-
Predicativity and Regions-Based ContinuaIn Gerhard Jäger & Wilfried Sieg (eds.), Feferman on Foundations: Logic, Mathematics, Philosophy, Springer. 2017.
-
2Mathematical StructuralismCambridge 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
-
56Carnap* RepliesThe 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
-
18Varieties of Continua: From Regions to Points and BackOxford University Press. 2017.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.
-
36Hilary Putnam’s Contributions to Mathematics, Logic, and the Philosophy ThereofThe Harvard Review of Philosophy 24 117-119. 2017.
-
Mathematics without Numbers. Towards a Modal-Structural InterpretationTijdschrift Voor Filosofie 53 (4): 726-727. 1991.
-
194Mathematics Without Numbers: Towards a Modal-Structural InterpretationOxford University Press. 1989.Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
-
174Physicalism: Ontology, determination and reductionJournal of Philosophy 72 (October): 551-64. 1975.
-
180Structuralism without structuresPhilosophia 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
-
45In 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
-
8Supervenience/Determination a Two-way Street? Yes, But One of the Ways Is the Wrong Way!Journal of Philosophy 89 (1): 42-47. 1992.
-
13Review: Stewart Shapiro, Second-Order Languages and Mathematical Practice (review)Journal of Symbolic Logic 54 (1): 291-293. 1989.
-
82EPR, bell, and collapse: A route around "stochastic" hidden variablesPhilosophy 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
-
81Quantum Logic and MeaningPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1980. 1980.Quantum logic as genuine non-classical logic provides no solution to the "paradoxes" of quantum mechanics. From the minimal condition that synonyms be substitutable salva veritate, it follows that synonymous sentential connectives be alike in point of truth-functionality. It is a fact of pure mathematics that any assignment Φ of (0, 1) to the subspaces of Hilbert space (dim. ≥ 3) which guarantees truth-preservation of the ordering and truth-functionality of QL negation, violates truth-functional…Read more
-
263The new Riddle of radical translationPhilosophy of Science 41 (3): 227-246. 1974.This paper presents parts of a theory of radical translation with applications to the problem of construing reference. First, in sections 1 to 4 the general standpoint, inspired by Goodman's approach to induction, is set forth. Codification of sound translational practice replaces the aim of behavioral reduction of semantic notions. The need for a theory of translational projection (manual construction on the basis of a finite empirical correlation of sentences) is established by showing the ano…Read more
-
67Mathematics without Numbers: Towards a Modal-Structural InterpretationPhilosophical Review 101 (4): 919. 1992.
-
79Mathematical constructivism in spacetimeBritish Journal for the Philosophy of Science 49 (3): 425-450. 1998.To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As th…Read more
-
6Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general condit…Read more
-
41Responses to Maher, and to Kelly, Schulte, and JuhlPhilosophy of Science 64 (2): 317-322. 1997.None
-
68Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern…Read more
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |