•  23
    Philosophy of Mathematics
    In Peter Clark & Katherine Hawley (eds.), Philosophy of Science Today, Clarendon Press. 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
  •  8
    Coping with fear through suppression and avoidance of threatening information
    with Jesper Nielsen
    Journal of Experimental Psychology: Applied 15 (3): 258-274. 2009.
  •  72
    The Nature and Limits of Abstraction (review)
    Philosophical Quarterly 54 (214). 2004.
    This article is an extended critical study of Kit Fine’s The limits of abstraction, which is a sustained attempt to take the measure of the neo-logicist program in the philosophy and foundations of mathematics, founded on abstraction principles like Hume’s principle. The present article covers the philosophical and technical aspects of Fine’s deep and penetrating study.
  •  100
    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
  •  2
    Mathematics Without Numbers (review)
    Noûs 27 (4): 522-525. 1993.
  •  162
    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.
  •  120
    Frege Meets Aristotle: Points as Abstracts
    Philosophia Mathematica. 2015.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at sta…Read more
  •  24
    Acceptable notation
    Notre Dame Journal of Formal Logic 23 (1): 14-20. 1982.
  •  76
    Set-Theoretic Foundations
    The Proceedings of the Twentieth World Congress of Philosophy 2000 183-196. 2000.
    Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematica…Read more
  •  179
    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
  •  36
    Varieties of Logic
    Oxford University Press. 2014.
    Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.
  • Mathematics and Objectivity
    In John Polkinghorne (ed.), Meaning in Mathematics, Oxford University Press. 2011.
  •  29
    Understanding the Infinite
    with Mary Ellen Smircich and Shaughan Lavine
    Philosophical Review 105 (2): 256. 1996.
    Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.
  •  113
  •  49
    Reasoning, logic and computation
    Philosophia Mathematica 3 (1): 31-51. 1995.
    The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism…Read more
  •  102
    Author index — volume 7
    Philosophia Mathematica 7 (3): 351-352. 1999.
  •  219
    Identity, indiscernibility, and Ante Rem structuralism: The tale of I and –I
    Philosophia Mathematica 16 (3): 285-309. 2008.
    Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one o…Read more
  •  33
    The Classical Continuum without Points – CORRIGENDUM
    with G. Hellman
    Review of Symbolic Logic 6 (3): 571-571. 2013.
  •  1
    Thinking about Mathematics
    Tijdschrift Voor Filosofie 63 (1): 189-190. 2001.
  •  9
    Foundations Without Foundationalism: A Case for Second-Order Logic
    Philosophical Quarterly 44 (174): 127-129. 1994.
  •  31
    On the notion of effectiveness
    History and Philosophy of Logic 1 (1-2): 209-230. 1980.
    This paper focuses on two notions of effectiveness which are not treated in detail elsewhere. Unlike the standard computability notion, which is a property of functions themselves, both notions of effectiveness are properties of interpreted linguistic presentations of functions. It is shown that effectiveness is epistemically at least as basic as computability in the sense that decisions about computability normally involve judgments concerning effectiveness. There are many occurrences of the pr…Read more
  •  3
    Second-Order Logic, Foundations, and Rules
    Journal of Philosophy 87 (5): 234. 1990.
  •  7
    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.