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The arithmetization of metamathematics in a philosophical setting (*)Revue Internationale de Philosophie 34 (1): 268-292. 1980.
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171Poincaré against the logiciansSynthese 90 (3). 1992.Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no prin…Read more
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88Löb'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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13Duality, 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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53The mechanization of reasonPhilosophia Mathematica 3 (1). 1995.Introduction to a special issue of Philosophia Mathematica on the mechanization of reasoning. Authors include: M. Detlefsen, D. Mundici, S. Shanker, S. Shapiro, W. Sieg and C. Wright.
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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
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85Poincaré vs. Russell on the rôle of logic in mathematicstPhilosophia Mathematica 1 (1): 24-49. 1993.In the early years of this century, Poincaré and Russell engaged in a debate concerning the nature of mathematical reasoning. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical in character. Russell urged the contrary view, maintaining that (i) the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that (ii) with the aid of recent developments in logic, it is possible t…Read more
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139On interpreting Gödel's second theoremJournal of Philosophical Logic 8 (1). 1979.In this paper I have considered various attempts to attribute significance to Gödel's second incompleteness theorem (G2 for short). Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are false. Two others (an argument suggested by Beth, Cohen and ??? and Resnik's Interpretation), I argue, are groundless.
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1Hilbert's formalismRevue Internationale de Philosophie 47 (186): 285-304. 1993.Various parallels between Kant's critical program and Hilbert's formalistic program for the philosophy of mathematics are considered.
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17Review of J. Folina, Poincare and the Philosophy of Mathematics (review)Philosophia Mathematica 3 (2): 208-218. 1995.
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29Essay ReviewHistory and Philosophy of Logic 9 (1): 93-105. 1988.S. SHAPIRO (ed.), Intensional Mathematics (Studies in Logic and the Foundations of Mathematics, vol. 11 3). Amsterdam: North-Holland, 1985. v + 230 pp. $38.50/100Df
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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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67Mind in the shadowsStudies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 29 (1): 123-136. 1998.This is a review of Penrose's trilogy, The Emperor's New Mind, Shadows of the Mind and The Large the Small and the Human Mind.
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D. MIÉVILLE . "Kurt Gödel: Actes du Colloque, Neuch'tel 13-14 juin 1991" (review)History and Philosophy of Logic 15 (1): 135. 1994.
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184What 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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2Aleksandar Pavković, ed., Contemporary Yugoslav Philosophy: The Analytic Approach Reviewed byPhilosophy in Review 9 (12): 492-496. 1989.
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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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154Proof and Knowledge in Mathematics (edited book)Routledge. 1992.These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is _a priori_ or _a posteriori_ in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification
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9Introduction to the Fiftieth Anniversary IssuesNotre Dame Journal of Formal Logic 50 (4): 363-364. 2009.
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4Janet Folina, Poincaré and the Philosophy of Mathematics (review)Philosophia Mathematica 3 (2): 208-218. 1995.
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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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201The four-color theorem and mathematical proofJournal of Philosophy 77 (12): 803-820. 1980.I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore not…Read more
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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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116Proof, Logic and Formalization (edited book)Routledge. 1992.The mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.
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192On an alleged refutation of Hilbert's program using gödel's first incompleteness theoremJournal of Philosophical Logic 19 (4). 1990.It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for selecting be…Read more
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73FormalismIn 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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88Wright on the non-mechanizability of intuitionist reasoningPhilosophia Mathematica 3 (1): 103-119. 1995.Crispin Wright joins the ranks of those who have sought to refute mechanist theories of mind by invoking Gödel's Incompleteness Theorems. His predecessors include Gödel himself, J. R. Lucas and, most recently, Roger Penrose. The aim of this essay is to show that, like his predecessors, Wright, too, fails to make his case, and that, indeed, he fails to do so even when judged by standards of success which he himself lays down.
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464Brouwerian intuitionismMind 99 (396): 501-534. 1990.The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerat…Read more
Michael Detlefsen
(1948 - 2019)
Notre Dame, Indiana, United States of America
Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Mathematics |