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67Stochastic Einstein-locality and the bell theoremsSynthese 53 (3). 1982.Standard proofs of generalized Bell theorems, aiming to restrict stochastic, local hidden-variable theories for quantum correlation phenomena, employ as a locality condition the requirement of conditional stochastic independence. The connection between this and the no-superluminary-action requirement of the special theory of relativity has been a topic of controversy. In this paper, we introduce an alternative locality condition for stochastic theories, framed in terms of the models of such a th…Read more
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8From Constructive to Predicative MathematicsIn John Earman & John Norton (eds.), The Cosmos of Science, University of Pittsburgh Press. pp. 6--153. 1997.
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103Quantum mechanical unbounded operators and constructive mathematics – a rejoinder to bridgesJournal of Philosophical Logic 26 (2): 121-127. 1997.As argued in Hellman (1993), the theorem of Pour-El and Richards (1983) can be seen by the classicist as limiting constructivist efforts to recover the mathematics for quantum mechanics. Although Bridges (1995) may be right that the constructivist would work with a different definition of 'closed operator', this does not affect my point that neither the classical unbounded operators standardly recognized in quantum mechanics nor their restrictions to constructive arguments are recognizable as ob…Read more
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17On the Scope and Force of Indispensability ArgumentsPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992 456-464. 1992.Three questions are highlighted concerning the scope and force of indispensability arguments supporting classical, infinitistic mathematics. The first concerns the need for non-constructive reasoning for scientifically applicable mathematics; the second concerns the need for impredicative set existence principles for finitistic and scientifically applicable mathematics, respectively; and the third concerns the general status of such arguments in light of recent work in mathematical logic, especi…Read more
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232Three varieties of mathematical structuralismPhilosophia Mathematica 9 (2): 184-211. 2001.Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects a…Read more
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52Symbol systems and artistic stylesJournal of Aesthetics and Art Criticism 35 (3): 279-292. 1977.
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59Real analysis without classesPhilosophia Mathematica 2 (3): 228-250. 1994.This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by…Read more
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228Does category theory provide a framework for mathematical structuralism?Philosophia Mathematica 11 (2): 129-157. 2003.Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recoveri…Read more
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49Pluralism and the Foundations of MathematicsIn ¸ Itekellersetal:Sp, . pp. 65--79. 2006.A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects …Read more
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90Bayes and beyondPhilosophy of Science 64 (2): 191-221. 1997.Several leading topics outstanding after John Earman's Bayes or Bust? are investigated further, with emphasis on the relevance of Bayesian explication in epistemology of science, despite certain limitations. (1) Dutch Book arguments are reformulated so that their independence from utility and preference in epistemic contexts is evident. (2) The Bayesian analysis of the Quine-Duhem problem is pursued; the phenomenon of a "protective belt" of auxiliary statements around reasonably successful theor…Read more
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40The Classical Continuum without Points – CORRIGENDUMReview of Symbolic Logic 6 (3): 571-571. 2013.
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139Dualling: A critique of an argument of Popper and MillerBritish Journal for the Philosophy of Science 37 (2): 220-223. 1986.
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53Interpretations of Probability in Quantum Mechanics: A Case of “Experimental Metaphysics”In Wayne C. Myrvold & Joy Christian (eds.), Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, Springer. pp. 211--227. 2009.
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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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91On 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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123What 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, Clarendon Press. 2006.
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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
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |