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Predicativity and Regions-Based ContinuaIn Gerhard Jäger & Wilfried Sieg (eds.), Feferman on Foundations: Logic, Mathematics, Philosophy, Springer. 2017.
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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
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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
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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.
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36Hilary Putnam’s Contributions to Mathematics, Logic, and the Philosophy ThereofThe Harvard Review of Philosophy 24 117-119. 2017.
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Mathematics without Numbers. Towards a Modal-Structural InterpretationTijdschrift Voor Filosofie 53 (4): 726-727. 1991.
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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 ...
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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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104Quantum 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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243Three 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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61Real 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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234Does 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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94Bayes 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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141Dualling: 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
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |