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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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77Predicativism as a Form of PotentialismReview of Symbolic Logic 16 (1): 1-32. 2023.In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.
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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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59Frege's context principle and reference to natural numbersIn Sten Lindström (ed.), Logicism, Intuitionism, and Formalism: What Has Become of Them, Springer. 2009.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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55Early analytic philosophy: Frege, Russell, Wittgenstein (review)Philosophical Review 109 (1): 98-101. 2000.
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54Potential Infinity and De Re Knowledge of Mathematical ObjectsIn Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner, Springer. pp. 79-98. 2023.Our first goal here is to show how one can use a modal language to explicate potentiality and incomplete or indeterminate domains in mathematics, along the lines of previous work. We then show how potentiality bears on some longstanding items of concern to Mark Steiner: the applicability of mathematics, explanation, and de re propositional attitudes toward mathematical objects.
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51New 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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47Sets, properties, and unrestricted quantificationIn Gabriel Uzquiano & Agustin Rayo (eds.), Absolute Generality, Oxford University Press. pp. 149--178. 2006.Call a quantifier unrestricted if it ranges over absolutely all things: not just over all physical things or all things relevant to some particular utterance or discourse but over absolutely everything there is. Prima facie, unrestricted quantification seems to be perfectly coherent. For such quantification appears to be involved in a variety of claims that all normal human beings are capable of understanding. For instance, some basic logical and mathematical truths appear to involve unrestricte…Read more
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47Philosophy 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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46Peacocke on magnitudes and numbersPhilosophical Studies 178 (8): 2717-2729. 2020.Peacocke’s recent The Primacy of Metaphysics covers a wide range of topics. This critical discussion focuses on the book’s novel account of extensive magnitudes and numbers. First, I further develop and defend Peacocke’s argument against nominalistic approaches to magnitudes and numbers. Then, I argue that his view is more Aristotelian than Platonist because reified magnitudes and numbers are accounted for via corresponding properties and these properties’ application conditions, and because the…Read more
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43RepliesTheoria 89 (3): 393-406. 2023.Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes.
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42Eklund, Maximalism, and the Problem of Incompatible Objects.
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39Early 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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35No Easy Road to Impredicative DefinabilismPhilosophia Mathematica 32 (1): 21-33. 2024.Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and p…Read more
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33Bob Hale. Essence and Existence: Selected EssaysPhilosophia Mathematica 29 (3): 420-427. 2021.Essence and Existence: Selected Essays brings together fifteen essays by Bob Hale, mostly written between the publication of his last book, Necessary Beings, in.
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26Introduction to special issue on ‘critical views of logic’Inquiry: An Interdisciplinary Journal of Philosophy 65 (6): 631-637. 2022.Critical views of logic are presented. These are views that are critical of logic in a sense akin to the way in which Kant is critical rather than dogmatic about traditional metaphysics. Such approaches differ from the Fregean ‘logic-first’ view. In accordance with the latter, logic is often regarded as epistemologically and methodologically fundamental. Hence, all disciplines – including mathematics – are considered as answerable to logic, rather than vice versa. In critical views of logic, by …Read more
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25PrécisTheoria 89 (3): 247-255. 2023.Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes.
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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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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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18Freges oppfatning av logikk: fra Kant til GrundgesetzeNorsk Filosofisk Tidsskrift 48 (3-4): 219-228. 2013.I first argue that Frege started out with a conception of logic that is closer to Kant’s than is generally recognized, after which I analyze Frege’s reasons for gradually rejecting this view. Although conceding that the demands posed by Frege’s logicism played some role, I argue that his increasingly vehement anti-psychologism provides a deeper and more interesting reason for rejecting his earlier view.
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1726 Potential Infinity, Paradox, and the Mind of God: Historical SurveyIn Mirosław Szatkowski (ed.), Ontology of Divinity, De Gruyter. pp. 531-560. 2024.
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15Chapter Four. Hilbert’s ProgramIn Philosophy of Mathematics, Princeton University Press. pp. 56-72. 2017.
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14Chapter Eight. Mathematical IntuitionIn Philosophy of Mathematics, Princeton University Press. pp. 116-125. 2017.
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11Chapter Three. Formalism and DeductivismIn Philosophy of Mathematics, Princeton University Press. pp. 38-55. 2017.
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10Chapter 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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8Chapter Six. Empiricism about MathematicsIn Philosophy of Mathematics, Princeton University Press. pp. 88-100. 2017.
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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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6Chapter One. Mathematics as a Philosophical ChallengeIn Philosophy of Mathematics, Princeton University Press. pp. 4-20. 2017.
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Science, Logic, and Mathematics |
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