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Models in Geometry and Logic: 1870-1920

In Seppälä Niniiluoto (ed.), Logic, Methodology and Philosophy of Science - Proceedings of the 15th International Congress. College Publications. pp. 41-61 (2017)

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  1. Conceptual cartography.Robert Smithson - 2021 - Inquiry: An Interdisciplinary Journal of Philosophy 64 (1-2):97-122.
    ABSTRACT Certain features of our conceptual scheme seem necessary for subjects with our basic nature: we cannot imagine humans accomplishing their basic projects without having a conceptual scheme with these features. Other aspects of our conceptual scheme seem more contingent: we can imagine communities effectively using a somewhat different conceptual scheme. Conceptual cartography is the project of investigating the necessity and contingency of the various features of conceptual schemes. The project of conceptual cartography has not received much explicit methodological attention. (...)
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  • What analytic metaphysics can do for scientific metaphysics.Chanwoo Lee - 2023 - Ratio 36 (3):192-203.
    The apparent chasm between two camps in metaphysics, analytic metaphysics and scientific metaphysics, is well recognized. I argue that the relationship between them is not necessarily a rivalry; a division of labour that resembles the relationship between pure mathematics and science is possible. As a case study, I look into the metaphysical underdetermination argument for ontic structural realism, a well‐known position in scientific metaphysics, together with an argument for the position in analytic metaphysics known as ontological nihilism. I argue that (...)
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  • What are Implicit Definitions?Eduardo N. Giovannini & Georg Schiemer - 2019 - Erkenntnis 86 (6):1661-1691.
    The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely as specifications of the meaning of the primitive terms of a theory and as definitions of higher-order mathematical concepts or structures. We analyze these (...)
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  • Projective duality and the rise of modern logic.Günther Eder - 2021 - Bulletin of Symbolic Logic 27 (4):351-384.
    The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Specifically, we (...)
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  • Frege and the origins of model theory in nineteenth century geometry.Günther Eder - 2019 - Synthese 198 (6):5547-5575.
    The aim of this article is to contribute to a better understanding of Frege’s views on semantics and metatheory by looking at his take on several themes in nineteenth century geometry that were significant for the development of modern model-theoretic semantics. I will focus on three issues in which a central semantic idea, the idea of reinterpreting non-logical terms, gradually came to play a substantial role: the introduction of elements at infinity in projective geometry; the study of transfer principles, especially (...)
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  • Peano’s structuralism and the birth of formal languages.Joan Bertran-San-Millán - 2022 - Synthese 200 (4):1-34.
    Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...)
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