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2With 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
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How to Godel a Frege-RussellIn Andrew D. Irvine (ed.), Bertrand Russell: Critical Assessments, Routledge. pp. 154. 1998.
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24Realist principlesPhilosophy 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
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13Quantum logic and the projection postulatePhilosophy 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
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1Beyond definitionism—but not too far beyondIn Matthias Schirn (ed.), The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993, Clarendon Press. 1998.
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15On nominalismPhilosophy 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
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2The Many Worlds Interpretation of Set TheoryPSA: 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
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39In 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
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30Stochastic Locality and the Bell TheoremsPSA: 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
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8Finitude, infinitude, and isomorphism of interpretations in some nominalistic calculiNoûs 3 (4): 413-425. 1969.
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7Randomness and RealityPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1978 79-97. 1978.
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22Constructivist Mathematics, Quantum Physics and QuantifiersAristotelian Society Supplementary Volume 66 (1). 1992.
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10Frege Meets Aristotle: Points as AbstractsPhilosophia 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
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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
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4Supervenience/Determination a Two-way Street? Yes, But One of the Ways Is the Wrong Way!Journal of Philosophy 89 (1): 42-47. 1992.
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |