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)

Luminance and color are strong and self-sufficient cues to pictorial depth in visual scenes and images. The present study investigates the conditions Under which luminance or color either strengthens or overrides geometric depth cues. We investigated how luminance contrasts associated with color contrast interact with relative height in the visual field, partial occlusion, and interposition in determining the probability that a given figure is perceived as ‘‘nearer’’ than another. Latencies of ‘‘near’’ responses were analyzed to test for effects of attentional (...) selection. Figures in a pair were supported by luminance contrast or isoluminant color contrast and combined with one of the three geometric cues. The results of Experiment 1 show that luminance contrasts associated with hue, when it does not interact with other hues, produces the same effects as achromatic luminance contrasts: The probability of‘‘near’’ increases with luminance contrast while the latencies for ‘‘near’’ responses decrease. Partial occlusion is found to be a strong enough pictorial cue to support a weaker red luminance contrast. Interposition cues lose out against cues of spatial position and partial occlusion. The results of Experiment 2, with isoluminant displays of varying color contrast, reveal that red color contrast on a light background supported by any of the three geometric cues wins over green or white supported by any of the three geometric cues. On a dark background, red color contrast supported by the interposition cue loses out against green or white color contrast supported by partial occlusion. These findings reveal that color is not an independent depth cue, but is strongly influenced by luminance contrast and stimulus geometry. Systematically shorter response latencies for stronger ‘‘near’’ percepts demonstrate that selective visual attention reliably detects the most likely depth cue combination in a given configuration. (shrink)

We prove that, in the framework of ordered geometry, the inner form of the Pasch axiom does not imply its outer form . We also show that OP can be properly split into IP and the weak Pasch axiom.

The city has featured as a central image in utopian thought. In planning the foundation of the new and ideal city there is a close interconnection between ideas about urban form and the vision of the moral good. The spatial structure of the ideal city in these visions is a framing device that embodies and articulates not only political philosophy but is itself an articulation of moral and cosmological systems. This paper analyses three different utopian moments in three different (...) historical epochs – Tommaso Campanella’s City of the Sun (1602), the Choson dynasty foundation of the city of Seoul (1395) and the modernist utopian urbanism of the controversial Le Corbusier (1887–1965). In this analysis attention is drawn to the cosmological and moral visions articulated in these three ‘images of the city’ (Lynch). The opposition between rationalistic/mechanistic and religious/traditional urban design can prove to be an oversimplification that obscures the complex interrelations between the moral geometry and the natural alignments of the ideal urban form. (shrink)

The aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of (...) geometry. I will try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein. (shrink)

Planar cell polarity (PCP) refers to the polarization of a field of cells within the plane of a cell sheet. This form of polarization is required for diverse cellular processes in vertebrates, including convergent extension (CE), the establishment of PCP in epithelial tissues and ciliogenesis. Perhaps the most distinct example of vertebrate PCP is the uniform orientation of stereociliary bundles at the apices of sensory hair cells in the mammalian auditory sensory organ. The establishment of PCP in the (...) mammalian cochlea occurs concurrently with CE in this ciliated epithelium, therefore linking three cellular processes regulated by the vertebrate PCP pathway in the same tissue and emerging as a model system for dissecting PCP signaling. This review summarizes the morphogenesis of this model system to assist the interpretation of the emerging data and proposes molecular mechanisms underlying PCP signaling in vertebrates. BioEssays 29: 120–132, 2007. © 2007 Wiley Periodicals, Inc. (shrink)

That Wittgenstein in the Tractatus likens logic to geometry has been noticed; however, the extent and force of the analogy he develops between logical form and a broadly Kantian account of geometry has not been sufficiently appreciated. In this paper, I trace Wittgenstein's analogy in detail by looking closely at the relevant texts. I then suggest that we regard the fact that Wittgenstein develops his account of logical form by analogy with a Kantian account of (...) class='Hi'>geometry as evidence for the bold thesis that Wittgenstein belongs within a Kantian epistemological tradition. Finally, I supply two small pieces of the interpretive puzzle needed to support the larger thesis: first, evidence that Wittgenstein's concern with logical form involves a crucial epistemological component; second, a sketch of how Wittgenstein's account of logical and mathematical knowledge can be viewed as continuing a Kantian tradition. (shrink)

When fcc single crystals with high-symmetry crystal orientations are deformed to moderate strains by rolling, tension or channel die compression, long dislocation boundaries inclined to the extension axis form. Similarly, long dislocation boundaries are often found in grains embedded in polycrystals deformed in the same manner. These extended planar boundaries are characteristically - 30-40° from the extension direction and contain the transverse specimen axis. The objective of the present article is to demonstrate that EPBs formed during plane strain (...) deformation are parallel to equivalent slip planes, a pair of hypothetical slip systems used for analyses of the strain and crystal rotation components in place of the larger number of physical slip systems. The coincidence of EPBs and equivalent slip plane inclinations is shown to account for persistent observations of EPBs in the angle range - 30-40° from the rolling direction, in rolled single crystals of various initial orientations. The tendency of EPBs towards tilt or twist boundary character can also be rationalized on the basis of the equivalent slip system concept and consideration of the dislocation types available to be incorporated into EPBs. (shrink)

Euclidean geometry, statics, and classical mechanics, being in some sense the simplest physical theories based on a full-fledged mathematical apparatus, are well suited to a historico-philosophical analysis of the way in which a physical theory differs from a purely mathematical theory. Through a series of examples including Newton’s Principia and later forms of mechanics, we will identify the interpretive substructure that connects the mathematical apparatus of the theory to the world of experience. This substructure includes models of experiments, models (...) of measurement, and modular connections with partial theories. It evolves during the life of a theory as physicists learn how to apply it in various contexts. It should nevertheless be regarded as an integral part of a genuine physical theory since the theory would otherwise degenerate into pure mathematics. (shrink)

The geometry of the state space of a finite-dimensional quantum mechanical system, with particular reference to four dimensions, is studied. Many novel features, not evident in the two-dimensional space of a single spin, are found. Although the state space is a convex set, it is not a ball, and its boundary contains mixed states in addition to the pure states, which form a low-dimensional submanifold. The appropriate language to describe the role of the observer is that of flag (...) manifolds. (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)

I discuss the debate between dynamical versus geometrical approaches to spacetime theories, in the context of both special and general relativity, arguing that the debate takes a substantially different form in the two cases; different versions of the geometrical approach—only some of which are viable—should be distinguished; in general relativity, there is no difference between the most viable version of the geometrical approach and the dynamical approach. In addition, I demonstrate that what have previously been dubbed two ‘miracles’ of (...) general relativity admit of no resolution from within general relativity, on either the dynamical or ‘qualified’ geometrical approaches, modulo some possible hints that the second ‘miracle’ may be resolved by appeal to recent results regarding the ‘geodesic principle’ in GR. (shrink)

L’œuvre mathématique de Leibniz a ceci d’intéressant qu’au travers des innombrables manuscrits de travail dont nous disposons dans ses archives à Hanovre, le philosophe nous a confié le matériel nécessaire pour rétablir ses divers cheminements de recherche ainsi que ses méthodes de découvertes à l’origine de ses créations mathématiques. L’exemple des exposants que nous allons traiter permet d’éclairer utilement la façon dont Leibniz apprend les mathématiques et change progressivement de posture et de démarche. Ainsi, dans sa première année parisienne, la (...) primauté est-elle donnée aux relations tabulaires, plutôt qu’aux formes et aux similitudes. Ces enjeux épistémologiques expliquent la manière dont Leibniz généralise partiellement la notion d’exposant. (shrink)

In this paper, we will make explicit the relationship that exists between geometric objects and geometric figures in planar Euclidean geometry. That will enable us to determine basic features regarding the role of geometric figures and diagrams when used in the context of pure and applied planar Euclidean geometry, arising due to this relationship. By taking into account pure geometry, as developed in Euclid’s Elements, and practical geometry, we will establish a relation between geometric (...) objects and figures. Geometric objects are defined in terms of idealizations of the corresponding figures of practical geometry. We name the relationship between them as a relation of idealization. This relation, existing between objects and figures, is what enables figures to have a role in pure and applied geometry. That is, we can use a figure or diagram as a representation of geometric objects or composite geometric objects because the relation of idealization corresponds to a resemblance-like relationship between objects and figures. Moving beyond pure geometry, we will defend that there are two other ‘layers’ of representation at play in applied geometry. To show that, we will consider Euclid’s Optics. (shrink)

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = f for the graph Laplacian Δθ, potential functions q, p built from the (...) probabilities, and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ıψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations. (shrink)

This article is primarily concerned with Aristotle’s theory of the syllogistic, and the investigation of the hypothesis that logical symbolism and methodology were in these early stages of a geometrical nature; with the gradual algebraization that occurred historically being one of the main reasons that some of the earlier passages on logic may often appear enigmatic. The article begins with a brief introduction that underlines the importance of geometric thought in ancient Greek science, and continues with a short exposition of (...) Aristotle’s views and methods in regard to logic. We then offer an interpretation of syllogisms, as well as of the main proof methods that are utilized by Aristotle, in terms of diagrams that can be seen as analogous to those of Euclidean geometry; where the various proofs proceed by appropriately manipulating an initially drawn diagram, in a way parallel to constructions that occur in Euclid’s Elements. In this way, logic is presented as following, to a large degree, methodological tenets established by the practice of geometry. Finally, we present a diagrammatic decision procedure that allows one to directly read off the validity of syllogisms from a corresponding diagram, in a way such that no further manipulation of it is necessitated. (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto (...) intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. (...) However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism. (shrink)

The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The (...) paper focuses mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation. (shrink)

In recent years the magnificent world of fractals has been revealed. Some of the fractal images resemble natural forms so closely that Benoit Mandelbrot's hypothesis, that the fractal geometry is the geometry of natural objects, has been accepted by scientists and non-scientists alike. The present paper critically examines Mandelbrot's hypothesis. It first analyzes the concept of a fractal. The analysis reveals that fractals are endless geometrical processes, and not geometrical forms. A comparison between fractals and irrational numbers shows (...) that the former are ontologically and epistemologically even more problematic than the latter. Therefore, it is argued, a proper understanding of the concept of fractal is inconsistent with ascribing a fractal structure to natural objects. Moreover, it is shown that, empirically, the so-called fractal images disconfirm Mandelbrot's hypothesis. It is conceded that the fractal geometry can be used as a useful rough approximation, but this fact has no bearing on the physical theory of natural forms. (shrink)

Les architectures molles, sculptées, transparentes, immatérielles prétendent se libérer des contraintes géométriques, comme si la géométrie ne revendiquait que la droite et la forme orthogonale ou le cercle! Certains architectes s'abandonnent aux " hasards " informatiques et construisent des édifices à la géométrie chahutée par un logiciel. Des urbanistes opposent encore le plan radioconcentrique au plan en damier en ce qui concerne l'expansion des villes et, refusant d'imaginer d'autres morphologies, laissent faire la promotion immobilière, les opportunités foncières et le chacun (...) pour soi. La géométrie, dans notre culture marquée par la philosophie grecque, est constitutive et de l'architecture et de l'urbanisme. Elle est mise en débat par le jeu extraordinairement varié des formes et de leurs agencements, aussi bien que par des régulations qui donnent une mesure au monde et suscitent des questionnements quant à ce qui est à la mesure de l'existence. Depuis le simple pas jusqu'aux théories les plus sophistiquées, la géométrie - qui n'a jamais cessé de se complexifier depuis Pythagore ou Euclide jusqu'aux géométries algébrique, infinitésimale et variable - se rappelle à nous. C'est ce rappel qu'il nous faut entendre, comme une invitation à penser aussi bien notre corps que le paysage, aussi bien la maison que la ville et la cité. Cet ouvrage collectif veut questionner géométriquement et philosophiquement l'urbain contemporain et les architectures qu'il provoque. En d'autres termes, il espère saisir à partir de la confrontation entre mathématiciens, géomètres, historiens, architectes, urbanistes, paysagistes et philosophes l'expérience existentielle de l'espace-temps des lieux. (shrink)

Cet article explore la signification de l’espace comme un « lieu pratiqué » selon la notion reprise à Michel de Certeau, en examinant la construction d’un gymnase et ses effets sur les relations sociales et les réseaux disciplinaires. Tout comme le laboratoire ou le théâtre, le gymnase a été spécifiquement pensé pour permettre certaines actions et en témoigner, en reflétant des conceptions de l’entraînement et de l’éducation corporelle. Ses divers agencements de l’espace y favorisent une incorporation de la race, du (...) lieu, du genre et de l’identité. Le War Memorial Gymnasium de l’Université de Colombie britannique dans l’Ouest du Canada, construit en 1851, est utilisé pour illustrer la manière dont, sous des formes complexes, les géométries du pouvoir – ces multiples configurations de l’espace et des relations sociales à l’intérieur et autour du gymnase – produisent dans ses murs une culture didactique et sportive aux enjeux véritablement stratégiques et tactiques, qui active et articule les division de genre. (shrink)

Most commentators agree that (part of what) Kant means by characterizing the propositions of geometry as synthetic is that they are not true merely in virtue of logic or meaning, and that this characterization has something to do with his views about the construction of geometrical concepts in intuition. Many commentators regard construction in intuition as an essential part of geometrical proofs on Kant’s view. On this reading, the propositions of geometry are synthetic because the geometrical theorems cannot (...) be proved in purely conceptual or logical terms. Other commentators see the main role of pure intuition and the figures constructed in pure intuition in that they provide a model for Euclidean geometry. On views of this kind, the propositions of geometry are synthetic because the geometrical axioms are substantive truths about one of our forms of intuition. On the interpretation proposed in this essay, what Kant means by claiming that the propositions of geometry are synthetic is not only that the Euclidean axioms and theorems cannot be reduced to tautologies or logical truths, but also that they apply to really possible objects. Construction in intuition plays no essential role in (what we now call) ‘pure’ geometry on Kant’s view. But the fact that the concepts of geometry can be constructed in intuition is of crucial importance in the context of Kant’s transcendental philosophy of geometry, because, among other things, it allows him to explain how Euclidean geometry is possible as an a priori synthetic science in the sense just indicated. (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)

I present an argument against a relational theory of spacetime that regards spacetime as a ‘structural quality of the field’. The argument takes the form of a trilemma. To make the argument, I focus on relativistic worlds in which there exist just two fields, an electromagnetic field and a gravitational field. Then there are three options: either spacetime is a structural quality of each field separately, both fields together, or one field but not the other. I argue that the (...) first option founders on a problem of geometric coordination and that the second and third options collapse into substantivalism. In particular, on the third option it becomes clear that the relationalist’s path to Leibniz equivalence is no simpler or more straightforward than the substantivalist’s. 1Introduction 2Background 2.1Relational theories of spacetime 2.2Brief remarks on general relativity 3First Horn: Geometric Coordination 4Second Horn: Problems of Coincidence 5Third Horn: General Relativity and Gauge 5.1Preventing collapse 5.2The revised shift argument 6Conclusion. (shrink)

For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems (...) for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems. (shrink)

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)

Summary In his Theoremata de lumine, et umbre (1521), Francesco Maurolyco (1494–1575) discussed, inter alia, the problem of the pinhole camera. Maurolyco outlined a framework based on Euclidean geometry in which he applied the rectilinear propagation of light to the casting of shadow on a screen behind a pinhole. We limit our discussion to the problem of how the image behind an aperture is formed, and follow the way Maurolyco combined theory with instrument to solve the problem of the (...) projection of light through small apertures. We show that Maurolyco not only reformed the classical sources which, he thought, were no longer the authoritative code of textual knowledge, but also established with the dioptra a novel linkage of method, theory, and instrument. He thereby demonstrated the importance of optics to the science of astronomy. (shrink)

After sketching the historical development of “emergence” and noting several recent problems relating to “emergent properties”, this essay proposes that properties may be either “emergent” or “mergent” and either “intrinsic” or “extrinsic”. These two distinctions define four basic types of change: stagnation, permanence, flux, and evolution. To illustrate how emergence can operate in a purely logical system, the Geometry of Logic is introduced. This new method of analyzing conceptual systems involves the mapping of logical relations onto geometrical figures, following (...) either an analytic or a synthetic pattern (or both together). Evolution is portrayed as a form of discontinuous change characterized by emergent properties that take on an intrinsic quality with respect to the object(s) or proposition(s) involved. Causal leaps, not continuous development, characterize the evolution of human life in a developing foetus, of a thought out of certain brain states, of a new idea (or insight) out of ordinary thoughts, and of a great person out of a set of historical experiences. The tendency to assume that understanding evolutionary change requires a step-by-step explanation of the historical development that led to the appearance of a certain emergent property is thereby discredited. (shrink)

In the Quattrocento and Cinquecento the rise of linear perspective caused many polemics which opposed the supporters of an artificial geometrisation of sight to those who were praising the qualities of the drawing according to nature, or were invoking some arguments on a physiological basis. These debates can be grouped according to the four alternatives that form their central concerns: restricted vs. broad field of vision; ocular immobility vs. mobility; curvilinear vs. planar picture; monocular vs. binocular vision. By (...) retaining the first terms of these four alternatives, the history of perspective eliminated many heterodox constructions. From the viewpoint of mathematisation the interest of these debates is that they succeeded, rather than preceded, the adoption of a perspective system defined by the intersection of the visual pyramid. Thus the history of linear perspective constitutes a genuine case of a posteriori justification, or, put differently, it gives us a case of upside down mathematisation. (shrink)

Historians of modern philosophy have been paying increasing attention to contemporaneous scientific developments. Isaac Newton's Principia is of course crucial to any discussion of the influence of scientific advances on the philosophical currents of the modern period, and two philosophers who have been linked especially closely to Newton are John Locke and Immanuel Kant. My dissertation aims to shed new light on the ties each shared with Newtonian science by treating Newton, Locke, and Kant simultaneously. I adopt Newton's philosophy of (...) geometry as the starting point of investigation, for here I believe we have a constructive means by which to assess Locke and Kant's relationship to Newton, In particular, I defend the thesis that the justification Newton, Locke and Kant offer for applying geometrical principles to nature is central to understanding their respective ties to a Newtonian science characterized by the intermingling of mathematics and experiment, Although little is said by Locke in regard to a mathematical approach to nature, I hope to show that his interpretation of the origins of our geometrical ideas has a close affinity to Newton's own characterization of geometry, leading us to reexamine the extent of Locke's 'Newtonianism.' Kant famously attempts to bridge the gap between geometry and the empirical world by establishing space as a "pure form of intuition." My discussion of Kant's Newtonian approach to nature centers on the imagination, and I argue that the mediating work completed by this faculty in geometrical construction and experience in general is equally important to understanding Kant's application of geometry to the empirical realm, In the end, I hope my treatment of the strategies employed by Newton, Locke, and Kant to account for a mathematical-experimental method of natural philosophy sheds further light on the importance of Newton to the progress of modern philosophy. (shrink)

There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are “the other way around”). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood (...) as involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators—Post’s and Kleene’s—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

In the study of Hiberno-Saxon art, three key monuments stand out: the Book of Durrow, the Lindisfarne Gospels, and the Book of Kells. They form an impressive trilogy. The earliest, the Book of Durrow, represents a developed but still “archaic” early stage; accomplished, but colored by a certain primitivism, it boldly reveals its sources in the art of pre-Christian Celtic and Germanic peoples and perhaps the late antique art of Coptic Egypt. These foundations are still evident in the latest (...) in date of the three, the Book of Kells, but they are shrouded in a world of decorative richness, intense technical virtuosity, and “baroque” complexity. Between these two in time, the Lindisfarne Gospels represents the “classical” manifestation of Hiberno-Saxon illumination — highly ordered in its ornamental mazes, sharp and clear in the statement of its forms. An aura of timeless perfection is felt especially in the majestic stability of its five cross-carpet pages. (shrink)

Richard Jeffrey and Michael Goldstein have both introduced systematic approaches to the structure of opinion changes. For both approaches there are theorems which indicate great generality and width of scope. The main questions addressed here will be to what extent the basic forms of representation are intertranslatable, and how we can conceive of such programs in general.

Richard Jeffrey and Michael Goldstein have both introduced systematic approaches to the structure of opinion changes. For both approaches there are theorems which indicate great generality and width of scope. The main questions addressed here will be to what extent the basic forms of representation are intertranslatable, and how we can conceive of such programs in general.

The essay records the changes in family organization for the importance of grandparents in these years of crisis. Their contribution is made in three areas: significant economic aid, organizational support, emotional support. It is an extraordinary contribution that has alleviated the consequences of the collapse, not just financial, of our country. But led by the generation that is usually defined as ‘lucky', a heavy existential commitment. The presence of grandparents, essential in cases of family separation to ensure security, continuity and (...) identity to the grandchildren, would entail the risk that their children, especially when they return to live in the family of origin, regress to a state of infantile dependence. The problems are not lacking, but the ethics of solidarity and generosity that are testifying is a heritage rich in possibilities and promises a view to a redefinition of "family form". (shrink)

ArgumentAccording to Hermann von Helmholtz, free mobility of bodies seemed to be an essential condition of geometry. This free mobility can be interpreted either as matter of fact, as a convention, or as a precondition making measurements in geometry possible. Since Henri Poincaré defined conventions as principles guided by experience, the question arises in which sense experiential data can serve as the basis for the constitution of geometry. Helmholtz considered muscular activity to be the basis on which (...) the form of space could be construed. Yet, due to the problem of illusion inherent in the subject’s self-assessment of muscular activity, this solution yielded new difficulties, in that if the manifold is abstracted from rigid bodies which serve as empirical justification of the geometrical notion of space, then illusionary bodies will produce fictive manifolds. The present article is meant to disentangle these difficulties. (shrink)

We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...) brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)

We explore the intimate connection between spacetime geometry and electrodynamics. This link is already implicit in the constitutive relations between the field strengths and excitations, which are an essential part of the axiomatic structure of electromagnetism, clearly formulated via integration theory and differential forms. We review the foundations of classical electromagnetism based on charge and magnetic flux conservation, the Lorentz force and the constitutive relations. These relations introduce the conformal part of the metric and allow the study of electrodynamics (...) for specific spacetime geometries. At the foundational level, we discuss the possibility of generalizing the vacuum constitutive relations, by relaxing the fixed conditions of homogeneity and isotropy, and by assuming that the symmetry properties of the electro-vacuum follow the spacetime isometries. The implications of this extension are briefly discussed in the context of the intimate connection between electromagnetism and the geometry of spacetime. (shrink)

This paper forms part of a wider campaign: to deny pointillisme. That is the doctrine that a physical theory's fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts. More specifically, this paper argues against pointillisme about the structure of space and-or spacetime itself, especially a paper by Bricker (1993). (...) A companion paper argues against pointillisme in mechanics, especially about velocity; it focusses on Tooley, Robinson and Lewis. To avoid technicalities, I conduct the argument almost entirely in the context of ``Newtonian'' ideas about space and time. But both the debate and my arguments carry over to relativistic, and even quantum, physics. (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 (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)

The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of (...) such theories as projective geometry; and not only as they were practiced in Kant’s time, but also as architects use them today. (shrink)