The role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the “geometry of visibles”, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s “geometry of visibles” and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject (...) to a choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles. (shrink)

In a brief but remarkable section of the Inquiry into the Human Mind, Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid—theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, R. B. Angell, and others with discovering non-Euclidean geometry over half a century before the mathematicians—sixty years before Lobachevsky and ninety years before Riemann. I believe that Reid does indeed (...) have a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid’s geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid’s case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known—his direct realist theory of perception. (shrink)

In a brief but remarkable section of the Inquiry into the Human Mind, Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid—theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, R. B. Angell, and others with discovering non-Euclidean geometry over half a century before the mathematicians—sixty years before Lobachevsky and ninety years before Riemann. I believe that Reid does indeed (...) have a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid’s geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid’s case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known—his direct realist theory of perception. (shrink)

The subject of this investigation is the role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the ‘geometry of visibles’. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s ‘geometry of visibles’ and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject (...) to a choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles.While Thomas Reid is well-known as the leading exponent of the Scottish ‘common-sense’ school of philosophy, his role in the history of geometry has only recently been drawing the attention of the scholarly community. In particular, several influential works, by N. Daniels and R. B. Angell, have claimed Reid as the discoverer of non-Euclidean geometry; an achievement, moreover, that pre-dates the geometries of Lobachevsky, Bolyai, and Gauss by over a half century. Reid’s alleged discovery appears within the context of his analysis of the geometry of the visual field, which he dubs the ‘geometry of visibles’. In summarizing the importance of Reid’s philosophy in this area, Daniels is led to conclude that ‘there can remain little doubt that Reid intends the geometry of visibles to be an alternative to Euclidean geometry’;1 while Angell, similarly inspired by Reid, draws a much stronger inference: ‘The geometry which precisely and naturally fits the actual configurations of the visual field is a non-Euclidean, two-dimensional, elliptical geometry. In substance, this thesis was advanced by Thomas Reid in 1764...’, 2 The significance of these findings has not gone unnoticed in mathematical and scientific circles, moreover, for Reid’s name is beginning to appear more frequently in historical surveys of the development of geometry and the theories of space., 3Implicit in the recent work on Reid’s ‘geometry of visibles’, or GOV, one can discern two closely related but distinct arguments: first, that Reid did in fact formulate a non-Euclidean geometry, and second, that the GOV is non-Euclidean. This essay will investigate mainly the latter claim, although a lengthy discussion will be accorded to the first. Overall, in contrast to the optimistic reports of a non-Euclidean GOV, it will be argued that there is a great deal of conceptual freedom in the construction of any geometry pertaining to the visual field. Rather than single out a non-Euclidean structure as the only geometry consistent with visual phenomena, an examination of Reid, Daniels, and Angell will reveal the crucial role of geometric ‘conventions’, especially of the metric sort, in the formulation of the GOV. Consequently, while a non-Euclidean geometry is consistent with Reid’s GOV, it is only one of many different geometrical structures that a GOV can possess. Angell’s theory that the GOV can only be construed as non-Euclidean, is thus incorrect. After an exploration of Reid’s theory and the alleged non-Euclidean nature of the GOV, in 1 and 2 respectively, the focus will turn to the tacit role of conventionalism in Daniels’ reconstruction of Reid’s GOV argument, and in the contemporary treatment of a non-Euclidean visual geometry offered by Angell. Finally, in the conclusion, a suggestion will be offered for a possible reconstruction of Reid’s GOV that does not violate his avowed ‘direct realist’ theory of perception, since this epistemological thesis largely prompted his formulation of the GOV. (shrink)

In his 'Inquiry', Reid claims, against Berkeley, that there is a science of the perspectival shapes of objects ('visible figures'): they are geometrically equivalent to shapes projected onto the surfaces of spheres. This claim should be understood as asserting that for every theorem regarding visible figures there is a corresponding theorem regarding spherical projections; the proof of the theorem regarding spherical projections can be used to construct a proof of the theorem regarding visible figures, and vice versa. I reconstruct Reid's (...) argument for this claim, and expose its mathematical underpinnings: it is successful, and depends on no empirical assumptions to which he was not entitled about the workings of the human eye. I also argue that, although Reid may or may not have been aware of it, the geometry of spherical projections is not the only geometry of visible figure. (shrink)

Thomas Reid (1710–1796) presented a two-dimensional geometry of the visual field in his Inquiry into the human mind (1764), whose axioms are different from those of Euclidean plane geometry. Reid’s ‘geometry of visibles’ is the same as the geometry of the surface of the sphere, described without reference to points and lines outside the surface itself. Interpreters of Reid seem to be divided in evaluating the significance of his geometry of visibles in the (...) history of the discovery of non-Euclidean geometries. The question will be reexamined with particular attention given to his unpublished manuscripts. These include comments on Saccheri’s work and Reid’s repeated attempts to derive Euclid’s parallel postulate from the axioms of incidence. (shrink)

An explication is offered of Reid’s claim (discussed recently by Yaffe and others) that the geometry of the visual field is spherical geometry. It is shown that the sphere is the only surface whose geometry coincides, in a certain strong sense, with the geometry of visibles.

Chapter I: The Geometry of Visibles 1 . The N on- Euclidean Geometry of Visibles In the chapter "The Geometry of Visibles" in Inquiry into the Human Mind, ...

Thomas Reid argued that the geometrical properties of visible figures equal the geometrical properties of their projections on the inside of a sphere centred around the eye. In recent scholarship there are only a few suggestions of which sources might have inspired Reid. I point to a widely ignored body of early eighteenth-century literature – introductions into projective geometry, the use of celestial globes and astronomy – in which the model of the eye in the centre of a sphere (...) was immensely popular. Moreover, I argue that Reid's account results naturally from some astronomical doctrines in conjunction with George Berkeley's theory of vision. (shrink)

Thomas Reid claims to have learned of Idomenians, “an order of beings” in “sublunary regions” whose visual system is very much like ours except that they could detect only the direction of rays reaching their eyes, not the distance of origin. The properties of Idomenian vision are here examined in the light of the physiological optics of Reid’s time and of the scientific developments that have since augmented our knowledge of the discipline.

Drawing on work done by Helmholtz, I argue that Reid was in no position to infer that objects appear as if projected on the inner surface of a sphere, or that they have the geometric properties of such projections even though they do not look concave towards the eye. A careful consideration of the phenomena of visual experience, as further illuminated by the practice of visual artists, should have led him to conclude that the sides of visible appearances either look (...) straight, in which case their angles appear to approximate Euclidean measures, or their angles do not appear to approximate Euclidean measures for straight line figures, in which case their sides do not look straight. (shrink)

A meta-analysis and an experiment show that the degree of compression of the in-depth dimension of visual space relative to the frontal dimension increases quickly as a function of the distance between the stimulus and the observer at first, but the rate of change slows beyond 7 m from the observer, reaching an apparent asymptote of about 50 %. In addition, the compression of visual space is greater for monocular and reduced cue conditions. The pattern of compression of the in-depth (...) dimension as a function of distance is similar to the ratio of in-depth to frontal visual angles of stimuli, but is not as extreme as this ratio would suggest, implying that observers are incapable of fully ignoring size information provided by cues to depth. Size and distance judgments may be described by an Affine transformation of physical space; however, the compression parameter in this model changes as a function of distance from the observer and other experimental conditions. (shrink)

Thomas Reid's Geometry of Visibles, according to which the geometrical properties of an object's perspectival appearance equal the geometrical properties of its projection on the inside of a sphere with the eye in its centre allows for two different interpretations. It may (1) be understood as a theory about phenomenal visual space – i.e. an account of how things appear to human observers from a certain point of view – or it may (2) be seen as a mathematical (...) model of viewpoint-relative but mind-independent relational properties of objects. This paper makes a systematic and a historical claim. I shall argue, first, that given certain features of the human visual system phenomenal visual space differs in several aspects from Reidean visual space. Secondly, I suggest that, since Reid was aware of some of these empirical facts, we should interpret Reid as endorsing the second interpretation of the Geometry of Visibles. (shrink)

Abstract: In this paper I consider recent attempts to establish that the geometry of visual experience is a spherical geometry. These attempts, offered by Gideon Yaffe, James van Cleve and Gordon Belot, follow Thomas Reid in arguing for an equivalency of a geometry of ‘visibles’ and spherical geometry. I argue that although the proposed equivalency is successfully established by the strongest form of the argument, this does not warrant any conclusion about the geometry of (...) visual experience. I argue, firstly, that the resistance of this contemporary argument to empirical considerations counts against its plausibility. Moreover, I argue that the contemporary approach provides no compelling reason for supposing that the geometry offered as the geometry of ‘visibles’ is the correct geometrical description of visual experience. (shrink)

Un enjeu majeur pour les recherches actuelles dans les sciences de la vision consiste à mettre au point une approche dépendante de l’observateur – une science des apparences visuelles située au-delà de leur véridicité. L’espace dont nous faisons l’expérience subjective est en réalité hautement « illusoire», et les éléments de base du champ visuel sont des structures qualitatives, contextuelles et relationnelles, et non des indices métriques et dépendants du stimulus. Sur la base de nombreux résultats disponibles dans la littérature traitant (...) de la manière dont fonctionnent les divers constituants de l’espace, l’article décrit les éléments qualitatifs de base d’un tel espace et pose la question de la « géométrie» des apparences visuelles. Il formule enfin un ensemble de propositions pour d’éventuelles recherches poursuivant l’examen de l’espace visuel d’un point de vue expérimental. (shrink)

Un enjeu majeur pour les recherches actuelles dans les sciences de la vision consiste à mettre au point une approche dépendante de l’observateur – une science des apparences visuelles située au-delà de leur véridicité. L’espace dont nous faisons l’expérience subjective est en réalité hautement « illusoire», et les éléments de base du champ visuel sont des structures qualitatives, contextuelles et relationnelles, et non des indices métriques et dépendants du stimulus. Sur la base de nombreux résultats disponibles dans la littérature traitant (...) de la manière dont fonctionnent les divers constituants de l’espace (formes, surfaces, etc.), l’article décrit les éléments qualitatifs de base d’un tel espace et pose la question de la « géométrie» des apparences visuelles. Il formule enfin un ensemble de propositions pour d’éventuelles recherches poursuivant l’examen de l’espace visuel d’un point de vue expérimental. (shrink)

Independently of any eighteenth century work on the geometry of parallels, Thomas Reid discovered the non-euclidean "geometry of visibles" in 1764. Reid's construction uses an idealized eye, incapable of making distance discriminations, to specify operationally a two dimensional visible space and a set of objects, the visibles. Reid offers sample theorems for his doubly elliptical geometry and proposes a natural model, the surface of the sphere. His construction draws on eighteenth century theory of vision for (...) some of its technical features and is motivated by Reid's desire to defend realism against Berkeley's idealist treatment of visual space. (shrink)

The work of David R. Lachterman, The Ethics of Geometry, subtitled A Genealogy of Modernity, concerns essentially the status of geometry in Euclid’s Elements and in Descartes’s Geometry. It is a remarkable work, at once by the declared breadth of its ambitions and by the very great precision of its analyses, which are always supported by a prodigious philosophical culture. David Lachterman’s concern is to grasp, by way of an in-depth commentary of certain, particularly crucial passages of (...) these two foundational works, the change in geometry’s status from the one to the other, and this, as a sort of symptom of what the Moderns, since Descartes, will always experience as a necessary, inaugural rupture with regard to tradition. This change is underlaid, according to the author, by a profound change in the philosophical ethos, and in particular, in the geometer’s ethos. One must understand ethos, Lachterman explains, in the manner of Aristotle: those characteristic means that human beings have by which to act in the world or to behave in relation to one another, or to themselves. There is thus an “ethics” of geometry, namely, in the manner and the style of doing geometry, of behaving as a mathematician both toward apprentices and toward the veritable nature of the “entities” which are to be taught or learnt, and which give their discipline its name. That there is a difference of ethos between Euclid and Descartes implies, according to Lachterman, that there is also a difference in the source of intelligibility of the “mathematical,” understood in the most general sense. This in turn implies a deeper difference in the mode of being in general: it suffices to note that, in the ancient case, the source of intelligibility of the geometric figure lies in the nature of the figure itself, and that in the modern case, the same source is found in the “strategies” and “tactics” suited to bringing the figure to its visibility or to its embodiment, 315 in order to notice that this change in the mode of access to intelligibility must have had profound repercussions upon philosophical thought. It is in this sense that the author’s design is also, at the same time, not genetic, but genealogical. The break between ancient and modern mathematics pleas in favor of a consideration of the rupture of modern thought relative to ancient thought; thus, implicitly—and it is one of the great merits of this book that it shows this—against all homogenizing levelling, in Heidegger’s fashion, of the history of thought down to the history of “the” discipline of metaphysics. If there is something Heideggerian, in the best sense of the term, in Lachterman’s way of considering ethos as coextensive with modes of being, there is, on the other hand, and we ought to be very glad of it, something anti-Heideggerian in his original manner of bringing out the effective novelty of modern thought. For, according to him, modern thought is no longer to be defined, explicitly or exclusively, by the so-called “metaphysics of subjectivity.” Instead, it is defined by a self-regulated, operative, and constructive productivity of thought—and this as much with regard to itself as to its objects. To put it briefly, whereas ancient thought is rather polarized by the logico-eidetic, modern thought is polarized by a sort of methodical constructivism, the status of which is, in reality, very complex. If Descartes speaks of “the construction of a problem,” while Leibniz speaks of the “construction of an equation,” and Kant of the “construction of a concept,” everything depends—as we might guess—upon the multiple meanings that the term ‘construction’ may take on. Is it to be taken in the sense of an absolute creation, a quasi ex nihilo creation, which would render mathematics divine? Is it the creation of artefacts, or again, is it the methodical exploration of problems posed to thought by the existence of objects or corresponding ideas, themselves supposed to be somewhere in the divine understanding? We already discern the breadth and the depth of the debate, and we sense that the essential will be played out in Lachterman’s commentaries on the famous Cartesian solution of the problem of Pappus—the birth certificate, as we know, of analytic geometry. (shrink)

Two theses appear to be central to Reid’s view of the visual field. By sight, we do not originally perceive depth or linear distance from the eye. By sight, we originally perceive the position that points on the surface of objects have with regard to the centre of the eye. In different terms, by sight, we originally perceive the compass direction and degree of elevation of points on the surface of objects with reference to the centre of the eye. I (...) consider various problems about distance perception and perception of position with regard to the centre of the eye raised by Reid’s texts. These questions are relevant to Reid’s description of the geometry of visibles as the “intrinsic” geometry of the surface of the sphere. (shrink)

In the chapter “The Geometry of Visibles” in his ‘Inquiry into the Human Mind’, Thomas Reid constructs a special space, develops a special geometry for that space, and offers a natural model for this geometry. In doing so, Reid “discovers” non-Euclidean Geometry sixty years before the mathematicians. This paper examines this “discovery” and the philosophical motivations underlying it. By reviewing Reid’s ideas on visible space and confronting him with Kant and Berkeley, I hope, moreover, to (...) resolve an alleged impasse in Reid’s philosophy concerning the contradictory characteristics of Reid’s tangible and visible space. (shrink)

This thesis is directed towards a philosophy of music by attention to conceptions and perceptions of space. I focus on melody and harmony, and do not emphasise rhythm, which, as far as I can tell, concerns time rather than space. I seek a metaphysical account of Western Classical music in the diatonic tradition. More specifically, my interest is in wordless, untitled music, often called 'absolute' music. My aim is to elucidate a spatial approach to the world combined with a curiosity (...) about the nature of the Pythagorean intervals. Thus one question I seek to answer is: What do the Pythagorean intervals import to music? In their original formulation by Pythagoras, and further, by Plato, their import seemed mystical and analogous to a 'harmony of the spheres.' In this spatial approach, the Pythagorean intervals are indicative of infinite depth. The faculty of hearing alone does not ordinarily spatially locate sources. Hearing is ordinarily combined with sight in order to spatially locate a sound. Yet we talk about music as if it were spatially located, that is, as if it were a visible object with a surface and themes or 'objects'. I explore the faculty of vision and consider in what ways the processes of vision might be similar to and distinct from audition. My source for this approach is David Marr's book 'Vision' and his theory of the 2 1/2-dimensional sketch. In a medium in which there is no spatial location, it is important to situate the listener. Newton solved the problem of location by demonstrating the effects of gravity. P. F. Strawson claims that an analogy of distance is required in hearing to situate the listener, that is, a sense of nearer to and further from a source or object that permits a perception of distance. I claim that in music, the key signature and the scalar relations of musical themes provide this analogy of distance. The works of Plato, Aristotle, Descartes, Leibniz, and Newton are studied for their accounts of motion, bodies, and space. From these metaphysicians can be gleaned elements of music that include form, motion, timbre, dynamics, and attraction. By incorporating motion, I aim to show that although we may not know the true nature of space, we can learn from hearing music that space is perceptible in other than three dimensions. (shrink)

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...) this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

For each vertex of a simple polygon P an integer valued weight is given. We consider the path p1, p2, ..., pk in P which is created according to the following strategy: p1 is a designated start vertex s and pi+1 is obtained by choosing the vertex with smallest weight among all vertices visible from pi and different from p1, p2, ..., pi. If there is no such vertex the path is finished. This path is called geometric lexicographic dead end (...) path. We shall prove the problem of determining whether a distinguished vertex t of P is on the geometric lexicographic dead end path or not to be P-complete. (shrink)

Thomas Reid presented a two-dimensional geometry of the visual field in his Inquiry into the Human Mind (1764). The axioms of this geometry are different from those of Euclidean plane geometry. The ‘geometry of visibles’ is the same as the geometry of the surface of the sphere, described without reference to points and lines outside the surface itself. In a recent article, James Van Cleve has argued that Reid can secure a non-Euclidean geometry (...) of visibles only at the cost of abandoning his direct realist theory of perception, and reintroducing sense-data. The question will be reexamined by considering two aspects of Reid’s theory of vision: the claim that we do not directly perceive distance by sight and Reid’s characterization of visible figure as a partial notion of an external object. (shrink)

This piece, included in the drift special issue of continent., was created as one step in a thread of inquiry. While each of the contributions to drift stand on their own, the project was an attempt to follow a line of theoretical inquiry as it passed through time and the postal service from October 2012 until May 2013. This issue hosts two threads: between space & place and between intention & attention. The editors recommend that to experience the drifiting thought (...) that attention be paid to the contributions as they entered into conversation one after another. This particular piece is from the BETWEEN SPACE & PLACE thread: April Vannini, Those Between the Common * Laura Dean & Jesse McClelland, Ballard: A Portrait of Placemaking * Amara Hark Weber, Crossroad * Isaac Linder & Berit Soli-Holt, The Call of the Wild: Terror Modulations * Ashley D. Hairston, Momma taught us to keep a clean house * Sean Smith, The Garage * * * * Instead of beginning with radical doubt, we start from naiveté. —Graham Harman, The Quadruple Object Deep in the forest a call was sounding, and as often as he heard this call, mysteriously thrilling and luring, he felt compelled to turn his back upon the fire and the beaten earth around it, and to plunge into the forest, and on and on, he knew not where or why; nor did he wonder where or why, the call sounding imperiously, deep in the forest. —Jack London, The Call of the Wild The figure of the feral remains a perpetual enigma, but the parameters remain relatively consistent. A person, usually a child, enters civilization after having been raised by wolves or kept in some kind of cruel captivity. The outsider perspective on domestication ensuing in an edge of a culture's self-recognition of its clumsier attributes, what has been taken for granted becomes apparent, is brought to the foreground with the stranger and made questionable. Amusement follows naïve questions or observations such as Kaspar Hauser in the Herzog film, The Enigma of Kaspar Hauser, when Kaspar notes that while in his room he is engulfed by it, but when he looks at the tower he can turn away and it disappears. Ergo, the room is larger than the tower. How entertaining. The aberrant one destabilizes the comforting cultural normative. Places become seen as mere impressions out of space, a patterning, a rut that not everyone lives in like us. This is one figure of the feral. The naiveté that begs all the questions. As a figure for a certain philosophical disposition, the rapidity of one’s saccade scans the environment, intuits it’s space, not from an initial thaumazein or a Critchlean sense of disappointment, but from a child-like naiveté bent on survival. To serve the naïve is merely one form of critique, and it is not nearly used enough in lieu of the critique that provides answers. How dull. It is not necessary to be an outsider to entrench a critique with naiveté. After having forced to suffer in the most parched and rocky terror, itself for so long rooted upwards of fifty feet into the ground upon which it grows, even a grapevine can spontaneously produce a white grape on a red vine. The curious feral can arise from within, and like pinot grigio, it adds variety without admonishing its roots. There is also the feral dog. Not raised by wolves, but humans. Founded in place the figure of this feral denies this place. The trajectory of this feral roves from the cultivated to uncultivated, or in speaking of plants from controlled to volunteer, finding the necessary nutrients and survival patterns on its own. Finding other places, reaching out into space testing its fertility. And when introduced into a foreign environment, it withers or flourishes. We would like to attempt a thesis at this juncture and to accept neither feral figure in its entirety, but to argue for the intimate conjunction between a cultivated place and its resonance with the space it procures for its nest and kin. I'm not a biter, I'm a writer for myself and others. —Jay-Z, What More Can I Say? I am writing for myself and strangers. This is the only way that I can do it. Everybody is a real one to me, everybody is like some one else too to me. No one of them that I know can want to know it and so I write for myself and strangers. —Gertrude Stein, The Making of Americans There is no subjective disposition outlining an unambiguous individual of the para-academy. There is no para-academic. We all have day jobs. 'Para-academic' seeped through the cracks as an adjective in the call to frame publishing dedicated to the critical rigor expected by academic publishing, but to deny the limitations of guarded legitimation through capital means. Open-access holds hands with this parasitic descriptor. Para-academic publishing's refusal to adhere to the valuation of locked access of a site, the site “of a desperate initiation to the empty form of value,” 1 seeks to recognize not merely an inclusive interpretation of significance, but the significance of thinking practice. The practice inside the paywalls of academic 'education' is held in a deathgrip by its infatuation with value and information, both empty without the apprehension of human experience, the barbaric yawp. “I can't breathe in here.” It is not that a para-academic practice leads one to the childish wonder of Kaspar Hauser who wonders about the spatiality of his room. It is the academic legitimation that distorts that one can hold the understanding of both in a constellation of place and space. Led to believe there is only a place for things, we are led to disillusionment. It is also not that a para-academic practice relinquishes itself to the invasive growth outside of careful cultivation, an abandonment of pleasantries for the toothy growl of a predator. It is the academy's fear that thought does not require capital to signify value. Some of the most nutritious meals can be foraged. Defining a para-academic practice is not outlining a place of accreditation of the practice, it is the recognition that any place is subject to modulation by the space it inhabits as well as creates. The para-academic practice keeps an eye of the creation of spaces, follows those paths that eat themselves in the name of academia. This is not unlike Red Peter's report to the academy, only successful if we report in idle idiosyncratic banalities that we have once again become victorious in our acculturation and nullification within the confines of accredited mush and our trajectory of wild rigor is defeated in our desire for recognition as recognizable in this place. Weeds are integral to the functioning of a large ecosystem. The manicured garden is entirely reliant on its keeper. The pansy can also go wild once neglected, the daisy definitely does.... a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow from the original act of severance. The act is itself already remembered, even if unconsciously, as our first attempt to distinguish different things in a world where, in the first place, the boundaries can be drawn anywhere we please. At this stage the universe cannot be distinguished from how we act upon it, and the world may seem like shifting sand beneath our feet. —George Spencer-Brown, Laws of Form Two edges are created: an obedient, conformist, plagiarizing edge, and another edge, mobile, blank, which is never anything but the site of its effect: the place where the death of language is glimpsed. These two edges, the compromise they bring about, are necessary. —Roland Barthes, The Pleasure of the Text Between these two epigraphs, interminable questions of where and questions of happening, gesture, and interface. To stay buoyed between a site of visible happening and haptic perspicacity. Bounded by one or the other leads to a desiccation of potential knowledge. The tumbleweed tumbles until met with mud, a bare structure moving but not movement. A tumbleweed tumbleweeds, propagates only at a place. It becomes significant again, continues. Significance, the site where meaning is made known through kinesthetic apprehension. 2 The feral founds a gestural horizon; an outsider’s scrawl-becoming-law; Deleuze teaching Meno’s dog geometry. Place as marked, outlined, recognized, territorialized. The academies marked by their peculiar disciplines, outlined by their rigid boundaries, recognized as factories of value. This far from ensures complete purchase on the space of thought, but it has made an undeniably elaborate means of making work significant. The academy is a muddy spot, it is fertile, but its gates are high and its dogs are barking. The coordinates of concept and experience. Already claimed by a stabilizing suspension, the terms enter specificity of ‘this is this’. Another correlation: activity and the individual. The individual, a placeholder in the crosshairs of juridical identification. Activity, what expands and surrounds this location, but utterly indebted to the node of “one who”. What's happening in this oscillation of nature and nurture is practice. Practice, as Stengers tells us “is not the activity of an individual or the product of that activity. It is the ingredient without which neither that activity nor this product would exist as such.” 3 Moving outward from our own honing, we're curious about the ingredient creating the place for holding conceptual and experiential engagements in each hand. And we'd like to argue that this place is not a limiting specification, but a practice undulating daily, by the minute. And we call this practice the para-academic practice. I repeat: there was no attraction for me in imitating human beings; I imitated them because I needed a way out, and for no other reason. —Franz Kafka, A Report to an Academy In order to exist, man must rebel, but rebellion must respect the limit it discovers in itself—a limit where minds meet and, in meeting, begin to exist. — Albert Camus, The Rebel Anyone is a para-academic or practicer of such means. The academic can, and we argue should, be active para-academically, to escape the bounds, recognizing no site specific place as a place to rest on or the place to grab the Kafka's top and wonder at its disobedience not to continue. Yet, the para-academic practice must maintain the desire for rigor in scholarship. Indeed desiring past itself to claim a more naïve rigor, one that does not take its form for granted. Without a para-academic practice, the scholar spends half the time merely working on behalf of a hierarchy, to maintain it, and the other measly amounts of time are in the name of thinking, but only in name. Not to mention the amount of debt it takes to attend the halls of higher education. Not to mention the snoring tenures. Not to mention the barely scraping by adjuncts. Not to mention the materials that shake the very force of producible theory. Not to mention when swimming in texts becomes slogging through data. Academia is a barbaric food chain and it is our claim that there is, as always, an imperative for thought to move, with Heidegger, beyond the logics of calculation and planning, to a time of its own. The path, into the panic of the dark wood of this space can be followed by any; any who let the silence and the rigor enter the play. Where the little theater is larger when inhabited ; where the data of the tutor asymptotically refutes; and where, as much as one wouldn’t expect it here, ballet may turn out to be the most feral of forms… NOTES Jean Baudrillard, “Value's Last Tango” Simulacra and Simulation trans. Sheila Faria Glaser,. “What is significance? It is meaning, insofar as it is sensually produced.” Roland Barthes in The Pleasure of the Text. Isabelle Stengers, “The Science Wars” Cosmopolitics I trans. Robert Bononno, 47. (shrink)

RésuméDans l'article qui précède, l'auteur s'efforce, à l'intention surtout de ceux qui enseignent les Eléments, de mettre en lumière la signification et l'importance de deux ouvrages concernant la géométrie. Le court écrit de M. G. Bouligand fait apparaǐtre la structure algébrique et logique de cette science et présente une ȧxiomatique introduisant les notions d'ensemble et de groupe de transformation. Ainsi s'élabore une classification progressive des problèmes selon le genre des solutions qui leur conviennent. Le livre beaucoup plus étendu de M. (...) F. Gonseth analyse les aspects intuitif, expérimental et théorique de ce qui constituait l'élémentaire dans les traités classiques, puis expose les recherches suscitées par le postulat d'Euclide qui ont amené au développement de la méthode axiomatique. Le triomphe de l'atomisme, découvrant un monde où les notions élémentaires de la géométrie ne s'applíquent plus intégralement, vient rendre plus aigu encore le problème du fondement de la vérité géométrique et celui de la nature de l'espace. Grǎce à l'introduction de notions telles que celle de correspondance schématique et celle de modèle, la géométrie apparaǐt comme une science rationnelle en évolution et à laquelle l'axiomatique sert de critère. Mais bien loin de présenter les caractères d'une pure rationalité, cette axiomatique retient certaines données intuitives, měme dans son application aux géométries non euclidiennes; elle organise un contrǒle de nos intuitions les plus élémentaires les unes par les autres, et assure par leur intermédiaire la portée expérimentale et la cohérence de la géométrie. On assiste ainsi à une véritable mutation de l'élémentaire; mais celle‐ci n'est pas due au hasard. La méthode qui a présidé à cette transformation, dont les étapes historiques sont décrites, est clairement dégagée et il est visible qu'elle régit également les formes nouvelles de la physique dont la constitution théorique ne diffère pas essentiellement de la géométrie considérée comme la science rationnelle de l'espace. Cette méthode est le statut d'une science ouverte. Ainsi se trouve pleinement légitimé un exposé comme celui de M. Bouligand. En s'illustrant l'un l'autre, les deux ouvrages offrent des horizons nouveaux à l'enseignement, měme élémentaire, de la géométrie. La portée philosophique de ces ouvrages n'est pas moindre que leur portée pédagogique.ZusammenfassungIn der voranstehenden Besprechung möchte der Rezensent insbesondere diejenigen unter den Lesern, die die Elemente unterrichten, auf zwei bedeutende Geometriewerke hinweisen. G. Bouligands knapp gehaltene Schrift stellt den algebraisch‐logischen Aufbau dieser Wissenschaft ins Licht und bietet eine Axiomatik, die die Begriffe der Menge und der Transformations‐gruppe einführt. Ausgehend von der passenden Lösungsart gelangt der Verfasser zu einer fortschreitenden Klassifizierung der Aufgaben. In seinem viel breiter angelegten Werk untersucht F. Gonseth das in den klassischen Lehrbüchern als das Elementare angesehene vom intuitiven, experimentellen und theoretischen Gesichtspunkte aus und macht daran anschliessend mit den von Euklids Postulat ausgehenden Nachforschungen, die zur Entwicklung der axiomatischen Methode geführt haben, bekannt. Der eigentliche Siegeszug des Atomismus und die mit dieser Wissenschaft verbundene Entdeckung einer Welt, in der die grundsätzlichen Begriffe der Geometrie ihre volle Bestätigung nicht mehr erfahren, macht die Frage nach den Grundlagen der Geometrie und die nach dem Wesen des Raums besonders brennend. Die Einführung von Begriffen wie Schema und Modell lässt die Geometrie als eine in Entwicklung begriffene Rationalwissenschaft erscheinen, welcher die Axiomatik als Prüfstein dient. Letztere trägt aber durchaus nicht die Züge reiner Vernunftmässigkeit, sie hält vielmehr, selbst in ihrer Anwendung auf die nichteuklidischen Geometrien, an gewissen intuitiven Voraussetzungen fest; sie ermöglicht eine wechselseitige Kritik unserer elementarsten intuitiven Erkenntnisse unter sich und gewährleistet so die experimentelle Tragweite und die Kohärenz der Geometrie. Es findet also eine eigentliche Mutation des Elementaren statt, doch bleibt dieser Wandel nicht dem Zufall überlassen. Die Planmässigkeit, mit der er sich vollzog, wird unter Beschreibung der geschichtlichen Entwicklungsstufen klar dargelegt, womit erkennbar wird, dass er auch die neueren Formen der Physik bestimmt, die sich ja nach ihrem theoretischen Grundgesetz von der Geometrie als der rationalen Wissenschaft vom Raume nicht wesentlich unterscheidet. Eine solche Methode erfüllt das Lebensgesetz einer offenen Wissenschaft. Durch sie finden Ausführungen wie die Bouligands ihre volle Rechtfertigung. Sich gegenseitig ergänzend und erläuternd eröffnen beide Werke selbst dem Elementarunterricht in der Geometrie neue Möglichkeiten. Der pädagogischen kommt ihre philosophische Bedeutung gleich.In the foregoing article the author endeavours—especially for all who are teaching the Elements—to set forth the significance and importance of two books about geometry. Mr. G. Bouligand's short writing brings out the algebraic and logical structure of that science and presents an »axiomatic« introducing both the notions of »whole« and »group of transformation«. A progressive classification of problems is thus elaborated according to the kinds of solutions that suit them. Mr. F. Gonseth, in his far more comprehensive work, analyses the intuitive, experimental and theoretical aspects of what used to constitute the elementary in classical treatises, then states the researches raised up by Euclid's postulate and leading to the development of the axiomatic method. The triumph of atomism, discovering a world to which the elementary notions of geometry do no longer apply fully, makes it still more difficult to solve the problems of the basing of both geometrical truth and the nature of space. Thanks to his introducing such ideas as »schematic correspondence« and »model«, geometry appears as a rational science in progress using the »axiomatic« as its criterion. But far from offering the characteristics of pure rationality, such an »axiomatic« retains some intuitive bases, even when applied to non‐Euclidian geometries; it organizes a control of our most elementary intuitions by setting them against one another, and secures through their medium the experimental significance and coherence of geometry. Thus a real mutation of the elementary takes place; but it is not owing to mere chance. The method which has presided over that transformation, the historical changes and stages of which are described, is clearly shown, and it is made evident that it also determines the new forms of physics, the theoretical constitution of which does not differ essentially from geometry considered as the rational science of space. That method is the statute of an open science. So Mr. Bouligand's account is fully justified. By elucidating each other, both books open new horizons to the teaching—even elementary—of geometry. The philosophical significance of those books is of no less importance than the pedagogical one. (shrink)

7 lectures, Torquay, UK, August 12-20, 1924 (CW 311) These seven intimate, aphoristic talks were presented to a small group on Steiner's final visit to England. Because they were given to "pioneers" dedicated to opening a new Waldorf school, these talks are often considered one of the best introductions to Waldorf education. Steiner shows the necessity for teachers to work on themselves first, in order to transform their own inherent gifts. He explains the need to use humor to keep their (...) teaching lively and imaginative. Above all, he stresses the tremendous importance of doing everything in the knowledge that children are citizens of both the spiritual and the earthly worlds. And, throughout these lectures, he continually returns to the practical value of Waldorf education. These talks are filled with practical illustrations and revolve around certain themes--the need for observation in teachers; the dangers of stressing the intellect too early; children's need for teaching that is concrete and pictorial; the education of children's souls through wonder and reverence; the importance of first presenting the "whole," then the parts, to the children's imagination. Here is one of the best introductions to Waldorf education, straight from the man who started it all. German source: Die Kunst des Erziehens aus dem Erfassen der Menschenwesenhiet (GA 311). ∞ ∞ ∞ SYNOPSIS OF THE LECTURES LECTURE 1: The need for a new art of education. The whole of life must be considered. Process of incarnation as a stupendous task of the spirit. Fundamental changes at seven and fourteen. At seven, the forming of the "new body" out of the "model body" inherited at birth. After birth, the bodily milk as sole nourishment. The teacher's task to give "soul milk" at the change of teeth and "spiritual milk" at puberty. LECTURE 2: In first epoch of life child is wholly sense organ. Nature of child's environment and conduct of surrounding adults of paramount importance. Detailed observation of children and its significance. In second epoch, seven to fourteen, fantasy and imagination as life blood of all education, e.g., in teaching of writing and reading, based on free creative activity of each teacher. The child as integral part of the environment until nine. Teaching about nature must be based on this. The "higher truths" in fairy tales and myths. How the teacher can guide the child through the critical moment of the ninth year. LECTURE 3: How to teach about plants and animals (seven to fourteen). Plants must always be considered, not as specimens, but growing in the soil. The plant belongs to the earth. This is the true picture and gives the child an inward joy. Animals must be spoken of always in connection with humans. All animal qualities and physical characteristics are to be found, in some form, in the human being. Humans as synthesis of the whole animal kingdom. Minerals should not be introduced until twelfth year. History should first be presented in living, imaginative pictures, through legends, myths, and stories. Only at eleven or twelve should any teaching be based on cause and effect, which is foreign to the young child's nature. Some thoughts on punishment, with examples. LECTURE 4: Development of imaginative qualities in the teacher. The story of the violet and the blue sky. Children's questions. Discipline dependent on the right mood of soul. The teacher's own preparation for this. Seating of children according to temperament. Retelling of stories. Importance of imaginative stories that can be recalled in later school life. Drawing of diagrams, from ninth year. Completion and metamorphosis of simple figures, to give children feeling of form and symmetry. Concentration exercises to awaken an active thinking as basis of wisdom for later life. Simple color exercises. A Waldorf school timetable. The "main lesson." LECTURE 5: All teaching matter must be intimately connected with life. In counting, each different number should be connected with the child or what the child sees in the environment. Counting and stepping in rhythm. The body counts. The head looks on. Counting with fingers and toes is good (also writing with the feet). The ONE is the whole. Other numbers proceed from it. Building with bricks is against the child's nature, whose impulse is to proceed from whole to parts, as in medieval thinking. Contrast atomic theory. In real life we have first a basket of apples, a purse of coins. In teaching addition, proceed from the whole. In subtraction, start with minuend and remainder; in multiplication, with product and one factor. Theorem of Pythagoras (eleven-twelve years). Details given of a clear, visual proof, based on practical thinking. This will arouse fresh wonder every time. LECTURE 6: In first seven years etheric body is an inward sculptor. After seven, child has impulse to model and to paint. Teacher must learn anatomy by modeling the organs. Teaching of physiology (nine to twelve years) should be based on modeling. Between seven and fourteen astral body gradually draws into physical body, carrying the breathing by way of nerves, as playing on a lyre. Importance of singing. Child's experience of well being like that of cows chewing the cud. Instrumental music from beginning of school life, wind or strings. Teaching of languages; up to nine through imitation, then beginnings of grammar, as little translation as possible. Vowels are expression of feeling, consonants are imitation of external processes. Each language expresses a different conception. Compare head, Kopf, testa. The parts of speech in relation to the life after death. If language is rightly taught, out of feeling, eurythmy will develop naturally, expressing inner and outer experiences in ordered movements--"visible speech." Finding relationship to space in gymnastics. LECTURE 7: Between seven and fourteen soul qualities are paramount. Beginnings of science teaching from twelfth year only, and connected with real phenomena of life. The problem of fatigue. Wrong conceptions of psychologists. The rhythmic system, predominant in second period, never tires. Rhythm and fantasy. Composition. Sums from real life, not abstractions. Einstein's theory. The kindergarten--imitation of life. Teachers' meetings, the heart of the school. Every child to be in the right class for its age. Importance of some knowledge of trades, e.g., shoemaking, handwork, and embroidery. Children's reports-- characterization, but no grading. Contact with the parents. QUESTIONS AND ANSWERS: The close relationship of Multiplication and Division. How to deal with both together. Transition from the concrete to the abstract in Arithmetic. Not before the ninth year. Healthiness of English weights and measures as related to real life. Decimal system as an intellectual abstraction. Drawing. Lines have no reality in drawing and painting, only boundaries. How to teach children to draw a tree in shading, speaking only of light and color. (Illustration). Line drawing belongs only to geometry. Gymnastics and Sport. Sport is of no educational value, but necessary as belonging to English life. Gymnastics should be taught by demonstration. Religious Instruction. Religion lessons in the Waldorf school given by Catholic priest and Protestant pastor. "Free" religion lessons provided for the other children. Plan of such teaching described, of which the fundamental aim is an understanding of Christianity. The Sunday services. Modern Language Lessons. Choice of languages must be guided by the demands of English life. These can be introduced at an early age. Direct method in language teaching. Closing words by Dr. Steiner on the seriousness of this first attempt to found a school in England. (shrink)

George Berkeley argues that vision is a language of God, that the immediate objects of vision are arbitrary signs for tactile objects and that there is no necessary connection between what we see and what we touch. Thomas Reid, on the other hand, aims to establish a geometrical connection between visible and tactile figures. Consequently, although Reid and Berkeley's theories of vision share important elements, Reid explicitly rejects Berkeley's idea that visible figures are merely arbitrary signs for tangible bodies. But (...) is he right in doing so? I show that many passages in Berkeley's work on vision suggest that he acknowledges a geometrical connection between visibles and tangibles. So the opposition between the arbitrariness Berkeley defends and a geometrical connection cannot be as universal as Reid thinks. This paper seeks to offer a plausible reading of Berkeley's theory of vision in this regard and an explanation of why Reid interprets Berkeley differently. (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)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Rhetoric 36.4 (2003) 379-383 [Access article in PDF] Husserl at the Limits of Phenomenology: Including Texts by Edmund Husserl. Maurice Merleau-Ponty. Ed. Leonard Lawlor with Bettina Bergo. Trans. Leonard Lawlor. Evanston, IL: Northwestern University Press, 2002. Pp. 192. $19.95 pbk. The most striking characteristic of this volume is the manner that it presents layers of interpretation to the reader, particularly in that the writing is not intended (...) as a postmodern fête, and has limited value as a hermeneutic enterprise. Ultimately the book's true contribution is the seventy-nine pages devoted to the translation and printing of lecture notes used by [End Page 379] Merleau-Ponty in a class he gave at College de France, shortly before his death. However, the explanation of why this is so will have to be withheld until the end of this review.The course that Merleau-Ponty gave was organized around three essays published by Edmund Husserl. These essays are included in the volume reviewed here and must be briefly dealt with, as they are the core of whatever thinking occurs within and around this book. Interesting essays all, they deal with three longstanding issues of philosophical debate. The first essay challenges the classical notion of a transcendent ideal. Husserl attempts to show, through an examination of geometry, that an ideal is first called into existence when it is called into the realm of human experience, and he does this without ever resorting to the quality of detached being that Plato attributes to ideals. Then, in his second essay, Husserl attempts to defeat modernist cosmology, arguing, against Galileo, that the earth is a fixed point in the universe. Finally, his third essay hearkens back to Heraclitus and Parmenides, examining how human experience is able to extract a sense of atemporal quiddity in a world that is constantly falling into a past. A reader might be baffled that an intellectual could make such claims, but they are made from within the framework of phenomenology, and, within this paradigm as Husserl explains it, the claims he makes do not seem ridiculous at all. "The Origins of Geometry," "The Originary Ark, The Earth, Does Not Move," and "The World of the Living Present,"— these are the three essays that Merleau-Ponty forms his lecture course around. At sixty pages within this volume, they constitute one third of the text itself. For intellectual worth, I would say that these essays are the finest contribution in the text, but that claim is discounted because much of the essay material was already available in English translation. Nevertheless, it only makes sense to have it included in this volume in order to conveniently work back and forth between Merleau-Ponty's notes and Husserl's actual essays.However, that brings us to a significant problem within the text itself. If readers are like me, they will realize that it is often difficult to draw conclusions from the notes of others. Like secretarial shorthand, notes rely on a reader's ability to read both what has been written and what has been withheld in an expressive passage. There is no code that will allow us to recover what Merleau-Ponty recovered as he looked down on his podium at these notes and then raised his head to speak to his class. All that we have are the fragments, and they do not make a coherent argument, lacking both the clarity and the painstaking thoroughness that we find in books [End Page 380] such as The Phenomenology of Perception and The Visible and the Invisible. Certainly one cannot fault Merleau-Ponty for this—he did not write these notes for us, but for himself. His course might have been magnificent, even transcendent, but we will not know that from these fragments.I suppose it could be argued that the ambiguity inherent in such fragments liberates the text and opens it for some sort of radical hermeneutic exegesis; however, I sigh at the very thought of such an argument. Unless we intend to transform... (shrink)

This paper investigates Hume's theory of the perception of spatial magnitude or size as developed in the _Treatise<D>, as well as its relation to his concepts of space and geometry. The central focus of the discussion is Hume's espousal of the 'composite' hypothesis, which holds that perceptions of spatial magnitude are composed of indivisible sensible points, such that the total magnitude of a visible figure is a derived by-product of its component parts. Overall, it will be argued that a (...) straightforward reading of this hypothesis fails to do full justice to the complexity of Hume's theory of spatial perception and geometry, and that a more adequate treatment must also admit an important role for the more direct process of spatial magnitude perception which he dubs 'intuition'. (shrink)

This paper begins by pointing to an obvious difficulty in Merleau-Ponty’s late philosophy: undoing the decisive separation between linguistic connotation and the denotated, undoing the decisive separation between linguistic meaning and the sensible world. This difficulty demands that we understand how the sensible and the symbolic have a sort of spontaneous relation. How can this be? The history of this problem is then traced back to Husserl, and in particular to his The Origin of Geometry. For Husserl, ‘abstract (...) class='Hi'>geometry’ is understood to be a self-constituting field that cuts itself off from a ‘pre-geometry.’ The latter sort of geometry has to do with a symbolic and phantasy consciousness that makes differences in an otherwise uniform sensible plenum. Merleau-Ponty is at odds with the idea of an abstract geometry that constitutes its own field. His phenomenology would have this abstract geometry reduced to pre-geometry, and this pre-geometry further reduced to the sensible, which is then not uniform at all. The problem becomes more precisely how the separations of the sensible and the differences between things are already symbolic.The paper suggests that a way to think through Merleau-Ponty’s problem can be found in a January 1960 Working Note to The Visible and the Invisible called “The Invisible, the negative, vertical Being.” This note, with its references to “joints and members” and “disjunction and dis-membering” is very likely a reference to, and correction of, Heidegger’s 1943 essay on Heraclitus. The point for Merleau-Ponty seems to be a radicalization of the Heraclitean coinstantiation of opposites. His reading allows him to resist understanding the sensible as a self-identity and replace it instead with disarticulation, disintegration, or even perhaps disruption. To show this very disruption is, the paper finally argues, to show the sensible with its phantasmagorical character. Finally, by way of conclusion, the paper takes a turn to sculpture precisely to discuss how we might grasp a sensible that is not uniform but always being shaped and differentiated; and how the symbolic is exactly in, enjambed with, this spacing apart. Ce texte commence par signaler une difficulté évidente dans l’oeuvre tardive de Merleau-Ponty : défaire la séparation décisive entre la connotation linguistique et la dénotation, défaire la séparation décisive entre la signification linguistique et le monde sensible. Cette difficulté requiert de comprendre comment le sensible et le symbolique se trouvent dans une sorte de relation spontanée. Comment cela se fait-il? L’histoire de ce problème remonte à Husserl et à son Origine de la géométrie. Pour Husserl, la « géométrie abstraite » est comprise comme un champ qui se constitue lui-même et qui se détache d’une « pré-géométrie ». Cette dernière sorte de géométrie est liée à une conscience symbolique et imaginative qui fait des différences dans un espace sensible autrement uniforme. Merleau-Ponty n’est pas favorable à l’idée d’une géométrie abstraite qui constitue son propre champ. Sa phénoménologie vise à réduire la géométrie abstraite à la pré-géométrie, et cette dernière au sensible, qui dès lors n’est plus du tout uniforme. Le problème consiste alors à montrer comment les séparations du sensible et les différences entre les choses sont toujours déjà symboliques.La contribution ici suggère qu’une solution peut être trouvée dans la note de travail du Visible et l’invisible de janvier 1960, intitulée « L’Invisible, le négatif, l’Être vertical ». Cette note, qui parle de « jointure et membrure » et de « dis-jonction et dé-membrement », est sans doute une référence à, et une correction l’essai de Heidegger de 1943 sur Héraclite. Le propos de Merleau-Ponty semble être une radicalisation de la co-instantiation héraclitéenne des opposés. Son interprétation lui permet de résister à la conception du sensible comme identité à soi et de la remplacer par la désarticulation, la désintégration ou même peut-être le bouleversement (disruption). Le fait de montrer ce bouleversement consiste précisément à montrer le sensible avec son caractère fantasmatique. Finalement, en guise de conclusion, le texte se tourne vers la sculpture pour discuter une notion du sensible comme étant non uniforme mais toujours déjà formé et différencié, et une notion de symbolique comme étant exactement dans cet espacement. Questo testo inizia portando l’attenzione su un evidente problema nella filosofia dell’ultimo Merleau-Ponty: annullare la separazione decisiva tra connotazione linguistica e ciò che viene denotato, annullare la separazione decisiva tra significato linguistico e mondo sensibile. Questo problema richiede una comprensione della relazione spontanea tra sensibile e simbolico. Come ciò è possibile? La storia di questo problema risale fino ad Husserl, ed in particolare al suo L’origine della geometria. Per Husserl, la ‘geometria astratta’ è da concepire come un campo che si costituisce da sé, distaccandosi da una ‘pre-geometria’. L’ultima tipologia di geometria ha a che fare con una coscienza simbolica e immaginativa, che crea delle differenze in un altrimenti uniforme plenum sensibile. Merleau-Ponty non concorda con l’idea di una geometria astratta che costituisca il proprio campo. La sua fenomenologia intende piuttosto riportare la geometria astratta alla pre-geometria, e quest’ultima al sensibile, che quindi è tutto fuorché uniforme. Il problema diventa allora, più precisamente, come mostrare che le separazioni del sensibile e delle differenze tra le cose siano già simboliche. Questo testo suggerisce che una soluzione per pensare questo problema in Merleau-Ponty possa essere trovata in una nota di lavoro al Visibile e l’invisibile del gennaio 1960, intitolata “L’invisibile, il negativo, l’essere verticale”. Questa nota, con i suoi riferimenti a “spazio topologico e il tempo di congiunzione e di membratura”, e ancora “di dis-giunzione e dis-membramento”, è molto probabilmente un riferimento, ed insieme una correzione, al saggio di Heidegger su Eraclito del 1943. Il punto, per Merleau-Ponty, sembra essere una radicalizzazione del concetto eracliteo di compresenza degli opposti. La sua lettura gli permette di resistere alla comprensione del sensibile come un’auto-identità, per rimpiazzarla invece con una disarticolazione, disintegrazione, o perfino interruzione. Mostrare quest’interruzione è, come argomentiamo in questo testo, mostrare il sensibile con il suo carattere fantasmagorico. Infine, tirando le conclusioni, questo testo si rivolge alla scultura, per discutere come sia possibile afferrare un sensibile che non sia uniforme, ma sempre in-formato e in differenziazione, e come la simbolica sia sempre in – e intrecciata con – questa spaziatura, questa distanziazione. (shrink)

"TAKE A LINE cut in two unequal sections, one for the kind that is seen, the other for the kind that is thought, and go on and cut each section in the same ratio". In order to follow this request, not only must one know geometry, which treats linear magnitudes; one must also know the relations between geometry and the art which treats kinds. The problem of the first cut in the line is the problem of determining what (...) ratio of two parts of a line is the same as the ratio of the two kinds: visible and thinkable. In order to solve this problem, a procedure seems to be needed for deciding when a ratio of magnitudes, a geometric ratio, is the same as a ratio of kinds, or at least what ratio of magnitudes is the same as the ratio of visible to thinkable. (shrink)

The artistic work of photographer Gudio Baselgia focuses on landscapes formed by nature s forces and, more recently, on the sky with the stellar and solar movements and phenomena as we see them from earth. Celestial mechanics have fascinated mankind in all known cultures, the Babylonians and ancient Egyptians as well as the Greek and Celts, the Maya, or the ancient Indians and Chinese. Until the present day we look at the sky and keep being amazed, and try to read (...) what it tells us. Many artists throughout history have been captivated by the spectacle we observe above us day and night. The modern term astrodynamics describes all movements of celestial bodies, in particular the solar system including the moon and other satellites, asteroids and comets, but also movements of stars within a stellar system or galaxy, or of galaxies towards each other. They are well understood today and depicted in coordinate systems and elaborate visualizations. Guido Baselgia s artistic project on astrodynamics and celestial phenomena has no scientific or didactic ambition. His analogue camera is used as a recorder inscribing the movement of stars on the light-sensitive surface of photographic paper. Thus Baselgia s images make traceable the trajectory of celestial bodies invisible to the human eye and show us astounding occurrences of light and shadow. Baselgia has been captivated in particular also by the phenomenon of the umbra, planet earth s shadow thrown into space. It becomes visible occasionally on a clear evening at sunset when a slight mist lies at the horizon: looking in opposite direction to the sun, a dark and sharply marked band of shadow can be seen rising while sun sets behind the observer. But also by recording sunrise and sunset at the polar circle or the tropic, Baselgia visualizes the geometry of celestial mechanics and the concurrence of forces, as well as the miracle of light as such that leaves us awestruck today as much as it did our ancestors. The new book "Guido Baselgia Light Fall" presents 80 outstanding black-and-white images from the artist s Light Fall project taken in Norway, the Tierra del Fuego archipelago in Argentina, in Ecuador, and the Swiss Alps. The brilliant tritone plates are complemented with essays by the German scholar Andrea Gnam and Swiss photography critic Nadine Olonetzky. ". (shrink)

"TAKE A LINE cut in two unequal sections, one for the kind that is seen, the other for the kind that is thought, and go on and cut each section in the same ratio". In order to follow this request, not only must one know geometry, which treats linear magnitudes; one must also know the relations between geometry and the art which treats kinds. The problem of the first cut in the line is the problem of determining what (...) ratio of two parts of a line is the same as the ratio of the two kinds: visible and thinkable. In order to solve this problem, a procedure seems to be needed for deciding when a ratio of magnitudes, a geometric ratio, is the same as a ratio of kinds, or at least what ratio of magnitudes is the same as the ratio of visible to thinkable. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

In this thesis, I both analyze the phenomenology of vision from a geometrical point of view, and also develop certain connections between that geometrical analysis and the mind body problem. In order to motivate the need for such an analysis, I first show, by means of a refutation of direct realism, that visual space is never identical with any of the physical objects being indirectly "seen" by constituting color arrangements in it. It thus follows that the geometry of visual (...) space may be quite different from the Euclidean geometry of physical space, and I proceed to analyze that geometry. ;I argue that topologically, visual space is two dimensional, inasmuch as regions of it are capable of being bounded by a line, such as the borders around the various objects constituted in it. An apparent paradox arises here though, inasmuch as we posess phenomenal depth perception, which is particularly striking during binocular vision, and thus the question arises as to how the binocular depth cue of retinal disparity is registered phenomenally in a two dimensional space. I resolve this apparent paradox by arguing that the internal metric struture of this space can be apprehended phenomenally, and can serve as such a phenomenal depth cue. It is shown that holistically, this metric structure is elliptical, since for example marginal distortions in wide-angle photography are not present in visual space, and it is also noted that there is a tendency towards size constancy in visual perception. It is shown from these geometrical considerations that visual space posseses a variable curvature, with that curvature being determined by the physical depths of objects constituted in the space. ;Finally, I investigate what bearing the preceding geometrical conclusions have on the question of how events in visual space may be causally determined by neural events in the brain. The classical isomorphic theory of Gestalt psychology is then reinterpreted in light of my analysis of the geometry of visual space. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion (...) that was first conceived by ancient Greek mathematicians. (shrink)