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50Open Texture and MathematicsNotre Dame Journal of Formal Logic 62 (1): 173-191. 2021.The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.
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49Ineffability within the limits of abstraction aloneIn Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism: Essays in Philosophy of Mathematics, Oxford University Press Uk. 2016.The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that our abstraction based theory can be mod…Read more
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46Understanding the InfinitePhilosophical Review 105 (2): 256. 1996.Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.
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46Groups, sets, and paradoxLinguistics and Philosophy 45 (6): 1277-1313. 2022.Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups rese…Read more
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44Group nouns and pseudo‐singularityThought: A Journal of Philosophy 10 (1): 66-77. 2021.Thought: A Journal of Philosophy, EarlyView.
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43On the notion of effectivenessHistory and Philosophy of Logic 1 (1-2): 209-230. 1980.This paper focuses on two notions of effectiveness which are not treated in detail elsewhere. Unlike the standard computability notion, which is a property of functions themselves, both notions of effectiveness are properties of interpreted linguistic presentations of functions. It is shown that effectiveness is epistemically at least as basic as computability in the sense that decisions about computability normally involve judgments concerning effectiveness. There are many occurrences of the pr…Read more
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41Stephen C. Kleene. Origins of recursive function theory. Annals of the history of computing, vol. 3 , pp. 52– 67. - Martin Davis. Why Gödel didn't have Church's thesis. Information and control, vol. 54 , pp. 3– 24. - Stephen C. Kleene. Reflections on Church's thesis. Notre Dame journal of formal logic, vol. 28 , pp. 490– 498 (review)Journal of Symbolic Logic 55 (1): 348-350. 1990.
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40Robert Lorne Victor Hale FRSE May 4, 1945 – December 12, 2017Philosophia Mathematica 26 (2): 266-274. 2018.
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40The Classical Continuum without Points – CORRIGENDUMReview of Symbolic Logic 6 (3): 571-571. 2013.
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39Computability, Notation, and de re Knowledge of NumbersPhilosophies 7 (1): 20. 2022.Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of _which number_ is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of _notation_. The purpose of this article is to explore the relationship b…Read more
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39Deflation and conservationIn Volker Halbach & Leon Horsten (eds.), Principles of Truth, Dr. Hänsel-hohenhausen. pp. 103-128. 2002.
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37Comparing implicit and explicit memory for brand names from advertisementsJournal of Experimental Psychology: Applied 2 (2): 147. 1996.
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36Priest, Graham. An Introduction to Non-classical Logic (review)Review of Metaphysics 56 (3): 670-672. 2003.
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34Mechanism, Mentalism and Metamathematics: An Essay on FinitismJournal of Symbolic Logic 51 (2): 472. 1980.
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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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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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32The History of Continua: Philosophical and Mathematical Perspectives (edited book)Oxford University Press. 2020.Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.
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31Inconsistency and Incompleteness, RevisitedIn Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer Verlag. pp. 469-479. 2019.Graham Priest introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might fo…Read more
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31John CorcoranHistory and Philosophy of Logic 42 (3): 201-223. 2021.We present a memorial summary of the professional life and contributions to logic of John Corcoran. We also provide a full list of his many publications.Courtesy of Lynn Corcoran.
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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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28The governance of identityIn Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 164--173. 2006.
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27I—Stewart ShapiroSupplement to the Proceedings of the Aristotelian Society 79 (1): 147-165. 2005.
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27Ontology via semantics? Introduction to the special issue on the semantics of cardinalsLinguistics and Philosophy 40 (4): 321-329. 2017.As introduction to the special issue on the semantics of cardinals, we offer some background on the relevant literature, and an overview of the contributions to this volume. Most of these papers were presented in earlier form at an interdisciplinary workshop on the topic at The Ohio State University, and the contributions to this issue reflect that interdisciplinary character: the authors represent both fields in the title of this journal.
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27Expressive completeness and decidabilityNotre Dame Journal of Formal Logic 31 (4): 576-579. 1990.
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27Introduction to Special Issue: The Emergence of StructuralismPhilosophia Mathematica 27 (3): 299-302. 2019.
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25Unwarranted philosophical assumptions in research on ANSBehavioral and Brain Sciences 44. 2021.Clarke and Beck import certain assumptions about the nature of numbers. Although these are widespread within research on number cognition, they are highly contentious among philosophers of mathematics. In this commentary, we isolate and critically evaluate one core assumption: the identity thesis.
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25Consumer memory for intentions: A prospective memory perspectiveJournal of Experimental Psychology: Applied 5 (2): 169. 1999.
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22Mereological Singularism and ParadoxErkenntnis 88 (1): 1-20. 2021.The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting thr…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 |