-
270Supervenience/determination a two-way street? Yes, but one of the ways is the wrong way!Journal of Philosophy 89 (1): 42-47. 1992.
-
262The new Riddle of radical translationPhilosophy of Science 41 (3): 227-246. 1974.This paper presents parts of a theory of radical translation with applications to the problem of construing reference. First, in sections 1 to 4 the general standpoint, inspired by Goodman's approach to induction, is set forth. Codification of sound translational practice replaces the aim of behavioral reduction of semantic notions. The need for a theory of translational projection (manual construction on the basis of a finite empirical correlation of sentences) is established by showing the ano…Read more
-
244The classical continuum without pointsReview of Symbolic Logic 6 (3): 488-512. 2013.We develop a point-free construction of the classical one- dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually ex…Read more
-
235Three 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
-
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
-
219Aristotelian ContinuaPhilosophia Mathematica 24 (2): 214-246. 2016.In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, and we show th…Read more
-
187Mathematics 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 ...
-
178Structuralism 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
-
172Physicalism: Ontology, determination and reductionJournal of Philosophy 72 (October): 551-64. 1975.
-
156Frege 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
-
140Dualling: A critique of an argument of Popper and MillerBritish Journal for the Philosophy of Science 37 (2): 220-223. 1986.
-
132Determination and logical truthJournal of Philosophy 82 (November): 607-16. 1985.Some remarks on determination, physicalism, model theory, and logical truth.//An attempt to defend physicalism against objections that its bases are indeterminate.
-
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
-
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
-
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.
-
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
-
100Hartry Field. Science Without Numbers: A Defense of Nominalism 2nd ed (review)Philosophia Mathematica 27 (1): 139-148. 2019.FieldHartry. Science Without Numbers: A Defense of Nominalism 2nd ed.Oxford University Press, 2016. ISBN 978-0-19-877792-2. Pp. vi + 56 + vi + 111.
-
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
-
91Carnap* 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
-
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
-
80EPR, bell, and collapse: A route around "stochastic" hidden variablesPhilosophy of Science 54 (4): 558-576. 1987.Two EPR arguments are reviewed, for their own sake, and for the purpose of clarifying the status of "stochastic" hidden variables. The first is a streamlined version of the EPR argument for the incompleteness of quantum mechanics. The role of an anti-instrumentalist ("realist") interpretation of certain probability statements is emphasized. The second traces out one horn of a central foundational dilemma, the collapse dilemma; complex modal reasoning, similar to the original EPR, is used to deri…Read more
-
80Quantum Logic and MeaningPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1980. 1980.Quantum logic as genuine non-classical logic provides no solution to the "paradoxes" of quantum mechanics. From the minimal condition that synonyms be substitutable salva veritate, it follows that synonymous sentential connectives be alike in point of truth-functionality. It is a fact of pure mathematics that any assignment Φ of (0, 1) to the subspaces of Hilbert space (dim. ≥ 3) which guarantees truth-preservation of the ordering and truth-functionality of QL negation, violates truth-functional…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 |