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Axiomatics, empiricism, and Anschauung in Hilbert's conception of geometry: Between arithmetic and general relativity

In José Ferreirós Domínguez & Jeremy Gray (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford, England: Oxford University Press. pp. 133--156 (2006)

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  1. The normative structure of mathematization in systematic biology.Beckett Sterner & Scott Lidgard - 2014 - Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences 46 (1):44-54.
    We argue that the mathematization of science should be understood as a normative activity of advocating for a particular methodology with its own criteria for evaluating good research. As a case study, we examine the mathematization of taxonomic classification in systematic biology. We show how mathematization is a normative activity by contrasting its distinctive features in numerical taxonomy in the 1960s with an earlier reform advocated by Ernst Mayr starting in the 1940s. Both Mayr and the numerical taxonomists sought to (...)
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  • The Frege–Hilbert controversy in context.Tabea Rohr - 2023 - Synthese 202 (1):1-30.
    This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century (...)
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  • Geometría, formalismo e intuición: David Hilbert y el método axiomático formal.Eduardo N. Giovannini - 2014 - Revista de Filosofía (Madrid) 39 (2):121-146.
    El artículo presenta y analiza un conjunto de notas manuscritas de clases para cursos sobre geometría, dictados por David Hilbert entre 1891 y 1905. Se argumenta que en estos cursos el autor elabora la concepción de la geometría que subyace a sus investigaciones axiomáticas en Fundamentos de la geometría . Por un lado, afirmo que lo que caracteriza esta concepción de la geometría es: i) una posición axiomática abstracta o formal; ii) una posición empirista respecto del origen de la geometría (...)
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  • The place of probability in Hilbert’s axiomatization of physics, ca. 1900–1928.Lukas M. Verburgt - 2016 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 53:28-44.
    Although it has become a common place to refer to the ׳sixth problem׳ of Hilbert׳s (1900) Paris lecture as the starting point for modern axiomatized probability theory, his own views on probability have received comparatively little explicit attention. The central aim of this paper is to provide a detailed account of this topic in light of the central observation that the development of Hilbert׳s project of the axiomatization of physics went hand-in-hand with a redefinition of the status of probability theory (...)
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  • A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography.Karin Usadi Katz & Mikhail G. Katz - 2012 - Foundations of Science 17 (1):51-89.
    We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.
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  • On the Procedural Character of Hilbert’s Axiomatic Method.Giambattista Formica - 2019 - Quaestio 19:459-482.
    Hilbert’s methodological reflection has certainly shaped a new image of the axiomatic method. However, the discussion on the procedural character of the method is still open, with commentators subs...
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  • Epistemology Versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf.Peter Dybjer, Sten Lindström, Erik Palmgren & Göran Sundholm (eds.) - 2012 - Dordrecht, Netherland: Springer.
    This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...)
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