Results for 'foundations of geometry'

956 found
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  1. Vigier III.Spin Foam Spinors & Fundamental Space-Time Geometry - 2000 - Foundations of Physics 30 (1).
  2.  52
    Electrodynamics and Spacetime Geometry: Foundations.Francisco Cabral & Francisco S. N. Lobo - 2017 - Foundations of Physics 47 (2):208-228.
    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 (...)
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  3.  76
    Point-free Foundation of Geometry and Multivalued Logic.Cristina Coppola, Giangiacomo Gerla & Annamaria Miranda - 2010 - Notre Dame Journal of Formal Logic 51 (3):383-405.
    Whitehead, in two basic books, considers two different approaches to point-free geometry: the inclusion-based approach , whose primitive notions are regions and inclusion relation between regions, and the connection-based approach , where the connection relation is considered instead of the inclusion. We show that the latter cannot be reduced to the first one, although this can be done in the framework of multivalued logics.
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  4. Mereological foundations of point-free geometry via multi-valued logic.Cristina Coppola & Giangiacomo Gerla - 2015 - Logic and Logical Philosophy 24 (4):535-553.
    We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting (...)
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  5. Frege on the Foundation of Geometry in Intuition.Jeremy Shipley - 2015 - Journal for the History of Analytical Philosophy 3 (6).
    I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at (...)
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  6. The epistemological foundations of geometry in 19 th century.Ladislav Kvasz - 1998 - Philosophia Scientiae 3 (2):183-202.
     
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  7.  55
    Roshdi Rashed, geometry and dioptrics in classical Islam. London: Al-furqan Islamic heritage foundation, 2005. Pp. XII+1178. Isbn 1873992998. £50.00. [REVIEW]Nader El-Bizri - 2007 - British Journal for the History of Science 40 (1):124-126.
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  8.  24
    Moral geometry, natural alignments and utopian urban form.Jean-Paul Baldacchino - 2018 - Thesis Eleven 148 (1):52-76.
    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 (...)
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  9.  30
    Space, geometry and aesthetics: through Kant and towards Deleuze.Peg Rawes - 2008 - New York: Palgrave-Macmillan.
    Peg Rawes examines a "minor tradition" of aesthetic geometries in ontological philosophy. Developed through Kant’s aesthetic subject she explores a trajectory of geometric thinking and geometric figurations--reflective subjects, folds, passages, plenums, envelopes and horizons--in ancient Greek, post-Cartesian and twentieth-century Continental philosophies, through which productive understandings of space and embodies subjectivities are constructed. Six chapters, explore the construction of these aesthetic geometric methods and figures in a series of "geometric" texts by Kant, Plato, Proclus, Spinoza, Leibniz, Bergson, Husserl and Deleuze. In (...)
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  10.  50
    Space, Number, and Geometry From Helmholtz to Cassirer.Francesca Biagioli - 2016 - Cham: Springer Verlag.
    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. (...)
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  11. Geometry as a Universal mental Construction.Véronique Izard, Pierre Pica, Danièle Hinchey, Stanislas Dehane & Elizabeth Spelke - 2011 - In Stanislas Dehaene & Elizabeth Brannon, Space, Time and Number in the Brain: Searching for the Foundations of Mathematical Thought. 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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  12.  7
    Advances in Geometry and Lie Algebras from Supergravity.Pietro Giuseppe Frè - 2018 - Cham: Imprint: Springer.
    This book aims to provide an overview of several topics in advanced Differential Geometry and Lie Group Theory, all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry/supergravity. The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in (...)
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  13.  11
    (2 other versions)oyce on The Relation of the Principles of Logic to the Foundations of Geometry[REVIEW]Theodore De Laguna - 1906 - Journal of Philosophy 3 (13):357.
  14. 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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  15.  67
    Physical Geometry.James P. Binkoski - 2016 - Dissertation, University of Massachusetts, Amherst
    All physical theories, from classical Newtonian mechanics to relativistic quantum field theory, entail propositions concerning the geometric structure of spacetime. To give an example, the general theory of relativity entails that spacetime is curved, smooth, and four-dimensional. In this dissertation, I take the structural commitments of our theories seriously and ask: how is such structure instantiated in the physical world? Mathematically, a property like 'being curved' is perfectly well-defined insofar as we know what it means for a mathematical space to (...)
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  16. Whitehead's pointfree geometry and diametric posets.Giangiacomo Gerla & Bonaventura Paolillo - 2010 - Logic and Logical Philosophy 19 (4):289-308.
    This note is motivated by Whitehead’s researches in inclusion-based point-free geometry as exposed in An Inquiry Concerning the Principles of Natural Knowledge and in The concept of Nature. More precisely, we observe that Whitehead’s definition of point, based on the notions of abstractive class and covering, is not adequate. Indeed, if we admit such a definition it is also questionable that a point exists. On the contrary our approach, in which the diameter is a further primitive, enables us to (...)
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  17.  88
    Finsler Geometry and Relativistic Field Theory.R. G. Beil - 2003 - Foundations of Physics 33 (7):1107-1127.
    Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of “gauge” transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.
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  18.  26
    Geometry, null hypersurfaces and new variables.David C. Robinson - 2003 - In A. Ashtekar, Revisiting the Foundations of Relativistic Physics. Springer. pp. 349--360.
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  19.  13
    Foundations of Geometry.Bertrand Russell - 1996 - Routledge.
    The Foundations of Geometry was first published in 1897, and is based on Russell's Cambridge dissertation as well as lectures given during a journey through the USA. This is the first reprint, complete with a new introduction by John Slater. It provides both an insight into the foundations of Russell's philosophical thinking and an introduction to the philosophy of mathematics and logic. As such it will be an invaluable resource not only for students of philosophy, but also (...)
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  20.  30
    Extracting Geometry from Quantum Spacetime: Obstacles Down the Road.Yuri Bonder, Chryssomalis Chryssomalakos & Daniel Sudarsky - 2018 - Foundations of Physics 48 (9):1038-1060.
    Any acceptable quantum gravity theory must allow us to recover the classical spacetime in the appropriate limit. Moreover, the spacetime geometrical notions should be intrinsically tied to the behavior of the matter that probes them. We consider some difficulties that would be confronted in attempting such an enterprise. The problems we uncover seem to go beyond the technical level to the point of questioning the overall feasibility of the project. The main issue is related to the fact that, in the (...)
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  21.  61
    Arithmetizing the geometry from inside: David Hilbert's segment calculus.Eduardo Nicolás Giovannini - 2015 - Scientiae Studia 13 (1):11-48.
    Sobre la base que aportan las notas manuscritas de David Hilbert para cursos sobre geometría, el artículo procura contextualizar y analizar una de las contribuciones más importantes y novedosas de su célebre monografía Fundamentos de la geometría, a saber: el cálculo de segmentos lineales. Se argumenta que, además de ser un resultado matemático importante, Hilbert depositó en su aritmética de segmentos un destacado significado epistemológico y metodológico. En particular, se afirma que para Hilbert este resultado representaba un claro ejemplo de (...)
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  22. Les originations de la géométrie. Les problèmes posés par le singulier dans le titre de Husserl.Julien Bernard - 2025 - Philosophia Scientiae 29-1 (29-1):15-52.
    The programme of geometrical archaeology initiated by the last Husserlian texts merits being taken further with renewed emphasis on historical discontinuities. The revolutions that geometry has gone through have led to profound changes in meaning. The arch-foundations of geometry were not actually definitively set out at the time of the first factual institution of geometry, namely Greek geometry. Instead, the full plurality of the historical forms of geometry does not just consist of a movement (...)
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  23.  69
    Review: Hans Georg Steiner, Frege und die Grundlagen der Geometrie; G. Frege, M. E. Szabo, E. D. Klemke, On the Foundations of Geometry[REVIEW]Michael Resnik - 1971 - Journal of Symbolic Logic 36 (1):155-155.
  24. On alternative geometries, arithmetics, and logics; a tribute to łukasiewicz.Graham Priest - 2003 - Studia Logica 74 (3):441 - 468.
    The paper discusses the similarity between geometry, arithmetic, and logic, specifically with respect to the question of whether applied theories of each may be revised. It argues that they can - even when the revised logic is a paraconsistent one, or the revised arithmetic is an inconsistent one. Indeed, in the case of logic, it argues that logic is not only revisable, but, during its history, it has been revised. The paper also discusses Quine's well known argument against the (...)
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  25.  97
    Topics in Noncommutative Geometry Inspired Physics.Rabin Banerjee, Biswajit Chakraborty, Subir Ghosh, Pradip Mukherjee & Saurav Samanta - 2009 - Foundations of Physics 39 (12):1297-1345.
    In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.
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  26.  42
    Distance geometry and geometric algebra.Andreas W. M. Dress & Timothy F. Havel - 1993 - Foundations of Physics 23 (10):1357-1374.
    As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore (...)
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  27. I—Tim Maudlin: Time, Topology and Physical Geometry.Tim Maudlin - 2010 - Aristotelian Society Supplementary Volume 84 (1):63-78.
    The standard mathematical account of the sub-metrical geometry of a space employs topology, whose foundational concept is the open set. This proves to be an unhappy choice for discrete spaces, and offers no insight into the physical origin of geometrical structure. I outline an alternative, the Theory of Linear Structures, whose foundational concept is the line. Application to Relativistic space-time reveals that the whole geometry of space-time derives from temporal structure. In this sense, instead of spatializing time, Relativity (...)
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  28. Natural number and natural geometry.Elizabeth S. Spelke - 2011 - In Stanislas Dehaene & Elizabeth Brannon, Space, Time and Number in the Brain: Searching for the Foundations of Mathematical Thought. Oxford University Press. pp. 287--317.
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  29. Kant on Euclid: Geometry in Perspective.Stephen R. Palmquist - 1990 - Philosophia Mathematica (1-2):88-113.
    There is a common assumption among philosophers, shared even by many Kant scholars, that Kant had a naive faith in the absolute valid­ity of Euclidean geometry, Aristotelian logic, and Newtonian physics, and that his primary goal in the Critique of Pure Reason was to pro­vide a rational foundation upon which these classical scientific theories could be based. This, it might be thought, is the essence of his attempt to solve the problem which, as he says in a footnote to (...)
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  30.  95
    Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach.Eduardo N. Giovannini - 2016 - Synthese 193 (1):31-70.
    The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry , namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central (...)
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  31.  21
    Hilbert, completeness and geometry.Giorgio Venturi - 2011 - Rivista Italiana di Filosofia Analitica Junior 2 (2):80-102.
    This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring (...)
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  32.  52
    Philosophie und Geometrie. Zur jüngeren Protophysik-Kritik.Peter Janich - 2008 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 39 (1):121-130.
    The critique of my protophysical approaches to operational foundation of geometry by Lucas Amiras (Journal for General Philosophy of Science Vol. 34 (2003)) concerns my first publication from 1976 but not the further 30 years of work. It does not offer any argument leading from the (erroneous) judgement “lacking success” to the conclusion “impossible”. And it is, in general, based on a philosophical defect: it ignores the principle of methodical order as leading for constructivist protophysics.
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  33.  40
    Groups and Plane Geometry.Victor Pambuccian - 2005 - Studia Logica 81 (3):387-398.
    We show that the first-order theory of a large class of plane geometries and the first-order theory of their groups of motions, understood both as groups with a unary predicate singling out line-reflections, and as groups acting on sets, are mutually inter-pretable.
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  34.  47
    Hume on Geometry and Infinite Divisibility in the Treatise.H. Mark Pressman - 1997 - Hume Studies 23 (2):227-244.
    In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume XXIII, Number 2, November 1997, pp. 227-244 Hume on Geometry and Infinite Divisibility in the Treatise H. MARK PRESSMAN Scholars have recognized that in the Treatise "Hume seeks to find a foundation for geometry in sense-experience."1 In this essay, I examine to what extent Hume succeeds in his attempt to ground geometry visually. I argue that the geometry Hume describes in the (...)
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  35. Space and Geometry.Henri Poincaré - forthcoming - Foundations of Science.
  36. Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm, Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. New York: Birkhäuser. 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 (...)
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  37.  30
    Über den produktiv-operativen ansatz zur begründung der geometrie in der protophysik.Lucas Amiras - 2003 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 34 (1):133-158.
    On the productive-operative approach to the foundations of geometry in protophysics. Attempts to establish a foundation to elementary geometry as a theory of spatial figures in Protophysics are surveyed in Section 1. An idea suggested by H. Dingler was to extract the basic properties of the geometrical primitive notions from descriptions of the operations performed in the ‘first’ production of the corresponding objects. P. Janich presents this ‘productive-operative’ approach as a succesful methodical alternative to the ‘geometry (...)
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  38. Spacetime theory as physical geometry.Robert Disalle - 1995 - Erkenntnis 42 (3):317-337.
    Discussions of the metaphysical status of spacetime assume that a spacetime theory offers a causal explanation of phenomena of relative motion, and that the fundamental philosophical question is whether the inference to that explanation is warranted. I argue that those assumptions are mistaken, because they ignore the essential character of spacetime theory as a kind of physical geometry. As such, a spacetime theory does notcausally explain phenomena of motion, but uses them to construct physicaldefinitions of basic geometrical structures by (...)
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  39.  31
    Serres and Foundations.Turo-Kimmo Lehtonen - 2020 - Theory, Culture and Society 37 (3):3-22.
    While Michel Serres’ work has become relatively well-known among social theoreticians in recent years, his explicit thematization of the foundations of human collectives has gained surprisingly little attention. This article claims that Serres’ approach to the theme of foundations can be clarified by scrutinizing the way in which he poses and answers the following three questions: How are we together? What and whom do we exclude from our togetherness and how? Who are we today? Instead of starting with (...)
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  40.  54
    Weyl's geometry and physics.Nathan Rosen - 1982 - Foundations of Physics 12 (3):213-248.
    It is proposed to remove the difficulty of nonitegrability of length in the Weyl geometry by modifying the law of parallel displacement and using “standard” vectors. The field equations are derived from a variational principle slightly different from that of Dirac and involving a parameter σ. For σ=0 one has the electromagnetic field. For σ<0 there is a vector meson field. This could be the electromagnetic field with finite-mass photons, or it could be a meson field providing the “missing (...)
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  41.  29
    Analyse et géométrie, histoire des courbes gauches De Clairaut à Darboux.Jean Delcourt - 2011 - Archive for History of Exact Sciences 65 (3):229-293.
    RésuméCet article est consacré à l’histoire de la théorie locale des courbes “à double courbure”. Initiée par Clairaut en 1731, cette théorie se développe en parallèle à la théorie des surfaces et trouve son achèvement avec les formules de Serret et Frenet et leur interprétation par Darboux, en 1887. Au delà de l’analyse des contributions de nombreux mathématiciens, parmi lesquels Monge bien sûr mais aussi Fourier, Lagrange et Cauchy, notre étude donne un regard particulier sur l’évolution conjointe de l’Analyse et (...)
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  42.  54
    Differential Sheaves and Connections: A Natural Approach to Physical Geometry.Anastasios Mallios & Elias Zafiris - 2015 - World Scientific.
    This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is (...)
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  43.  24
    The Foundations of Geometry and Induction.Jean Nicod - 1930 - Humana Mente 5 (19):455-460.
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  44. Frege’s ‘On the Foundations of Geometry’ and Axiomatic Metatheory.Günther Eder - 2016 - Mind 125 (497):5-40.
    In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a (...)
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  45. L. E. J. BROUWER: "Collected Works Volume I, Philosophy and Foundations of Mathematics." Edited by A. Heyting, 1975, Volume 2, "Geometry, Analysis, Topology and Mechanics." Edited by H. Freudenthal, 1976. [REVIEW]M. Dummett - 1980 - Mind 89:606.
     
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  46.  25
    Affine Geometry and Relativity.Božidar Jovanović - 2023 - Foundations of Physics 53 (3):1-29.
    We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincaré transformations of a plane. In particular, a 1-parametric family of natural deformations of the Poincaré (...)
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  47.  78
    Charge, Geometry, and Effective Mass.Gerald E. Marsh - 2008 - Foundations of Physics 38 (3):293-300.
    Charge, like mass in Newtonian mechanics, is an irreducible element of electromagnetic theory that must be introduced ab initio. Its origin is not properly a part of the theory. Fields are then defined in terms of forces on either masses—in the case of Newtonian mechanics, or charges in the case of electromagnetism. General Relativity changed our way of thinking about the gravitational field by replacing the concept of a force field with the curvature of space-time. Mass, however, remained an irreducible (...)
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  48.  17
    Geometry and analysis in Anastácio da Cunha’s calculus.João Caramalho Domingues - 2023 - Archive for History of Exact Sciences 77 (6):579-600.
    It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions (...)
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  49.  76
    Charge, Geometry, and Effective Mass in the Kerr-Newman Solution to the Einstein Field Equations.Gerald E. Marsh - 2008 - Foundations of Physics 38 (10):959-968.
    It has been shown that for the Reissner-Nordström solution to the vacuum Einstein field equations charge, like mass, has a unique space-time signature (Marsh, Found. Phys. 38:293–300, 2008). The presence of charge results in a negative curvature. This work, which includes a discussion of effective mass, is extended here to the Kerr-Newman solution.
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  50.  27
    Geometry and analysis in Euler’s integral calculus.Giovanni Ferraro, Maria Rosaria Enea & Giovanni Capobianco - 2017 - Archive for History of Exact Sciences 71 (1):1-38.
    Euler developed a program which aimed to transform analysis into an autonomous discipline and reorganize the whole of mathematics around it. The implementation of this program presented many difficulties, and the result was not entirely satisfactory. Many of these difficulties concerned the integral calculus. In this paper, we deal with some topics relevant to understand Euler’s conception of analysis and how he developed and implemented his program. In particular, we examine Euler’s contribution to the construction of differential equations and his (...)
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