•  238
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or …Read more
  •  418
    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
  •  103
    Sets and Abstracts – Discussion
    Philosophical Studies 122 (3): 315-332. 2005.
  •  52
    Vagueness in Context
    Oxford University Press UK. 2006.
    Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professional basketball players. The main feature of S…Read more
  •  132
    Author index — volume 7
    Philosophia Mathematica 7 (3): 351-352. 1999.
  •  126
    Tarski’s Theorem and the Extensionality of Truth
    Erkenntnis 78 (5): 1197-1204. 2013.
  •  1
    Intensional Mathematics
    Philosophy of Science 56 (1): 177-178. 1989.
  •  54
    Review of Michael P. Lynch, Truth as One and Many (review)
    Notre Dame Philosophical Reviews 2009 (9). 2009.
  •  83
    The George Boolos memorial symposium II
    Philosophia Mathematica 9 (1): 3-4. 2001.
  •  228
    Higher-Order Logic or Set Theory: A False Dilemma
    Philosophia Mathematica 20 (3): 305-323. 2012.
    The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?
  •  57
    Turing projectability
    Notre Dame Journal of Formal Logic 28 (4): 520-535. 1987.
  •  224
    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
  •  182
    So truth is safe from paradox: now what?
    Philosophical Studies 147 (3): 445-455. 2010.
    The article is part of a symposium on Hartry Field’s “Saving truth from paradox”. The book is one of the most significant intellectual achievements of the past decades, but it is not clear what, exactly, it accomplishes. I explore some alternatives, relating the developed view to the intuitive, pre-theoretic notion of truth.
  •  16
    Do Not Claim Too Much: Second-order Logic and First-order Logic
    Philosophia Mathematica 6 (3): 42-64. 1998.
    The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.
  •  15
    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
  •  200
  •  21
    Book reviews (review)
    with Timo Airaksinen and W. Stephen Croddy
    Philosophia 14 (3-4): 427-467. 1984.
  •  9
    Structure and Ontology
    Philosophical Topics 17 (2): 145-171. 1989.
  •  6
    A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐…Read more
  •  111
    Famously, Michael Dummett argues that considerations concerning the role of language in communication lead to the rejection of classical logic in favor of intuitionistic logic. Potentially, this results in massive revisions of established mathematics. Recently, Neil Tennant (“The law of excluded middle is synthetic a priori, if valid”, Philosophical Topics 24 (1996), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle groun…Read more
  •  204
    Modality and ontology
    Mind 102 (407): 455-481. 1993.
  •  16
    Review: The Nature and Limits of Abstraction (review)
    Philosophical Quarterly 54 (214). 2004.
  •  17
    Book reviews (review)
    Mind 101 (402): 225-250. 1992.
  •  279
    New V, ZF and Abstraction
    with Alan Weir
    Philosophia Mathematica 7 (3): 293-321. 1999.
    We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New …Read more