Mathematical practice and naturalist epistemology : structures with potential for interaction

Van Kerkhove, Bart; Van Bendegem, Jean Paul

Philosophia Scientiae, Aperçus philosophiques en logique et en mathématiques, Tome 9 (2005) no. 2, pp. 61-78.

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology.

[1] Ascher, Marcia 1998.- Ethnomathematics. A Multicultural View of Mathematical Ideas, Boca Raton: Chapman & Hall / CRC. | MR | Zbl

[2] Ascher, Marcia 2002.- Mathematics Elsewhere. An Exploration of Ideas Across Cultures, Princeton - Oxford: Princeton University Press. | MR | Zbl

[3] Bell, E.T. 1992.- The Development of Mathematics, New York: Dover Publications. (unaltered from the 1945 second edition published by McGraw-Hill.) 1991 Knowledge and Social Imagery, Chicago - London: The University of Chicago Press. (second edition) | MR | Zbl

[4] Browder, Felix E. 2002.- Reflections on the Future of Mathematics, Notices of the AMS, 49(6), 658-62. | MR | Zbl

[5] Butterworth, Brian 2000.- The Mathematical Brain, London: Papermac. (reprint of the 1999 original published by Macmillan)

[6] Cheng Don, Pan & Cheng Biao, Pan 1992.- Goldbach Conjecture, Beijing: Science Press. | MR

[7] Code, Lorraine 1996.- What is Natural about Epistemology Naturalized?, American Philosophical Quarterly, 33(1), 1-22.

[8] Corry, Leo 1992.- Nicolas Bourbaki and the Concept of Mathematical Structure, Synthese, 92, 315-48. | MR | Zbl

[9] Cox, David A. 1994.- Introduction to Fermat's Last Theorem, American Mathematical Monthly, 101(1), 3-14. | MR | Zbl

[10] Courant, Richard & Robbins, Herbert 1996.- What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford - New York: Oxford University Press. (second, revised edition by Ian Stewart of the 1941 original) | MR | Zbl

[11] Davis, Philip J. 1972.- Fidelity in Mathematical Discourse: Is One and One Really Two?, American Mathematical Monthly, 79(3), 252-63. | MR | Zbl

[12] Dehaene, Stanislas 1998.- The Number Sense. How the Mind Creates Mathematics, London - New York: Allen Lane - The Penguin Press. (first published in 1997 by Oxford University Press, New York) | Zbl

[13] Devlin, Keith 1992.- The Death of Proof?, Notices of the AMS, 40, 1352.

[14] Duda, Roman 1997.- Mathematics: Essential Tensions, Foundations of Science 2(1), 11-9. | Zbl

[15] Ernest, Paul 1998.- Social Constructivism as a Philosophy of Mathematics, Albany: State University of New York Press. | MR

[16] Fritsch, Rudolf & Fritsch, Gerda 1998.- The Four-Color Theorem. History, Topological Foundations and Idea of Proof, New York: Springer-Verlag. (translated from the German original text by Julie Peschke) | MR | Zbl

[17] Gelbart, Stephen 1984.- An Elementary Introduction to the Langlands Programme, Bulletin (New Series) of the AMS, 10(2), 177-219. | MR | Zbl

[18] Gillies, Donald (ed.) 1992.- Revolutions in Mathematics, Oxford: Oxford University Press. | MR | Zbl

[19] Goodman, Nicolas D. 1990.- Mathematics as Natural Science, Journal of Symbolic Logic, 55(1), 182-93. | MR | Zbl

[20] Heintz, Bettina 2000.- Die Innenwelt der Mathematik. Zur Kultur und Praxis einer beweisenden Disziplin, Vienna - New York: Springer. | MR | Zbl

[21] Hilbert, David 2002.- Mathematical Problems, Bulletin (New Series) of the AMS, 8, 437-79.

[22] Jaffe, Arthur & Quinn, Frank 1993.- Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics, Bulletin (New Series) of the AMS, 29(1), 1-13. | MR | Zbl

[23] Kline, Morris 1990.- Mathematical Thought from Ancient to Modern Times, New York: Oxford University Press. (three volumes, second print of the 1972 original edition) | MR | Zbl

[24] Koetsier, Teun 1991.- Lakatos' Philosophy of Mathematics. A Historical Approach, Amsterdam: North-Holland. | MR | Zbl

[25] Kornblith, Hilary 1997.- Naturalizing Epistemology, Cambridge, MA - London: The MIT Press.

[26] Kuhn, Thomas S. 1962.- The Structure of Scientific Revolutions, Chicago - London: The University of Chicago Press.

[27] Lakatos, Imre 1976.- Proofs and Refutations. The Logic of Mathematical Discovery, Cambridge, MA: Cambridge University Press. (edited by J. Worrall and E. Zahar) | MR | Zbl

[28] Livingston, Eric 1986.- The Ethnomethodological Foundations of Mathematics, London: Routledge & Kegan Paul. 1997 Naturalism in Mathematics, Oxford: Clarendon Press. | MR

[29] Maddy, Penelope 1998.- How to Be a Naturalist about Mathematics, H.G. Dales and G. Oliveri (eds.), Truth in Mathematics, Oxford: Clarendon Press, 161-80. | MR | Zbl

[30] Otte, Michael 1999.- Mathematical Creativity and the Character of Mathematical Objects, Logique et Analyse, 167-168, 387-410. | MR | Zbl

[31] Parshall, Karen Hunger & Rice, Adrian C. (eds.) 2002.- Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800-1945. History of Mathematics Vol.23, Providence (RI) - London: American Mathematical Society. | MR | Zbl

[32] Peressini, Anthony 1999.- Confirming Mathematical Theories: An Ontologically Agnostic Stance, Synthese, 118(2), 257-77. | MR | Zbl

[33] Polya, George 1973.- Mathematics and Plausible Reasoning. Two Vols, Princeton (NJ): Princeton University Press. (eight print of the 1954 original) | Zbl

[34] Putnam, Hilary 2002.- The Collapse of the Fact/Value Dichotomy and Other Essays, Cambridge (MA) - London: Harvard University Press.

[35] Resnik, Michael 1998.- Holistic Mathematics, Matthias Schirn (ed.), The Philosophy of Mathematics Today, Oxford: Clarendon Press, 227-46. | MR | Zbl

[36] Restivo, Sal 1992.- Mathematics in Society and History. Episteme Vol.20, Dordrecht - Boston - London: Kluwer Academic Publishers. | MR | Zbl

[37] Rotman, Brian 2000.- Mathematics as Sign: Writing, Imagining, Counting, Stanford: Stanford University Press. | MR | Zbl

[38] Solomon, Ronald 2001.- A Brief History of the Classification of the Finite Simple Groups, Bulletin (New Series) of the AMS, 38(3), 315-52. | MR | Zbl

[39] Van Bendegem, Jean Paul 1998.- What, If Anything, Is an Experiment in Mathematics?, Dionysios Anapolitanos (ed.), Philosophy and the Many Faces of Science, Lanham: Rowman & Littlefield, Lanham, 172-82.

[40] Van Bendegem, Jean Paul 1999.- The Creative Growth of Mathematics, Philosophica, 63, 119-52.

[41] Van Bendegem, Jean Paul 2000.- Analogy and Metaphor as Essentials Tools for the Working Mathematician, Fernand Hallyn, ed., Metaphor and Analogy in the Sciences, Dordrecht - Boston: Kluwer Academic Publishers, 105-23.

[42] Van Kerkhove, Bart 2002.- Guises of Naturalism in the Foundations of Mathematics Debate, Proceedings of the Canadian Society for the History and Philosophy of Mathematics, 15, 169-82.

[43] Van Kerkhove, Bart 2004.- Review of [Heintz 2000], Philosophia Mathematica (III), 12(1), 69-74.

[44] Wang, Hao 1974.- From Mathematics to Philosophy, London: Routledge & Kegan Paul. | Zbl

[45] Weinberg, Steven 1993.- Dreams of a Final Theory. The Search for the Fundamental Laws of Nature, London: Vintage. (First published by Hutchinson Radius.)