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162Mechanism, truth, and Penrose's new argumentJournal of Philosophical Logic 32 (1): 19-42. 2003.Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). I…Read more
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250Where in the (world wide) web of belief is the law of non-contradiction?Noûs 41 (2). 2007.It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analytic-synthetic distinction. This logic-friendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so…Read more
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58EffectivenessIn Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 37--49. 2006.
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331We hold these truths to be self-evident: But what do we mean by that?: We hold these truths to be self-evidentReview of Symbolic Logic 2 (1): 175-207. 2009.At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are u…Read more
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Mathematics and ObjectivityIn John Polkinghorne (ed.), Meaning in mathematics, Oxford University Press. 2011.
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107Set-Theoretic FoundationsThe Proceedings of the Twentieth World Congress of Philosophy 6 183-196. 2000.Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematica…Read more
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26Consumer memory for intentions: A prospective memory perspectiveJournal of Experimental Psychology: Applied 5 (2): 169. 1999.
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202Logical consequence: Models and modalityIn Matthias Schirn (ed.), The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993, Clarendon Press. pp. 131--156. 1998.
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18Review: Wilfried Sieg, Step by Recursive Step: Church's Analysis of Effective Calculability (review)Journal of Symbolic Logic 64 (1): 398-399. 1999.
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1Burali-Forti's revengeIn J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox, Oxford University Press. 2007.
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21Vagueness and LogicIn Giuseppina Ronzitti (ed.), Vagueness: A Guide, Springer Verlag. pp. 55--81. 2011.
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346Identity, indiscernibility, and Ante Rem structuralism: The tale of I and –IPhilosophia Mathematica 16 (3): 285-309. 2008.Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one o…Read more
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88Reasoning, logic and computationPhilosophia Mathematica 3 (1): 31-51. 1995.The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism…Read more
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277The Objectivity of MathematicsSynthese 156 (2): 337-381. 2007.The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.
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 |