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Constructive ReasoningIn B. Van Rootselaar & J. F. Staal (eds.), Logic, Methodology and Philosophy of Science III, North-holland. pp. 185-99. 1968.
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14J. P. Mayberry. The foundations of mathematics in the theory of sets. Encyclopedia of mathematics and its applications, vol. 82. Cambridge University Press, Cambridge 2000, New York 2001, etc., xx + 424 pp (review)Bulletin of Symbolic Logic 8 (3): 424-426. 2002.
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Frege versus Cantor and Dedekind: On the Concept of NumberIn Matthias Schirn (ed.), Frege: importance and legacy, Walter De Gruyter. pp. 70-113. 1996.
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1What Hilbert and Bernays Meant by "Finitism"In Gabriele Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium, De Gruyter. pp. 249-261. 2019.
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23Meeting of the Association for Symbolic Logic, Chicago 1975Journal of Symbolic Logic 41 (2): 551-560. 1976.
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To appear in the Proceedings of Logic Colloquium 2006. (32 pages).
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38Review: J. P. Mayberry, The Foundations of Mathematics in the Theory of Sets (review)Bulletin of Symbolic Logic 8 (3): 424-426. 2002.
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282The myth of the mindTopoi 21 (1-2): 65-74. 2002.Of course, I do not mean by the title of this paper to deny the existence of something called
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101Beyond the axioms: The question of objectivity in mathematicsPhilosophia Mathematica 9 (1): 21-36. 2001.This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a m…Read more
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42Finite Definability of Number-Theoretic Functions and Parametric Completeness of Equational CalculiZeitschrift fur mathematische Logik und Grundlagen der Mathematik 7 (1-5): 28-38. 1961.
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9Review: A. Grzegorczyk, Some Proofs of Undecidability of Arithmetic (review)Journal of Symbolic Logic 23 (1): 46-47. 1958.
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19Mayberry J. P.. The foundations of mathematics in the theory of sets. Encyclopedia of mathematics and its applications, vol. 82. Cambridge University Press, Cambridge 2000, New York 2001, etc., xx + 424 pp (review)Bulletin of Symbolic Logic 8 (3): 424-426. 2002.
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128Gödel on intuition and on Hilbert's finitismIn Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial, Association For Symbolic Logic. 2010.There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primar…Read more
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11Chicago 1967 meeting of the Association for Symbolic LogicJournal of Symbolic Logic 36 (2): 359-368. 1971.
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51The five questionsIn V. F. Hendricks & Hannes Leitgeb (eds.), Philosophy of Mathematics: Five Questions, Automatic Press/vip. 2007.1. A Road to Philosophy of Mathematics l became interested in philosophy and mathematics at more or less the same time, rather late in high school; and my interest in the former certainly influenced my attitude towards the latter, leading me to ask what mathematics is really about at a fairly early stage. I don ’t really remember how it was that I got interested in either subject. A very good math teacher came to my school when I was in 9th grade and I got caught up in his course on solid geomet…Read more
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132Against intuitionism: Constructive mathematics is part of classical mathematics (review)Journal of Philosophical Logic 12 (2). 1983.
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23Review: S. C. Kleene, Extension of an Effectively Generated Class of Functions by Enumeration (review)Journal of Symbolic Logic 25 (3): 279-280. 1960.
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30Meeting of the Association for Symbolic Logic, Chicago, 1977Journal of Symbolic Logic 43 (3). 1978.
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22Orey Steven. On ω-consistency and related propertiesJournal of Symbolic Logic 23 (1): 40-41. 1958.
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78Kant and FinitismJournal of Philosophy 113 (5/6): 261-273. 2016.An observation and a thesis: The observation is that, whatever the connection between Kant’s philosophy and Hilbert’s conception of finitism, Kant’s account of geometric reasoning shares an essential idea with the account of finitist number theory in “Finitism”, namely the idea of constructions f from ‘arbitrary’ or ‘generic’ objects of various types. The thesis is that, contrary to a substantial part of contemporary literature on the subject, when Kant referred to number and arithmetic, he was …Read more
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151There can be no doubt about the value of Frege's contributions to the philosophy of mathematics. First, he invented quantification theory and this was the first step toward making precise the notion of a purely logical deduction. Secondly, he was the first to publish a logical analysis of the ancestral R* of a relation R, which yields a definition of R* in second-order logic.1 Only a narrow and arid conception of philosophy would exclude these two achievements. Thirdly and very importantly, the …Read more
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40William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gdel
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154The completeness of Heyting first-order logicJournal of Symbolic Logic 68 (3): 751-763. 2003.Restricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃x : A.F (x) is understood as disjoint union, are the projections, and these do not preserve firstorderedness. This note shows, however, that the Curry-Howard theory is conservative over Heyting’s system.
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19Early Analytic Philosophy: Frege, Russell, Wittgenstein : Essays in Honor of Leonard LinskyOpen Court Publishing Company. 1997.These essays present new analyzes of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.
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