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Synthese

, Volume 137, Issue 3, pp 369–419 | Cite as

Dedekind's Structuralism: An Interpretation and Partial Defense

  • Erich H. Reck
Article

Abstract

Various contributors to recent philosophy of mathematics havetaken Richard Dedekind to be the founder of structuralismin mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the main contemporary variants. In addition, I argue that his writings contain philosophical insights that are worth reexamining and reviving. The discussion focusses on Dedekind's classic essay “Was sind und was sollen die Zahlen?”, supplemented by evidence from “Stetigkeit und irrationale Zahlen”, his scientific correspondence, and his Nachlaß.

Keywords

Structuralist View Recent Philosophy Philosophical Insight Contemporary Variant Classic Essay 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. Awodey, S. and E. Reck: 2001, 'Completeness and Categoricity: 19th Century Aximomatics to 21st Century Semantics', Technical Report CMU-PHIL-118, Carnegie Mellon University. Forthcoming, in two parts, in History and Philosophy of Logic.Google Scholar
  2. Benacerraf, P.: 1965, 'What Numbers Could Not Be', Philosophical Review 74, 47–73. Reprinted in P. Benacerraf and H. Putnam, Philosophy of Mathematics, 2nd edn, Cambridge University Press, Cambridge, 1983, pp. 272–294.Google Scholar
  3. Bernays, P. 1950, 'Mathematische Existenz und Wiederspruchsfreiheit', in Abhandlungen zur Philosophie der Mathematik, Wissenschaftliche Buchgesellschaft, Darmstadt, pp. 92–106.Google Scholar
  4. Burgess, J.: 1999, 'Review of (Shapiro 1997)', Notre Dame Journal of Formal Logic 40, 283–291.Google Scholar
  5. Cantor, G.: 1883, Grundlagen einer allgemeinen Mannigfaltigkeitslehre, Teubner, Leipzig.Google Scholar
  6. Cassirer, E.: 1910, Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik, B. Cassirer, Berlin.Google Scholar
  7. Dedekind, R.: 1854, 'Ñber die Einführung neuer Funktionen in der Mathematik', Reprinted in R. Fricke et al. (eds.), Gesammelte Mathematische Werke, Vol. 3, Vieweg, Braunschweig, 1932, pp. 428–438.Google Scholar
  8. Dedekind, R.: 1872, 'Stetigkeit und irrationale Zahlen', Reprinted in R. Fricke et al. (eds.), Gesammelte Mathematische Werke, Vol. 3, Vieweg, Braunschweig, 1932, pp. 315–334. Engl. trans. 'Continuity and Irrational Numbers', in R. Dedekind, Essays on the Theory of Numbers, Dover, New York, 1963, pp. 1–27. W. W. Behman, trans.Google Scholar
  9. Dedekind, R.: 1876a, 'Brief an Lipschitz 1', in R. Fricke et al. (eds.), Gesammelte Mathematische Werke, Vol. 3, Vieweg, Braunschweig, 1932, pp. 468–474.Google Scholar
  10. Dedekind, R.: 1876b, 'Brief an Lipschitz 2', in R. Fricke et al. (eds.), Gesammelte Mathematische Werke, Vol. 3, Vieweg, Braunschweig, 1932, pp. 474–479.Google Scholar
  11. Dedekind, R.: 1887, 'Was sind und was sollen die Zahlen?, dritter Entwurf, mit zugehörigen Notizen', unpublished manuscript, Dedekind Nachlaß, University of Göttingen, Germany, item Cod. Ms. Dedekind III, 1 (III).Google Scholar
  12. Dedekind, R.: 1888a. 'Brief an Weber', in R. Fricke et al. (eds.), Gesammelte Mathematischen Werke, Vol. 3, Vieweg, Braunschweig, 1932, pp. 488–490.Google Scholar
  13. Dedekind, R.: 1888b, 'Was sind und was sollen die Zahlen?', Reprinted in R. Fricke et al. (eds.), Gesammelte Mathematische Werke, Vol. 3, Vieweg, Braunschweig, 1932, pp. 335–391. Engl. trans. 'The Nature and Meaning of Numbers', in R. Dedekind, Essays on the Theory of Numbers, Dover, New York, 1963, pp. 29–115. W. W. Behman, trans.Google Scholar
  14. Dedekind, R.: 1890, 'Letter to Keferstein', in J. van Heijenoort (ed.), From Frege to Gödel, Harvard University Press, Cambridge MA, 1967, pp. 98–103. S. Bauer-Mengelberg, trans.Google Scholar
  15. Dedekind, R.: 1932, Gesammelte Mathematischen Werke, Volumes 1–3, Vieweg, Braunschweig.Google Scholar
  16. Dedekind, R.: 1963, Essays on the Theory of Numbers, Dover, New York. W. W. Behman, trans.Google Scholar
  17. Frege, G.: 1884, Die Grundlagen der Arithmetik, Koebner, Breslau. Engl. trans. The Foundations of Arithmetic. Northwestern University Press, Evanston, 1968, J. L. Austin, trans.Google Scholar
  18. Friedman, M.: 2000, A Parting of the Ways: Carnap, Cassirer, and Heidegger, Open Court, Chicago.Google Scholar
  19. Hellman, G.: 1989, Mathematics Without Numbers, Oxford University Press, Oxford.Google Scholar
  20. Hellman, G.: 2001, 'Three Varieties of Mathematical Structuralism', Philosophia Mathematica 9, 184–211.Google Scholar
  21. Hilbert, D.: 1900, 'Ñber den Zahlbegriff', Jahresbericht der Deutschen Mathematiker-Vereinigung 8, 180–84. Engl. trans. 'On the Concept of Number', in W. Ewald (ed.), From Frege to Hilbert, Vol. 2, Oxford University Press, Oxford, 1996, pp. 1089–1095. W. Ewald, trans.Google Scholar
  22. Keränen, J.: 2001, 'The Identity Problem for Realist Structuralism', Philosophia Mathematica 9, 308–330.Google Scholar
  23. Kitcher, P.: 1986, 'Frege, Dedekind, and the Philosophy of Mathematics', in L. Haaparanta and J. Hintikka (eds.), Frege Synthesized, Reidel, Dordrecht, pp. 299–343.Google Scholar
  24. Kusch, M.: 1995, Psychologism: A Case Study in the Sociology of Knowledge, Routledge, London.Google Scholar
  25. McCarty, D.: 1995, 'The Mysteries of Richard Dedekind', in J. Hintikka (ed.), Essays on the Development of the Foundations of Mathematics, Kluwer, Dordrecht, pp. 53–96.Google Scholar
  26. Parsons, C.: 1990, 'The Structuralist View of Mathematical Objects', Synthese 84, 303–346.Google Scholar
  27. Peano, G.: 1889, Arithmetices principia nova methodo exposita, Bocca, Turin. Engl. trans. 'The Principles of Arithmetic, Presented by a New Method', in H. Kennedy (ed. and trans.), Selected Works of Giuseppe Peano, London, Allen and Unwin, 1973, pp. 101–135.Google Scholar
  28. Rath, M.: 1994, Der Psychologismusstreit in der deutschen Philosophie, Alber, Freiburg.Google Scholar
  29. Reck, E. and M. Price: 2000, 'Structures and Structuralism in Contemporary Philosophy of Mathematics', Synthese 125, 341–383.Google Scholar
  30. Russell, B.: 1903, Principles of Mathematics, Allen and Unwin, London. Reprinted by Cambridge University Press, Cambridge, 1937.Google Scholar
  31. Schlimm, D.: 2000, 'Richard Dedekind: Axiomatic Foundations of Mathematics', Master's thesis, Carnegie Mellon University, Pittsburgh.Google Scholar
  32. Shapiro, S.: 1997, Philosophy of Mathematics: Structure and Ontology, Oxford University Press, Oxford.Google Scholar
  33. Sieg, W.: 2002, 'Beyond Hilbert's Reach?', in D. Malament (ed.), Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics, Open Court, Chicago, pp. 363–405.Google Scholar
  34. Stein, H.: 1988, 'Logos, Logic, Logistiké: Some Philosophical Remarks on Nineteenth Century Transformations in Mathematics', in W. Aspray and P. Kitcher (eds.), History and Philosophy of Mathematics, University of Minnesota Press, Minneapolis, pp. 238–259.Google Scholar
  35. Tait, W. W.: 1986, 'Truth and Proof: The Platonism of Mathematics', Synthese 69, 341–370.Google Scholar
  36. Tait, W. W.: 1997, 'Cantor versus Frege and Dedekind: On the Concept of Number', in W. W. Tait (ed.), Early Analytic Philosophy, Open Court, Chicago, pp. 213–48.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Erich H. Reck
    • 1
  1. 1.Department of PhilosophyUniversity of CaliforniaRiversideU.S.A. E-mail

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