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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.
  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. 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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  9. 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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  10. 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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  11. Katherine Brading (2008). Leo Corry. David Hilbert and the Axiomatization of Physics (1898–1918). Philosophia Mathematica 16 (1):113-129.
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  12. James Robert Brown (2007). Siobhan Roberts. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. Philosophia Mathematica 15 (3):386-388.
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  13. James E. Broyles (1966). Geometry, Semantics and Space. Philosophia Mathematica (1-2):9-16.
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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. Lawrence Foss (1967). Modern Geometries and the “Transcendental Aesthetic”. Philosophia Mathematica (1-2):35-45.
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  22. 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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  23. Hardy Grant (1989). Geometry and Medicine: Mathematics in the Thought of Galen of Pergamum. Philosophia Mathematica (1):29-34.
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  24. Emily Grosholz (2004). Critical Studies / Book Reviews. Philosophia Mathematica 12 (1):79-80.
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  25. 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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  26. Richard J. Hall (1965). A Philosophy of Geometry. Philosophia Mathematica (1):13-31.
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  27. 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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  28. 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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  29. 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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  30. C. A. Hooker (1981). Graham Nerlich: The Shape of Space. Dialogue 20 (04):783-798.
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  31. 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.
  32. 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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  33. René Jagnow, Geometry and Spatial Intuition: A Genetic Approach.
    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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  34. 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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  35. 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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  36. 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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  37. Ulrich Majer (1995). Geometry, Intuition and Experience: From Kant to Husserl. [REVIEW] Erkenntnis 42 (2):261 - 285.
  38. 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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  39. 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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  40. Ryszard Stanislaw Michalski (1978). A Planar Geometrical Model for Representing Multidimensional Discrete Spaces and Multiple-Valued Logic Functions. Dept. Of Computer Science, University of Illinois at Urbana-Champaign.
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  41. Ian Mueller (1981/2006). Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Dover Publications.
    A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.
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  42. John Mumma (2008). Nathaniel Miller. Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry. Csli Studies in the Theory and Applications of Diagrams. Philosophia Mathematica 16 (2):256-264.
  43. John O. Nelson (1975). Some Experiential Incoherencies of Riemannian Space. Philosophia Mathematica (1):66-75.
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  44. Graham Nerlich (2009). Incongruent Counterparts and the Reality of Space. Philosophy Compass 4 (3):598-613.
    Left and right hands are incongruent counterparts. Yet each replicates the intrinsic properties of the other. This suggests that differing relations to space make the difference. Kant's and Weyl's discussions of the problem are critically discussed. It emerges that spatial relationism fails to explain how its relations may be interpreted. An excursion into visual geometry explains the basis of handedness in the orientable structure of space.
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  45. Graham Nerlich (1979). What Can Geometry Explain? British Journal for the Philosophy of Science 30 (1):69-83.
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  46. Keith K. Niall (2002). Visual Imagery and Geometric Enthymeme: The Example of Euclid I. Behavioral and Brain Sciences 25 (2):202-203.
    Students of geometry do not prove Euclid's first theorem by examining an accompanying diagram, or by visualizing the construction of a figure. The original proof of Euclid's first theorem is incomplete, and this gap in logic is undetected by visual imagination. While cognition involves truth values, vision does not: the notions of inference and proof are foreign to vision.
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  47. V. Pambuccian (2013). David Hilbert. David Hilbert's Lectures on the Foundations of Geometry, 1891–1902. Michael Hallett and Ulrich Majer, Eds. David Hilbert's Foundational Lectures; 1. Berlin: Springer-Verlag, 2004. ISBN 3-540-64373-7. Pp. Xxviii + 661. [REVIEW] Philosophia Mathematica 21 (2):255-277.
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  48. Victor Pambuccian (2002). Axiomatizations of Hyperbolic Geometry: A Comparison Based on Language and Quantifier Type Complexity. Synthese 133 (3):331 - 341.
    Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type forallexistsforall, while the axiom system based on congruence and order can beformulated using only forallexists-axioms.
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  49. Marco Panza (2012). The Twofold Role of Diagrams in Euclid's Plane Geometry. Synthese 186 (1):55-102.
    Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams (...)
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  50. Moritz Pasch (2010). Essays on the Foundations of Mathematics. Springer.
    Translator's introduction -- Fundamental questions of geometry -- The decidability requirement -- The origin of the concept of number -- Implicit definition and the proper grounding of mathematics -- Rigid bodies in geometry -- Prelude to geometry : the essential ideas -- Physical and mathematical geometry -- Natural geometry -- The concept of the differential -- Reflections on the proper grounding of mathematics I -- Concepts and proofs in mathematics -- Dimension and space in mathematics -- Reflections on the proper (...)
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