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241Categories, 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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62Vagueness 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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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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54Review of Michael P. Lynch, Truth as One and Many (review)Notre Dame Philosophical Reviews 2009 (9). 2009.
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233Higher-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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226Philosophy 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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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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183So 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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17Do 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.
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 |