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1Vagueness, Metaphysics, and ObjectivityIn Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and Clouds: Vaguenesss, its Nature and its Logic, Oxford University Press. 2010.
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33Life on the Ship of NeurathCroatian Journal of Philosophy 9 (2): 149-166. 2009.Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics.
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140Structure and identityIn Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 34--69. 2006.According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of crit…Read more
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29The Work of John Corcoran: An AppreciationHistory and Philosophy of Logic 20 (3-4): 149-158. 1999.
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162Incompleteness, mechanism, and optimismBulletin of Symbolic Logic 4 (3): 273-302. 1998.§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” wou…Read more
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7Review of Michael D. Resnik: Mathematics as a Science of Patterns_; Stewart Shapiro: _Philosophy of Mathematics: Structure and Ontology (review)British Journal for the Philosophy of Science 49 (4): 652-656. 1998.
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145An “I” for an I: Singular terms, uniqueness, and referenceReview of Symbolic Logic 5 (3): 380-415. 2012.There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in lingu…Read more
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28The governance of identityIn Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 164--173. 2006.
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123Introduction to special issue: Abstraction and Neo-LogicismPhilosophia Mathematica 8 (2): 97-99. 2000.
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34Philosophy of MathematicsIn Peter Clark & Katherine Hawley (eds.), Philosophy of Science Today, Oxford University Press Uk. 2003.Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle
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229The classical continuum without pointsReview of Symbolic Logic 6 (3): 488-512. 2013.We develop a point-free construction of the classical one- dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually ex…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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176Foundations of Mathematics: Metaphysics, Epistemology, StructurePhilosophical Quarterly 54 (214). 2004.Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to prov…Read more
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155Mechanism, 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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243Where 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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24Consumer memory for intentions: A prospective memory perspectiveJournal of Experimental Psychology: Applied 5 (2): 169. 1999.
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329We 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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106Set-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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19Vagueness and LogicIn Giuseppina Ronzitti (ed.), Vagueness: A Guide, Springer Verlag. pp. 55--81. 2011.
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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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17Review: 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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276The 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.
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341Identity, 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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192Truth, function and paradoxAnalysis 71 (1): 38-44. 2011.Michael Lynch’s Truth as One and Many is a contribution to the large body of philosophical literature on the nature of truth. Within that genre, advocates of truth-as-correspondence, advocates of truth-as-coherence, and the like, all hold that truth has a single underlying metaphysical nature, but they sharply disagree as to what this nature is. Lynch argues that many of these views make good sense of truth attributions for a limited stretch of discourse, but he adds that each of the contenders …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 |