
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

512Pluralities and SetsJournal of Philosophy 107 (3): 144164. 2010.Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?

496Identity and discernibility in philosophy and logicReview of Symbolic Logic 5 (1): 162186. 2012.Questions about the relation between identity and discernibility are important both in philosophy and in model theory. We show how a philosophical question about identity and dis cernibility can be ‘factorized’ into a philosophical question about the adequacy of a formal language to the description of the world, and a mathematical question about discernibility in this language. We provide formal definitions of various notions of discernibility and offer a complete classification of their logica…Read more

494Category theory as an autonomous foundationPhilosophia Mathematica 19 (3): 227254. 2011.Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in …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

284Actual and Potential InfinityNoûs 53 (1): 160191. 2019.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.

280Platonism in the Philosophy of MathematicsIn Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, . 2009.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.

274Structuralism and the notion of dependencePhilosophical Quarterly 58 (230): 5979. 2008.This paper has two goals. The ﬁrst goal is to show that the structuralists’ claims about dependence are more signiﬁcant to their view than is generally recognized. I argue that these dependence claims play an essential role in the most interesting and plausible characterization of this brand of structuralism. The second goal is to defend a compromise view concerning the dependence relations that obtain between mathematical objects. Two extreme views have tended to dominate the debate, namely the…Read more

253Metaontological MinimalismPhilosophy Compass 7 (2): 139151. 2012.Can there be objects that are ‘thin’ in the sense that very little is required for their existence? A number of philosophers have thought so. For instance, many Fregeans believe it suffices for the existence of directions that there be lines standing in the relation of parallelism; other philosophers believe it suffices for a mathematical theory to have a model that the theory be coherent. This article explains the appeal of thin objects, discusses the three most important strategies for articul…Read more

249Plural quantification exposedNoûs 37 (1). 2003.This paper criticizes George Boolos's famous use of plural quantification to argue that monadic secondorder 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.

248The potential hierarchy of setsReview of Symbolic Logic 6 (2): 205228. 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 equiinterpretable with Zermelo Fraenkel set theory but sheds new light on the settheoretic paradoxes and the foundations of set theory.

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’

244Superplurals 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

241Bad company tamedSynthese 170 (3). 2009.The neoFregean 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 wellfounded 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

223On 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 firstorder quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of nonstandard interpretations that confronts higherorder logics on their more traditional, setbased semantics. We challenge both claims. Our challenge is based on a Henkin…Read more

220What is the infinite?The Philosophers' Magazine 61 (61): 4247. 2013.This is an accessible introduction to the concept of infinity, its historical evolution, and mathematical and philosophical analysis.

209Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory (review)Philosophy 87 (1): 133137. 2012.

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

205Frege's conception of logic: From Kant to grundgesetzeManuscrito 26 (2): 235252. 2003.I shall make two main claims. My first main claim is that Frege started out with a view of logic that is closer to Kant’s than is generally recognized, but that he gradually came to reject this Kantian view, or at least totally to transform it. My second main claim concerns Frege’s reasons for distancing himself from the Kantian conception of logic. It is natural to speculate that this change in Frege’s view of logic may have been spurred by a desire to establish the logicality of the axiom syst…Read more

198Plural quantificationStanford Encyclopedia of Philosophy. 2008.Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it has been argued that we have good reason to ad…Read more

196Predicative fragments of Frege arithmeticBulletin of Symbolic Logic 10 (2): 153174. 2004.Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural deﬁnitions imply all of secondorder Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying secondorder logic—a…Read more

192On WitnessDiscernibility of Elementary ParticlesErkenntnis 78 (5): 11331142. 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 witnessdiscernibility. We inquire into how witnessdiscernibility relates to known kinds of discernibility. Our conclusion will be that for a wide variety of cases, including the intended quantummechanical ones, witnessdiscernibility collapses exten…Read more

187Platonism 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

186Term Models for Abstraction PrinciplesJournal of Philosophical Logic 45 (1): 123. 2016.Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the metho…Read more

184IntroductionSynthese 170 (3): 321329. 2009.NeoFregean 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.

183Burgess on plural logic and set theoryPhilosophia Mathematica 15 (1): 7993. 2007.John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail. I am grateful to Philip Welch, two anonymous referees, and especially Ignacio Jané for written comments on earlier versions of this paper, which have led to substantial improvements. Thanks also to the partici…Read more

179Aristotelian ContinuaPhilosophia Mathematica 24 (2): 214246. 2016.In previous work, Hellman and Shapiro present a regionsbased account of a onedimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, and we show th…Read more

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

174Two types of abstraction for structuralismPhilosophical Quarterly 64 (255): 267283. 2014.If numbers were identified with any of their standard settheoretic realizations, then they would have various nonarithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’…Read more

174Some Criteria for Acceptable AbstractionNotre Dame Journal of Formal Logic 52 (3): 331338. 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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