My main research interests are in the philosophies of logic and mathematics, metaphysics and the philosophy of language.
I am particularly interested in questions concerning ontology, individuation, essence, reference (especially to abstract objects),
necessity and knowledge of necessary truths.
My approach to these questions is broadly Fregean in that I relate them to questions in philosophical logic and the
philosophy of language.
Edited volumes

The bad company problem, special issue of Synthese, 170:3 (2009)
A special issue dedicated to the 'bad company problem' in abstractionist
accounts of mathematics, with contribution by Roy Cook, Philip Ebert and Stewart Shapiro, Matti Eklund,
Bob Hale and Crispin Wright, Øystein Linnebo, John MacFarlane, and Gabriel Uzquiano.

New Waves in Philosophy of Mathematics (coedited with Otávio Bueno), forthcoming,
Palgrave Macmillan, 2009.
Articles

'Identity and Discernibility in Philosophy and Logic' (with James Ladyman and Richard Pettigrew), forthcoming in Review of Symbolic Logic.
[ Preprint].

'Hierarchies Ontological and Ideological' (with Agustín Rayo), forthcoming in Mind.
[ Preprint].

'Metaontological Minimalism', forthcoming in Philosophy Compass.

'Category Theory as an Autonomous Foundation' (with Richard Pettigrew), Philosophia Mathematica 19(3): 2011, 227254.
[ Preprint] [ Journal link].

'Some Criteria for Acceptable Abstraction', Notre Dame Journal of Formal Logic 52(3): 2011, 331338.
[ Preprint] [ Journal link].

'Pluralities and Sets', Journal of Philosophy 107(3): 2010, 144164.
[ Preprint] [ Journal link].
Under what conditions do some things form a set? This paper investigates some arguments that can be given for and against the simple answer that, whenever there are some things, they form a set. I propose a way in which the simple answer can be accepted without paradox by means of a modal explication of the idea that sets are formed in stages.

'Which Abstraction Principles Are Acceptable? Some Limitative Results' (with Gabriel Uzquiano), British Journal for the Philosophy of Science 60(2): 2009, 239253.
[ Preprint] [ Journal link]
We argue that stabilitywhich is one of the more promising answers to the question in the title of our paperfails to provide a sufficient condition for an abstraction principle to be acceptable.

'The Individuation of the Natural Numbers', in New Waves in Philosophy of Mathematics, O. Bueno and Ø. Linnebo (eds.) (Palgrave, 2009).
[ Preprint].


'Bad Company Tamed', Synthese 170(3): 2009, 371391.
[ Preprint] [ Journal link]
One of the most serious problems facing neoFregean approaches to mathematics is
the 'bad company problem'. I develop a new solution to the problem based on the idea that
the individuation of entities must be wellfounded.

'Frege's Context Principle and Reference to Natural Numbers',
in Logicism, Intuitionism, and Formalism: What has become of them?,
eds. S. Lindström et al. (Springer, 2009), 4768. [ Preprint]
Frege proposed that his Context
Principlewhich says that a word has meaning only in the context
of a propositioncan 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 to
the resulting account of reference. 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.

'Superplurals in English' (with David Nicolas), Analysis 68(3): 2008.
[ Preprint] [ Journal link]
Are there terms that stand to ordinary plural terms the way ordinary plural terms stand to singular terms? We examine some earlier arguments for the existence of such terms and find these arguments to be either inconclusive or not sufficiently farreaching. Then we present some better examples.

'The Nature of Mathematical Objects',
in Proof and Other Dilemmas: Mathematics and Philosophy,
in B. Gold and R. Simons (eds.) (Washington: Mathematical Association of America, 2008), 205219. [ Preprint]
I first analyze Frege's argument for mathematical platonism and discuss two challenges to it,
having to do with "epistemic access" and ontological extravagance. To make progress, I propose that we ask what is required
for mathematical singular terms to have semantic values. This leads to an account of mathematical objects as very "thin."
I argue that this account allows us to respond to the two challenges to mathematical platonism.

'Structuralism and the Notion of Dependence', Philosophical Quarterly 58: 2008, 5979.
[ Preprint] [ Journal link]
I argue that dependence claims are more important to mathematical structuralism than is generally recognized. Then I defend a compromise view concerning the dependence relations between mathematical objects, according to which structuralists are right about some mathematical objects but wrong about others.

'Burgess on Plural Logic and Set Theory',
Philosophia Mathematica 15(1): 2007, 7993. [ Preprint] [ Journal link]
John Burgess's 'E pluribus unum' combines 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.

'Mending the Master' (Critical notice of John P. Burgess, Fixing Frege),
Philosophia Mathematica 14(3): 2006, 338351. [ Preprint] [ Journal link]

'Sets, Properties, and Unrestricted Quantification',
in Absolute Generality,
eds. A. Rayo and G. Uzquiano, Oxford University Press, 2006. [ Preprint]
I point out some problems with the most promising defense to date of the
coherence of unrestricted quantification. Then I attempt to develop a better defense, based on a sharp
distinction between sets and properties.

'Epistemological Challenges to Mathematical Platonism',
Philosophical Studies 129(3): 2006, 545574. [ Preprint] [ Journal link]
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 unduly assimilate
mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism.
I conclude with some suggestions about how this improved challenge may be met.

'To Be Is to Be an F', Dialectica 59(2): 2005, 201222 (special issue on Frege's "Julius Caesar Problem"). [ Preprint] [ Journal link]
I defend the view that our ontology divides into disjoint categories, each with its own
canonical way of identifying and distinguishing the objects it encompasses. I do this by arguing that reference
has a richer structure than normally assumed. I use this structure to give precise definitions of the important
but slippery notions of criterion of identity, sortal concept, and category.

'Frege's Proof of Referentiality', Notre Dame Journal of Formal Logic 45:2 (2004), 7398. [ Preprint] [ Journal link]
I present a novel interpretation of Frege's famous attempt at Grundgesetze I §§2931
to prove that every expression of his language has a unique reference. I argue that Frege's proof is based on a contextual
account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle.
Although Frege's proof is incorrect, I argue that the account of
reference on which it is based is of potential philosophical
value, and I analyze the class of cases to which it may successfully be applied.

Ordinary English contains two sorts of object quantifiers. In addition to the usual
singular quantifiers, as in 'There is an apple on the table', there are plural quantifiers, as in 'There
are some apples on the table'. This article provides a survey of recent discussions of the logic, semantics, and
metaphysics of plural quantification, as well as of its various philosophical applications.

'Predicative Fragments of Frege Arithmetic', Bulletin of Symbolic Logic
10:2 (2004), 153174. [ Preprint] [ Journal link]
Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume's Principle.
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 FAone having to do with Hume's
Principle, the other, with the underlying secondorder logicand investigates how much of Frege's Theorem
goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes
through, the most important exception being the axiom that every natural number has
a successor. Theorem 2 shows that this axiom cannot be proved in the theories that are predicative
in either dimension.

Critical Study of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology,
Philosophia Mathematica, 11:1 (2003), 92104. [ Preprint] [ Journal link]

'Plural Quantification Exposed',
Noûs, 37:1 (2003), 7192. [ Preprint] [ Journal link]
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.

'Frege's Conception of Logic: From Kant to Grundgesetze',
Manuscrito 26:2 (2003), 235252 (special issue on Frege edited by Marco Ruffino). [ Preprint]
I first argue that Frege started out with a conception of logic closer to Kant's than is generally recognized.
Then I analyze Frege's reasons for gradually rejecting this view. Although I concede that the demands imposed by Frege's logicism played
some role, I argue that his increasingly vehement antipsychologism provides a deeper and more interesting reason for rejecting his earlier view.
Reviews and Shorter Pieces

Review of Fraser MacBride (ed.), Identity and Modality,
Mind 117:467 (2008), 705708. [ Preprint]
[ Journal link]

Review of Kit Fine, Modality and Tense,
Philosophical Quarterly 57:227 (2007), 2947. [ Preprint]
[ Journal link]

Review of Kit Fine, The Limits of Abstraction,
Australasian Journal of Philosophy 82:4 (2004), 6536. [ Preprint] [ Journal link]

Introductory Note to Correspondence with Walter Pitts (with Charles Parsons),
in S. Feferman et al. (eds.), Kurt Gödel: Collected Works, Vol. V (Oxford:
Clarendon, 2003).

Review of Michael Potter, Reason's Nearest Kin, Mind, 110:439 (2001), 81013. [ Journal link]

Review of William Tait (ed.), Early Analytic Philosophy, Philosophical Review,
109:1 (2000), 98101. [ Journal link]
Work in Progress

'Ontology and the Concept of an Object' [Draft of January 2003.
Likely to be superseded by other work.]
Philosophers often reject certain classes of objects as problematic while
simultaneously acknowledging we have to go on speaking as if such objects existed; for instance,
they reject numbers but keep doing arithmetic. I argue against such views and propose a novel
account of objecthood which allows the objects in question to be so "thin" as to avoid the alleged problems.
I illustrate this view by sketching an account of pure mathematical objects.
