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An Essay in the Foundations of Geometry

Cambridge, England: Cambridge University Press (1897)

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  1. What did Russell learn from Leibniz?Nicholas Griffin - 2013 - Journal for the History of Analytical Philosophy 2 (1).
    Russell’s rejection in 1898 of the doctrine of internal relations — the view that all relations are grounded in the intrinsic properties of the terms related — was a decisive part of his break with Hegelianism and opened the way for his turn to analytic philosophy. Before rejecting it, Russell had given the doctrine little thought, though it played an essential role in the most intractable of the problems facing his attempt to construct a Hegelian dialectic of the sciences. I (...)
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  • Ernst Cassirer's Neo-Kantian Philosophy of Geometry.Jeremy Heis - 2011 - 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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  • Category theory and the foundations of mathematics: Philosophical excavations.Jean-Pierre Marquis - 1995 - Synthese 103 (3):421 - 447.
    The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 (...)
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  • Tautology: How not to use a word.Burton Dreben & Juliet Floyd - 1991 - Synthese 87 (1):23 - 49.
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  • Special-issue book review.Jean-Pierre Marquis - 1996 - Philosophia Mathematica 4 (2):202-205.
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  • Karl Popper’s Philosophical Breakthrough.Stefano Gattei - 2004 - Philosophy of Science 71 (4):448-466.
    Despite his well‐known deductivism, in his early (unpublished) writings, Popper held an inductivist position. Up to 1929 epistemology entered Popper's reflections only as far as the problem was that of the justification of the scientific character of these fields of research. However, in that year, while surveying the history of non‐Euclidean geometries, Popper explicitly discussed the cognitive status of geometry without referring to psycho‐pedagogical aspects, thus turning from cognitive psychology to the logic and methodology of science. As a consequence of (...)
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  • The problem of the invariance of dimension in the growth of modern topology, part I.Dale M. Johnson - 1979 - Archive for History of Exact Sciences 20 (2):97-188.
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  • Volume Introduction – Method, Science and Mathematics: Neo-Kantianism and Analytic Philosophy.Scott Edgar - 2018 - Journal for the History of Analytical Philosophy 6 (3):1-10.
    Introduction to the Special Volume, “Method, Science and Mathematics: Neo-Kantianism and Analytic Philosophy,” edited by Scott Edgar and Lydia Patton. At its core, analytic philosophy concerns urgent questions about philosophy’s relation to the formal and empirical sciences, questions about philosophy’s relation to psychology and the social sciences, and ultimately questions about philosophy’s place in a broader cultural landscape. This picture of analytic philosophy shapes this collection’s focus on the history of the philosophy of mathematics, physics, and psychology. The following essays (...)
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  • Articulating Space in Terms of Transformation Groups: Helmholtz and Cassirer.Francesca Biagioli - 2018 - Journal for the History of Analytical Philosophy 6 (3).
    Hermann von Helmholtz’s geometrical papers have been typically deemed to provide an implicitly group-theoretical analysis of space, as articulated later by Felix Klein, Sophus Lie, and Henri Poincaré. However, there is less agreement as to what properties exactly in such a view would pertain to space, as opposed to abstract mathematical structures, on the one hand, and empirical contents, on the other. According to Moritz Schlick, the puzzle can be resolved only by clearly distinguishing the empirical qualities of spatial perception (...)
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  • After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics.Janet Folina - 2018 - Journal for the History of Analytical Philosophy 6 (3).
    The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to (...)
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  • Molyneux’s Question in Berkeley’s Theory of Vision.Juan R. Loaiza - 2017 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 32 (2):231-247.
    I propose a reading of Berkeley's Essay towards a New Theory of Vision in which Molyneux-type questions are interpreted as thought experiments instead of arguments. First, I present the general argumentative strategy in the NTV, and provide grounds for the traditional reading. Second, I consider some roles of thought experiments, and classify Molyneux-type questions in the NTV as constructive conjectural thought experiments. Third, I argue that (i) there is no distinction between Weak and Strong Heterogeneity theses in the NTV; (ii) (...)
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  • Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic.Corey Edward Mulvihill - 2015 - Dissertation, University of Waterloo
    Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...)
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  • Geometry and generality in Frege's philosophy of arithmetic.Jamie Tappenden - 1995 - Synthese 102 (3):319 - 361.
    This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...)
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  • The philosophy of Hans Reichenbach.Wesley C. Salmon - 1977 - Synthese 34 (1):5 - 88.
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  • (1 other version)Avicenna on Syllogisms Composed of Opposite Premises.Behnam Zolghadr - 2021 - In Mojtaba Mojtahedi, Shahid Rahman & MohammadSaleh Zarepour (eds.), Mathematics, Logic, and their Philosophies: Essays in Honour of Mohammad Ardeshir. Springer. pp. 433-442.
    This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.
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  • Hegel and the hermeneutics of German idealism.Tom Rockmore - 1995 - International Journal of Philosophical Studies 3 (1):111 – 131.
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  • Jolen Galaugher, Russell’s Philosophy of Logical Analysis: 1897–1905. [REVIEW]Kevin C. Klement - 2015 - Journal for the History of Analytical Philosophy 3 (2).
    Review of Russell’s Philosophy of Logical Atomism 1897–1905, by Jolen Galaugher (Palgrave Macmillan 2013).
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  • ‘Metamathematics’ in Transition.Matthias Wille - 2011 - History and Philosophy of Logic 32 (4):333 - 358.
    In this paper, we trace the conceptual history of the term ?metamathematics? in the nineteenth century. It is well known that Hilbert introduced the term for his proof-theoretic enterprise in about 1922. But he was verifiably inspired by an earlier usage of the phrase in the 1870s. After outlining Hilbert's understanding of the term, we will explore the lines of inducement and elucidate the different meanings of ?metamathematics? in the final decades of the nineteenth century. Finally, we will investigate the (...)
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  • The significance of a non-reductionist ontology for the discipline of mathematics: A historical and systematic analysis. [REVIEW]D. F. M. Strauss - 2010 - Axiomathes 20 (1):19-52.
    A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...)
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  • Frege on intuition and objecthood in projective geometry.Günther Eder - 2021 - Synthese 199 (3-4):6523-6561.
    In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by focussing on his views on a field which occupied center stage in nineteenth century geometry, namely, projective geometry.
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  • Pasch's empiricism as methodological structuralism.Dirk Schlimm - 2020 - In Erich H. Reck & Georg Schiemer (eds.), The Pre-History of Mathematical Structuralism. Oxford: Oxford University Press. pp. 80-105.
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  • Russell, Meinong and the Origin of the Theory of Descriptions.Harm Boukema - 2007 - Russell: The Journal of Bertrand Russell Studies 27 (1):41-72.
    According to his own account, Russell was “led to” the Theory of Descriptions by “the desire to avoid Meinong’s unduly populous realm of being”. This “official view” has been subjected to severe criticism. However stimulating this criticism may be, it is too extreme and therefore not critical enough. It fails to fully acknowledge both the way it is itself opposed to Russell and the way Russell and Meinong were opposed to _their_ opponents. In order to avoid these failures, a more (...)
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  • José Echegaray: entre la ciencia, el teatro y la política.José Manuel Sánchez Ron - 2004 - Arbor 179 (707/708):601-688.
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  • From geometry to phenomenology.Mirja Helena Hartimo - 2008 - Synthese 162 (2):225-233.
    Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the winter 1901–1902. (...)
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  • Gravity and Gauge.Nicholas J. Teh - 2016 - British Journal for the Philosophy of Science 67 (2):497-530.
    Philosophers of physics and physicists have long been intrigued by the analogies and disanalogies between gravitational theories and gauge theories. Indeed, repeated attempts to collapse these disanalogies have made us acutely aware that there are fairly general obstacles to doing so. Nonetheless, there is a special case space-time dimensions) in which gravity is often claimed to be identical to a gauge theory. I subject this claim to philosophical scrutiny in this article. In particular, I analyse how the standard disanalogies can (...)
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  • The Dimensionality of Visual Space.William H. Rosar - 2016 - Topoi 35 (2):531-570.
    The empirical study of visual space has centered on determining its geometry, whether it is a perspective projection, flat or curved, Euclidean or non-Euclidean, whereas the topology of space consists of those properties that remain invariant under stretching but not tearing. For that reason distance is a property not preserved in topological space whereas the property of spatial order is preserved. Specifically the topological properties of dimensionality, orientability, continuity, and connectivity define “real” space as studied by physics and are the (...)
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  • Da geometria à topologia : filosofia do espaço métrico.Ramiro Délio Borges de Meneses - 2010 - Endoxa 25:185.
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  • Time travel, hyperspace and Cheshire Cats.Alasdair Richmond - 2018 - Synthese 195 (11):5037-5058.
    H. G. Wells’ Time Traveller inhabits uniform Newtonian time. Where relativistic/quantum travelers into the past follow spacetime curvatures, past-bound Wellsians must reverse their direction of travel relative to absolute time. William Grey and Robin Le Poidevin claim reversing Wellsians must overlap with themselves or fade away piecemeal like the Cheshire Cat. Self-overlap is physically impossible but ‘Cheshire Cat’ fades destroy Wellsians’ causal continuity and breed bizarre fusions of traveler-stages with opposed time-directions. However, Wellsians who rotate in higher-dimensional space can reverse (...)
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  • German Philosophy of Mathematics from Gauss to Hilbert.Donald Gillies - 1999 - Royal Institute of Philosophy Supplement 44:167-192.
    Suppose we were to ask some students of philosophy to imagine a typical book of classical German philosophy and describe its general style and character, how might they reply? I suspect that they would answer somewhat as follows. The book would be long and heavy, it would be written in a complicated style which employed only very abstract terms, and it would be extremely difficult to understand. At all events a description of this kind does indeed fit many famous works (...)
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  • Review. [REVIEW]Andreas Kamlah - 1986 - Erkenntnis 24 (2):235-252.
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  • (1 other version)Introdução a três textos de Einstein sobre a geometria, a teoria física e a experiência.Michel Paty - 2005 - Scientiae Studia 3 (4):641-662.
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