Notre Dame, Indiana, United States of America
  • 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.
  • 7.1 Purity as an ideal of proof
    In 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
  •  1
    Purity as an ideal of proof
    In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice, Oxford University Press. pp. 179--197. 2008.
  • Curtis, C. VV. 255
    with D. Von Dalen, M. Dehn, G. Deleuze, G. Desargues, P. G. L. Dirichlet, P. Dugac, M. Dummett, W. G. Dwyer, and M. Eckehardt
    In José Ferreirós Domínguez & Jeremy Gray (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy, Oxford University Press. 2006.
  •  220
    Purity of Methods
    Philosophers' 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
  •  9
    Medical Paternalism Reconsidered
    Pacific Philosophical Quarterly 62 (1): 95-98. 2017.
  •  46
    2000-2001 Spring Meeting of the Association for Symbolic Logic
    with Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints, and Richard Zach
    Bulletin of Symbolic Logic 7 (3): 413-419. 2001.
  • Book reviews (review)
    with Donald Rutherford, E. R. Grosholz, D. M. Clarke, A. D. Irvine, Gerhard Heinzmann, I. Jané, N. C. A. Da Costa, and Larry Hauser
    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
  •  4
    Proof 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
  •  26
    Formalism and Hilbert’s understanding of consistency problems
    Archive for Mathematical Logic 60 (5): 529-546. 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 empirico-semantic 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. Newton-Smith's The Rationality of Science (review)
    Revue Internationale de Philosophie 37 (146): 364-371. 1983.
  • Rigor, Reproof and Bolzano's Critical Program
    In Pierre Edouard Bour, Manuel Rebuschi & Laurent Rollet (eds.), Construction: A Festschrift for Gerhard Heinzmann, King's College Publications. pp. 171-184. 2010.
  • Discovery, Invention and Realism: Gödel and others on the Reality of Concepts
    In John Polkinghorne (ed.), Mathematics and its Significance, Oxford University Press. pp. 73-96. 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 mind-independent way although our minds have epistemic access to them.
  • Dedekind against Intuition: Rigor, Scope and the Motives of his Logicism
    In Carlo Cellucci, Emily Grosholz & Emiliano Ippoliti (eds.), Logic and Knowledge, Cambridge Scholars Publications. pp. 205-221. 2011.
  • Freedom and Consistency
    In Emily Goldblatt, B. Kim & R. Downey (eds.), Proceedings of the 12th Asian Logic Conference, World Scientific. pp. 89-111. 2013.
  •  1
    Completeness and the Ends of Axiomatization
    In Juliette Kennedy (ed.), Interpreting Gödel: Critical Essays, Cambridge University Press. pp. 59-77. 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 theory
    In Heinrich Wansing (ed.), Dag Prawitz on Proofs and Meaning, Springer. 2015.
  •  66
    Ian Hacking. Why Is There Philosophy of Mathematics At All?
    Philosophia Mathematica 25 (3): 407-412. 2017.
    © The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [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
  •  24
    Hilbert's Program
    Noûs 26 (4): 513-514. 1992.
  •  56
    Mind in the shadows
    Studies 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.
  •  188
    What 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
  •  49
    Poincaré versus Russell sur le rôle de la logique dans les mathématiques
    Les 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