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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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17Kleene S. C.. Extension of an effectively generated class of functions by enumeration. Colloquium mathematicum, vol. 6 , pp. 68–78 (review)Journal of Symbolic Logic 25 (3): 279-280. 1960.
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38The reduction of the lambda calculus to the theory of combinators in [Sch¨ onfinkel, 1924] applies to positive implicational logic, i.e. to the typed lambda calculus, where the types are built up from atomic types by means of the operation A −→ B, to show that the lambda operator can be eliminated in favor of combinators K and S of each type A −→ (B −→ A) and (A −→ (B −→ C)) −→ ((A −→ B) −→ (A −→ C)), respectively.1 I will extend that result to the case in which the types are built up by means o…Read more
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71Godel's interpretation of intuitionismPhilosophia Mathematica 14 (2): 208-228. 2006.Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting…Read more
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85Foundations of a General Theory of Manifolds [Cantor, 1883], which I will refer to as the Grundlagen, is Cantor’s first work on the general theory of sets. It was a separate printing, with a preface and some footnotes added, of the fifth in a series of six papers under the title of “On infinite linear point manifolds”. I want to briefly describe some of the achievements of this great work. But at the same time, I want to discuss its connection with the so-called paradoxes in set theory. There se…Read more
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47A counterexample to a conjecture of Scott and SuppesJournal of Symbolic Logic 24 (1): 15-16. 1959.
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92The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.
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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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41Finite 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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127Gödel 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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