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66Review of T. Franzen, Godel's theorem: An incomplete guide to its use and abuse (review)Philosophia Mathematica 14 (2): 262-264. 2006.This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience with the rigor of mathemat…Read more
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34Mechanism, Mentalism and Metamathematics: An Essay on FinitismJournal of Symbolic Logic 51 (2): 472. 1980.
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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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34Life 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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1""Bertrand Russell," On Denoting"(1905) and" Mathematical Logic as Based on the Theory of Types"(1908)In Jorge J. E. Gracia, Gregory M. Reichberg & Bernard N. Schumacher (eds.), The Classics of Western Philosophy: A Reader's Guide, Blackwell. pp. 460. 2003.
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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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153An “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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125Introduction 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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239The 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
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 |