The concept of geometry may evoke a world of pure platonic shapes, such as spheres and cubes, but a deeper understanding of visual experience demands insight into the perceptual organization of naturalistic form. Japanese gardens excel as designed environments where the complex fractal geometry of nature has been simplified to a structural core that retains the essential properties of the natural landscape, thereby presenting an ideal opportunity for investigating the geometry and perceptual significance of such naturalistic (...) characteristics. Here, fronto-parallel perspective, asymmetrical structuring of the ground plane, spatial arrangement of garden elements, tuning of textural qualities and choice of naturalistic form, are presented as a set of physical features that facilitate a systematic analysis of Japanese garden design per se, as well as the geometry of the particular naturalistic features that it aims to enhance. Comparison with Western landscape design before and after contact between Western and Eastern hemispheres illustrates the degree of naturalness achieved in the Japanese garden, and suggests how classical Western landscape design generally differs in this regard. It further reveals how modern Western gardens culminate in a different naturalistic geometry, thus also a distinctly different vision of the natural landscape, even if these designs were greatly influenced by the gardens of China and Japan. (shrink)

We cannot imagine two straight lines intersecting at two points even though they may do so. In this case our abilities to imagine depend upon our abilities to visualise.

This paper examines the interplay of changelessness and change, being and becoming, from an historical and dialectical standpoint. Topological paradox is employed to elucidate the dynamic interweaving of these ontological opposites. The essay concludes by exploring the relevance of the dialectic to the question of human freedom.

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as “natural”. We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry – i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3–34 years), and (...) 25 participants from the Amazon (age 5–67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in “sense”). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded. (shrink)

Some recently popular accounts of perception account for the phenomenal character of perceptual experience in terms of the qualities of objects. My concern in this paper is with naturalistic versions of such a phenomenal externalist view. Focusing on visual spatial perception, I argue that naturalistic phenomenal externalism conflicts with a number of scientific facts about the geometrical characteristics of visual spatial experience.

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and (...) class='Hi'>geometry of symmetry is controlled by numerical parameters, which act in analogy with a biological organism’s DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional (2D) design parameters, affine geometry and shape symmetry for visual or tactile shape sensation and perception-based decision making in populations of experts and non-experts. It will be shown that 2D fractal symmetry, referred to herein as the “symmetry of things in a thing”, results from principles very similar to those of affine projection. Results from experiments on aesthetic and visual preference judgments in response to 2D fractal trees with varying degrees of asymmetry are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (on a scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from “symmetry of things in a thing” significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion. (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)

Geometrical intuitions spontaneously drive visuo-spatial reasoning in human adults, children and animals. Is their emergence intrinsically linked to visual experience, or does it reflect a core property of cognition shared across sensory modalities? To address this question, we tested the sensitivity of blind-from-birth adults to geometrical-invariants using a haptic deviant-figure detection task. Blind participants spontaneously used many geometric concepts such as parallelism, right angles and geometrical shapes to detect intruders in haptic displays, but experienced difficulties with symmetry and complex (...) spatial transformations. Across items, their performance was highly correlated with that of sighted adults performing the same task in touch (blindfolded) and in vision, as well as with the performances of uneducated preschoolers and Amazonian adults. Our results support the existence of an amodal core-system of geometry that arises independently of visual experience. However, performance at selecting geometric intruders was generally higher in the visual compared to the haptic modality, suggesting that sensory-specific spatial experience may play a role in refining the properties of this core-system of geometry. (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)

With In Focus Sacred Geometry, learn the fascinating history behind this ancient tradition as well as how to decipher the geometrical symbols, formulas, and patterns based on mathematical patterns. People have searched for the meaning behind mathematical patterns for thousands of years. At its core, sacred geometry seeks to find the universal patterns that are found and applied to the objects surrounding us, such as the designs found in temples, churches, mosques, monuments, art, architecture, and nature. Learn the (...) fundamental principles behind: Interpreting the sacred symbolism and principles behind shapes Calculating numbers used in sacred geometry Applying the golden ratio and Fibonacci’s Sequence to objects Translating symbolic and geometrical letters and alphabets This accessible and beautifully designed guide to sacred geometry includes a frameable poster of the main sacred geometrical shapes and their unique, innate arrangements. The In Focus series applies a modern approach to teaching the classic body, mind, and spirit subjects. Authored by experts in their respective fields, these beginner's guides feature smartly designed visual material that clearly illustrates key topics within each subject. As a bonus, each book includes reference cards or a poster, held in an envelope inside the back cover, that give you a quick, go-to guide containing the most important information on the subject. (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)

In this paper I contrast the geometric structure of phenomenal visual space with that of photographic images. I argue that topologically both are two-dimensional and that both involve central projections of scenes being depicted. However, I also argue that the metric structures of the spaces differ inasmuch as two types of “apparent distortions”—marginal distortion in wide-angle photography and close-up distortions—which occur in photography do not occur in the corresponding visual experiences. In particular, I argue that the absence of (...) marginal distortions in vision is evidence for a holistic metric of visual space that is spherical, and that the absence of close-up distortions shows that the local metric structure possesses a dynamic variable curvature which is dependent upon the distance away of objects being viewed at a given time. (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)

In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal (...) logical inferences. By developing such an account of Euclid's geometry, I complete the "standard view" that geometry is either a formal system or an empirical science, which was developed mainly by the logical positivists and which is currently accepted by many mathematicians and philosophers. My thesis is divided into three parts. I use Hans Reichenbach's arguments against Kant and Edmund Husserl's genetic approach to the concept of space as a means of arguing that the "standard view" has to be supplemented by a concept of a geometry whose propositions have genuine spatial content. I then develop a coherent interpretation of Euclid's method by investigating both the subject matter of Euclid's geometry and the nature of geometric inferences. In the final part of this thesis, I modify Husserl's phenomenological analysis of the constitution of visual space in order to define a concept of spatial intuition that allows me not only to explain how Euclid's practice is grounded in visual space, but also to account for the apriority of its results. (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)

The empirical study of visual space has centered on determining its geometry, whether it is a perspective projection, flat or curved, Euclidean or non-Euclidean, whereas the topology of space consists of those properties that remain invariant under stretching but not tearing. For that reason distance is a property not preserved in topological space whereas the property of spatial order is preserved. Specifically the topological properties of dimensionality, orientability, continuity, and connectivity define “real” space as studied by physics and (...) are the spatial properties that characterize the physical universe as being an integral whole. By contrast the geometrical analysis of VS has taken little cognizance of its topology. Instead such properties have been presupposed a priori rather than being established a posteriori by empirical means, perhaps because these properties are self-evident. Applying the method of coordinative definition expounded by Hans Reichenbach for determining geometrical and topological properties of physical space, it can be shown that VS fulfills the topological criteria of being a “real” space sui generis. Though theorized to be produced by the brain, the topology of VS is not topologically equivalent with the structure and activity of the brain because, as will be shown, the topology of VS cannot be formed from the topology of the brain without tearing and/or cutting and pasting. (shrink)

Reviewed Works:Reuben Hersh, Proving is Convincing and Explaining.Philip J. Davis, Visual Theorems.Gila Hanna, H. Niels Jahnke, Proof and Application.Daniel Chazan, High School Geometry Students' Justification for Their Views of Empirical Evidence and Mathematical Proof.

Spatial perception and spatial representation are not less central to experimental psychology than to visual art. Geometry allows their description and formalization. Therefore, geometrical language can be considered as a kind of generative grammar, which is embedded in the human perceptual experience of space. The paper outlines the suggestion that Euclidean geometry, along with most perspective geometries, even when applied to geometrical problem solving, have phenomenal bases, since they emerge from direct experience of the world, and not (...) necessarily from higher order cognitive processes. (shrink)

In his Rules for the Direction of the Mind, Descartes elevates arithmetic and geometry to the status of paradigms for all the sciences, because of the potential for certainty in their results. This emphasis on certainty is present throughout the Cartesian corpus, but in the Rules and other early works the substance dualism characteristic of Cartesian philosophy is not as obvious. However, when several key concepts from this early work are considered together, it becomes clear that Cartesian dualism necessarily (...) follows. The most important of these concepts are: Descartes’s rejection of the Scholastic theory of sense data in favor of a telecommunicative theory of perception; his innovative reconceptualization of mathematics in which he treats number and magnitude as interchangeable, making use of the symbolism of algebra; his frequent use of visual metaphors to describe perception in general, as well as other cognitive activity. It is in consideration of this third point that Descartes’s treatment of the imagination shows its importance, for the relationship of the imagination to the intellect parallels that of geometric figures to algebraic formulae. Though the role of the imagination in the Rules seems to make the status of a dualist ontology in this work more ambiguous, this paper will argue that in fact it is precisely in the treatment of imagination that one can find the traces of a fully developed substance dualism. (shrink)

According to Kepler and Descartes, the geometry of the triangle formed by the two eyes when focused on a single point affords perception of the distance to that point. Kepler characterized the processes involved as associative learning. Descartes described the processes as a “ natural geometry.” Many interpreters have Descartes holding that perceivers calculate the distance to the focal point using angle-side-angle, calculations that are reduced to unnoticed mental habits in adult vision. This article offers a purely psychophysiological (...) interpretation of Descartes’s natural geometry. In his account of perceived limb position from the Treatise on Man, he envisioned a central brain state that controls ocular convergence and thereby co-varies with the distance from observer to object. A psychophysiological law relates the visual perception of distance to this brain state. Descartes also invokes more traditional theories of distance and size perception based on unnoticed judgments, yielding a hybrid account. (shrink)

Students of geometry do not prove Euclid's first theorem by examining an accompanying diagram, or by visualizing the construction of a figure. The original proof of Euclid's first theorem is incomplete, and this gap in logic is undetected by visual imagination. While cognition involves truth values, vision does not: the notions of inference and proof are foreign to vision.

In mathematics textbooks and special mathematical treatises, themes and theses of Arthur Schopenhauer's elementary geometry appear again and again. Since Schopenhauer's geometry or philosophy of geometry was considered exemplary in the 19th and early 20th centuries in its relation to figures and thus to the intuition, the two-hundred-year reception history sketched in this paper also follows the evaluation of intuition-related geometries, which depends on the mathematical paradigms.

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)

Picture geometry is often regarded as an area of technical knowledge that accompanies or provides useful information for basic research on visual culture and almost never as a methodological one. Despite the historical and conceptual connections between mathe­matics and the visual, even a basic geometric competence is by no means a common of image and visual culture researchers. At the same time, the overwhelming majority of this kind of work belong to the field of technical knowledge, (...) the history of mathemat­ics, or positivism based theories; in other words, they do not address the most important issues for humanities research. Rare non-technical works in this area do not find due recognition and dissemination among a wider community of specialists. The article offers the main result of the research, the subject of which was pictorial depth. Pictorial depth is the version of the specific experience of distance with which the image is traditionally associated. The problem of distance is a classical area of philosophical reflection on the image. The article presents an attempt to geometrically express one of its versions, namely pictorial and visual depth. At the same time, the important dimension of inacces­sibility and heterogeneity is preserved, that is, the experience of depth is not reduced to visual data or the literal geometry of a flat picture. By constructing a geometrically object that can be considered depth, it is possible to extract intuitively non-obvious properties that would not be available through direct analysis of experience and other methods of naked reflection. The article presents only some of the findings obtained during the study. (shrink)

The goal of this paper is to challenge the rather insouciant attitude that many investigators seem to adopt when they go about describing the items and events in their " visual fields". There are at least three distinct categories of interpretation of what these reports might mean, and only under one of those categories do those reports have anything resembling an observational character. The others demand substantive revisions in one's beliefs about what one sees. The ur-concept of a " (...) visual field" is that of the "sum of things seen", but one can interpret the latter in very different ways. The first is the "field of view", or the sum of physical things seen. The second is an array of visual impressions, whose spatial relations are distinct from those of physical phenomena in front of the eyes. The third is an intentional object: the world as it is represented visually. These three categories are described, and various locutions of vision science--such as "optic array", "retinocentric space", " visual geometry", "virtual object" and others--are analyzed and variously located within them. Finally, a recent argument purporting to necessitate the existence of a version two visual field is examined and shown wanting. (shrink)

This paper examines Helmholtz's attempt to use empirical psychology to refute certain of Kant's epistemological positions. Particularly, Helmholtz believed that his work in the psychology of visual perception showed Kant's doctrine of the a priori character of spatial intuition to be in error. Some of Helmholtz's arguments are effective, but this effectiveness derives from his arguments to show the possibility of obtaining evidence that the structure of physical space is non-Euclidean, and these arguments do not depend on his theory (...) of vision. Helmholtz's general attempt to provide an empirical account of the "inferences" of perception is regarded as a failure. (shrink)

Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of persuasive (...) visual routes to two simple theorems (Rolle, Bolzano) shows that they are not ways of discovering the theorems; the type of visual thinking involved can never be used to discover analytic theorems of a certain generality. The hypothesis that visual thinking is never a means of discovering the existence or nature of the limit of some infinite process is considered, and a likely counter-example is set out. It is still possible that restricted theorems can be discovered visually: an example from Littlewood is examined in detail and not found wanting. Even when visual thinking is not a means of discovery it can provide the idea for a proof in a direct way; an example is presented (Intermediate Value Theorem). In conclusion: it may be possible to discover theorems in elementary real analysis by visual means, but only theorems of a restricted kind; however, visual thinking in analysis can be very useful in other ways. (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)

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 square tabular array was introduced by R. C. Punnett in (1907) to visualize systematically and economically the combination of gametes to make genotypes according to Mendel’s theory. This mode of representation evolved and rapidly became standardized as the canonical way of representing like problems in genetics. Its advantages over other contemporary methods are discussed, as are ways in which it evolved to increase its power and efficiency, and responded to changing theoretical perspectives. It provided a natural visual decomposition (...) of a complex problem into a number of inter-related stages. This explains its computational and conceptual power, for one could simply “read off” answers to a wide variety of questions simply from the “right” visual representation of the problem, and represent multiple problems, and multiple layers of problems in the same diagram. I relate it to prior work on the evolution of Weismann diagrams by Griesemer and Wimsatt (What Philosophy of Biology Is, Martinus-Nijhoff, the Hague, 1989), and discuss a crucial change in how it was interpreted that midwifed its success. (shrink)

One often hears it said that our visual-perceptual contact with the world is “perspectival.” But this can mean quite different things. Three different senses in which our visual contact with the world is “perspectival” are distinguished. The first involves the detection or representation of behaviorally important relations, holding between a perceiving subject and the world. These include time to contact, body-scaled size, egocentric position, and direction of heading. The second perspective becomes at least explicitly manifest in taking up (...) the “proximal mode” and involves the representation of relational “appearance properties,” such as, perhaps , angular size and angular/projective shape at or from a viewpoint. The third perspective is associated with the pencil of rays structured by the position and geometry of the eyes. The relationships between these three perspectives are charted, and their connection with conscious experiential awareness discussed. So, e.g., there is no simple or direct route from the detection of the relations constitutive of perspective2 to detection of the relations constitutive of perspective1. The relations definitive of perspective3 are probably not themselves detected or represented by the visual system, but define the informational arena within which detection of the relations definitive of perspective1 and perspective2 takes place. (shrink)

This paper aims to show that in ancient Greek architecture, it is possible to find a genesis of the geometric modeling of visual perception present in propositions of Euclid's Optics, considering mathematical knowledge as a human wisdom expression. Let us start by emphasizing that mathematical thinking is not exclusively rooted in mathematical disciplines, but also includes the broad spectrum of human activities, including activities that come from everyday life. Based on this, we present a socio-cultural characterization of human experience (...) as the source and sustainer of mathematical knowledge. Thereafter, on the basis of a content and contextual analysis of _Euclid's Optics_, we explain the use and development of geometry in the study of various optical effects of visual perception in the architectural art embodied in the Acropolis of Athens; and we focus our analysis on one of the most important of these effects, the Acropolis of Athens. We focus our analysis on one of the most emblematic architectural works of ancient Greek culture: the Parthenon of Athens. (shrink)

The overarching guiding principle of Alan Turing’s work was directed towards modelling the human mind as a machine. It is extraordinary that Turing introduced, in his early papers, ideas that are only now beginning to be investigated. Throughout his life, he considered conjectures to be of great importance because they suggest useful lines of research. In my own conjecture, I am asking the question: what is the brain’s geometry? Can it ever be unravelled, or does its complexity defy any (...) form of visual representation? Recent research reveals that the detailed workings of the brain take place in a ‘global workspace’, where conscious contents appear to be disseminated globally to a great multitude of networks throughout the brain that are unconscious. During conscious tasks, neurons contributing to the global workspace enter into a coherent activation pattern by being tightly connected through long axons. Is it possible, therefore, that my proposed geometric pattern (an Islamic latticework) can be envisaged as a useful means of interconnecting multiple long axons in a unified global workspace dedicated to conscious thought? I share and communicate my idea through a process that combines visual imagery with musical performance (i.e. visual music). It is a method that can assist in clarifying concepts that might otherwise remain elusive or esoteric. (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)

As the word “optics” was understood from antiquity into and beyond the early modern period, it did not mean simply the physics and geometry of light, but meant the “theory of vision” and included what we should now call physiological and psychological aspects. From antiquity, these aspects were subject to geometrical analysis. Accordingly, the geometry of visual experience has long been an object of investigation. This chapter examines accounts of size and distance perception in antiquity (Euclid and (...) Ptolemy) and the Middle Ages (Ibn al-Haytham), before turning to "natural geometry" in Kepler and Descartes. It finds a purely mechanical realization of natural geometry in Descartes (primarily in his L'Homme, 1664), in which he conceives states of the visual system as varying in accordance with distance, much as his geometrical compasses vary their spatial relations in accordance with geometrical proportions. This reading challenges the notion that a two-dimensional sensation (without three-dimensional phenomenality) was universally accepted prior to the nineteenth century by positing a direct psychophysiological account of the experience of distance in Descartes. It thereby challenges the interpretations of Descartes on natural geometry of George Berkeley, Nancy Maull, Margaret Wilson, Erwin Panofsky, and Jonathan Crary. (shrink)

Cognitive impenetrability (CI) of a large part of visual perception is taken for granted by those of us in the field of computational vision who attempt to recover descriptions of space using geometry and statistics as tools. These tools clearly point out, however, that CI cannot extend to the level of structured descriptions of object surfaces, as Pylyshyn suggests. The reason is that visual space – the description of the world inside our heads – is a nonEuclidean (...) curved space. As a consequence, the only alternative for a vision system is to develop several descriptions of space–time; these are representations of reduced intricacy and capture partial aspects of objective reality. As such, they make sense in the context of a class of tasks/actions/plans/purposes, and thus cannot be cognitively impenetrable. (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.

The research was conducted at one of the universities in Banten. Even semester, a sample of all students majoring in mathematics education, which totaled 88 students, was taken. The data collection approach involves learning style questionnaires, geometric thinking skills exams, and interviews, and the research method is qualitative. Qualitative data analysis follows a three-stage flow model: data reduction, data presentation and conclusion, and verification. The research problems are: 1) how students learn during online learning, and 2) how students' geometry (...) thinking skills are reviewed from their learning style. The results of the study concluded that students more have auditorial learning style during online learning. Then based on the ability to think geometry, students are seen from the mastery of fundamental geometry skill indicators and reviewed from their learning style. Known students with visual learning styles master visual indicators and drawing but lack master verbal, logical, and applied indicators. Furthermore, students with an auditorial learning style show mastery of logic and applied indicators but lack mastering visual, verbal, and drawing indicators. Students with kinesthetic learning styles show mastery of drawing and applied indicators but lack mastering visual, verbal, and logic indicators. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume XXIII, Number 2, November 1997, pp. 227-244 Hume on Geometry and Infinite Divisibility in the Treatise H. MARK PRESSMAN Scholars have recognized that in the Treatise "Hume seeks to find a foundation for geometry in sense-experience."1 In this essay, I examine to what extent Hume succeeds in his attempt to ground geometry visually. I argue that the geometry Hume describes in the (...) Treatise faces a serious set of problems. Geometric Lines Hume maintains that ideas "are images" (T 6) which may be called up "when I shut my eyes" (T 3). "That we may fix the meaning of a word, figure," according to Hume, "we may revolve in our mind the ideas of circles, squares, parallelograms, triangles of different sizes and proportions, and may not rest on one image or idea" (T 22). As Arthur Pap notes, "'Idea' is in Hume's usage synonymous with 'mental image'." D.C.G. MacNabb also notes that "Hume thought of ideas as images, and primarily as visual images."2 When Hume argues, then, that "we have the idea of indivisible points, lines and surfaces conformable to the [geometer's] definition" (T 44), he is arguing that we have mental images of geometry's points, lines and planes. Consider geometric lines first. Geometers define a line "to be length without breadth or depth" (T 42). It is not apparent to everyone that we have mental images of such lines. According to "L'Art de penser" (T 43), published in 1662, it is impossible "to conceive a length without any breadth" (T 43). H. Mark Pressman is at the Department of Philosophy, University of California, Davis, 1238 Social Sciences and Humanities Building, Davis CA 95616-8673 USA. email: imn78@cvin.fresno.com 228 H. Mark Pressman We can only by an abstraction "consider the one without regarding the other" (T 43). Hume, however, offers "clear proof" (T 44), in the form of two arguments, that we actually possess ideas or mental images of geometry's lines. First, we have ideas or mental images of lengths with breadth, though these are supposed never to be without breadth and thus not geometric lines. (A mental image oflength is also a mental image with length, just as "to say the idea of extension agrees to any thing, is to say it is extended" [T 240].) Hume reduces to absurdity the view that our mental images of length are always with breadth and thus not images of geometric lines. If our length-images were always with breadth, then our mental images would be endlessly divisible in breadth, since they would always have some positive breadth (T 43). But no mental image is endlessly divisible, for the mind arrives "at an end in the division of its ideas, nor are there any possible means of evading the evidence of this conclusion" (T 27). We thus have an idea or image of length without (divisible) breadth conforming to the geometer's definition (T 44). Second, we can imagine or picture the termination of a geometric plane. But a geometric plane must terminate in a geometric line (T 44). (Proof: Assume L is a line with breadth which terminates plane P. Since L has breadth, L must contain at least two parts, w and y, exactly one of which actually terminates P. But whether it is w or y, L doesn't terminate P, which was the assumption.) Since we can picture the termination of a geometric plane, we can picture a geometric line, length without divisible breadth. It is one thing to have ideas or mental images of geometric lines which may be called up "when I shut my eyes" (T 3). It is another for such objects actually to exist "in nature." The conceptualist holds that though (a) geometry 's points, lines and planes are ideas, nevertheless (b) there are no such things in nature and that (c) there can't be such things in nature: [T]he objects of geometry... are mere ideas in the mind, and not only never did, but never can exist in nature. They never did exist; for no one will pretend to draw a line or make a... (shrink)