-
74Walter 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. 1998.
-
66Constructive existence claimsIn Matthias Schirn (ed.), The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993, Clarendon Press. pp. 1998--307. 1998.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 of this century---the intuitionist pro…Read more
-
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.
-
4Janet Folina, Poincaré and the Philosophy of Mathematics (review)Philosophia Mathematica 3 (2): 208-218. 1995.
-
17Review of J. Folina, Poincare and the Philosophy of Mathematics (review)Philosophia Mathematica 3 (2): 208-218. 1995.
-
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.
-
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.
-
Book reviews (review)History and Philosophy of Logic 15 (1): 127-147. 1994.Hide Ishiguro, Leibniz’s philosophy of logic and language. 2nd ed. Cambridge:Cambridge University Press, 1990. x + 246pp. £27.50/$49.50 ; £10.95/$16.95 Massimo Mugnai, Leibniz’ theory of relations. Stuttgart:Franz Steiner Verlag, 1992. 291 pp. 96 DM W. A. Wallace, Galileo’s logic of discovery and proof The background, content, and use of his appropriated treatises on Aristotle’s posterior analytics. Dordrecht, Boston, and London:Kluwer, 1992. xxiii + 323 pp. £84, $139, DF1240 W. A. Wallace, Gali…Read more
-
D. MIÉVILLE . "Kurt Gödel: Actes du Colloque, Neuch'tel 13-14 juin 1991" (review)History and Philosophy of Logic 15 (1): 135. 1994.
-
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.
-
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
-
1Proof and Knowledge in MathematicsRevue Philosophique de la France Et de l'Etranger 185 (1): 133-134. 1992.
-
117Proof, 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.
-
2Proof and Knowledge in Mathematics (edited book)Routledge. 1992.This volume of essays addresses the main problem confronting an epistemology for mathematics; namely, the nature and sources of mathematical justification. Attending to both particular and general issues, the essays, by 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 of what epistemological importance is t…Read more
-
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
-
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
-
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
-
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
-
2Aleksandar Pavković, ed., Contemporary Yugoslav Philosophy: The Analytic Approach Reviewed byPhilosophy in Review 9 (12): 492-496. 1989.
-
80Fregean hierarchies and mathematical explanationInternational Studies in the Philosophy of Science 3 (1). 1988.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 two models for such or…Read more
-
30Essay 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
-
96An 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 ...
-
Critical essay on W. P. Newton-Smith's The Rationality of Science (review)Revue Internationale de Philosophie 37 (146): 364-371. 1983.
-
57On a theorem of FefermanPhilosophical Studies 38 (2). 1980.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.
-
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
-
The arithmetization of metamathematics in a philosophical setting (*)Revue Internationale de Philosophie 34 (1): 268-292. 1980.
-
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.
Michael Detlefsen
(1948 - 2019)
Notre Dame, Indiana, United States of America
Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Mathematics |