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  1. The Development of Mathematical Logic From Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press.
    The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...)
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  • Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics. [REVIEW]I. Grattan-Guinness - 2011 - Logica Universalis 5 (1):21-73.
    A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, of (...)
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  • Frege’s ‘On the Foundations of Geometry’ and Axiomatic Metatheory.Günther Eder - 2016 - Mind 125 (497):5-40.
    In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a reconstruction of this (...)
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  • Space, Number and Structure: A Tale of Two Debates.Stewart Shapiro - 1996 - Philosophia Mathematica 4 (2):148-173.
    Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics (...)
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  • Methodological Practice and Complementary Concepts of Logical Consequence: Tarski's Model-Theoretic Consequence and Corcoran's Information-Theoretic Consequence.José M. Sagüillo - 2009 - History and Philosophy of Logic 30 (1):21-48.
    This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by (...)
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  • Analysis and Interpretation in the Exact Sciences: Essays in Honour of William Demopoulos.Melanie Frappier, Derek Brown & Robert DiSalle (eds.) - 2011 - Dordrecht and London: Springer.
    The essays in this volume concern the points of intersection between analytic philosophy and the philosophy of the exact sciences. More precisely, it concern connections between knowledge in mathematics and the exact sciences, on the one hand, and the conceptual foundations of knowledge in general. Its guiding idea is that, in contemporary philosophy of science, there are profound problems of theoretical interpretation-- problems that transcend both the methodological concerns of general philosophy of science, and the technical concerns of philosophers of (...)
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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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  • 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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  • On the Exhaustion of Mathematical Entities by Structures.Adrian Heathcote - 2014 - Axiomathes 24 (2):167-180.
    There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.
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  • Lingua Characterica and Calculus Ratiocinator: The Leibnizian Background of the Frege-Schröder Polemic.Joan Bertran-San Millán - 2021 - Review of Symbolic Logic 14 (2):411-446.
    After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain (...)
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  • Soft Axiomatisation: John von Neumann on Method and von Neumann's Method in the Physical Sciences.Miklós Rédei & Michael Stöltzner - 2006 - In Emily Carson & Renate Huber (eds.), Intuition and the Axiomatic Method. Springer. pp. 235--249.
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  • The Neo-Fregean Program in the Philosophy of Arithmetic.William Demopoulos - 2006 - In Emily Carson & Renate Huber (eds.), Intuition and the Axiomatic Method. Springer. pp. 87--112.
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  • On the Origin and Status of Our Conception of Number.William Demopoulos - 2000 - Notre Dame Journal of Formal Logic 41 (3):210-226.
    This paper concerns the epistemic status of "Hume's principle"--the assertion that for any concepts and , the number of s is the same as the number of s just in case the s and the s are in one-one correspondence. I oppose the view that Hume's principle is a stipulation governing the introduction of a new concept with the thesis that it represents the correct analysis of a concept in use. Frege's derivation of the basic laws of arithmetic from Hume's (...)
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  • 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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  • Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic.Sean Walsh - 2016 - Journal of Philosophical Logic 45 (3):277-326.
    This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms (...)
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  • Geometric Conventionalism and Carnap's Principle of Tolerance: We Discuss in This Paper the Question of the Scope of the Principle of Tolerance About Languages Promoted in Carnap's The Logical Syntax of Language and the Nature of the Analogy Between It and the Rudimentary Conventionalism Purportedly Exhibited in the Work of Poincaré and Hilbert. We Take It More or Less for Granted That Poincaré and Hilbert Do Argue for Conventionalism. We Begin by Sketching Coffa's Historical Account, Which Suggests That Tolerance Be Interpreted as a Conventionalism That Allows Us Complete Freedom to Select Whatever Language We Wish—an Interpretation That Generalizes the Conventionalism Promoted by Poincaré and Hilbert Which Allows Us Complete Freedom to Select Whatever Axiom System We Wish for Geometry. We Argue That Such an Interpretation Saddles Carnap with a Theory of Meaning That has Unhappy Consequences, a Theory We Believe He Did Not Hold. We Suggest That the Principle of Linguistic Tolerance In.David De Vidi & Graham Solomon - 1993 - Studies in History and Philosophy of Science Part A 25 (5):773-783.
    We discuss in this paper the question of the scope of the principle of tolerance about languages promoted in Carnap's The Logical Syntax of Language and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincaré and Hilbert. We take it more or less for granted that Poincaré and Hilbert do argue for conventionalism. We begin by sketching Coffa's historical account, which suggests that tolerance be interpreted as a conventionalism that allows us (...)
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  • Dedekind and Hilbert on the Foundations of the Deductive Sciences.Ansten Klev - 2011 - Review of Symbolic Logic 4 (4):645-681.
    We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, whereas (...)
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  • Logic in the 1930s: Type Theory and Model Theory.Georg Schiemer & Erich H. Reck - 2013 - Bulletin of Symbolic Logic 19 (4):433-472.
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  • Logical Contextuality in Frege.Brice Halimi - 2018 - Review of Symbolic Logic 11 (1):1-20.
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