Geometry

Edited by Nemi Boris Pelgrom (Ludwig Maximilians Universität, München)
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  1. Why did Fermat believe he had `a truly marvellous demonstration' of FLT?Bhupinder Singh Anand - manuscript
    Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...)
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  2. Euclidean Geometry is a Priori.Boris Culina - manuscript
    In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.
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  3. The Point or the Primary Geometric Object.ZERARI Fathi - manuscript
    The definition of a point in geometry is primordial in order to understand the different elements of this branch of mathematics ( line, surface, solids…). This paper aims at shedding fresh light on the concept to demonstrate that it is related to another one named, here, the Primary Geometric Object; both concepts concur to understand the multiplicity of geometries and to provide hints as concerns a new understanding of some concepts in physics such as time, energy, mass….
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  4. Conics and Quadric surfaces.Jonathan Taborda & Jaime Chica - manuscript
    There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What surface is in space? These two problems are posed and solved by applying the study of matrices and spectral theory.
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  5. Cónicas y Superficies Cuádricas.Jonathan Taborda & Jaime Chica - manuscript
    There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What surface is in space? These two problems are posed and solved by applying the study of matrices and spectral theory.
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  6. The Constitution of Weyl’s Pure Infinitesimal World Geometry.C. D. McCoy - 2022 - Hopos: The Journal of the International Society for the History of Philosophy of Science 12 (1):189–208.
    Hermann Weyl was one of the most important figures involved in the early elaboration of the general theory of relativity and its fundamentally geometrical spacetime picture of the world. Weyl’s development of “pure infinitesimal geometry” out of relativity theory was the basis of his remarkable attempt at unifying gravitation and electromagnetism. Many interpreters have focused primarily on Weyl’s philosophical influences, especially the influence of Husserl’s transcendental phenomenology, as the motivation for these efforts. In this article, I argue both that these (...)
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  7. La Neutro-Geometría y la Anti-Geometría como Alternativas y Generalizaciones de las Geometrías no Euclidianas.Florentin Smarandache - 2022 - Neutrosophic Computing and Machine Learning 20 (1):91-104.
    In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation of one (...)
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  8. The Homeomorphism of Minkowski Space and the Separable Complex Hilbert Space: The physical, Mathematical and Philosophical Interpretations.Vasil Penchev - 2021 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 14 (3):1-22.
    A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized (...)
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  9. NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries (revisited).Florentin Smarandache - 2021 - Neutrosophic Sets and Systems 46 (1):456-477.
    In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the (...)
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  10. NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries (revisited).Florentin Smarandache - 2021 - Neutrosophic Sets and Systems 46 (1):456-477.
    In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry.
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  11. From practical to pure geometry and back.Mario Bacelar Valente - 2020 - Revista Brasileira de História da Matemática 20 (39):13-33.
    The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some of which are basically (...)
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  12. The isomorphism of Minkowski space and the separable complex Hilbert space and its physical interpretation.Vasil Penchev - 2020 - Philosophy of Science eJournal (Elsevier:SSRN) 13 (31):1-3.
    An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the invariance to (...)
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  13. Geometrical objects and figures in practical, pure, and applied geometry.Mario Bacelar Valente - 2020 - Disputatio. Philosophical Research Bulletin 9 (15):33-51.
    The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.
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  14. Explicaciones Geométrico-Diagramáticas en Física desde una Perspectiva Inferencial.Javier Anta - 2019 - Revista Colombiana de Filosofía de la Ciencia 38 (19).
    El primer objetivo de este artículo es mostrar que explicaciones genuinamente geométricas/matemáticas e intrínsecamente diagramáticas de fenómenos físicos no solo son posibles en la práctica científica, sino que además comportan un potencial epistémico que sus contrapartes simbólico-verbales carecen. Como ejemplo representativo utilizaremos la metodología geométrica de John Wheeler (1963) para calcular cantidades físicas en una reacción nuclear. Como segundo objetivo pretendemos analizar, desde un marco inferencial, la garantía epistémica de este tipo de explicaciones en términos de dependencia sintáctica y semántica (...)
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  15. Geometry of motion: some elements of its historical development.Mario Bacelar Valente - 2019 - ArtefaCToS. Revista de Estudios de la Ciencia y la Tecnología 8 (2):4-26.
    in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion that was first (...)
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  16. Dediche tortuose. La Geometria morale di Vincenzo Viviani e gli imbarazzi dell’eredità galileiana.Sara Bonechi - 2019 - Noctua 6 (1–2):75-181.
    This study of the history and contents of a hitherto unedited work on geometry by Vincenzo Viviani seeks to present a picture of the scientific environment in Italy in the second half of the 17th century, with particular emphasis on Tuscany and the impact the condemnation of Galileo had on ongoing scholarship. Information derived from unedited or less well-known material serves to illuminate a range of prominent and marginal figures who adopted different strategies for the dissemination of Galileo’s thought and (...)
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  17. Francesca Biagioli: Space, Number, and Geometry from Helmholtz to Cassirer: Springer, Dordrecht, 2016, 239 pp, $109.99 , ISBN: 978-3-319-31777-9. [REVIEW]Lydia Patton - 2019 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 50 (2):311-315.
    Francesca Biagioli’s Space, Number, and Geometry from Helmholtz to Cassirer is a substantial and pathbreaking contribution to the energetic and growing field of researchers delving into the physics, physiology, psychology, and mathematics of the nineteenth and twentieth centuries. The book provides a bracing and painstakingly researched re-appreciation of the work of Hermann von Helmholtz and Ernst Cassirer, and of their place in the tradition, and is worth study for that alone. The contributions of the book go far beyond that, however. (...)
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  18. Fine-structure constant from Sommerfeld to Feynman.Michael A. Sherbon - 2019 - Journal of Advances in Physics 16 (1):335-343.
    The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The (...)
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  19. Perverted Space-Time Geodesy in Einstein’s Views on Geometry.Mario Bacelar Valente - 2018 - Philosophia Scientiae 22:137-162.
    A perverted space-time geodesy results from the idea of variable rods and clocks, whose length and rates are taken to be affected by the gravitational field. By contrast, what we might call a concrete geodesy relies on the idea of invariable unit-measuring rods and clocks. Indeed, this is a basic assumption of general relativity. Variable rods and clocks lead to a perverted geodesy, in the sense that a curved space-time may be seen as a result of a departure from the (...)
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  20. An Elementary System of Axioms for Euclidean Geometry Based on Symmetry Principles.Boris Čulina - 2018 - Axiomathes 28 (2):155-180.
    In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s (...)
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  21. A Survey of Geometric Algebra and Geometric Calculus.Alan Macdonald - 2017 - Advances in Applied Clifford Algebras 27:853-891.
    The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.
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  22. 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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  23. Fundamental Physics and the Fine-Structure Constant.Michael A. Sherbon - 2017 - International Journal of Physical Research 5 (2):46-48.
    From the exponential function of Euler’s equation to the geometry of a fundamental form, a calculation of the fine-structure constant and its relationship to the proton-electron mass ratio is given. Equations are found for the fundamental constants of the four forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation. Symmetry principles are then associated with traditional physical measures.
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  24. Imagination in mathematics.Andrew Arana - 2016 - In Amy Kind (ed.), Routledge Handbook on the Philosophy of Imagination. Routledge. pp. 463-477.
    This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.
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  25. Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics.Jean-Pierre Marquis - 2013 - Synthese 190 (12):2141-2164.
    In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...)
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  26. Review of Geometric Possibility. [REVIEW]Chris Smeenk - 2013 - Philosophia Mathematica 21 (3):416-421.
    Review of Geometric Possibility (2011), by Gordon Belot. Oxford and New York: Oxford University Press. x + 219 pp.
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  27. 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 aforementioned areas.
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  28. Kant's Views on Non-Euclidean Geometry.Michael Cuffaro - 2012 - Proceedings of the Canadian Society for History and Philosophy of Mathematics 25:42-54.
    Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that both (...)
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  29. Notes on Groups and Geometry, 1978-1986.Steven H. Cullinane - 2012 - Internet Archive.
    Typewritten notes on groups and geometry.
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  30. 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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  31. 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 spatial content of the (...)
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  32. 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 intuitions of space that (...)
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  33. 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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  34. Review of M. Giaquinto's Visual thinking in mathematics. [REVIEW]Andrew Arana - 2009 - Analysis 69 (2):401-403.
    Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” as (...)
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  35. The Aristotelian Explanation of the Halo.Monte Ransome Johnson - 2009 - Apeiron 42 (4):325-357.
    For an Aristotelian observer, the halo is a puzzling phenomenon since it is apparently sublunary, and yet perfectly circular. This paper studies Aristotle's explanation of the halo in Meteorology III 2-3 as an optical illusion, as opposed to a substantial thing (like a cloud), as was thought by his predecessors and even many successors. Aristotle's explanation follows the method of explanation of the Posterior Analytics for "subordinate" or "mixed" mathematical-physical sciences. The accompanying diagram described by Aristotle is one of the (...)
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  36. 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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  37. 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 of (...)
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  38. The impossibility of relations between non-collocated spatial objects and non-identical topological spaces.Jeffrey Grupp - 2005 - Axiomathes 15 (1):85-141.
    I argue that relations between non-collocated spatial entities, between non-identical topological spaces, and between non-identical basic building blocks of space, do not exist. If any spatially located entities are not at the same spatial location, or if any topological spaces or basic building blocks of space are non-identical, I will argue that there are no relations between or among them. The arguments I present are arguments that I have not seen in the literature.
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  39. La signification de Nicod pour la phénoménologie de Wittgenstein.Ludovic Soutif - 2005 - Revue de Métaphysique et de Morale 2 (2):215-243.
    Quoique l'on ne trouve qu'un nombre limité de références à Nicod dans les manuscrits de la période dite « intermédiaire » de Wittgenstein, une lecture attentive de La Géométrie dans le monde sensible s'avère pourtant décisive pour comprendre la nature du projet phénoménologique de Wittgenstein de la fin des années vingt. Nous nous proposons de montrer que la prise en compte ainsi que la reformulation du problème posé par Nicod en 1924, celui de la nature de la relation d'inclusion spatiale, (...)
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  40. The Idea of a Diagram.Desmond Fearnley-Sander - 1989 - In Hassan Ait-Kaci & Maurice Nivat (eds.), Resolution of Equations in Algebraic Structures. Academic Press.
    A detailed axiomatisation of diagrams (in affine geometry) is presented, which supports typing of geometric objects, calculation of geometric quantities and automated proof of theorems.
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  41. Cuádricas.Jonathan Taborda & Jaime Chica - manuscript
    There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What surface is in space? These two problems are posed and solved by applying the study of matrices and spectral theory.
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  42. Iter Italicum and Leibniz/Giordano correspondence.Francesco Tampoia - manuscript
    Letters exchanged by scientists are a crucial source by which to trace the process that accompanies their scientific evolution. In this paper -accomplished through a historical approach- I aim to throw new light on Leibniz's continuing interest in classical geometry and to stress the significance of his correspondence with the Italian mathematician Vitale Giordano.
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  43. The Point or the Primary geometric Object.Fathi ZERARI - unknown
    The definition of a point in geometry is primordial in order to understand the different elements of this branch of mathematics ( line, surface, solids…). This paper aims at shedding fresh light on the concept to demonstrate that it is related to another one named, here, the Primary Geometric Object; both concepts concur to understand the multiplicity of geometries and to provide hints as concerns a new understanding of some concepts in physics such as time, energy, mass….
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