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  1. Ernest W. Adams (1994). On the Method of Superposition. British Journal for the Philosophy of Science 45 (2):693-708.
  2. Irving H. Anellis (1987). Book-Review. Philosophia Mathematica (1):110-116.
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  3. Andrew Arana (2009). Review of M. Giaquinto's Visual Thinking in Mathematics. [REVIEW] Analysis 69:401-403.
    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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  4. Andrew Arana (2009). Visual Thinking in Mathematics • by Marcus Giaquinto. 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 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to 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 follows : " Mathematicians had for (...)
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  5. Andrew Arana & Paolo Mancosu (2012). On the Relationship Between Plane and Solid Geometry. 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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  6. Jeremy Avigad, Edward Dean & John Mumma (2009). A Formal System for Euclid's Elements. Review of Symbolic Logic 2 (4):700--768.
    We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
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  7. Peter Baofu (2009). The Future of Post-Human Geometry: A Preface to a New Theory of Infinity, Symmetry, and Dimensionality. Cambridge Scholars.
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  8. I. G. Bashmakova & G. S. Smirnova (2000). Geometry: The First Universal Language of Mathematics. In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge. Kluwer Academic Publishers. 331--340.
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  9. Martina Bečvářová (2005). Euclid's Elements in the Czech Lands. NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine 13 (3):156-167.
    This article is dedicated to Euclid’s Elements, to translations of this work into Czech, and to the translators who have taken on the task of translation. It contains a short overview of the results achieved during a three-year project supported by the Czech Grant Agency.We explored how Euclid’s Elements were spread around the Czech lands.We will try to describe the circumstances that lay behind attempts to translate the Elements into the Czech language.
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  10. J. L. Bell (1994). Introduction. Philosophia Mathematica 2 (1):4-4.
    Continuous as the stars that shine And twinkle on the milky way, They stretched in never-ending line Along the margin of a bay: Ten thousand saw I at a glance, Tossing their heads in sprightly dance.
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  11. David Bostock (2010). Whitehead and Russell on Points. Philosophia Mathematica 18 (1):1-52.
    This paper considers the attempts put forward by A.N. Whitehead and by Bertrand Russell to ‘construct’ points (and temporal instants) from what they regard as the more basic concept of extended ‘regions’. It is shown how what they each say themselves will not do, and how it should be filled out and amended so that the ‘construction’ may be regarded as successful. Finally there is a brief discussion of whether this ‘construction’ is worth pursuing, or whether it is better—as in (...)
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  12. Michael Boylan (1983). Book Review:Philosophy of Mathematics and Deductive Structure in Euclid's Elements Ian Mueller; The Beginnings of Greek Mathematics Arpad Szabo, A. M. Ungar. [REVIEW] Philosophy of Science 50 (4):665-.
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  13. Katherine Brading (2008). Leo Corry. David Hilbert and the Axiomatization of Physics (1898–1918). Philosophia Mathematica 16 (1):113-129.
    This book is a wonderful resource for historians and philosophers of mathematics and physics alike, not just for Hilbert's own work in physics, but also because Corry sets Hilbert in context, bringing out the people with whom Hilbert had contact, describing their work and possible links with Hilbert's work, and describing the activities going on around Hilbert. The historical thesis of this book is that Hilbert worked on a wide range of issues in physics for a period lasting more than (...)
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  14. James Robert Brown (2007). Siobhan Roberts. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. Philosophia Mathematica 15 (3):386-388.
    Donald Coxeter died in 2003, at a ripe old age of 96. Though I had regularly seen him at mathematics talks in Toronto for over twenty years, I never felt rushed to seek him out. It seemed he would go on forever. His death left me regretting my missed opportunity and Siobhan Robert's excellent book makes me regret it even more. Like any good biography of an intellectual, King of Infinite Space contains personal details and mathematical achievements in some detail. (...)
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  15. James E. Broyles (1966). Geometry, Semantics and Space. Philosophia Mathematica (1-2):9-16.
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  16. Ivor Bulmer-Thomas (1983). S. M. Taisbak: Coloured Quadrangles. A Guide to the Tenth Book of Euclid's Elements. (Opuscula Graecolatina, 24.) Pp. 78; Mathematical Diagrams. Copenhagen: Museum Tusculanum, 1982. Paper, Dan. Kr. 45. [REVIEW] The Classical Review 33 (01):143-144.
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  17. V. C. C. (1956). An Essay on the Foundations of Geometry. Review of Metaphysics 10 (2):369-369.
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  18. Carlos Augusto Casanocva G. (2006). Metaphysical Notes Concerning Hilbert and His Studies on Non-Euclidean an Non-Archimedean Geometries. Teorema 25 (2):73-93.
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  19. C. Cellucci (2004). L. Gaeta. Segni Del Cosmo. Logica E Geometria in Whitehead [Signs of the Cosmos. Logic and Geometry in Whitehead]. Philosophia Mathematica 12 (3):289-290.
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  20. Jing-Ling Chen & Abraham A. Ungar (2002). The Bloch Gyrovector. Foundations of Physics 32 (4):531-565.
    Hyperbolic vectors are called gyrovectors. We show that the Bloch vector of quantum mechanics is a gyrovector. The Bures fidelity between two states of a qubit is generated by two Bloch vectors. Treating these as gyrovectors rather than vectors results in our novel expression for the Bures fidelity, expressed in terms of its two generating Bloch gyrovectors. Taming the Thomas precession of Einstein's special theory of relativity led to the advent of the theory of gyrogroups and gyrovector spaces. Gyrovector spaces, (...)
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  21. Michael E. Cuffaro (2012). Kant's Views on Non-Euclidean Geometry. 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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  22. Graciela De Pierris (2012). Hume on Space, Geometry, and Diagrammatic Reasoning. Synthese 186 (1):169-189.
    Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...)
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  23. Stanislas Dehaene, Véronique Izard, Pierre Pica & Elizabeth Spelke (2006). Core Knowledge of Geometry in an Amazonian Indigene Group. 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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  24. Stanislas Dehaene, Véronique Izard, Pierre Pica & Elizabeth Spelke (2006). Examining Knowledge of Geometry : Response to Wulf and Delson. Science 312 (5778):1309-1310.
    La connaissances noyau de la géométrie euclidienne est liée au raisonnement déductif et non à la reconnaissance de motifs perceptuels.
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  25. F. T. Falciano, M. Novello & J. M. Salim (2010). Geometrizing Relativistic Quantum Mechanics. Foundations of Physics 40 (12):1885-1901.
    We propose a new approach to describe quantum mechanics as a manifestation of non-Euclidean geometry. In particular, we construct a new geometrical space that we shall call Qwist. A Qwist space has a extra scalar degree of freedom that ultimately will be identified with quantum effects. The geometrical properties of Qwist allow us to formulate a geometrical version of the uncertainty principle. This relativistic uncertainty relation unifies the position-momentum and time-energy uncertainty principles in a unique relation that recover both of (...)
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  26. Lawrence Foss (1967). Modern Geometries and the “Transcendental Aesthetic”. Philosophia Mathematica (1-2):35-45.
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  27. D. H. Fowler (1983). Investigating Euclid's Elements. [REVIEW] British Journal for the Philosophy of Science 34 (1):57-70.
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  28. Sébastien Gandon (2009). Toward a Topic-Specific Logicism? Russell's Theory of Geometry in the Principles of Mathematics. Philosophia Mathematica 17 (1):35-72.
    Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was sustained (...)
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  29. Hardy Grant (1989). Geometry and Medicine: Mathematics in the Thought of Galen of Pergamum. Philosophia Mathematica (1):29-34.
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  30. Emily Grosholz (2004). Critical Studies / Book Reviews. Philosophia Mathematica 12 (1):79-80.
  31. Jeffrey Grupp (2005). The Impossibility of Relations Between Non-Collocated Spatial Objects and Non-Identical Topological Spaces. 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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  32. Richard J. Hall (1965). A Philosophy of Geometry. Philosophia Mathematica (1):13-31.
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  33. Jeremy Heis (2011). Ernst Cassirer's Neo-Kantian Philosophy of Geometry. British Journal for the History of Philosophy 19 (4):759 - 794.
    One of the most important philosophical topics in the early twentieth century and a topic that was seminal in the emergence of analytic philosophy was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of (...)
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  34. Chris Heunen, Klaas Landsman & Bas Spitters, The Principle of General Tovariance.
    We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining (...)
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  35. B. J. Hiley (2010). On the Relationship Between the Wigner-Moyal and Bohm Approaches to Quantum Mechanics: A Step to a More General Theory? [REVIEW] Foundations of Physics 40 (4):356-367.
    In this paper we show that the three main equations used by Bohm in his approach to quantum mechanics are already contained in the earlier paper by Moyal which forms the basis for what is known as the Wigner-Moyal approach. This shows, contrary to the usual perception, that there is a deep relation between the two approaches. We suggest the relevance of this result to the more general problem of constructing a quantum geometry.
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  36. T. A. Hirst (1878). Logic and the Elements of Geometry. Mind 3:564.
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  37. C. A. Hooker (1981). Graham Nerlich: The Shape of Space. Dialogue 20 (04):783-798.
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  38. P. L. Huddleston, M. Lorente & P. Roman (1975). Contractions of Space-Time Groups and Relativistic Quantum Mechanics. Foundations of Physics 5 (1):75-87.
    The relation of the conformal group to various earlier proposed relativistic quantum mechanical dynamical groups (and other related groups) is studied in the framework of projective geometry, by explicitly constructing the contractions of the six-dimensional coordinate transformations. Five-dimensional realizations are then derived. An attempt is made to improve our physical insight through geometry.
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  39. Véronique Izard, Pierre Pica, Danièle Hinchey, Stanislas Dehane & Elizabeth Spelke (2011). Geometry as a Universal Mental Construction. In Stanislas Dehaene & Elizabeth Brannon (eds.), Space, Time and Number in the Brain. Oxford University Press.
  40. René Jagnow (2006). Edmund Husserl on the Applicability of Formal Geometry. In Emily Carson & Renate Huber (eds.), Intuition and the Axiomatic Method. Springer. 67--85.
    In this paper, I reconstruct Edmund Husserl's view on the relationship between formal inquiry and the life-world, using the example of formal geometry. I first outline Husserl's account of geometry and then argue that he believed that the applicability of formal geometry to intuitive space (the space of everyday-experience) guarantees the conceptual continuity between different notions of space.
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  41. Rene Jagnow (2003). Geometry and Spatial Intuition: A Genetic Approach. Dissertation, Mcgill University (Canada)
    In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal logical inferences. By developing (...)
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  42. Monte Ransome Johnson (2009). The Aristotelian Explanation of the Halo. 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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  43. Joongol Kim (2006). Concepts and Intuitions in Kant's Philosophy of Geometry. Kant-Studien 97 (2):138-162.
    This paper is an exposition and defense of Kant’s philosophy of geometry. The main thesis is that Euclidean geometry investigates the properties of geometrical objects in an inner space that is given to us a priori (pure space) and hence is a priori and synthetic. This thesis is supported by arguing that Euclidean geometry requires certain intuitive objects (Sect. 1), that these objects are a priori constructions in pure space (Sect. 2), and finally that the role of geometrical construction is (...)
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  44. Alexey Kryukov, Geometric Derivation of Quantum Uncertainty.
    Quantum observables can be identified with vector fields on the sphere of normalized states. Consequently, the uncertainty relations for quantum observables become geometric statements. In the Letter the familiar uncertainty relation follows from the following stronger statement: Of all parallelograms with given sides the rectangle has the largest area.
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  45. L. Kvasz (2011). Kant's Philosophy of Geometry--On the Road to a Final Assessment. Philosophia Mathematica 19 (2):139-166.
    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; (...)
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  46. David Rapport Lachterman (1989). The Ethics of Geometry: A Genealogy of Modernity. Routledge.
    The Ethics of Geometry is a study of the relationship between philosophy and mathematics. Essential differences in the ethos of mathematics, for example, the customary ways of undertaking and understanding mathematical procedures and their objects, provide insight into the fundamental issues in the quarrel of moderns with ancients. Two signal features of the modern ethos are the priority of problem-solving over theorem-proving, and the claim that constructability by human minds or instruments establishes the existence of relevant entities. These figures are (...)
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  47. Ulrich Majer (1995). Geometry, Intuition and Experience: From Kant to Husserl. [REVIEW] Erkenntnis 42 (2):261 - 285.
  48. Kenneth Manders (2008). The Euclidean Diagram. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. 80--133.
    This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...)
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  49. Jean-Pierre Marquis (2013). Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics. 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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  50. T. Merrick (2006). What Frege Meant When He Said: Kant is Right About Geometry. 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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