Synthese

, Volume 171, Issue 1, pp 157–173 | Cite as

Logical structuralism and Benacerraf’s problem

Article

Abstract

There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called “Benacerraf’s Problem”, which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a different way than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf’s problem.

Keywords

Philosophy of mathematics Structuralism Dedekind Benacerraf’s problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Benacerraf P. (1965) What numbers could not be. Philosophical Review 74: 47–73CrossRefGoogle Scholar
  2. Benacerraf P. (1970) Mathematical truth. Journal of Philosophy 70: 661–679CrossRefGoogle Scholar
  3. Carnap, R. (1956). Empiricism, semantics, and ontology. In Meaning and necessity. University of Chicago Press.Google Scholar
  4. Curry, H. (1983). Remarks on the definition and nature of mathematics. In P. Benacerraf H. Putnam (Eds.), Philosophy of mathematics. Cambridge University Press.Google Scholar
  5. Dedekind, R. (1872). Continuity and irrational numbers. In W. Ewald (Ed.), From Kant to Hilbert: A sourcebook in the foundations of mathematics (Vol. II). Clarendon Press.Google Scholar
  6. Dedekind, R. (1888a). Letter to Heinrich Weber. In W. Ewald (Ed.), From Kant to Hilbert: A sourcebook in the foundations of mathematics (Vol. II). Clarendon Press.Google Scholar
  7. Dedekind, R. (1888b). Was sind und was sollen die Zahlen. In W. Ewald (Ed.), From Kant to Hilbert: A sourcebook in the foundations of mathematics (Vol. II). Clarendon Press.Google Scholar
  8. Dedekind, R. (1890). Letter to Keferstein. In J. van Heijenoort (Ed.), From Frege to Gödel: A sourcebook in mathematical logic (2nd ed., pp. 1879–1931). Harvard University Press.Google Scholar
  9. Ferreirós, J. (2007). Labyrinth of thought: A history of set theory and its role in modern mathematics (2nd ed.). Birkhauser Verlag.Google Scholar
  10. Field, H. (1989). Realism, mathematics, and modality. Blackwell.Google Scholar
  11. Kitcher, P. (1986). Frege, Dedekind, and the philosophy of mathematics. In L. Haaparanta & J. Hintikka (Eds.), Frege synthesized. D. Reidel Publishing Company.Google Scholar
  12. Maddy, P. (1990). Realism in mathematics. Clarendon Press.Google Scholar
  13. Parsons C. (1990) The structuralist view of mathematical objects. Synthese 84: 303–346CrossRefGoogle Scholar
  14. Reck E. (2003) Dedekind’s structuralism: An interpretation and partial defense. Synthese 137: 369–419CrossRefGoogle Scholar
  15. Resnik M. (1981) Mathematics as a science of patterns: Ontology and reference. Nous 14(4): 529–550Google Scholar
  16. Resnik M. (1982) Mathematics as a science of patterns: Epistemology. Nous 16(1): 95–105Google Scholar
  17. Shapiro, S. (2005). Philosophy of mathematics and its logic: Introduction. In S. Shapiro (Ed.), The Oxford handbook of philosophy of mathematics and logic. Oxford University Press.Google Scholar
  18. Sieg W., Schlimm D. (2005) Dedekind’s analysis of number: Systems and axioms. Synthese 147: 121–170CrossRefGoogle Scholar
  19. Tait, W. (1996). Frege versus Cantor and Dedekind: On the concept of number. In W. Tait (Ed.), Frege, Russell, Wittgenstein: Essays in early analytic philosophy (in honor of Leonard Linsky). Open Court Press.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.University of VictoriaVictoriaCanada

Personalised recommendations