
Discovery, Invention and Realism: Gödel and others on the Reality of ConceptsIn John Polkinghorne (ed.), Meaning in mathematics, Oxford University Press. 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 mindindependent way although our minds have epistemic access to them.

7.1 Purity as an ideal of proofIn Paolo Mancosu (ed.), The Philosophy of Mathematical Practice, Oxford University Press. pp. 179. 2008.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 ought to or at least might ideally…Read more

1Purity as an ideal of proofIn Paolo Mancosu (ed.), The Philosophy of Mathematical Practice, Oxford University Press. pp. 179197. 2008.

Curtis, C. VV. 255In José Ferreirós Domínguez & Jeremy Gray (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy, Oxford University Press. 2006.

220Purity of MethodsPhilosophers' Imprint 11. 2011.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 …Read more

4620002001 Spring Meeting of the Association for Symbolic LogicBulletin of Symbolic Logic 7 (3): 413419. 2001.

Book reviews (review)History and Philosophy of Logic 15 (1): 127147. 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

4Proof 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

26Formalism and Hilbert’s understanding of consistency problemsArchive for Mathematical Logic 60 (5): 529546. 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 empiricosemantic 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

Critical essay on W. P. NewtonSmith's The Rationality of Science (review)Revue Internationale de Philosophie 37 (146): 364371. 1983.

Proof: Its Nature and SignificanceIn Bonnie Gold & Roger A. Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy, Maa. pp. 332. 2008.

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. 171184. 2010.

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. 7071. 2011.

Discovery, Invention and Realism: Gödel and others on the Reality of ConceptsIn John Polkinghorne (ed.), Mathematics and its Significance, Oxford University Press. pp. 7396. 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 mindindependent way although our minds have epistemic access to them.

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. 205221. 2011.

Freedom and ConsistencyIn Emily Goldblatt, B. Kim & R. Downey (eds.), Proceedings of the 12th Asian Logic Conference, World Scientific. pp. 89111. 2013.

1Completeness and the Ends of AxiomatizationIn Juliette Kennedy (ed.), Interpreting Gödel: Critical Essays, Cambridge University Press. pp. 5977. 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

On the motives for proof theoryIn Heinrich Wansing (ed.), Dag Prawitz on Proofs and Meaning, Springer. 2015.

Gentzen's antiformalist ideasIn Reinhard Kahle & Michael Rathjen (eds.), Gentzen's Centenary: The Quest for Consistency, Springer. pp. 2544. 2015.

Abstraction, Axiomatization and Rigor: Pasch and HilbertIn John Burgess (ed.), Hilary Putnam on Logic and Mathematics, Springer Verlag. 2018.

66Ian Hacking. Why Is There Philosophy of Mathematics At All?Philosophia Mathematica 25 (3): 407412. 2017.© The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please email: [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

56Mind in the shadowsStudies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 29 (1): 123136. 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.

D. MIÉVILLE . "Kurt Gödel: Actes du Colloque, Neuch'tel 1314 juin 1991" (review)History and Philosophy of Logic 15 (1): 135. 1994.

2Aleksandar Pavković, ed., Contemporary Yugoslav Philosophy: The Analytic Approach Reviewed byPhilosophy in Review 9 (12): 492496. 1989.

188What does Gödel's second theorem say?Philosophia Mathematica 9 (1): 3771. 2001.We consider a seemingly popular justification (we call it the Reflexivity Defense) for the third derivability condition of the HilbertBernaysLöb 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

49Poincaré 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
Michael Detlefsen
(1948  2019)
Notre Dame, Indiana, United States of America
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 