Results for 'Geometry'

171 found
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  1. Core Knowledge of Geometry in an Amazonian Indigene Group.Stanislas Dehaene, Véronique Izard, Pierre Pica & Elizabeth Spelke - 2006 - Science 311 (5759)::381-4.
    Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.
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  2. Flexible Intuitions of Euclidean Geometry in an Amazonian Indigene Group.Pierre Pica, Véronique Izard, Elizabeth Spelke & Stanislas Dehaene - 2011 - Pnas 23.
    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 (...)
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  3. Carnap’s Conventionalism in Geometry.Stefan Lukits - 2013 - Grazer Philosophische Studien 88 (1):123-138.
    Against Thomas Mormann's argument that differential topology does not support Carnap's conventionalism in geometry we show their compatibility. However, Mormann's emphasis on the entanglement that characterizes topology and its associated metrics is not misplaced. It poses questions about limits of empirical inquiry. For Carnap, to pose a question is to give a statement with the task of deciding its truth. Mormann's point forces us to introduce more clarity to what it means to specify the task that decides between competing (...)
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  4. What Frege Meant When He Said: Kant is Right About Geometry.Teri Merrick - 2006 - Philosophia Mathematica 14 (1):44-75.
    This paper argues that Frege's notoriously long commitment to Kant's thesis that Euclidean geometry is synthetic _a priori_ is best explained by realizing that Frege uses ‘intuition’ in two senses. Frege sometimes adopts the usage presented in Hermann Helmholtz's sign theory of perception. However, when using ‘intuition’ to denote the source of geometric knowledge, he is appealing to Hermann Cohen's use of Kantian terminology. We will see that Cohen reinterpreted Kantian notions, stripping them of any psychological connotation. Cohen's defense (...)
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  5.  80
    Geometry as a Universal Mental Construction.Véronique Izard, Pierre Pica, Danièle Hinchey, Stanislas Dehane & Elizabeth Spelke - 2011 - In Stanislas Dehaene & Elizabeth Brannon (eds.), Space, Time and Number in the Brain. Oxford University Press.
    Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The (...)
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  6. Deleuze, Leibniz and Projective Geometry in the Fold.Simon Duffy - 2010 - Angelaki 15 (2):129-147.
    Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...)
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  7. ARISTOTELIAN LOGIC AND EUCLIDEAN GEOMETRY.John Corcoran - 2014 - Bulletin of Symbolic Logic 20 (1):131-2.
    John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from the (...)
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  8. Emergence, Evolution, and the Geometry of Logic: Causal Leaps and the Myth of Historical Development. [REVIEW]Stephen Palmquist - 2007 - Foundations of Science 12 (1):9-37.
    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 (...)
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  9. Natural Philosophy, Deduction, and Geometry in the Hobbes-Boyle Debate.Marcus P. Adams - 2017 - Hobbes Studies 30 (1):83-107.
    This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of (...)
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  10.  59
    David Hyder. The Determinate World: Kant and Helmholtz on the Physical Meaning of Geometry. Viii + 229 Pp., Bibl., Index. Berlin/New York: Walter de Gruyter, 2009. $105. [REVIEW]Gary Hatfield - 2012 - Isis 103 (4):769-770.
    David Hyder.The Determinate World: Kant and Helmholtz on the Physical Meaning of Geometry. viii + 229 pp., bibl., index. Berlin/New York: Walter de Gruyter, 2009.
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  11.  36
    The Language of Geometry : Fast Comprehension of Geometrical Primitives and Rules in Human Adults and Preschoolers.Pierre Pica & Mariano Sigman & Stanislas Dehaene With Marie Amalric, Liping Wang - 2017 - PLoS Biology 10.
    Article Authors Metrics Comments Media Coverage Abstract Author Summary Introduction Results Discussion Supporting information Acknowledgments Author Contributions References Reader Comments (0) Media Coverage (0) Figures Abstract During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a “geometrical language” with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to (...)
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  12. The Epistemology of Geometry I: The Problem of Exactness.Anne Newstead & Franklin James - 2010 - Proceedings of the Australasian Society for Cognitive Science 2009.
    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 (...)
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  13. Spatial Perception and Geometry in Kant and Helmholtz.Gary Hatfield - 1984 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1984:569 - 587.
    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 (...)
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  14.  50
    The Ontology of Knowledge, Logic, Arithmetic, Sets Theory and Geometry.Jean-Louis Boucon - 2019
    At ordinary scales, the ontological model proposed by Ontology of Knowledge (OK) does not call into question the representation of the world elaborated by common sense or science. This is not the world such as it appears to us and as science describes it that is challenged by the OK but the way it appears to the knowing subject and science. In spite of the efforts made to separate scientific reasoning and metaphysical considerations, in spite of the rigorous construction of (...)
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  15. Fine-Structure Constant From Golden Ratio Geometry.Michael A. Sherbon - 2018 - International Journal of Mathematics and Physical Sciences Research 5 (2):89-100.
    After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the (...)
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  16.  19
    From McTaggart to AdS_5 Geometry V. 3.Paul Merriam - manuscript
    The purpose of this note is to show how an 'AB-series' interpretation of time leads, surprisingly, apparently, to AdS_5 geometry. This is not a theory of 2 time dimensions. Rather, it is a theory of 1 time dimension that has both A-series and B-series characteristics. To summarize the result, a spacetime in terms of (1) the earlier-to-later aspect of time, and (2) the (related) future-present-past aspect of time, and (3) 3-d space, it would seem, gives us the AdS_5 (...). (shrink)
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  17. Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Springer. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that (...)
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  18. Geometry for a Brain. Optimal Control in a Network of Adaptive Memristors.Ignazio Licata & Germano Resconi - 2013 - Adv. Studies Theor. Phys., (no.10):479-513.
    In the brain the relations between free neurons and the conditioned ones establish the constraints for the informational neural processes. These constraints reflect the systemenvironment state, i.e. the dynamics of homeocognitive activities. The constraints allow us to define the cost function in the phase space of free neurons so as to trace the trajectories of the possible configurations at minimal cost while respecting the constraints imposed. Since the space of the free states is a manifold or a non orthogonal space, (...)
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  19. Einstein’s Physical Geometry at Play: Inertial Motion, the Boostability Assumption, the Lorentz Transformations, and the so-Called Conventionality of the One-Way Speed of Light.Mario Bacelar Valente - manuscript
    In this work, Einstein’s view of geometry as physical geometry is taken into account in the analysis of diverse issues related to the notions of inertial motion and inertial reference frame. Einstein’s physical geometry enables a non-conventional view on Euclidean geometry (as the geometry associated to inertial motion and inertial reference frames) and on the uniform time. Also, by taking into account the implications of the view of geometry as a physical geometry, it (...)
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  20.  77
    Application of Natural Deduction in Renaissance Geometry.Mirek Ryszard - 2014 - Argument: Biannual Philosophical Journal 4 (2):425-438.
    my goal here is to provide a detailed analysis of the methods of inference that are employed in De prospectiva pingendi. For this purpose, a method of natural deduction is proposed. the treatise by Piero della Francesca is a manifestation of a union between the ne arts and the mathematical sciences of arithmetic and geometry. He de nes painting as a part of perspective and, speaking precisely, as a branch of geometry, which is why we nd advanced geometrical (...)
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  21. Notes on Groups and Geometry, 1978-1986.Steven H. Cullinane - 2012 - Internet Archive.
    Typewritten notes on groups and geometry.
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  22. On Explanations From 'Geometry of Motion'.Juha Saatsi - 2018 - British Journal for the Philosophy of Science 69 (1):253–273.
    This paper examines explanations that turn on non-local geometrical facts about the space of possible configurations a system can occupy. I argue that it makes sense to contrast such explanations from "geometry of motion" with causal explanations. I also explore how my analysis of these explanations cuts across the distinction between kinematics and dynamics.
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  23.  95
    On the Relationship Between Plane and Solid Geometry.Andrew Arana & Paolo Mancosu - 2012 - Review of Symbolic Logic 5 (2):294-353.
    Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the (...)
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  24. The Euclidean Mousetrap: Schopenhauer’s Criticism of the Synthetic Method in Geometry.Jason M. Costanzo - 2008 - Idealistic Studies 38 (3):209-220.
    In his doctoral dissertation On the Principle of Sufficient Reason, Arthur Schopenhauer there outlines a critique of Euclidean geometry on the basis of the changing nature of mathematics, and hence of demonstration, as a result of Kantian idealism. According to Schopenhauer, Euclid treats geometry synthetically, proceeding from the simple to the complex, from the known to the unknown, “synthesizing” later proofs on the basis of earlier ones. Such a method, although proving the case logically, nevertheless fails to attain (...)
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  25. Physical Geometry and Fundamental Metaphysics.Cian Dorr - 2011 - Proceedings of the Aristotelian Society 111 (1pt1):135-159.
    I explore some ways in which one might base an account of the fundamental metaphysics of geometry on the mathematical theory of Linear Structures recently developed by Tim Maudlin (2010). Having considered some of the challenges facing this approach, Idevelop an alternative approach, according to which the fundamental ontology includes concrete entities structurally isomorphic to functions from space-time points to real numbers.
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  26.  89
    Affine Geometry, Visual Sensation, and Preference for Symmetry of Things in a Thing.Birgitta Dresp-Langley - 2016 - Symmetry 127 (8).
    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 (...) 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)
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  27.  51
    Spinoza's Geometry of Power. [REVIEW]John Morrison - 2013 - British Journal for the History of Philosophy 21 (3):610-613.
    A book review of Valtteri Viljanen's "Spinoza’s Geometry of Power".
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  28.  42
    A 2-Dimensional Geometry for Biological Time.Francis Bailly, Giuseppe Longo & Maël Montévil - 2011 - Progress in Biophysics and Molecular Biology 106:474 - 484.
    This paper proposes an abstract mathematical frame for describing some features of biological time. The key point is that usual physical (linear) representation of time is insufficient, in our view, for the understanding key phenomena of life, such as rhythms, both physical (circadian, seasonal …) and properly biological (heart beating, respiration, metabolic …). In particular, the role of biological rhythms do not seem to have any counterpart in mathematical formalization of physical clocks, which are based on frequencies along the usual (...)
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  29.  24
    Golden Ratio Geometry and the Fine-Structure Constant.Michael A. Sherbon - 2019 - Journal of Advances in Physics 16 (1):362 -368.
    The golden ratio is found to be related to the fine-structure constant, which determines the strength of the electromagnetic interaction. The golden ratio and classical harmonic proportions with quartic equations give an approximate value for the inverse fine-structure constant the same as that discovered previously in the geometry of the hydrogen atom. With the former golden ratio results, relationships are also shown between the four fundamental forces of nature: electromagnetism, the weak force, the strong force, and the force of (...)
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  30. The Geometry of Visual Space and the Nature of Visual Experience.Farid Masrour - 2015 - Philosophical Studies 172 (7):1813-1832.
    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.
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  31.  63
    Is Geometry Analytic?David Mwakima - 2017 - Dianoia 1 (4):66 - 78.
    In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.
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  32. 2D Geometry Predicts Perceived Visual Curvature in Context-Free Viewing.Birgitta Dresp-Langley - 2015 - Computational Intelligence and Neuroscience 2015 (708759):1-9.
    Planar geometry was exploited for the computation of symmetric visual curves in the image plane, with consistent variations in local parameters such as sagitta, chordlength, and the curves’ height-to-width ratio, an indicator of the visual area covered by the curve, also called aspect ratio. Image representations of single curves (no local image context) were presented to human observers to measure their visual sensation of curvature magnitude elicited by a given curve. Nonlinear regression analysis was performed on both the individual (...)
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  33. Visual Geometry.James Hopkins - 1973 - Philosophical Review 82 (1):3-34.
    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.
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  34. Berkeley and Proof in Geometry.Richard J. Brook - 2012 - Dialogue 51 (3):419-435.
    Berkeley in his Introduction to the Principles of Human knowledge uses geometrical examples to illustrate a way of generating “universal ideas,” which allegedly account for the existence of general terms. In doing proofs we might, for example, selectively attend to the triangular shape of a diagram. Presumably what we prove using just that property applies to all triangles.I contend, rather, that given Berkeley’s view of extension, no Euclidean triangles exist to attend to. Rather proof, as Berkeley would normally assume, requires (...)
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  35. Synthetic Geometry and Aufbau.Thomas Mormann - 2003 - In Thomas Bonk (ed.), Language, Truth and Knowledge. Kluwer Academic Publishers. pp. 45--64.
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  36. Remarks on the Geometry of Complex Systems and Self-Organization.Luciano Boi - 2012 - In Vincenzo Fano, Enrico Giannetto, Giulia Giannini & Pierluigi Graziani (eds.), Complessità e Riduzionismo. © ISONOMIA – Epistemologica, University of Urbino. pp. 28-43.
    Let us start by some general definitions of the concept of complexity. We take a complex system to be one composed by a large number of parts, and whose properties are not fully explained by an understanding of its components parts. Studies of complex systems recognized the importance of “wholeness”, defined as problems of organization (and of regulation), phenomena non resolvable into local events, dynamics interactions in the difference of behaviour of parts when isolated or in higher configuration, etc., in (...)
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  37. Review of Fenstad's "Grammar, Geometry & Brain". [REVIEW]Erich Rast - 2014 - Studia Logica 102 (1):219-223.
    In this small book logician and mathematician Jens Erik Fenstad addresses some of the most important foundational questions of linguistics: What should a theory of meaning look like and how might we provide the missing link between meaning theory and our knowledge of how the brain works? The author’s answer is twofold. On the one hand, he suggests that logical semantics in the Montague tradition and other broadly conceived symbolic approaches do not suffice. On the other hand, he does not (...)
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  38. Kallikles i geometria. Przyczynek do Platońskiej koncepcji sprawiedliwości [Callicles and Geometry: On Plato’s Conception of Justice].Marek Piechowiak - 2013 - In Zbigniew Władek (ed.), Księga życia i twórczości. Księga pamiątkowa dedykowana Profesorowi Romanowi A. Tokarczykowi. Wydawnictwo Polihymnia. pp. vol. 5, 281-291.
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  39.  63
    The Cognitive Geometry of War.Barry Smith - 1997 - In Peter Koller & Klaus Puhl (eds.), Current Issues in Political Philosophy: Justice in Society and World Order. Vienna: Hölder-Pichler-Tempsky. pp. 394--403.
    When national borders in the modern sense first began to be established in early modern Europe, non-contiguous and perforated nations were a commonplace. According to the conception of the shapes of nations that is currently preferred, however, nations must conform to the topological model of circularity; their borders must guarantee contiguity and simple connectedness, and such borders must as far as possible conform to existing topographical features on the ground. The striving to conform to this model can be seen at (...)
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  40.  33
    Geometry and Geography of Morality.Jovan Babić - 2017 - Metascience 26 (3):475-479.
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  41.  21
    From McTaggert to AdS_5 Geometry.Paul Merriam - manuscript
    The purpose of this note is to show how an 'AB-series' interpretation of time, given in a companion paper, leads, surprisingly, directly to the physicists' important AdS5 geometry. This is not a theory of 2 time dimensions. Rather, it is a theory of 1 time dimension that has both B-series and A-series characteristics. -/- To summarize the result, a spacetime in terms of 1. the earlier-to-later aspect of time, and 2. the related future-present-past aspect of time, and 3. 3-d (...)
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  42.  19
    From McTaggart to AdS_5 Geometry 2.Paul Merriam - manuscript
    The purpose of this note is to show how an 'AB-series' interpretation of time, given in a companion paper, leads, surprisingly, to AdS_5 geometry. This is not a theory of 2 time dimensions. Rather, it is a theory of 1 time dimension that has both A-series and B-series characteristics.
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  43. The Kinds of Truth of Geometry Theorems.Michael Bulmer, Desmond Fearnley-Sander & Tim Stokes - 2001 - In Jürgen Richter-Gebert & Dongming Wang (eds.), LNCS: Lecture Notes In Computer Science. Springer Verlag. pp. 129-142.
    Proof by refutation of a geometry theorem that is not universally true produces a Gröbner basis whose elements, called side polynomials, may be used to give inequations that can be added to the hypotheses to give a valid theorem. We show that (in a certain sense) all possible subsidiary conditions are implied by those obtained from the basis; that what we call the kind of truth of the theorem may be derived from the basis; and that the side polynomials (...)
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  44.  69
    In General Relativity, Gravity is Effect of Coordinates with Change of Geometry of Spacetime.Alfonso Leon Guillen Gomez - manuscript
    Einstein structured the theoretical frame of his work on gravity under the Special Relativity and Minkowski´s spacetime using three guide principles: The strong principle of equivalence establishes that acceleration and gravity are equivalents. Mach´s principle explains the inertia of the bodies and particles as completely determined by the total mass existent in the universe. And, general covariance searches to extend the principle of relativity from inertial motion to accelerated motion. Mach´s principle was abandoned quickly, general covariance resulted mathematical property of (...)
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  45. Pure and Applied Geometry in Kant.Marissa Bennett - manuscript
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  46. What Are Logical Notions?John Corcoran & Alfred Tarski - 1986 - History and Philosophy of Logic 7 (2):143-154.
    In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.
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  47. Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics.Vladimir Kanovei, Mikhail G. Katz & Thomas Mormann - 2013 - Foundations of Science 18 (2):259-296.
    We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...)
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  48. Hobbes on Natural Philosophy as "True Physics" and Mixed Mathematics.Marcus P. Adams - 2016 - Studies in History and Philosophy of Science Part A 56:43-51.
    I offer an alternative account of the relationship of Hobbesian geometry to natural philosophy by arguing that mixed mathematics provided Hobbes with a model for thinking about it. In mixed mathematics, one may borrow causal principles from one science and use them in another science without there being a deductive relationship between those two sciences. Natural philosophy for Hobbes is mixed because an explanation may combine observations from experience (the ‘that’) with causal principles from geometry (the ‘why’). My (...)
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  49. Interaction of Color and Geometric Cues in Depth Perception: When Does Red Mean "Near"?Christophe Guibal & Birgitta Dresp - 2004 - Psychological Research 69:30-40.
    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 (...)
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  50. Grassmann’s Epistemology: Multiplication and Constructivism.Paola Cantù - 2010 - In Hans-Joachim Petsche (ed.), From Past to Future: Graßmann's Work in Context.
    The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the (...)
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