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Kant's space and modern mathematics

Mind 2 (5):38-46 (1877)

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  1. Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments?Jamie Tappenden - 2000 - Notre Dame Journal of Formal Logic 41 (3):271-315.
    It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...)
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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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  • Kant's formulation of the laws of motion.Robert Palter - 1972 - Synthese 24 (1-2):96 - 116.
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  • The evolution of empiricism: Hermann Von helmholtz and the foundations of geometry.Joan L. Richards - 1977 - British Journal for the Philosophy of Science 28 (3):235-253.
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  • Bertrand Russell's Essay on the Foundations of Geometry and the Cambridge Mathematical Tradition.Joan Richards - 1988 - Russell: The Journal of Bertrand Russell Studies 8 (1):59.
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  • Euclides ab omni naevo vindicatus.J. R. Lucas - 1969 - British Journal for the Philosophy of Science 20 (1):1-11.
    The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all (...)
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  • What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”?: Helmholtz’s Claim in Context.Francesca Biagioli - 2014 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 45 (1):1-21.
    In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be (...)
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