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Profile: Michael Detlefsen (University of Notre Dame)
  1. Proof, Logic and Formalization.Michael Detlefsen (ed.) - 2015 - Routledge.
    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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  2. Purity of Methods.Michael Detlefsen & Andrew Arana - 2011 - Philosophers' Imprint 11 (2).
    Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...)
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  3.  22
    Hilbert's Program: An Essay on Mathematical Instrumentalism.Michael Detlefsen - 1986 - Reidel.
    An Essay on Mathematical Instrumentalism M. Detlefsen. THE PHILOSOPHICAL FUNDAMENTALS OF HILBERT'S PROGRAM 1. INTRODUCTION In this chapter I shall attempt to set out Hilbert's Program in a way that is more revealing than ...
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  4.  6
    Formalism.Michael Detlefsen - 2005 - In Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 236--317.
    A comprehensive historical overview of formalist ideas in the philosophy of mathematics.
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  5. Brouwerian Intuitionism.Michael Detlefsen - 1990 - Mind 99 (396):501-534.
    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 considerations. The (...)
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  6.  72
    The Four-Color Theorem and Mathematical Proof.Michael Detlefsen & Mark Luker - 1980 - Journal of Philosophy 77 (12):803-820.
    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 so novel as he (...)
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  7.  79
    What Does Gödel's Second Theorem Say.Michael Detlefsen - 2001 - Philosophia Mathematica 9 (1):37-71.
    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 justification of this fourth condition faces serious obstacles. We (...)
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  8.  48
    Poincaré Vs. Russell on the Rôle of Logic in Mathematicst.Michael Detlefsen - 1993 - Philosophia Mathematica 1 (1):24-49.
    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 to demonstrate its falsity. This (...)
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  9.  64
    Poincaré Against the Logicians.Michael Detlefsen - 1992 - Synthese 90 (3):349 - 378.
    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 principles of (...)
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  10. On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem.Michael Detlefsen - 1990 - Journal of Philosophical Logic 19 (4):343 - 377.
    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 (...)
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  11.  44
    Fregean Hierarchies and Mathematical Explanation.Michael Detlefsen - 1988 - International Studies in the Philosophy of Science 3 (1):97 – 116.
    There is a long line of thinkers in the philosophy of mathematics who have sought to base an account of proof on what might be called a 'metaphysical ordering' of the truths of mathematics. Use the term 'metaphysical' to describe these orderings is intended to call attention to the fact that they are regarded as objective and not subjective and that they are conceived primarily as orderings of truths and only secondarily as orderings of beliefs. -/- I describe and consider (...)
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  12.  43
    Wright on the Non-Mechanizability of Intuitionist Reasoning.Michael Detlefsen - 1995 - Philosophia Mathematica 3 (1):103-119.
    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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  13.  43
    Constructive Existence Claims.Michael Detlefsen - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. pp. 1998--307.
    It is a commonplace of constructivist thought that a claim that an object of a certain kind exists is to be backed by an explicit display or exhibition of an object that is manifestly of that kind. Let us refer to this requirement as the exhibition condition. The main objective of this essay is to examine this requirement and to arrive at a better understanding of its epistemic character and the role that it plays in the two main constructivist philosophies (...)
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  14.  33
    Proof: Its Nature and Significance.Michael Detlefsen - 2008 - In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 1.
    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 cannot have a (...)
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  15.  66
    On Interpreting Gödel's Second Theorem.Michael Detlefsen - 1979 - Journal of Philosophical Logic 8 (1):297 - 313.
    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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  16.  42
    Mind in the Shadows.Michael Detlefsen - 1998 - Studies in History and Philosophy of Science Part B 29 (1):123-136.
  17.  58
    Walter van Stigt. Brouwer's Intuitionism. Amsterdam: North-Holland Publishing Co., 1990. Pp. Xxvi + 530. ISBN 0-444-88384-3 (Cloth). [REVIEW]M. Detlefsen - 1998 - Philosophia Mathematica 6 (2):235-241.
  18. Hilbert's Program: An Essay on Mathematical Instrumentalism.Michael Detlefsen - 1991 - Journal of Philosophy 88 (6):331-336.
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  19.  3
    Duality, Epistemic Efficiency and Consistency.Michael Detlefsen - 2014 - In G. Link (ed.), Formalism & Beyond. De Gruyter. pp. 1-24.
    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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  20.  51
    Löb's Theorem as a Limitation on Mechanism.Michael Detlefsen - 2002 - Minds and Machines 12 (3):353-381.
    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 (...)
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  21. Purity as an Ideal of Proof.Michael Detlefsen - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 179-197.
    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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  22. W. H. Newton-smith, "the Rationality Of Science".Michael Detlefsen - 1983 - Revue Internationale de Philosophie 37 (3=146):364.
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  23.  36
    The Concept of Logical Consequence.Michael Detlefsen - 1993 - Philosophical Books 34 (1):1-10.
  24.  15
    Poincaré versus Russell sur le rôle de la logique dans les mathématiques.Michael Detlefsen - 2011 - Les Etudes Philosophiques 97 (2):153.
    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 Kant, argued that mathematical reasoning was characteristically non-logical in (...)
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  25.  36
    The Mechanization of Reason.Michael Detlefsen - 1995 - Philosophia Mathematica 3 (1).
    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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  26.  12
    The Minneapolis Hyatt Regency, Minneapolis, Minnesota May 3–4, 2001.Michael Detlefsen, Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints & Richard Zach - 2001 - Bulletin of Symbolic Logic 7 (3).
  27. Hilbert's Formalism.Michael Detlefsen - 1993 - Revue Internationale de Philosophie 47 (186):285-304.
    Various parallels between Kant's critical program and Hilbert's formalistic program for the philosophy of mathematics are considered.
     
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  28.  18
    Essay Review.M. Detlefsen - 1988 - History and Philosophy of Logic 9 (1):93-105.
    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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  29.  25
    On a Theorem of Feferman.Michael Detlefsen - 1980 - Philosophical Studies 38 (2):129 - 140.
    In this paper I argue that Feferman's theorem does not signify the existence of skeptic-satisfying consistency proofs. However, my argument for this is much different than other arguments (most particularly Resnik's) for the same claim. The argument that I give arises form an analysis of the notion of 'expression', according to which the specific character of that notion is seen as varying from one context of application (of a result of arithmetic metamathematics) to another.
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  30.  6
    2000-2001 Spring Meeting of the Association for Symbolic Logic.Michael Detlefsen, Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints & Richard Zach - 2001 - Bulletin of Symbolic Logic 7 (3):413-419.
  31. Aleksandar Pavković, Ed., Contemporary Yugoslav Philosophy: The Analytic Approach Reviewed By.Michael Detlefsen - 1989 - Philosophy in Review 9 (12):492-496.
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  32.  2
    Mind in the Shadows.Michael Detlefsen - 1998 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 29 (1):123-136.
  33.  8
    Introduction to Logicism and the Paradoxes: A Reappraisal.Michael Detlefsen - 2000 - Notre Dame Journal of Formal Logic 41 (3):185-185.
  34.  7
    Introduction to the Fiftieth Anniversary Issues.Michael Detlefsen - 2009 - Notre Dame Journal of Formal Logic 50 (4):363-364.
  35.  7
    Review of J. Folina, Poincare and the Philosophy of Mathematics[REVIEW]Michael Detlefsen - 1995 - Philosophia Mathematica 3 (2):208-218.
  36.  1
    Introduction to the Fiftieth Anniversary Issues.Michael Detlefsen - 2010 - Notre Dame Journal of Formal Logic 51 (1):1-2.
  37.  3
    Introduction to Special Issue on George S. Boolos.Michael Detlefsen - 1999 - Notre Dame Journal of Formal Logic 40 (1):1-2.
  38. Logic From a to Z: The Routledge Encyclopedia of Philosophy Glossary of Logical and Mathematical Terms.John B. Bacon, Michael Detlefsen & David Charles McCarty - 1999 - Routledge.
    First 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 to help comprehension and add (...)
     
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  39. Curtis, C. VV. 255.D. Von Dalen, M. Dehn, G. Deleuze, G. Desargues, M. Detlefsen, P. G. L. Dirichlet, P. Dugac, M. Dummett, W. G. Dwyer & M. Eckehardt - 2006 - In José Ferreirós Domínguez & Jeremy Gray (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford University Press.
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  40. BURGESS, JP and ROSEN, G.-A Subject with No Object.M. Detlefsen - 2000 - Philosophical Books 41 (3):153-162.
    Review of John Burgess' and Gideon Rosen's A Subject with no Object.
     
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  41. Book Review. [REVIEW]Michael Detlefsen - 1983 - Revue Internationale de Philosophie 37 (146):364.
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  42. Book Reviews. [REVIEW]Michael Detlefsen - 1995 - Philosophia Mathematica 3 (2):208-218.
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  43. Duality, Epistemic Efficiency & Consistency.Michael Detlefsen - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. De Gruyter. pp. 1-24.
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  44. Discovery, Invention and Realism: Gödel and Others on the Reality of Concepts.Michael Detlefsen - 2011 - In John Polkinghorne (ed.), Meaning in Mathematics. Oxford University Press.
    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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  45. D. MIÉVILLE . "Kurt Gödel: Actes du Colloque, Neuch'tel 13-14 Juin 1991". [REVIEW]M. Detlefsen - 1994 - History and Philosophy of Logic 15 (1):135.
  46.  11
    Logic From a to Z.Michael Detlefsen - 1999 - Routledge.
    First published as part of the renowned Routledge Encyclopedia of Philosophy , this complete glossary of logical terms and notation is the definitive handbook for students of the subject. Logic from A to Z features over 500 short definitional entries on terms used in the most highly technical areas of philosophy--philosophical logic and the philosophy of mathematics. Readers will find key terms such as Argument; Turing Machine; Isomorphism; Function; Algorithm; Variable ; and much more, plus a complete table of logical (...)
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  47.  86
    Proof and Knowledge in Mathematics.Michael Detlefsen (ed.) - 1992 - Routledge.
    Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael Detlefsen has brought together (...)
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  48.  1
    Proof and Knowledge in Mathematics.Michael Detlefsen (ed.) - 2014 - Routledge.
    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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  49. Proof and Knowledge in Mathematics.Michael Detlefsen (ed.) - 2005 - Routledge.
    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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  50. Proof and Knowledge in Mathematics.Michael Detlefsen - 1995 - Revue Philosophique de la France Et de l'Etranger 185 (1):133-134.
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