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  1. Mathematics and Set Theory:数学と集合論.Sakaé Fuchino - 2018 - Journal of the Japan Association for Philosophy of Science 46 (1):33-47.
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  • More Infinity for a Better Finitism.Sam Sanders - 2010 - Annals of Pure and Applied Logic 161 (12):1525-1540.
    Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed in around 1995 by Patrick Suppes and Richard Sommer. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes. Here, we discuss the inherent problems and limitations of the classical nonstandard framework and propose a much-needed refinement of ERNA, called , in the spirit of Karel Hrbacek’s stratified set theory. We study the metamathematics of and its (...)
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  • The Metamathematics of Ergodic Theory.Jeremy Avigad - 2009 - Annals of Pure and Applied Logic 157 (2-3):64-76.
    The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns (...)
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  • Critical Study of Michael Potter’s Reason’s Nearest Kin. [REVIEW]Richard Zach - 2005 - Notre Dame Journal of Formal Logic 46 (4):503-513.
    Critical study of Michael Potter, Reason's Nearest Kin. Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000. x + 305 pages.
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  • Reliability of Mathematical Inference.Jeremy Avigad - forthcoming - Synthese:1-23.
    Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of (...)
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  • A New Psychologism in Logic? Reflections From the Point of View of Belief Revision.Hans Rott - 2008 - Studia Logica 88 (1):113-136.
    This paper addresses the question whether the past couple of decades of formal research in belief revision offers evidence of a new psychologism in logic. In the first part I examine five potential arguments in favour of this thesis and find them all wanting. In the second part of the paper I argue that belief revision research has climbed up a hierarchy of models for the change of doxastic states that appear to be clearly normative at the bottom, but are (...)
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  • What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory.Colin McLarty - 2010 - Bulletin of Symbolic Logic 16 (3):359-377.
    This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.
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  • What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory.Colin McLarty - 2010 - Bulletin of Symbolic Logic 16 (3):359-377.
    This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.
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  • Carroll’s Infinite Regress and the Act of Diagramming.John Mumma - 2019 - Topoi 38 (3):619-626.
    The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.
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  • Hilbert's Program Then and Now.Richard Zach - 2007 - In Dale Jacquette (ed.), Philosophy of Logic. Amsterdam: North Holland. pp. 411–447.
    Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...)
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  • Proof Theory.Jeremy Avigad - unknown
    At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...)
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  • La théorie des nombres chez Herbrand et Lautman.Yvon Gauthier - 2010 - Philosophiques 37 (1):149-161.
    Dans cet article, je compare les vues de Lautman et Herbrand sur la théorie des nombres et la philosophie de l’arithmétique. Je montre que, bien que Lautman eût avoué avoir été marqué par l’influence de Herbrand, les postures fondationnelles des deux amis divergent considérablement. Alors que Lautman versait dans un réalisme platonicien, Herbrand est resté fidèle au finitisme hilbertien. Il est vrai que Lautman était philosophe et que Herbrand était avant tout arithméticien et logicien, mais il demeure que l’oeuvre de (...)
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  • Understanding, Formal Verification, and the Philosophy of Mathematics.Jeremy Avigad - 2010 - Journal of the Indian Council of Philosophical Research 27:161-197.
    The philosophy of mathematics has long been concerned with deter- mining the means that are appropriate for justifying claims of mathemat- ical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary math- ematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The as- sociated values are often loosely classied as (...)
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  • Sur la philosophie analytique de la logique et des mathématiques. À propos de Infini, logique, géométrie de Paolo Mancosu.Yvon Gauthier - 2016 - Dialogue 55 (1):193-201.
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  • Fermat's Last Theorem and Catalan's Conjecture in Weak Exponential Arithmetics.Petr Glivický & Vítězslav Kala - 2017 - Mathematical Logic Quarterly 63 (3-4):162-174.
    We study Fermat's last theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions math formula of models of arithmetical theories by a binary function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms math formula. We construct a model math formula and a substructure math formula with e total (...)
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  • Proof Theory in Philosophy of Mathematics.Andrew Arana - 2010 - Philosophy Compass 5 (4):336-347.
    A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
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