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34Hilary 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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34Stewart 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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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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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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311995–1996 annual meeting of the association for symbolic logicBulletin of Symbolic Logic 2 (4): 448-472. 1996.
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30Stochastic Locality and the Bell TheoremsPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1982 601-615. 1982.After some introductory remarks on "experimental metaphysics", a brief survey of the current situation concerning the major types of hidden-variable theories and the inexistence proofs is presented. The category of stochastic, contextual, local theories remains open. Then the main features of a logical analysis of "locality" are sketched. In the deterministic case, a natural "light-cone determination" condition helps bridge the gap that has existed between the physical requirements of the specia…Read more
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26The Many Worlds Interpretation of Set TheoryPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988 445-455. 1988.Standard presentations of axioms for set theory as truths simpliciter about actual-objects the sets-confront a number of puzzles associated with platonism and foundationalism. In his classic, Zermelo suggested an alternative "many worlds" view. Independently, Putnam proposed something similar, explicitly incorporating modality. A modal-structural synthesis of these ideas is sketched in which obstacles to their formalization are overcome. Extendability principles are formulated and used to motiva…Read more
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22Constructivist Mathematics, Quantum Physics and QuantifiersAristotelian Society Supplementary Volume 66 (1). 1992.
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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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17Reflections 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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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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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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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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13Review: Stewart Shapiro, Second-Order Languages and Mathematical Practice (review)Journal of Symbolic Logic 54 (1): 291-293. 1989.
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10Quantum 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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8Supervenience/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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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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6Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general condit…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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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
Areas of Specialization
Aesthetics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Physical Science |
Areas of Interest
17th/18th Century Philosophy |