Philosophy of Mathematics (i) — general issues

Forthcoming. ‘On the Alleged Free Acceptability of Reflection Principles for Peano Arithmetic’. In an anthology on reflection principles, edited by Henry Galinon, Nikolaj Pederson, Daniel Waxman, and Matteo Zicchetti. Routledge.

2025. Review of Peter van Inwagen and William Lane Craig’s Do Numbers Exist? A Debate about Abstract Objects. Notre Dame Philosophical Reviews. [url]

2025. ‘Reflections on Wittgenstein on the Normativity of Arithmetic’. In New Essays on Normative Realism, edited by Paul Boghossian and Christopher Peacocke. Oxford University Press. [.pdf]

2008. Contribution to Philosophy of Mathematics: 5 Questions, edited by Vincent Hendriks and Hannes Leitgeb. Automatic Press. [.pdf]

(with Shapiro, Stewart). 2006. ‘All Things Indefinitely Extensible’. In Absolute Generality, edited by Agustin Rayo and Gabriel Uzquiano. Oxford University Press. [.pdf]

1995. ‘Intuitionists are not (Turing) Machines’. Philosophia Mathematica, 3 (3): 86-102. [.pdf]

1994. ‘About “The Philosophical Significance of Gödel’s Theorem”: Some Issues’. In The Philosophy of Michael Dummett, edited by Brian McGuinness and Gianluigi Oliveri. Kluwer Academic Publishers. [.pdf]

(with Hale, Bob). 1994. ‘A Reductio ad Surdum? Field on the Contingency of Mathematical Objects’. Mind 103: 169-84. [.pdf]

(with Hale, Bob). 1992. ‘Nominalism and the Contingency of Abstract Objects’. Journal of Philosophy 89: 111-35. [.pdf]

1991. ‘Wittgenstein on Mathematical Proof’. Royal Institute of Philosophy Supplement 28: 79-99. [.pdf]

1988. ‘Why Numbers Can Believably Be: A Reply to Hartry Field’. Special number on Philosophy of Mathematics, Revue Internationale de Philosophie 42: 425-73. [.pdf]

(with Benacerraf, Paul). 1985. ‘Skolem and the Sceptic’. Supplement, Proceedings of the Aristotelian Society 59: 117-37. [.pdf]

1982. ‘Strict Finitism’. Synthese 51: 203-82. [.pdf]