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24Formalism and Hilbert’s understanding of consistency problemsArchive for Mathematical Logic 60 (5): 529-546. 2021.Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism, game formalism and instrumental formalism. After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilber…Read more
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Abstraction, Axiomatization and Rigor: Pasch and HilbertIn Roy Cook & Geoffrey Hellman (eds.), Hilary Putnam on Logic and Mathematics, Springer Verlag. 2018.
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64Ian Hacking. Why Is There Philosophy of Mathematics At All?Philosophia Mathematica 25 (3): 407-412. 2017.© The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] author makes clear that he does not see this book as a contribution to the philosophy of mathematics as traditionally understood. He takes it instead to be an essay about the philosophy of mathematics, one whose purpose is to explain its existence and to make clear the limited extent to which its current and past forms are properly regarded as philosophi…Read more
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Gentzen's anti-formalist ideasIn Reinhard Kahle & Michael Rathjen (eds.), Gentzen's Centenary: The Quest for Consistency, Springer. pp. 25-44. 2015.
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On the motives for proof theoryIn Heinrich Wansing (ed.), Dag Prawitz on Proofs and Meaning, Springer. 2015.
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21Duality, Epistemic Efficiency and ConsistencyIn G. Link (ed.), Formalism & Beyond, De Gruyter. pp. 1-24. 2014.Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I consider possible arguments for this view. Major elements of this argument are out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They also require a special form of consistency requirement.
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12Duality, Epistemic Efficiency & ConsistencyIn Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse, De Gruyter. pp. 1-24. 2014.Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I attempt to construct a serious argument for this view. Certain major elements of this argument are then considered at length. They’re found to be out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They’re also found to require a special form of consistency requirement.
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1Completeness and the Ends of AxiomatizationIn Juliette Cara Kennedy (ed.), Interpreting Gödel, Cambridge University Press. pp. 59-77. 2014.The type of completeness Whitehead and Russell aimed for in their Principia Mathematica was what I call descriptive completeness. This is completeness with respect to the propositions that have been proved in traditional mathematics. The notion of completeness addressed by Gödel in his famous work of 1930 and 1931 was completeness with respect to the truths expressible in a given language. What are the relative significances of these different conceptions of completeness for traditional mathemat…Read more
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Freedom and ConsistencyIn Emily Goldblatt, B. Kim & R. Downey (eds.), Proceedings of the 12th Asian Logic Conference, World Scientific. pp. 89-111. 2013.
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Discovery, Invention and Realism: Gödel and others on the Reality of ConceptsIn John Polkinghorne (ed.), Mathematics and its Significance, Oxford University Press. pp. 73-96. 2011.The general question considered is whether and to what extent there are features of our mathematical knowledge that support a realist attitude towards mathematics. I consider, in particular, reasoning from claims such as that mathematicians believe their reasoning to be part of a process of discovery (and not of mere invention), to the view that mathematical entities exist in some mind-independent way although our minds have epistemic access to them.
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41Poincaré versus Russell sur le rôle de la logique dans les mathématiquesLes Etudes Philosophiques 97 (2): 153. 2011.Au début du XXe siècle, Poincaré et Russell eurent un débat à propos de la nature du raisonnement mathématique. Poincaré, comme Kant, défendait l’idée que le raisonnement mathématique était de caractère non logique. Russell soutenait une conception contraire et critiquait Poincaré. Je défends ici l’idée que les critiques de Russell n’étaient pas fondées.In the early twentieth century, Poincare and Russell engaged in a discussion concerning the nature of mathematical reasoning. Poincare, like Kan…Read more
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Sensing objectivity: A comment on Mary Leng's "Creation and Discovery in Mathematics"In John Polkinghorne (ed.), Mathematics and its Significance, Oxford University Press. pp. 70-71. 2011.
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Dedekind against Intuition: Rigor, Scope and the Motives of his LogicismIn Carlo Cellucci, Emily Grosholz & Emiliano Ippoliti (eds.), Logic and Knowledge, Cambridge Scholars Publications. pp. 205-221. 2011.
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3Introduction to the Fiftieth Anniversary IssuesNotre Dame Journal of Formal Logic 51 (1): 1-2. 2010.
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Rigor, Reproof and Bolzano's Critical ProgramIn Pierre Edouard Bour, Manuel Rebuschi & Laurent Rollet (eds.), Construction: A Festschrift for Gerhard Heinzmann, King's College Publications. pp. 171-184. 2010.
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Proof: Its Nature and SignificanceIn Bonnie Gold & Roger A. Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy, Maa. pp. 3-32. 2009.
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9Introduction to the Fiftieth Anniversary IssuesNotre Dame Journal of Formal Logic 50 (4): 363-364. 2009.
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4Purity as an ideal of proofIn Paolo Mancosu (ed.), The Philosophy of Mathematical Practice, Oxford University Press. pp. 179-197. 2008.Various ideals of purity are surveyed and discussed. These include the classical Aristotelian ideal, as well as certain neo-classical and contemporary ideals. The focus is on a type of purity ideal I call topical purity. This is purity which emphasizes a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its content.
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105Proof: Its nature and significanceIn Bonnie Gold & Roger A. Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Association of America. pp. 1. 2008.I focus on three preoccupations of recent writings on proof. I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of which we c…Read more
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71FormalismIn Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press. pp. 236--317. 2005.A comprehensive historical overview of formalist ideas in the philosophy of mathematics.
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86Löb's theorem as a limitation on mechanismMinds and Machines 12 (3): 353-381. 2002.We argue that Löb's Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löb's Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to be the theorem-set. This gives, …Read more
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432000-2001 Spring Meeting of the Association for Symbolic LogicBulletin of Symbolic Logic 7 (3): 413-419. 2001.
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181What does Gödel's second theorem say?Philosophia Mathematica 9 (1): 37-71. 2001.We consider a seemingly popular justification (we call it the Re-flexivity Defense) for the third derivability condition of the Hilbert-Bernays-Löb generalization of Godel's Second Incompleteness Theorem (G2). We argue that (i) in certain settings (rouglily, those where the representing theory of an arithmetization is allowed to be a proper subtheory of the represented theory), use of the Reflexivity Defense to justify the tliird condition induces a fourth condition, and that (ii) the justificat…Read more
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21Introduction to Logicism and the Paradoxes: A ReappraisalNotre Dame Journal of Formal Logic 41 (3): 185-185. 2000.
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BURGESS, JP and ROSEN, G.-A Subject with No ObjectPhilosophical Books 41 (3): 153-162. 2000.Review of John Burgess' and Gideon Rosen's A Subject with no Object.
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14Introduction to Special Issue on George S. BoolosNotre Dame Journal of Formal Logic 40 (1): 1-2. 1999.
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29First published in the most ambitious international philosophy project for a generation; the _Routledge Encyclopedia of Philosophy_. _Logic from A to Z_ is a unique glossary of terms used in formal logic and the philosophy of mathematics. Over 500 entries include key terms found in the study of: * Logic: Argument, Turing Machine, Variable * Set and model theory: Isomorphism, Function * Computability theory: Algorithm, Turing Machine * Plus a table of logical symbols. Extensively cross-referenced…Read more
Michael Detlefsen
(1948 - 2019)
Notre Dame, Indiana, United States of America
Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Mathematics |