.

This book is about what makes law possible. A stranger to contemporary legal practice might think such a book unnecessary, but the eight authors of this book share the view that what makes law possible is under siege today. The authors also share the hope that by exploring how law is a humanistic practice that involves whole persons, the siege will be reversed. The pathbreaking work of University of Michigan Law professor Joseph Vining provides the authors' focus for their varied (...) analyses of how law works not through force but, instead, through affinity.Vining's four books and other writings, spanning four decades, reveal the hidden connections by which men and women freely create and sustain a world of meaning through the phenomena we associate with law. Drawing on legal philosophy, theology, musicology, and other humanistic disciplines, the authors join Vining in discovering how law is, as Vining has written, “evidence of view and belief far stronger than academic statement or introspection can provide.” Law as Vining and the other authors reveal it is evidence of our better selves, not of the totalizing and brutalizing selves humans are capable of becoming, sometimes even under cover of law.In addition to the three editors, the book's authors are Joseph Vining, Rev. John McCausland, Hon. John T. Noonan, Jr., Steven Smith, and James Boyd White.Lawyers and all who care about law, the human future, and what human freedom can do to connect person to person as valued will find much to ponder in the chapters of this book. By avoiding jargon and the cliché, the authors follow Vining's lead in illuminating the deep springs of law's vitality and authority. (shrink)

We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ∀∃∀, but that there is an axiom system, all of whose axioms are ∀∃∀∃ sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields.

Phase models for affine linear logic were independently devised by Lafont [10] and Piazza [15], although foreshadowed by Ono [14]. However, the existing semantics either contain no explicit directions for the construction of models in the general case, or else are forced to resort to additional conditions extending Girard's semantics. We dispense with these extra postulates - at least for the subexponential fragment of this logic - considering structures where the set of antiphases is concretely constructed. Moreover, we show (...) the equivalence of our phase models and the algebraic models of affine linear logic by means of a representation theorem. (shrink)

Several authors have suggested that a more parsimonious and conceptually elegant treatment of everyday mereological and topological reasoning can be obtained by adopting a spatial ontology in which regions, not points, are the primitive entities. This paper challenges this suggestion for mereotopological reasoning in two-dimensional space. Our strategy is to define a mereotopological language together with a familiar, point-based interpretation. It is proposed that, to be practically useful, any alternative region-based spatial ontology must support the same sentences in our language (...) as this familiar interpretation. This proposal has the merit of transforming a vague, open-ended question about ontologies for practical mereotopological reasoning into a precise question in model theory. We show that (a version of) the familiar interpretation is countable and atomic, and therefore prime. We conclude that useful alternative ontologies of the plane are, if anything, less parsimonious than the one which they are supposed to replace. (shrink)

In the Critique of Pure Reason Kant introduced both transcendental idealism and transcendental arguments into philosophy. Transcendental arguments in general aim to establish conditions necessary for our having self-conscious experience at all. Transcendental idealism holds that such conditions do not hold independently of human subjects; those conditions obtain or are satisfied because they are generated or fulfilled by the structure or functioning of the subject’s cognitive capacities. Is transcendental idealism the only possible explanation of such conditions? I pursue this question (...) by exploring a widely neglected issue, the transcendental affinity of the manifold of intuition. I argue the following: (1) This issue remains vital to the second edition of the Critique, even though many passages on the topic were omitted from that edition. (2) Kant’s link between transcendental idealism and transcendental arguments is substantive, not methodological. (3) Kant’s views on transcendental affinity show that there are non-subjective, transcendental material conditions for the possibility of unified self-conscious experience. (4) These conditions and Kant’s arguments for them directly undermine Kant’s own arguments for transcendental idealism. This criticism of Kant’s arguments for transcendental idealism is entirely internal to the first Critique. (5) These points reveal some serious flaws in Henry Allison’s defense of Kant’s idealism. (6) Realists, naturalists, and pragmatists have much to learn and to borrow from Kant’s transcendental analysis of the conditions of unified self-conscious experience. (shrink)

This essay juxtaposes concepts created by Gilles Deleuze and Félix Guattari with worlds imagined by Ursula Le Guin in a performance of ‘rhizosemiotic play’ that explores some possible ways of generating and sustaining what William Pinar calls ‘complicated conversation’ within the regime of signs that constitutes an increasingly internationalized curriculum field. Deleuze and Guattari analyze thinking as flows or movements across space. They argue, for example, that every mode of intellectual inquiry needs to account for the plane of immanence (...) upon which it operates—the preconceptual field presupposed by the concepts that inquiry creates. Curriculum inquiry currently operates on numerous nationally distinctive planes of immanence. I argue that the internationalization of curriculum studies should not presume a singular transnational plane of immanence but, rather, envisage a process performed by curriculum scholars with the capacities and competencies to change planes—to move between one plane of immanence and another and/or to transform their own planes. My essay is a ‘narrative experiment’ that takes seriously Deleuze’s argument that a work of philosophy should be, in part, a kind of science fiction, and also takes inspiration from Le Guin’s science fictional stories of ‘changing planes’ to generate productive and disruptive transnational agendas in curriculum inquiry. (shrink)

In this paper I offer a reading of the 1790 Introduction to the Critique of Judgement intended to show that the Critique of Judgement itself attempts to make good a serious deficit in the argumentation of the Critique of Pure Reason. In effect, the conditions outlined in the Critique of Pure Reason could be fulfilled without experience being constituted. There must therefore be additional conditions for the possibility of experience. And an account of these is to be found in the (...) Critique of Judgement, despite the repeated insistence in the text itself that its strictures operate on the level of a merely 'reflective' judgement, quite distinct from the 'determinant' judgements underwriting the time-determination of logical operations in the first Critique. The new transcendental elements for the condition of possibility of experience turn out, through an analysis of the concept of affinity, to table a novel, transcendental, conception of matter irreducible to the experiential understanding of matter and conditioning its possibility. In conclusion, I suggest that the structure of reflective judgement can itself be understood as a description of the operation of this transcendental matter in the special case of the transcendental spontaneity required by the Kantian subject to constitutive experience. (shrink)