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238Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematicsPhilosophia Mathematica 13 (1): 61-77. 2005.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 …Read more
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162010 winter meeting of the association for symbolic logicBulletin of Symbolic Logic 16 (3): 438-444. 2010.
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418Mathematics and realityPhilosophy of Science 50 (4): 523-548. 1983.The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) sur…Read more
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89Book Review: John P. Burgess and Gideon Rose. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics (review)Notre Dame Journal of Formal Logic 39 (4): 600-612. 1998.
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52Vagueness in ContextOxford University Press UK. 2006.Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professional basketball players. The main feature of S…Read more
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54Review of Michael P. Lynch, Truth as One and Many (review)Notre Dame Philosophical Reviews 2009 (9). 2009.
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228Higher-Order Logic or Set Theory: A False DilemmaPhilosophia Mathematica 20 (3): 305-323. 2012.The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?
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224Philosophy of mathematics: structure and ontologyOxford University Press. 1997.Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests re…Read more
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182So truth is safe from paradox: now what?Philosophical Studies 147 (3): 445-455. 2010.The article is part of a symposium on Hartry Field’s “Saving truth from paradox”. The book is one of the most significant intellectual achievements of the past decades, but it is not clear what, exactly, it accomplishes. I explore some alternatives, relating the developed view to the intuitive, pre-theoretic notion of truth.
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16Do Not Claim Too Much: Second-order Logic and First-order LogicPhilosophia Mathematica 6 (3): 42-64. 1998.The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.
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15Essay ReviewHistory and Philosophy of Logic 6 (1): 215-221. 1985.D. GABBAY and F. GUENTHNER (eds.), Handbook of philosophical logic. Volume 1: Elements of classical logic. Dordrecht, Boston, and Lancaster: D. Reidel Publishing Company, 1983. xiv + 497 pp. Dfl225/$98.00
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6Classical logic II: Higher-order logicIn Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell. pp. 33--54. 2001.A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐…Read more
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111Why anti-realists and classical mathematicians cannot get alongTopoi 20 (1): 53-63. 2001.Famously, Michael Dummett argues that considerations concerning the role of language in communication lead to the rejection of classical logic in favor of intuitionistic logic. Potentially, this results in massive revisions of established mathematics. Recently, Neil Tennant (“The law of excluded middle is synthetic a priori, if valid”, Philosophical Topics 24 (1996), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle groun…Read more
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2Vagueness and ConversationIn J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox, Clarendon Press. 2004.
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279New V, ZF and AbstractionPhilosophia Mathematica 7 (3): 293-321. 1999.We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New …Read more
Columbus, Ohio, United States of America
Areas of Specialization
Philosophy of Language |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Areas of Interest
Philosophy of Language |
Logic and Philosophy of Logic |
Philosophy of Mathematics |