Abstract
In this paper, as part of an argument for the of revolutions in mathematics, I argue that there in incommensurability in Mathematics. After Devising A Framework Sensitive To Meaning Change And To Changes In The Extension Of Mathematical Predicates, I Consider Two Case Studies That Illustrate Different Ways In Which Incommensurability Emerge In Mathematical Practice. The Most Detailed Case Involves Nonstandard Analysis, And The Existence Of Different Notions Of The Continuum. But I Also Examine How Incommensurability Found Its Way Into Set Theory. I Conclude By Examining Some Consequences That Incommensurability Has To Mathematical Practice.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Azzouni, J. [1994]: Metaphysical Myths, Mathematical Practice. Cambridge: Cambridge University Press.
Azzouni, J. [1997]: “Thick Epistemic Access: Distinguishing the Mathematical from the Empirical”, Journal of Philosophy 94, pp. 472-484.
Balaguer, M. [1998]: Platonism and Anti-Platonism in Mathematics. New York: Oxford University Press.
Bell, J.L., and Machover, M. [1977]: A Course in Mathematical Logic. Amsterdam: North-Holland.
Berkeley, G. [1734]: “The Analyst”, in Berkeley [1951], vol.4., pp. 65-102.
Berkeley, G. [1951]: The Works of George Berkeley Bishop of Cloyne. (Edited by A.A. Luce and T.E. Jessop.) London: Thomas Nelson and Sons.
Bernstein, A., and Robinson, A. [1966]: “Solution of an Invariant Subspace Problem of K.T. Smith and P.R. Halmos”, Pacific Journal of Mathematics 16, pp. 421-431. (Reprinted in Robinson [1979b], pp. 88-98.)
Bueno, O. [2000]: “Empiricism, Mathematical Change and Scientific Change”, Studies in History and Philosophy of Science 31, pp. 269-296.
Bueno, O. [2002]: “Mathematical Change and Inconsistency: A Partial Structures Approach”, in Meheus (ed.) [2002], pp. 59-79.
Cantor, G. [1883]: “Über unendliche, lineare punktmannigfaltigkeiten. V”, Mathematische Annalen 21, pp. 545-591.
Dauben, J.W. [1995]: Abraham Robinson: The Creation of Nonstandard Analysis, a Personal and Mathematical Odyssey. Princeton, NJ: Princeton University Press.
Gillies, D. (ed.) [1992]: Revolutions in Mathematics. Oxford: Clarendon Press.
Hallett, M. [1984]: Cantorian Set Theory and Limitation of Size. Oxford: Clarendon Press.
Halmos, P. [1966]: “Invariant Subspaces of Polynomially Compact Operators”, Pacific Journal of Mathematics 16, pp. 433-437.
Halmos, P. [1985]: I Want to Be a Mathematician. An Automathography. New York: Springer-Verlag.
Henson, C.W., and Keisler, H.J. [1986]: “On the Strength of Nonstandard Analysis”, Journal of Symbolic Logic 51, pp. 377-386.
Kanamori, A. [1996]: “The Mathematical Development of Set Theory From Cantor to Cohen”, Bulletin of Symbolic Logic 2, pp. 1-71.
Kanamori, A. [1997]: “The Mathematical Import of Zermelo’s Well-Ordering Theorem”, Bulletin of Symbolic Logic 3, pp. 281-311.
Kuhn, T. [1962]: The Structure of Scientific Revolutions. Chicago: University of Chicago Press. (A second edition was published in 1970.)
Lakatos, I. (ed.) [1967]: Problems in the Philosophy of Mathematics. Amsterdam: North-Holland.
Lakatos, I. [1976]: Proofs and Refutations. Cambridge: Cambridge University Press.
Lakatos, I. [1978a]: “Cauchy and the Continuum: The Significance of Non-Standard Analysis for the History and Philosophy of Mathematics”, in Lakatos [1978c], pp. 43-60.
Lakatos, I. [1978b]: “What Does a Mathematical Proof Prove?”, in Lakatos [1978c], pp. 61-69.
Lakatos, I. [1978c]: Mathematics, Science and Epistemology. Cambridge: Cambridge University Press.
Lavine, S. [1994]: Understanding the Infinite. Cambridge, Mass.: Harvard University Press.
Leibniz, G.W. [1701]: “Mémoire de M.G.G. Leibniz touchant son sentiment sur le calcul différentiel. Journal de Trévoux”, in Leibniz [1858], p. 350.
Leibniz, G.W. [1716]: “Letter to Dangicourt, sur les monades et le calcul infinitésimal etc. (September 11)”, in Leibniz [1789], vol. 3, pp. 499-502.
Leibniz, G.W. [1789]: Opera Omnia. (Edited by L. Dutens.) Geneva.
Leibniz, G.W. [1858]: Mathematische Schriften. (Volume 6 of Die Philosophischen Scriften von G.W. Leibniz, edited by C.I. Gerhardt.) Berlin.
Luxemburg, W.A.J. [1962]: Nonstandard Analysis. Lectures on A. Robinson’s Theory of Infinitesimals and Infinitely Large Numbers. Pasadena: Mathematics Department, California Institute of Technology. (Second corrected edition, 1964.)
Machover, M. [1967]: “Non-Standard Analysis without Tears: An Easy Introduction to A. Robinson’s Theory of Infinitesimals”, Technical Report no. 27, Office of Naval Research, Information Systems Branch. Jerusalem: The Hebrew University of Jerusalem.
Meheus, J. (ed.) [2002]: Inconsistency in Science. Dordrecht: Kluwer Academic Publishers.
Moore, G.H. [1982]: Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. New York: Springer-Verlag.
Resnik, M. [1997]: Mathematics as a Science of Patterns. Oxford: Clarendon Press.
Robinson, A. [1967]: “The Metaphysics of the Calculus”, in Lakatos (ed.) [1967], pp. 27-46. (Reprinted in Robinson [1979b], pp. 537-555.)
Robinson, A. [1974]: Non-Standard Analysis. (First edition, 1966.) Amsterdam: North-Holland. (Reprinted edition Princeton, NJ: Princeton University Press, 1996.)
Robinson, A. [1979a]: Selected Papers of Abraham Robinson. Volume 1: Model Theory and Algebra. (Edited by H.J. Keisler, S. Körner, W.A.J. Luxemburg and A.D. Young.) New Haven: Yale University Press.
Robinson, A. [1979b]: Selected Papers of Abraham Robinson. Volume 2: Nonstandard Analysis and Philosophy. (Edited by H.J. Keisler, S. Körner, W.A.J. Luxemburg and A.D. Young.) New Haven: Yale University Press.
Shapiro, S. [1991]: Foundations Without Foundationalism: A Case for Second-order Logic. Oxford: Clarendon Press.
van Fraassen, B.C. [1980]: The Scientific Image. Oxford: Clarendon Press.
van Heijenoort, J. (ed.) [1967]: From Frege to Gödel. Cambridge, Mass.: Harvard University Press.
Zermelo, E. [1904]: “Beweis, dass jede Menge wohlgeordnet werden kann (Aus einem an Herrn Hilbert gerichteten Briefe)”, Mathematische Annalen 59, pp. 514-516. (English translation in van Heijenoort (ed.) [1967], pp. 139-141.)
Zermelo, E. [1908a]: “Neuer Beweis für die Möglichkeit einer Wohlordnung”, Mathematische Annalen 65, pp. 107-128. (English translation in van Heijenoort (ed.) [1967], pp. 183-198.)
Zermelo, E. [1908b]: “Untersuchungen über die Grundlagen der Mengenlehre. I”, Mathematische Annalen 65, pp. 261-281. (English translation in van Heijenoort (ed.) [1967], pp. 199-215.)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
Bueno, O. (2007). Incommensurability In Mathematics. In: van Kerkhove, B., van Bendegem, J.P. (eds) Perspectives On Mathematical Practices. Logic, Epistemology, and the Unity of Science, vol 5. Springer, Dordrecht. https://doi.org/10.1007/1-4020-5034-8_5
Download citation
DOI: https://doi.org/10.1007/1-4020-5034-8_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-5033-6
Online ISBN: 978-1-4020-5034-3
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)