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.
  •  31
    Proof, 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.
  •  195
    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
  •  81
    Formalism
    In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press. pp. 236--317. 2005.
    A comprehensive historical overview of formalist ideas in the philosophy of mathematics.
  •  89
    Wright on the non-mechanizability of intuitionist reasoning
    Philosophia 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.
  •  468
    Brouwerian intuitionism
    Mind 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