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52Symbol systems and artistic stylesJournal of Aesthetics and Art Criticism 35 (3): 279-292. 1977.
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60Real 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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230Does 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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140Dualling: A critique of an argument of Popper and MillerBritish Journal for the Philosophy of Science 37 (2): 220-223. 1986.
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54Interpretations 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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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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78Regions-based two dimensional continua: The Euclidean caseLogic and Logical Philosophy 24 (4). 2015.
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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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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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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
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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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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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69Realist 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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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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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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125On 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
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |