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Extendability and ParadoxIn John Burgess (ed.), Hilary Putnam on Logic and Mathematics, Springer Verlag. 2018.
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1Possibility and Reality in Mathematics: A Review of Realism, Mathematics, and Modality (review)British Journal for the Philosophy of Science 43 (2): 245-262. 1992.
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11Quantum Measurement: Beyond Paradox (edited book)University of Minnesota Press. 1998.Together with relativity theory, quantum mechanics stands as the conceptual foundation of modern physics. It forms the basis by which we understand the minute workings of the subatomic world. But at its core lies a paradox--it is unmeasurable. This book presents a powerful and energetic new approach to the measurement dilemma.
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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
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101Hartry 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.
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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.
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270Supervenience/determination a two-way street? Yes, but one of the ways is the wrong way!Journal of Philosophy 89 (1): 42-47. 1992.
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69Bell-type inequalities in the nonideal case: Proof of a conjecture of bellFoundations of Physics 22 (6): 807-817. 1992.Recently Bell has conjectured that, with “epsilonics,” one should be able to argue, à la EPR, from “almost ideal correlations” (in parallel Bohm-Bell pair experiments) to “almost determinism,” and that this should suffice to derive an approximate Bell-type inequality. Here we prove that this is indeed the case. Such an inequality—in principle testable—is derived employing only weak locality conditions, imperfect correlation, and a propensity interpretation of certain conditional probabilities. O…Read more
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35Hilary Putnam on Logic and Mathematics (edited book)Springer Verlag. 2018.This book explores the research of Professor Hilary Putnam, a Harvard professor as well as a leading philosopher, mathematician and computer scientist. It features the work of distinguished scholars in the field as well as a selection of young academics who have studied topics closely connected to Putnam’s work. It includes 12 papers that analyze, develop, and constructively criticize this notable professor's research in mathematical logic, the philosophy of logic and the philosophy of mathemati…Read more
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3Randomness and RealityPSA Proceedings of the Biennial Meeting of the Philosophy of Science Association 1978 (2): 79-97. 1978.In previous technical work ([1] and [2]) on which his present paper [3] draws, Benioff has presented results conforming with the following argument-scheme:First, if we construe Quantum Mechanics as making claims to the effect that infinite outcome sequences (generated by repeated measurements on a system for a given observable in a given state) be random; and second, if a strong definition of “random” is adopted in this construal, then certain models of Zermelo-Fraenkel set theory (ZF) cannot be…Read more
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39Penelope Rush.* Ontology and the Foundations of Mathematics: Talking Past Each OtherPhilosophia Mathematica 30 (3): 387-392. 2022.This compact volume, belonging to the Cambridge Elements series, is a useful introduction to some of the most fundamental questions of philosophy and foundations of mathematics. What really distinguishes realist and platonist views of mathematics from anti-platonist views, including fictionalist and nominalist and modal-structuralist views?1 They seem to confront similar problems of justification, presenting tradeoffs between which it is difficult to adjudicate. For example, how do we gain acces…Read more
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18Reflections on Reflection in a MultiverseIn Erich H. Reck (ed.), Logic, Philosophy of Mathematics, and Their History: Essays in Honor of W. W. Tait, College Publications. pp. 77-90. 2018.
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32The History of Continua: Philosophical and Mathematical Perspectives (edited book)Oxford University Press. 2020.Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.
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15Mathematics and its Logics: Philosophical EssaysCambridge University Press. 2020.In these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman's modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which se…Read more
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36Stewart Shapiro. Second-order languages and mathematical practice. The journal of symbolic logic, vol. 50 , pp. 714–742 (review)Journal of Symbolic Logic 54 (1): 291-293. 1989.
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19The Statue within: An Autobiography. François Jacob, F. Philip (review)Philosophy of Science 58 (1): 132-132. 1991.
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49Robert L. Martin and Peter W. Woodruff. On representing ‘true-in-L' in L. Philosophia , vol. 5 no. 3 , pp. 213–217. - Saul Kripke. Outline of a theory of truth. The journal of philosophy, vol. 72 , pp. 690–716. - Anil Gupta. Truth and paradox. Journal of philosophical logic, vol. 11 , pp. 1–60. - Hans G. Herzberger. Notes on naive semantics. Journal of philosophical logic, vol. 11 , pp. 61–102 (review)Journal of Symbolic Logic 50 (4): 1068-1071. 1985.
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33Stewart Shapiro. Philosophy of mathematics. Structure and ontology. Oxford University Press, New York and Oxford 1997, x + 279 pp (review)Journal of Symbolic Logic 64 (2): 923-926. 1999.
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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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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
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16Varieties 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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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 ...
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174Physicalism: Ontology, determination and reductionJournal of Philosophy 72 (October): 551-64. 1975.
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179Structuralism 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
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |