•  249
    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
  •  47
    The Company Kept by Cut Abstraction (and its Relatives)
    Philosophia Mathematica 19 (2): 107-138. 2011.
    This article concerns the ongoing neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections . The overall plan is to develop branches of established mathematics using abstraction principles in the form: Formula where a and b are variables of a g…Read more
  •  93
    Frege meets dedekind: A neologicist treatment of real analysis
    Notre Dame Journal of Formal Logic 41 (4): 335--364. 2000.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of ra…Read more
  •  16
    The Lindenbaum construction and decidability
    Notre Dame Journal of Formal Logic 29 (2): 208-213. 1988.
  •  25
    Expressive completeness and decidability
    with George F. Schumm
    Notre Dame Journal of Formal Logic 31 (4): 576-579. 1990.
  •  15
    Second-Order Logic, Foundations, and Rules
    Journal of Philosophy 87 (5): 234. 1990.
  •  39
    Deflation and conservation
    In Volker Halbach & Leon Horsten (eds.), Principles of Truth, Dr. Hänsel-hohenhausen. pp. 103-128. 2002.
  •  65
    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
  •  1
    Vagueness, Metaphysics, and Objectivity
    In Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and Clouds: Vaguenesss, its Nature and its Logic, Oxford University Press. 2010.
  •  33
    Life on the Ship of Neurath
    Croatian Journal of Philosophy 9 (2): 149-166. 2009.
    Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics.
  •  134
    Structure and identity
    In Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 34--69. 2006.
    According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of crit…Read more
  •  136
    An “I” for an I: Singular terms, uniqueness, and reference
    Review of Symbolic Logic 5 (3): 380-415. 2012.
    There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in lingu…Read more
  •  27
    The Work of John Corcoran: An Appreciation
    with Stewart Shapiro and Michael Scanlan
    History and Philosophy of Logic 20 (3-4): 149-158. 1999.
  •  151
    Incompleteness, mechanism, and optimism
    Bulletin of Symbolic Logic 4 (3): 273-302. 1998.
    §1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” wou…Read more
  •  22
    The governance of identity
    In Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 164--173. 2006.
  •  87
  •  33
    Philosophy of Mathematics
    In Peter Clark & Katherine Hawley (eds.), Philosophy of Science Today, Oxford University Press Uk. 2003.
    Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle
  •  214
    The classical continuum without points
    Review of Symbolic Logic 6 (3): 488-512. 2013.
    We develop a point-free construction of the classical one- dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually ex…Read more
  •  58
    Effectiveness
    In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 37--49. 2006.
  •  167
    Foundations of Mathematics: Metaphysics, Epistemology, Structure
    Philosophical Quarterly 54 (214). 2004.
    Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to prov…Read more
  •  152
    Mechanism, truth, and Penrose's new argument
    Journal of Philosophical Logic 32 (1): 19-42. 2003.
    Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). I…Read more
  •  238
    It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analytic-synthetic distinction. This logic-friendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so…Read more
  •  22
    Consumer memory for intentions: A prospective memory perspective
    with H. Shanker Krishnan
    Journal of Experimental Psychology: Applied 5 (2): 169. 1999.