
Frege versus Cantor and Dedekind: On the Concept of NumberIn Matthias Schirn (ed.), Frege: Importance and Legacy, Walter De Gruyter. pp. 70113. 1996.

What Hilbert and Bernays Meant by "Finitism"In Gabriele M. Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics, De Gruyter. pp. 249261. 2019.

1Review: A. Grzegorczyk, Some Proofs of Undecidability of Arithmetic (review)Journal of Symbolic Logic 23 (1): 4647. 1958.

30Finite Definability of NumberTheoretic Functions and Parametric Completeness of Equational CalculiZeitschrift fur mathematische Logik und Grundlagen der Mathematik 7 (15): 2838. 1961.

5Mayberry 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): 424426. 2002.

94Gödel 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Essays for His Centennial, Association For Symbolic Logic. 2010.There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between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 doublereversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primar…Read more

6Chicago 1967 meeting of the Association for Symbolic LogicJournal of Symbolic Logic 36 (2): 359368. 1971.

36The 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

106Against intuitionism: Constructive mathematics is part of classical mathematics (review)Journal of Philosophical Logic 12 (2). 1983.

17Review: S. C. Kleene, Extension of an Effectively Generated Class of Functions by Enumeration (review)Journal of Symbolic Logic 25 (3): 279280. 1960.

5Orey Steven. On ωconsistency and related propertiesJournal of Symbolic Logic 23 (1): 4041. 1958.

17Meeting of the Association for Symbolic Logic, Chicago, 1977Journal of Symbolic Logic 43 (3). 1978.

39Kant and FinitismJournal of Philosophy 113 (5/6): 261273. 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

118There 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 secondorder logic.1 Only a narrow and arid conception of philosophy would exclude these two achievements. Thirdly and very importantly, the …Read more

25William 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

140The completeness of Heyting firstorder logicJournal of Symbolic Logic 68 (3): 751763. 2003.Restricted to ﬁrstorder formulas, the rules of inference in the CurryHoward type theory are equivalent to those of ﬁrstorder predicate logic as formalized by Heyting, with one exception: ∃elimination in the CurryHoward 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 CurryHoward theory is conservative over Heyting’s system.

REVIEWS: E. MenzlerTrottLogic's lost genius: The life of Gerhard Gentzen (review)Bulletin of Symbolic Logic 16 (2). 2010.

50Noesis: Plato on exact scienceIn David B. Malament (ed.), Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics, Open Court. pp. 1131. 2002.

14The Hilton New York Hotel New York, NY December 27–29, 2005Bulletin of Symbolic Logic 12 (3). 2006.

117Gödel's reformulation of Gentzen's first consistency proof for arithmetic: The nocounterexample interpretationBulletin of Symbolic Logic 11 (2): 225238. 2005.The last section of “Lecture at Zilsel’s” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s ﬁrst version of his consistency proof for P A [8], reformulating it as what has come to be called the nocounterexample interpretation. I will describe Gentzen’s result (in gametheoretic terms), ﬁll in the details (with some corrections) of Godel's reformulation, and discuss the relation between the two proofs.

4Zermelo's Conception of Set Theory and Reflection PrinciplesIn Matthias Schirn (ed.), The Philosophy of Mathematics Today, Clarendon Press. 1998.

16Extensional Equality in the Classical Theory of TypesVienna Circle Institute Yearbook 3 219234. 1995.The classical theory of types in question is essentially the theory of MartinLöf [1] but with the law of double negation elimination. I am ultimately interested in the theory of types as a framework for the foundations of mathematics and, for this purpose, we need to consider extensions of the theory obtained by adding ‘wellordered types,’ for example the type N of the finite ordinals; but the unextended theory will suffice to illustrate the treatment of extensional equality

3A Nonconstructive Proof of Gentzen's Hauptsatz for Second Order Predicate LogicJournal of Symbolic Logic 33 (2): 289290. 1968.

16Review: Steven Orey, On $omega$Consistency and Related Properties (review)Journal of Symbolic Logic 23 (1): 4041. 1958.

22Plato's Second Best MethodReview of Metaphysics 39 (3). 1986.AT PHAEDO 96AC Plato portrays Socrates as describing his past study of "the kind of wisdom known as περὶ φυσέως ἱστορία." At 96c97b, Socrates says that this study led him to realize that he had an inadequate understanding of certain basic concepts which it involved. In consequence, he says at 97b, he abandoned this method and turned to a method of his own. But at this point in the dialogue, instead of proceeding immediately to describe his method, Plato has him interjecting a complaint concern…Read more
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