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407A subject with no object: strategies for nominalistic interpretation of mathematicsOxford University Press. 1997.Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obsc…Read more
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42Philosophy of Mathematics in the Twentieth Century: Selected EssaysHistory and Philosophy of Logic 36 (1): 93-95. 2015.The second volume of Charles Parsons’ selected papers, dedicated to Solomon Feferman, Wilfred Sieg, and William Tait, collects eleven mainly historical essays and reviews on philosophy and philosop...
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Mathematics, Models, and Modality: Selected Philosophical EssaysCambridge University Press. 2008.John Burgess is the author of a rich and creative body of work which seeks to defend classical logic and mathematics through counter-criticism of their nominalist, intuitionist, relevantist, and other critics. This selection of his essays, which spans twenty-five years, addresses key topics including nominalism, neo-logicism, intuitionism, modal logic, analyticity, and translation. An introduction sets the essays in context and offers a retrospective appraisal of their aims. The volume will be o…Read more
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32Axioms for tense logic. II. Time periodsNotre Dame Journal of Formal Logic 23 (4): 375-383. 1982.
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73Discussion—Soames on EmpiricismPhilosophical Studies 129 (3): 619-626. 2006.Philosophical Analysis in the Twentieth Century by Scott Soames reminds me of nothing so much as Lectures on Literature by Vladimir Nabokov. Both are works that arose immediately out of the needs of undergraduate teaching, yet each manages to say much of significance to knowledgeable professionals. Each indirectly provides an outline of the history of its field, through a presentation of selected major works, taken in chronological order and including items that are generally recognized as marki…Read more
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11Review: Nicholas Rescher, A Probabilistic Approach to Modal Logic (review)Journal of Symbolic Logic 35 (4): 583-583. 1970.
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29In this era when results of empirical scientific research are being appealed to all across philosophy, when we even find moral philosophers invoking the results of brain scans, many profess to practice "naturalized epistemology," or to be "epistemological naturalists." Such phrases derive from the title of a well-known essay by Quine,[1] but Paul Gregory's thesis in the work under review is that there is less connection than is usually assumed between Quine's variety of naturalized epistemology …Read more
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24Proofs about Proofs: a defense of classical logic. Part I: the aims of classical logicIn Michael Detlefsen (ed.), Proof, Logic and Formalization, Routledge. 1992.
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32From preference to utility: A problem of descriptive set theoryNotre Dame Journal of Formal Logic 26 (2): 106-114. 1985.
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289Mathematics and bleak housePhilosophia Mathematica 12 (1): 18-36. 2004.The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers
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76Thomas McKay. Plural predicationPhilosophia Mathematica 16 (1): 133-140. 2008.This work, the first book-length study of its topic, is an important contribution to the literature of philosophical logic and philosophy of language, with implications for other branches of philosophy, including philosophy of mathematics. However, five of the book's ten chapters , including many of the author's most original contributions, are devoted to issues about natural language, and lie pretty well outside the scope of this journal, not to mention that of the reviewer's competence. For th…Read more
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Read, Stephen, "Relevant Logic: A Philosophical Examination of Inference" (review)Mind 99 (n/a): 140. 1990.
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11Chapter Three. Modal LogicIn J. W. Davis (ed.), Philosophical logic, D. Reidel. pp. 40-70. 1969.
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39The completeness of intuitionistic propositional calculus for its intended interpretationNotre Dame Journal of Formal Logic 22 (1): 17-28. 1981.
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174Alan Weir. Truth through Proof: A Formalist Foundation for Mathematics. Oxford: Clarendon Press, 2010. ISBN 978-0-19-954149-2. Pp. xiv+281: Critical Studies/Book Reviews (review)Philosophia Mathematica 19 (2): 213-219. 2011.Alan Weir’s new book is, like Darwin’s Origin of Species, ‘one long argument’. The author has devised a new kind of have-it-both-ways philosophy of mathematics, supposed to allow him to say out of one side of his mouth that the integer 1,000,000 exists and even that the cardinal ℵω exists, while saying out of the other side of his mouth that no numbers exist at all, and the whole book is devoted to an exposition and defense of this new view. The view is presented in the book in a way that can ma…Read more
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69One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.
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36No requirement of relevanceIn Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press. pp. 727--750. 2005.There are schools of logicians who claim that an argument is not valid unless the conclusion is relevant to the premises. In particular, relevance logicians reject the classical theses that anything follows from a contradiction and that a logical truth follows from everything. This chapter critically evaluates several different motivations for relevance logic, and several systems of relevance logic, finding them all wanting.
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16Careful choices---a last word on Borel selectorsNotre Dame Journal of Formal Logic 22 (3): 219-226. 1981.
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47KripkePolity. 2012.Saul Kripke has been a major influence on analytic philosophy and allied fields for a half-century and more. His early masterpiece, _Naming and Necessity_, reversed the pattern of two centuries of philosophizing about the necessary and the contingent. Although much of his work remains unpublished, several major essays have now appeared in print, most recently in his long-awaited collection _Philosophical Troubles_. In this book Kripke’s long-time colleague, the logician and philosopher John P. B…Read more
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32Synthetic mechanics revisitedJournal of Philosophical Logic 20 (2). 1991.Earlier results on eliminating numerical objects from physical theories are extended to results on eliminating geometrical objects
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231E pluribus unum: Plural logic and set theoryPhilosophia Mathematica 12 (3): 193-221. 2004.A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory
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48Review of B. Hale and A. Hoffmann (eds.), Modality: Metaphysics, Logic, and Epistemology (review)Notre Dame Philosophical Reviews 2010 (10). 2010.
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135Charles Parsons. Mathematical thought and its objectsPhilosophia Mathematica 16 (3): 402-409. 2008.This long-awaited volume is a must-read for anyone with a serious interest in philosophy of mathematics. The book falls into two parts, with the primary focus of the first on ontology and structuralism, and the second on intuition and epistemology, though with many links between them. The style throughout involves unhurried examination from several points of view of each issue addressed, before reaching a guarded conclusion. A wealth of material is set before the reader along the way, but a revi…Read more
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