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183The modal logic of set-theoretic potentialism and the potentialist maximality principlesReview of Symbolic Logic 15 (1): 1-35. 2022.We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-…Read more
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72‘Just is’-Statements as Generalized IdentitiesInquiry: An Interdisciplinary Journal of Philosophy 57 (4): 466-482. 2014.Identity is ordinarily taken to be a relation defined on all and only objects. This consensus is challenged by Agustín Rayo, who seeks to develop an analogue of the identity sign that can be flanked by sentences. This paper is a critical exploration of the attempted generalization. First the desired generalization is clarified and analyzed. Then it is argued that there is no notion of content that does the desired philosophical job, namely ensure that necessarily equivalent sentences coincide in…Read more
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Putnam on Mathematics as Modal LogicIn John Burgess (ed.), Hilary Putnam on Logic and Mathematics, Springer Verlag. 2018.
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111Thin Objects: An Abstractionist AccountOxford University Press. 2018.Are there objects that are “thin” in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. This book attempts to develop the needed explanations by drawing on some F…Read more
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123Dummett on Indefinite ExtensibilityPhilosophical Issues 28 (1): 196-220. 2018.Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.
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78III-Reference by AbstractionProceedings of the Aristotelian Society 112 (1pt1): 45-71. 2012.Frege suggests that criteria of identity should play a central role in the explanation of reference, especially to abstract objects. This paper develops a precise model of how we can come to refer to a particular kind of abstract object, namely, abstract letter types. It is argued that the resulting abstract referents are ‘metaphysically lightweight’
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200Reply to Florio and ShapiroMind 123 (489): 175-181. 2014.Florio and Shapiro take issue with an argument in ‘Hierarchies Ontological and Ideological’ for the conclusion that the set-theoretic hierarchy is open-ended. Here we clarify and reinforce the argument in light of their concerns.
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253Epistemological Challenges to Mathematical PlatonismPhilosophical Studies 129 (3): 545-574. 2006.Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rathe…Read more
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415Hierarchies Ontological and IdeologicalMind 121 (482). 2012.Gödel claimed that Zermelo-Fraenkel set theory is 'what becomes of the theory of types if certain superfluous restrictions are removed'. The aim of this paper is to develop a clearer understanding of Gödel's remark, and of the surrounding philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages
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52New waves in philosophy of mathematics (edited book)Palgrave-Macmillan. 2009.Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration fr…Read more
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20To Be Is to Be an FDialectica 59 (2): 201-222. 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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58Philosophy of MathematicsPrinceton University Press. 2017.Mathematics is one of the most successful human endeavors—a paradigm of precision and objectivity. It is also one of our most puzzling endeavors, as it seems to deliver non-experiential knowledge of a non-physical reality consisting of numbers, sets, and functions. How can the success and objectivity of mathematics be reconciled with its puzzling features, which seem to set it apart from all the usual empirical sciences? This book offers a short but systematic introduction to the philosophy of m…Read more
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21Gottlob Frege: Utvalgte teksterNorsk Filosofisk Tidsskrift 52 (4): 187-192. 2017.This is a review (in Norwegian) of the first major translation of the works of Gottlob Frege into Norwegian.
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6Chapter Twelve. The Quest for New AxiomsIn Philosophy of Mathematics, Princeton University Press. pp. 170-182. 2017.
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11Chapter Three. Formalism and DeductivismIn Philosophy of Mathematics, Princeton University Press. pp. 38-55. 2017.
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2Chapter Two. Frege’s LogicismIn Philosophy of Mathematics, Princeton University Press. pp. 21-37. 2017.
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8Chapter Six. Empiricism about MathematicsIn Philosophy of Mathematics, Princeton University Press. pp. 88-100. 2017.
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11Chapter Ten. The Iterative Conception of SetsIn Philosophy of Mathematics, Princeton University Press. pp. 139-153. 2017.
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10Chapter Seven. NominalismIn Philosophy of Mathematics, Princeton University Press. pp. 101-115. 2017.
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3Chapter Eleven. StructuralismIn Philosophy of Mathematics, Princeton University Press. pp. 154-169. 2017.
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6Chapter One. Mathematics as a Philosophical ChallengeIn Philosophy of Mathematics, Princeton University Press. pp. 4-20. 2017.
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5Chapter Five. IntuitionismIn Philosophy of Mathematics, Princeton University Press. pp. 73-87. 2017.
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14Chapter Eight. Mathematical IntuitionIn Philosophy of Mathematics, Princeton University Press. pp. 116-125. 2017.
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4Chapter Nine. Abstraction ReconsideredIn Philosophy of Mathematics, Princeton University Press. pp. 126-138. 2017.
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15Chapter Four. Hilbert’s ProgramIn Philosophy of Mathematics, Princeton University Press. pp. 56-72. 2017.
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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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Gottlob Frege |
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