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521Actual and Potential InfinityNoûs 53 (1): 160-191. 2017.The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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173Critical studies/book reviewsPhilosophia Mathematica 11 (1): 92-104. 2003.This is a critical notice of Stewart Shapiro's 1997 book, Philosophy of Mathematics: Structure and Ontology.
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232What is the infinite?The Philosophers' Magazine 61 (61): 42-47. 2013.This is an accessible introduction to the concept of infinity, its historical evolution, and mathematical and philosophical analysis.
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62IntroductionSynthese 170 (3): 321-329. 2009.Neo-Fregean logicism seeks to base mathematics on abstraction principles. But the acceptable abstraction principles are surrounded by unacceptable ones. This is the "bad company problem." In this introduction I first provide a brief historical overview of the problem. Then I outline the main responses that are currently being debated. In the course of doing so I provide summaries of the contributions to this special issue.
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173To be is to be an FDialectica 59 (2). 2005.I defend the view that our ontology divides into categories, each with its own canonical way of identifying and distinguishing the objects it encompasses. For instance, I argue that natural numbers are identified and distinguished by their positions in the number sequence, and physical bodies, by facts having to do with spatiotemporal continuity. I also argue that objects belonging to different categories are ipso facto distinct. My arguments are based on an analysis of reference, which ascribes…Read more
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311Plural quantification exposedNoûs 37 (1). 2003.This paper criticizes George Boolos's famous use of plural quantification to argue that monadic second-order logic is pure logic. I deny that plural quantification qualifies as pure logic and express serious misgivings about its alleged ontological innocence. My argument is based on an examination of what is involved in our understanding of the impredicative plural comprehension schema.
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214On Witness-Discernibility of Elementary ParticlesErkenntnis 78 (5): 1133-1142. 2013.In the context of discussions about the nature of ‘identical particles’ and the status of Leibniz’s Principle of the Identity of Indiscernibles in Quantum Mechanics, a novel kind of physical discernibility has recently been proposed, which we call witness-discernibility. We inquire into how witness-discernibility relates to known kinds of discernibility. Our conclusion will be that for a wide variety of cases, including the intended quantum-mechanical ones, witness-discernibility collapses exten…Read more
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59Frege's context principle and reference to natural numbersIn Sten Lindstr©œm, Erik Palmgren, Krister Segerberg & Viggo Stoltenberg-Hansen (eds.), logicism, intuitionism, and formalism - What has become of them?, Springer. 2008.Frege proposed that his Context Principle—which says that a word has meaning only in the context of a proposition—can be used to explain reference, both in general and to mathematical objects in particular. I develop a version of this proposal and outline answers to some important challenges that the resulting account of reference faces. Then I show how this account can be applied to arithmetic to yield an explanation of our reference to the natural numbers and of their metaphysical status.
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1Against Limitation of SizeThe Baltic International Yearbook of Cognition, Logic and Communication 1. 2005.
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127Thin objectsIn Alexander Hieke & Hannes Leitgeb (eds.), Reduction, Abstraction, Analysis, Ontos. pp. 11--227. 2009.
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4Logic and PluralsIn Kirk Ludwig & Marija Jankovic (eds.), The Routledge Handbook of Collective Intentionality, Routledge. pp. 451-463. 2018.This chapter provides an overview of the philosophical and linguistic debate about the logic of plurals. We present the most prominent singularizing analyses of plurals as well as the main criticisms that such analyses have received. We then introduce an alternative approach to plurals known as plural logic, focusing on the question whether plural logic can count as pure logic.
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195Some Criteria for Acceptable AbstractionNotre Dame Journal of Formal Logic 52 (3): 331-338. 2011.Which abstraction principles are acceptable? A variety of criteria have been proposed, in particular irenicity, stability, conservativeness, and unboundedness. This note charts their logical relations. This answers some open questions and corrects some old answers
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355Platonism in the Philosophy of MathematicsIn Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, The Metaphysics Research Lab. 2014.Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.
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109Mending the master: John P. Burgess, fixing Frege. Princeton, N. J.: Princeton university press, 2005. ISBN 0-691-12231-8. Pp. XII + 257 (review)Philosophia Mathematica 14 (3): 338-351. 2006.Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege's writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well-informed overview provided by the first of its three chapters. Specialists will find the boo…Read more
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262Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory (review)Philosophy 87 (1): 133-137. 2012.
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61What is the infinite?The Philosophers' Magazine 61 42-47. 2013.The paper discusses some different conceptions of the infinity, from Aristotle to Georg Cantor (1845-1918) and beyond. The ancient distinction between actual and potential infinity is explained, along with some arguments against the possibility of actually infinite collections. These arguments were eventually rejected by most philosophers and mathematicians as a result of Cantor’s elegant and successful theory of actually infinite collections.
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180Entanglement and non-factorizabilityStudies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (3): 215-221. 2013.Quantum mechanics tells us that states involving indistinguishable fermions must be antisymmetrized. This is often taken to mean that indistinguishable fermions are always entangled. We consider several notions of entanglement and argue that on the best of them, indistinguishable fermions are not always entangled. We also present a simple but unconventional way of representing fermionic states that allows us to maintain a link between entanglement and non-factorizability.
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144The individuation of the natural numbersIn Ø. Linnebo O. Bueno (ed.), New Waves in Philosophy of Mathematics, Palgrave-macmillan. 2009.It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal part of this water? According to Quine, you must at least …Read more
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223Plurals and modalsCanadian Journal of Philosophy 46 (4-5): 654-676. 2016.Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.
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167Frege's proof of referentialityNotre Dame Journal of Formal Logic 45 (2): 73-98. 2004.I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which it may successf…Read more
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273Bad company tamedSynthese 170 (3). 2009.The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting w…Read more
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374The potential hierarchy of setsReview of Symbolic Logic 6 (2): 205-228. 2013.Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.
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267On the Innocence and Determinacy of Plural QuantificationNoûs 50 (3). 2016.Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-…Read more
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259Superplurals in EnglishAnalysis 68 (3). 2008.where ‘aa’ is a plural term, and ‘F’ a plural predicate. Following George Boolos (1984) and others, many philosophers and logicians also think that plural expressions should be analysed as not introducing any new ontological commitments to some sort of ‘plural entities’, but rather as involving a new form of reference to objects to which we are already committed (for an overview and further details, see Linnebo 2004). For instance, the plural term ‘aa’ refers to Alice, Bob and Charlie simultaneo…Read more
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276Platonism in the Philosophy of MathematicsStanford Encyclopedia of Philosophy. forthcoming.Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathemati…Read more
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79New Model NaturalismMetascience 18 (3): 433-436. 2009.This is a review of John P. Burgess, Mathematics, Models, and Modality: Selected Philosophical Essays.
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147Early Analytic Philosophy (review)Philosophical Review 109 (1): 98-101. 2000.Analytic philosophy has traditionally been little concerned with the history of philosophy, including that of analytic philosophy itself. But in recent years the study of the early period of the analytic tradition has become an active and lively branch of Anglo-American philosophy. Early Analytic Philosophy, a collection of papers presented in honor of professor Leonard Linsky at the University of Chicago in April 1992, is an example of this. The contributors, many of them leading scholars in th…Read more
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