
Freges oppfatning av logikk: fra Kant til GrundgesetzeNorsk Filosofisk Tidsskrift 48 (304): 219228. 2013.

206Which abstraction principles are acceptable? Some limitative resultsBritish Journal for the Philosophy of Science 60 (2): 239252. 2009.NeoFregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neoFregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these …Read more

39Riemann’s Scale: A Puzzle About InfinityErkenntnis 13. forthcoming.Ordinarily, the order in which some objects are attached to a scale does not affect the total weight measured by the scale. This principle is shown to fail in certain cases involving infinitely many objects. In these cases, we can produce any desired reading of the scale merely by changing the order in which a fixed collection of objects are attached to the scale. This puzzling phenomenon brings out the metaphysical significance of a theorem about infinite series that is well known by mathematic…Read more

19Peacocke on magnitudes and numbersPhilosophical Studies. forthcoming.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

39Early analytic philosophy: Frege, Russell, Wittgenstein (review)Philosophical Review 109 (1): 98101. 2000.

28Critical Plural LogicPhilosophia Mathematica 28 (2): 172203. 2020.What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.

69The modal logic of settheoretic potentialism and the potentialist maximality principlesReview of Symbolic Logic 136. forthcoming.

178‘Just is’Statements as Generalized IdentitiesInquiry: An Interdisciplinary Journal of Philosophy 57 (4): 466482. 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

44Thin ObjectsOxford 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

67Dummett on indefinite extensibilityPhilosophical Issues 28 (1): 196220. 2018.Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and antirealism, distinct from his more famous, meaningtheoretic 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.

247IIIReference by AbstractionProceedings of the Aristotelian Society 112 (1pt1): 4571. 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’

165Reply to Florio and ShapiroMind 123 (489): 175181. 2014.Florio and Shapiro take issue with an argument in ‘Hierarchies Ontological and Ideological’ for the conclusion that the settheoretic hierarchy is openended. Here we clarify and reinforce the argument in light of their concerns.

594Epistemological Challenges to Mathematical PlatonismPhilosophical Studies 129 (3): 545574. 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 truthpredicate rathe…Read more

293Hierarchies Ontological and IdeologicalMind 121 (482). 2012.Gödel claimed that ZermeloFraenkel 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 higherorder languages in other higherorder languages

4New Waves in Philosophy of Mathematics (edited book)PalgraveMacmillan. 2009.Thirteen upandcoming 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

9To Be Is to Be an FDialectica 59 (2): 201222. 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

2Philosophy 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 nonexperiential knowledge of a nonphysical 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

10Gottlob Frege: Utvalgte teksterNorsk Filosofisk Tidsskrift 52 (4): 187192. 2017.This is a review (in Norwegian) of the first major translation of the works of Gottlob Frege into Norwegian.

4Chapter Six. Empiricism about MathematicsIn Philosophy of Mathematics, Princeton University Press. pp. 88100. 2017.

6Chapter Ten. The Iterative Conception of SetsIn Philosophy of Mathematics, Princeton University Press. pp. 139153. 2017.

7Chapter Seven. NominalismIn Philosophy of Mathematics, Princeton University Press. pp. 101115. 2017.

3Chapter Twelve. The Quest for New AxiomsIn Philosophy of Mathematics, Princeton University Press. pp. 170182. 2017.

6Chapter Three. Formalism and DeductivismIn Philosophy of Mathematics, Princeton University Press. pp. 3855. 2017.

Chapter Two. Frege’s LogicismIn Philosophy of Mathematics, Princeton University Press. pp. 2137. 2017.

7Chapter Eight. Mathematical IntuitionIn Philosophy of Mathematics, Princeton University Press. pp. 116125. 2017.
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