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174Physicalism: Ontology, determination and reductionJournal of Philosophy 72 (October): 551-64. 1975.
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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
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78Regions-based two dimensional continua: The Euclidean caseLogic and Logical Philosophy 24 (4). 2015.
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39After some metatheoretic preliminaries on questions of justification and rational reconstruction, we lay out some key desiderata for foundational frameworks for mathematics, some of which reflect recent discussions of pluralism and structuralism. Next we draw out some implications (pro and con) bearing on set theory and category and topos therory. Finally, we sketch a variant of a modal-structural core system, incorporating elements of predicativism and the systems of reverse mathematics, and co…Read more
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52Constructive mathematics and quantum mechanics: Unbounded operators and the spectral theorem (review)Journal of Philosophical Logic 22 (3). 1993.
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124What is categorical structuralism?In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 151--161. 2006.
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Against 'Absolutely Everything'!In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute generality, Oxford University Press. 2006.
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98On the significance of the Burali-Forti paradoxAnalysis 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
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128Mathematical Pluralism: The Case of Smooth Infinitesimal AnalysisJournal 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
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62With 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 A. D. Irvine (ed.), Bertrand Russell: Critical Assessments, Routledge. pp. 154. 1999.
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75Realist 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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4Beyond 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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45Quantum 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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26The 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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127On 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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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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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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42Finitude, infinitude, and isomorphism of interpretations in some nominalistic calculiNoûs 3 (4): 413-425. 1969.
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38Randomness 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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157Frege 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
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |