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  1. Hilbert, Duality, and the Geometrical Roots of Model Theory.Günther Eder & Georg Schiemer - 2018 - Review of Symbolic Logic 11 (1):48-86.
    The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the (...)
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  • Model Theory and the Philosophy of Mathematical Practice: Formalization Without Foundationalism.John T. Baldwin - 2018 - Cambridge University Press.
    Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of (...)
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  • (1 other version)Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2002 - Philosophy and Phenomenological Research 65 (2):467-475.
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  • (1 other version)From Frege to Gödel. A Source Book in Mathematical Logic 1879-1931.Jean van Heijenoort - 1968 - Synthese 18 (2-3):302-305.
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  • (2 other versions)The Basic Laws of Arithmetic. [REVIEW]Aaron Sloman - 1966 - British Journal for the Philosophy of Science 17 (3):249-253.
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  • Frege’s Conception of Logic.Patricia Blanchette - 2012 - Oxford, England: Oup Usa.
    In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic.
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  • On Computable Numbers, with an Application to the Entscheidungsproblem.Alan Turing - 1936 - Proceedings of the London Mathematical Society 42 (1):230-265.
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  • Introduction to metamathematics.Stephen Cole Kleene - 1952 - Groningen: P. Noordhoff N.V..
    Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...)
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  • Metamathematics of First-Order Arithmetic.Petr Hajek & Pavel Pudlak - 1998 - Springer Verlag.
    People have always been interested in numbers, in particular the natural numbers. Of course, we all have an intuitive notion of what these numbers are. In the late 19th century mathematicians, such as Grassmann, Frege and Dedekind, gave definitions for these familiar objects. Since then the development of axiomatic schemes for arithmetic have played a fundamental role in a logical understanding of mathematics. There has been a need for some time for a monograph on the metamathematics of first-order arithmetic. The (...)
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  • From Kant to Hilbert: a source book in the foundations of mathematics.William Bragg Ewald (ed.) - 1996 - New York: Oxford University Press.
    This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...)
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  • Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1999 - Studia Logica 77 (1):129-129.
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  • The Frege-Hilbert controversy.Michael David Resnik - 1974 - Philosophy and Phenomenological Research 34 (3):386-403.
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  • (1 other version)Bemerkungen zur Philosophie der Mathematik.Paul Bernays - 1978 - Journal of Symbolic Logic 43 (1):151-152.
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  • The Reality of Mathematics and the Case of Set Theory.Daniel Isaacson - 2010 - In Zsolt Novák & András Simonyi (eds.), Truth, reference, and realism. New York: Central European University Press. pp. 1-76.
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  • Posthumous Writings.Gottlob Frege - 1982 - Revue Philosophique de la France Et de l'Etranger 172 (1):101-103.
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  • Philosophy of Mathematics in the Twentieth Century: Selected Essays.Charles Parsons - 2013 - Cambridge, Massachusetts: Harvard University Press.
    In these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: Brouwer, Hilbert, Bernays, Weyl, Gödel, Russell, Quine, Putnam, Wang, and Tait.
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  • Second-order logic and foundations of mathematics.Jouko Väänänen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
    We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically (...)
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  • Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics.Stewart Shapiro - 2005 - Philosophia Mathematica 13 (1):61-77.
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...)
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  • (1 other version)Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.
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  • Grundzüge der theoretischen Logik.D. Hilbert & W. Ackermann - 1928 - Annalen der Philosophie Und Philosophischen Kritik 7:157-157.
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  • A system of axiomatic set theory—Part I.Paul Bernays - 1937 - Journal of Symbolic Logic 2 (1):65-77.
    Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus (...)
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  • Hilbertian Structuralism and the Frege-Hilbert Controversy†.Fiona T. Doherty - 2019 - Philosophia Mathematica 27 (3):335-361.
    ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish (...)
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  • Hilbert's program relativized: Proof-theoretical and foundational reductions.Solomon Feferman - 1988 - Journal of Symbolic Logic 53 (2):364-384.
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  • (1 other version)Grundlagen der Mathematik II.D. Hilbert & P. Bernays - 1974 - Journal of Symbolic Logic 39 (2):357-357.
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  • (1 other version)On recursively enumerable and arithmetic models of set theory.Michael O. Rabin - 1958 - Journal of Symbolic Logic 23 (4):408-416.
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  • On an alleged refutation of Hilbert's program using gödel's first incompleteness theorem.Michael Detlefsen - 1990 - Journal of Philosophical Logic 19 (4):343 - 377.
    It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...)
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  • (1 other version)A note on the entscheidungsproblem.Alonzo Church - 1936 - Journal of Symbolic Logic 1 (1):40-41.
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  • Incompleteness Via Paradox and Completeness.Walter Dean - 2020 - Review of Symbolic Logic 13 (3):541-592.
    This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in (...)
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  • The Ways of Hilbert's Axiomatics: Structural and Formal.Wilfried Sieg - 2014 - Perspectives on Science 22 (1):133-157.
    It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates (...)
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  • (1 other version)``A Note on the Entcheidunsproblem".Alonzo Church - 1936 - Journal of Symbolic Logic 1 (1):40-41.
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  • (1 other version)Grundlagen der Mathematik I. Hilbert & Bernays - 1935 - Revue de Métaphysique et de Morale 42 (2):12-14.
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  • Frege and Hilbert.M. Hallett - 2010 - In Michael Potter, Joan Weiner, Warren Goldfarb, Peter Sullivan, Alex Oliver & Thomas Ricketts (eds.), The Cambridge Companion to Frege. New York: Cambridge University Press. pp. 413--464.
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  • (1 other version)Abhandlungen zur Philosophie der Mathematik.G. T. Kneebone & Paul Bernays - 1977 - Philosophical Quarterly 27 (106):72.
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  • Decidability and essential undecidability.Hilary Putnam - 1957 - Journal of Symbolic Logic 22 (1):39-54.
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  • (1 other version)Metamathematics of First-Order Arithmetic.P. Hájek & P. Pudlák - 2000 - Studia Logica 64 (3):429-430.
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  • (1 other version)Metamathematics of First-Order Arithmetic.Petr Hajék & Pavel Pudlák - 1994 - Studia Logica 53 (3):465-466.
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  • Basic laws of arithmetic.Gottlob Frege - 1893 - In Basic Laws of Arithmetic. Oxford, U.K.: Oxford University Press.
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  • Second-order Logic And Foundations Of Mathematics.Jouko V. "A. "An "Anen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
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  • Relative consistency and accessible domains.Wilfried Sieg - 1990 - Synthese 84 (2):259 - 297.
    Wilfred Sieg. Relative Consistency and Accesible Domains.
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  • (1 other version)On Recursively Enumerable and Arithmetic Models of Set Theory.Michael O. Rabin - 1963 - Journal of Symbolic Logic 28 (2):167-168.
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