•  1
    The Continuous (edited book)
    Oxford University Press. forthcoming.
  •  16
    The Statue within: An Autobiography. François Jacob, F. Philip (review)
    Philosophy of Science 58 (1): 132-132. 1991.
  •  1
    Mathematical Structuralism
    Cambridge University Press. 2018.
    The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as a…Read more
  •  67
    Carnap* Replies
    The Monist 101 (4): 388-393. 2018.
    In an imagined dialogue between two figures called “Carnap*” and “Quine*” that appeared in the Library of Living Philosophers volume in 1986, certain proposals and clarifications of the linguistic doctrine were offered by Carnap* answering Quinean objections, but these were brushed aside rather breezily in a reply to this dialogue in the same volume by Quine himself. After a brief summary of the questions at issue in that earlier dialogue, Carnap* is here allowed a final reply, introducing yet a…Read more
  • Hellman and Shapiro explore the development of the idea of the continuous, from the Aristotelian view that a true continuum cannot be composed of points to the now standard, entirely punctiform frameworks for analysis and geometry. They then investigate the underlying metaphysical issues concerning the nature of space or space-time.
  •  26
    Hilary Putnam’s Contributions to Mathematics, Logic, and the Philosophy Thereof
    The Harvard Review of Philosophy 24 117-119. 2017.
  • Mathematics without Numbers. Towards a Modal-Structural Interpretation
    Tijdschrift Voor Filosofie 53 (4): 726-727. 1991.
  • Steps in the Theory of Radical Translation
    Dissertation, Harvard University. 1973.
  •  106
    Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
  •  127
    Structuralism without structures
    Philosophia Mathematica 4 (2): 100-123. 1996.
    Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the c…Read more
  •  7
    Against bad method
    Metaphilosophy 10 (2). 1979.
  •  71
    On the significance of the Burali-Forti paradox
    Analysis 71 (4): 631-637. 2011.
    After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliter…Read more
  •  100
    What is categorical structuralism?
    In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 151--161. 2006.
  •  98
    Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis
    Journal of Philosophical Logic 35 (6): 621-651. 2006.
    A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis ('SIA'), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis ('CA') without resort to the method of limits. Formally, however, unlike Robinsonian 'nonstandard analysis', SIA conflicts with CA, deriving, e.g., 'not every quantity is either = 0 or not = 0.' Internally, consistency is maintained by using intuitionistic logic (without the law …Read more
  •  10
    1995–1996 annual meeting of the association for symbolic logic
    with Tomek Bartoszynski, Harvey Friedman, Bakhadyr Khoussainov, Phokion G. Kolaitis, Richard Shore, Charles Steinhorn, Mirna Dzamonja, Itay Neeman, and Slawomir Solecki
    Bulletin of Symbolic Logic 2 (4): 448-472. 1996.
  •  60
    With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the set-theoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulfilling certain structural relations among themselves and in relation to othe…Read more
  •  43
    EPR, bell, and collapse: A route around "stochastic" hidden variables
    Philosophy of Science 54 (4): 558-576. 1987.
    Two EPR arguments are reviewed, for their own sake, and for the purpose of clarifying the status of "stochastic" hidden variables. The first is a streamlined version of the EPR argument for the incompleteness of quantum mechanics. The role of an anti-instrumentalist ("realist") interpretation of certain probability statements is emphasized. The second traces out one horn of a central foundational dilemma, the collapse dilemma; complex modal reasoning, similar to the original EPR, is used to deri…Read more
  •  41
    Realist principles
    Philosophy of Science 50 (2): 227-249. 1983.
    We list, with discussions, various principles of scientific realism, in order to exhibit their diversity and to emphasize certain serious problems of formulation. Ontological and epistemological principles are distinguished. Within the former category, some framed in semantic terms (truth, reference) serve their purpose vis-a-vis instrumentalism (Part 1). They fail, however, to distinguish the realist from a wide variety of (constructional) empiricists. Part 2 seeks purely ontological formulatio…Read more
  •  8
    Book reviews (review)
    Philosophia Mathematica 1 (1): 75-88. 1993.
  •  28
    Quantum logic and the projection postulate
    Philosophy of Science 48 (3): 469-486. 1981.
    This paper explores the status of the von Neumann-Luders state transition rule (the "projection postulate") within "real-logic" quantum logic. The entire discussion proceeds from a reading of the Luders rule according to which, although idealized in applying only to "minimally disturbing" measurements, it nevertheless makes empirical claims and is not a purely mathematical theorem. An argument (due to Friedman and Putnam) is examined to the effect that QL has an explanatory advantage over Copenh…Read more
  •  98
    On nominalism
    Philosophy and Phenomenological Research 62 (3): 691-705. 2001.
    Probably there is no position in Goodman’s corpus that has generated greater perplexity and criticism than Goodman’s “nominalism”. As is abundantly clear from Goodman’s writings, it is not “abstract entities” generally that he questions—indeed, he takes sensory qualia as “basic” in his Carnap-inspired constructional system in Structure—but rather just those abstracta that are so crystal clear in their identity conditions, so fundamental to our thought, so prevalent and seemingly unavoidable in o…Read more
  •  21
    The Many Worlds Interpretation of Set Theory
    PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988 445-455. 1988.
    Standard presentations of axioms for set theory as truths simpliciter about actual-objects the sets-confront a number of puzzles associated with platonism and foundationalism. In his classic, Zermelo suggested an alternative "many worlds" view. Independently, Putnam proposed something similar, explicitly incorporating modality. A modal-structural synthesis of these ideas is sketched in which obstacles to their formalization are overcome. Extendability principles are formulated and used to motiva…Read more
  •  24