•  46
    The paradox of the Unexpected Hanging, related prediction paradoxes, and the Sorites paradoxes all involve reasoning about ordered collections of entities: days ordered by date in the case of the Unexpected Hanging; men ordered by the number of hairs on their heads the case of the bald man version of the Sorites. The reasoning then assigns each entity a value that depends on the previously assigned value of one of the neighboring entities. The final result is paradoxical because it conflicts wit…Read more
  •  2
  •  143
    _1. Philosophical background: iteration, ineffability, reflection._ There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice : the iterative conception and limitation of size. Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.
  •  14
    Essay Review
    History and Philosophy of Logic 6 (1): 215-221. 1985.
    D. GABBAY and F. GUENTHNER (eds.), Handbook of philosophical logic. Volume 1: Elements of classical logic. Dordrecht, Boston, and Lancaster: D. Reidel Publishing Company, 1983. xiv + 497 pp. Dfl225/$98.00
  •  20
    Book reviews (review)
    with Timo Airaksinen and W. Stephen Croddy
    Philosophia 14 (3-4): 427-467. 1984.
  •  116
  •  263
    At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are u…Read more
  •  339
    Mathematics and reality
    Philosophy of Science 50 (4): 523-548. 1983.
    The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) sur…Read more
  •  91
    Sets and Abstracts – Discussion
    Philosophical Studies 122 (3): 315-332. 2005.
  •  160
    Conservativeness and incompleteness
    Journal of Philosophy 80 (9): 521-531. 1983.
  •  13
    Vagueness and Logic
    In Giuseppina Ronzitti (ed.), Vagueness: A Guide, Springer Verlag. pp. 55--81. 2011.
  •  42
    Review of Michael P. Lynch, Truth as One and Many (review)
    Notre Dame Philosophical Reviews 2009 (9). 2009.
  •  11
    Arithmetic Sinn and Effectiveness
    Dialectica 38 (1): 3-16. 1984.
    SummaryAccording to Dummett's understanding of Frege, the sense of a denoting expression is a procedure for determining its denotation. The purpose of this article is to pursue this suggestion and develop a semi‐formal interpretation of Fregean sense for the special case of a first‐order language of arithmetic. In particular, we define the sense of each arithmetic expression to be a hypothetical process to determine the denoted number or truth value. The sense‐process is “hypothetical” in that t…Read more
  •  216
    The Objectivity of Mathematics
    Synthese 156 (2): 337-381. 2007.
    The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.
  •  1
    Intensional Mathematics
    Philosophy of Science 56 (1): 177-178. 1989.
  •  118
    Philosophy of Mathematics: Structure and Ontology
    Oxford University Press. 1997.
    Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests re…Read more
  • Understanding the Infinite
    with Shaughan Lavine
    Studia Logica 63 (1): 123-128. 1994.
  •  160
    Truth, function and paradox
    Analysis 71 (1): 38-44. 2011.
    Michael Lynch’s Truth as One and Many is a contribution to the large body of philosophical literature on the nature of truth. Within that genre, advocates of truth-as-correspondence, advocates of truth-as-coherence, and the like, all hold that truth has a single underlying metaphysical nature, but they sharply disagree as to what this nature is. Lynch argues that many of these views make good sense of truth attributions for a limited stretch of discourse, but he adds that each of the contenders …Read more
  •  14
    The Lindenbaum construction and decidability
    Notre Dame Journal of Formal Logic 29 (2): 208-213. 1988.
  •  23
    Expressive completeness and decidability
    with George F. Schumm
    Notre Dame Journal of Formal Logic 31 (4): 576-579. 1990.
  •  2
    Simple truth, contradiction, and consistency
    In Graham Priest, J. C. Beall & Bradley Armour-Garb (eds.), The Law of Non-Contradiction, Oxford University Press. 2004.
  •  249
    Epistemology of mathematics: What are the questions? What count as answers?
    Philosophical Quarterly 61 (242): 130-150. 2011.
    A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietis…Read more
  •  36
    Review of T. Franzen, Godel's theorem: An incomplete guide to its use and abuse (review)
    Philosophia Mathematica 14 (2): 262-264. 2006.
    This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience with the rigor of mathemat…Read more
  •  144
  •  1
    Structure and Ontology
    Philosophical Topics 17 (2): 145-171. 1989.