• Freges oppfatning av logikk: fra Kant til Grundgesetze
    Norsk Filosofisk Tidsskrift 48 (3-04): 219-228. 2013.
  •  206
    Which abstraction principles are acceptable? Some limitative results
    British Journal for the Philosophy of Science 60 (2): 239-252. 2009.
    Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans 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
  •  39
    Riemann’s Scale: A Puzzle About Infinity
    Erkenntnis 1-3. 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
  •  19
    Peacocke on magnitudes and numbers
    Philosophical 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
  •  28
    Critical Plural Logic
    Philosophia Mathematica 28 (2): 172-203. 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.
  •  15
    Modality and Tense: Philosophical Papers
    Philosophical Quarterly 57 (227): 294-297. 2007.
  •  178
    ‘Just is’-Statements as Generalized Identities
    Inquiry: 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
  •  44
    Thin Objects
    Oxford 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
  •  67
    Dummett on indefinite extensibility
    Philosophical 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.
  •  247
    III-Reference by Abstraction
    Proceedings 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’
  •  165
    Reply to Florio and Shapiro
    Mind 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.
  •  594
    Epistemological Challenges to Mathematical Platonism
    Philosophical 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
  •  293
    Hierarchies Ontological and Ideological
    with Agustin Rayo
    Mind 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
  •  4
    New 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
  •  9
    To Be Is to Be an F
    Dialectica 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
  •  2
    Philosophy of Mathematics
    Princeton 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
  •  2
    In Philosophy of Mathematics, Princeton University Press. pp. 199-204. 2017.
  •  2
    In Philosophy of Mathematics, Princeton University Press. pp. 1-3. 2017.
  •  10
    Gottlob Frege: Utvalgte tekster
    Norsk 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.
  •  1
    In Philosophy of Mathematics, Princeton University Press. 2017.
  •  4
    Chapter Six. Empiricism about Mathematics
    In Philosophy of Mathematics, Princeton University Press. pp. 88-100. 2017.
  •  6
    Chapter Ten. The Iterative Conception of Sets
    In Philosophy of Mathematics, Princeton University Press. pp. 139-153. 2017.
  •  7
    Chapter Seven. Nominalism
    In Philosophy of Mathematics, Princeton University Press. pp. 101-115. 2017.
  •  3
    Chapter Twelve. The Quest for New Axioms
    In Philosophy of Mathematics, Princeton University Press. pp. 170-182. 2017.
  •  6
    Chapter Three. Formalism and Deductivism
    In Philosophy of Mathematics, Princeton University Press. pp. 38-55. 2017.
  • Chapter Two. Frege’s Logicism
    In Philosophy of Mathematics, Princeton University Press. pp. 21-37. 2017.
  •  7
    Chapter Eight. Mathematical Intuition
    In Philosophy of Mathematics, Princeton University Press. pp. 116-125. 2017.