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Profile: Michael Detlefsen (University of Notre Dame)
  1. Michael Detlefsen (2014). Duality, Epistemic Efficiency and Consistency. In G. Link (ed.), Formalism & Beyond. 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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  2. Michael Detlefsen (2011). Discovery, Invention and Realism: Gödel and Others on the Reality of Concepts. In John Polkinghorne (ed.), Meaning in Mathematics. Oup Oxford.
    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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  3. Michael Detlefsen (2011). Poincaré versus Russell sur le rôle de la logique dans les mathématiques. Les Etudes Philosophiques 2 (2):153-178.
    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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  4. Michael Detlefsen & Andrew Arana (2011). Purity of Methods. 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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  5. Michael Detlefsen (2009). Introduction to the Fiftieth Anniversary Issues. Notre Dame Journal of Formal Logic 50 (4):363-364.
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  6. Mic Detlefsen (2008). Purity as an Ideal of Proof. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oup Oxford. 179--197.
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  7. Mic Detlefsen & Michael Hallett (2008). Purity of Methods. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oup Oxford.
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  8. Michael Detlefsen (2008). Proof: Its Nature and Significance. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. 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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  9. Michael Detlefsen (2008). 7.1 Purity as an Ideal of Proof. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oup Oxford. 179.
    This is a paper on a type of purity of proof I call topical purity. This is purity which, practically speaking, enforces 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. -/- For some, this has been regarded as an epistemic ideal concerning the type of knowledge that proof (...)
     
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  10. 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). Curtis, C. VV. 255. In José Ferreirós Domínguez & Jeremy Gray (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford University Press.
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  11. Michael Detlefsen (2005). Formalism. In Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. 236--317.
    A comprehensive historical overview of formalist ideas in the philosophy of mathematics.
     
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  12. Michael Detlefsen (2002). Löb's Theorem as a Limitation on Mechanism. 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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  13. Michael Detlefsen (2001). What Does Gödel's Second Theorem Say. 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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  14. Michael Detlefsen, Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints & Richard Zach (2001). 2000-2001 Spring Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 7 (3):413-419.
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  15. Michael Detlefsen, Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints & Richard Zach (2001). The Minneapolis Hyatt Regency, Minneapolis, Minnesota May 3–4, 2001. Bulletin of Symbolic Logic 7 (3).
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  16. M. Detlefsen (2000). BURGESS, JP and ROSEN, G.-A Subject with No Object. Philosophical Books 41 (3):153-162.
    Review of John Burgess' and Gideon Rosen's A Subject with no Object.
     
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  17. Michael Detlefsen (2000). Introduction to Logicism and the Paradoxes: A Reappraisal. Notre Dame Journal of Formal Logic 41 (3):185-185.
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  18. Michael Detlefsen (1999). Logic From a to Z. 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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  19. Michael Detlefsen (1999). Introduction to Special Issue on George S. Boolos. Notre Dame Journal of Formal Logic 40 (1):1-2.
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  20. M. Detlefsen (1998). Walter van Stigt. Brouwer's Intuitionism. Amsterdam: North-Holland Publishing Co., 1990. Pp. Xxvi + 530. ISBN 0-444-88384-3 (Cloth). [REVIEW] Philosophia Mathematica 6 (2):235-241.
  21. Michael Detlefsen (1998). Constructive Existence Claims. In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. 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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  22. Michael Detlefsen (1998). Mind in the Shadows. Studies in History and Philosophy of Science Part B 29 (1):123-136.
    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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  23. S. Macintyre, Ellaway aA & M. Detlefsen (1998). Mind in the Shadows. Studies in History and Philosophy of Science Part B 29 (1):123-136.
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  24. Michael Detlefsen (1995). Review of J. Folina, Poincare and the Philosophy of Mathematics. [REVIEW] Philosophia Mathematica 3 (2):208-218.
  25. Michael Detlefsen (1995). Book Reviews. [REVIEW] Philosophia Mathematica 3 (2):208-218.
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  26. Michael Detlefsen (1995). The Mechanization of Reason. 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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  27. Michael Detlefsen (1995). Wright on the Non-Mechanizability of Intuitionist Reasoning. 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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  28. M. Detlefsen (1994). D. MIÉVILLE . "Kurt Gödel: Actes du Colloque, Neuch'tel 13-14 Juin 1991". [REVIEW] History and Philosophy of Logic 15 (1):135.
     
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  29. Michael Detlefsen (1993). Hilbert's Formalism. 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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  30. Michael Detlefsen (1993). Poincaré Vs. Russell on the Rôle of Logic in Mathematicst. 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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  31. Michael Detlefsen (1993). The Concept of Logical Consequence. Philosophical Books 34 (1):1-10.
  32. Michael Detlefsen (ed.) (1992). Proof and Knowledge in Mathematics. 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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  33. Michael Detlefsen (1992). Poincaré Against the Logicians. 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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  34. Michael Detlefsen (ed.) (1992). Proof, Logic, and Formalization. Routledge.
    Proof, Logic and Formalization addresses the various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing on this issue in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience in proof and the (...)
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  35. Michael Detlefsen (1990). Brouwerian Intuitionism. 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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  36. Michael Detlefsen (1990). On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem. 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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  37. Michael Detlefsen (1989). Aleksandar Pavković, Ed., Contemporary Yugoslav Philosophy: The Analytic Approach Reviewed By. Philosophy in Review 9 (12):492-496.
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  38. M. Detlefsen (1988). S. SHAPIRO "Intensional Mathematics". History and Philosophy of Logic 9 (1):93.
     
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  39. M. Detlefsen (1988). Essay Review. 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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  40. Michael Detlefsen (1988). Fregean Hierarchies and Mathematical Explanation. 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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  41. Michael Detlefsen (1986). Hilbert's Program: An Essay on Mathematical Instrumentalism. 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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  42. Michael Detlefsen (1983). Book Review. [REVIEW] Revue Internationale de Philosophie 37 (146):364.
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  43. Michael Detlefsen (1983). W. H. Newton-smith, "the Rationality Of Science". Revue Internationale de Philosophie 37 (3=146):364.
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  44. Michael Detlefsen (1980). On a Theorem of Feferman. 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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  45. Michael Detlefsen (1980). The Arithmetization of Metamathematics in a Philosophical Setting (*). Revue Internationale de Philosophie 34 (1):268-292.
     
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  46. Michael Detlefsen & Mark Luker (1980). The Four-Color Theorem and Mathematical Proof. 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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  47. Michael Detlefsen (1979). On Interpreting Gödel's Second Theorem. 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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  48. Michael Detlefsen (1976). The Importance of Goedel's Second Incompleteness Theorem for the Foundations of Mathematics. Dissertation, The Johns Hopkins University
     
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