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34The 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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72A Note on Choice Principles in Second-Order LogicReview of Symbolic Logic 16 (2): 339-350. 2023.Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We als…Read more
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16Link’s Revenge: A Case Study in Natural Language MereologyIn Gabriele Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium, De Gruyter. pp. 3-36. 2019.Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends tha…Read more
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5Mathematics in Philosophy, Philosophy in Mathematics: Three Case StudiesIn Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics, Springer International Publishing. 2016.The interaction between philosophy and mathematics has a long and well articulated history. The purpose of this note is to sketch three historical case studies that highlight and further illustrate some details concerning the relationship between the two: the interplay between mathematical and philosophical methods in ancient Greek thought; vagueness and the relation between mathematical logic and ordinary language; and the study of the notion of continuity.
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34Inconsistency and Incompleteness, RevisitedIn Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency, Springer Verlag. pp. 469-479. 2019.Graham Priest introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might fo…Read more
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80Logical pluralism and normativityInquiry: An Interdisciplinary Journal of Philosophy 63 (3-4): 389-410. 2020.We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which po…Read more
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136Link's Revenge: A Case Study in Natural Language MereologyIn Gabriele Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium, De Gruyter. pp. 3-36. 2019.Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends tha…Read more
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28Introduction to Special Issue: The Emergence of StructuralismPhilosophia Mathematica 27 (3): 299-302. 2019.
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75Friedrich Waismann: The Open Texture of Analytic Philosophy (edited book)Palgrave Macmillan. 2019.This edited collection covers Friedrich Waismann's most influential contributions to twentieth-century philosophy of language: his concepts of open texture and language strata, his early criticism of verificationism and the analytic-synthetic distinction, as well as their significance for experimental and legal philosophy. In addition, Waismann's original papers in ethics, metaphysics, epistemology and the philosophy of mathematics are here evaluated. They introduce Waismann's theory of action a…Read more
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270Set Theory, Type Theory, and Absolute GeneralityMind 123 (489): 157-174. 2014.In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are ope…Read more
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6An Introduction to Non-classical Logic (review)Review of Metaphysics 56 (3): 670-671. 2003.This book is just what its title says: an introduction to nonclassical logic. And it is a very good one. Given the extensive interest in nonclassical logics, in various parts of the philosophical scene, it is a welcome addition to the corpus. Typical courses in logic, at all levels and in both philosophy departments and mathematics departments, focus exclusively on classical logic. Most instructors, and some textbooks, give some mention to some nonclassical systems, but usually few details are p…Read more
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17Philosophy of Mathematics: Structure and OntologyOxford University Press USA. 1997.Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.
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7Mathematics in philosophy, Selected essays, by Charles Parsons, Cornell University Press, Ithaca, N.Y., 1983, 365 pp (review)Journal of Symbolic Logic 53 (1): 320-329. 1988.
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27Stephen C. Kleene. Origins of recursive function theory. Annals of the history of computing, vol. 3 , pp. 52– 67. - Martin Davis. Why Gödel didn't have Church's thesis. Information and control, vol. 54 , pp. 3– 24. - Stephen C. Kleene. Reflections on Church's thesis. Notre Dame journal of formal logic, vol. 28 , pp. 490– 498 (review)Journal of Symbolic Logic 55 (1): 348-350. 1990.
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11Perspectives on the history of mathematical logic, edited by Thomas Drucker, Birkhäuser, Boston, Basel, and Berlin, 1991, xxiii + 195 pp. - John W. Dawson Jr. The reception of Gödel's incompleteness theorems. Pp. 84–100 (review)Journal of Symbolic Logic 57 (4): 1487-1489. 1992.
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10Wilfried Sieg. Step by recursive step: Church's analysis of effective calculability. The bulletin of symbolic logic, vol. 3 , pp. 154–180 (review)Journal of Symbolic Logic 64 (1): 398-399. 1999.
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123Does Logical Pluralism Imply, or Suggest, Truth Pluralism, or Vice Versa?Synthese 198 (Suppl 20): 4925-4936. 2019.The answers to the questions in the title depend on the kind of pluralism one is talking about. We will focus here on our own views. The purpose of this article is to trace out some possible connections between these kinds of pluralism. We show how each of them might bear on the other, depending on how certain open questions are resolved.
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Changing the Subject: Quine, Putnam and Waismann on Meaning-Change, Logic, and AnalyticityIn John Burgess (ed.), Hilary Putnam on Logic and Mathematics, Springer Verlag. 2018.
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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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58Ineffability within the limits of abstraction aloneIn Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism: Essays in Philosophy of Mathematics, Oxford University Press Uk. 2016.The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that our abstraction based theory can be mod…Read more
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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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521Actual and Potential InfinityNoûs 53 (1): 160-191. 2017.The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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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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94Oxford Handbook of Philosophy of Mathematics and Logic (edited book)Oxford University Press. 2005.This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book include…Read more
Columbus, Ohio, United States of America
Areas of Specialization
Philosophy of Language |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Areas of Interest
Philosophy of Language |
Logic and Philosophy of Logic |
Philosophy of Mathematics |