•  2
    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
  •  56
    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
  •  18
    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.
  •  36
    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.
  •  194
    Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
  •  180
    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
  •  45
    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
  •  124
    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
  •  26
    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
  •  166
    Predicative foundations of arithmetic
    with Solomon Feferman
    Journal of Philosophical Logic 24 (1). 1995.
  •  39
    In the …rst part of this paper, the origins of modal-structuralism are traced from Hilary Putnam’s seminal article, "Mathematics without Foundations" (1967) to its transformation and development into the author’s modal-structural approach. The addition of a logic of plurals is highlighted for its recovery (in combination with the resources of mereology) of full, second-order logic, essential for articulating a good theory of mathematical structures. The second part concentrates on the motivation…Read more
  •  31
    Stochastic Locality and the Bell Theorems
    PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1982 601-615. 1982.
    After some introductory remarks on "experimental metaphysics", a brief survey of the current situation concerning the major types of hidden-variable theories and the inexistence proofs is presented. The category of stochastic, contextual, local theories remains open. Then the main features of a logical analysis of "locality" are sketched. In the deterministic case, a natural "light-cone determination" condition helps bridge the gap that has existed between the physical requirements of the specia…Read more
  •  6
  •  38
    Randomness and reality
    In Peter D. Asquith & Ian Hacking (eds.), PSA 1978, University of Chicago Press. pp. 79--97. 1978.
  •  22
  •  157
    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
  •  12
    Against bad method
    Metaphilosophy 10 (2). 1979.
  •  45
    In a recent paper, while discussing the role of the notion of analyticity in Carnap’s thought, Howard Stein wrote: “The primitive view–surely that of Kant–was that whatever is trivial is obvious. We know that this is wrong; and I would put it that the nature of mathematical knowledge appears more deeply mysterious today than it ever did in earlier centuries – that one of the advances we have made in philosophy has been to come to an understanding of just ∗I am grateful to audiences at the Steinf…Read more
  •  22
    Introduction
    Noûs 18 (4): 557-567. 1984.
  •  82
    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