Structure and Structures

  • Reinhard KahleEmail author
Part of the Boston Studies in the Philosophy and History of Science book series (BSPS, volume 334)


In this paper we critically evaluate the notion of the structure of the natural numbers with respect to the question how the internal structure of such a structure might be specified.


  1. Antos, Carolin, Sy-David Friedman, Radek Honzik, and Claudio Ternullo. 2015. Multiverse conceptions in set theory. Synthese 192(8): 2463–2488.CrossRefGoogle Scholar
  2. Bernays, Paul. 1950. Mathematische Existenz und Widerspruchsfreiheit. In Etudes de Philosophie des Sciences, 11–25. Neuchâtel: Éditions du Griffon. Reprinted in Bernays (1976), p. 92–106.Google Scholar
  3. Bernays, Paul. 1976. Abhandlungen zur Philosophie der Mathematik. Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
  4. Bishop, Errett. 1967. Foundations of constructive analysis. New York: McGraw-Hill.Google Scholar
  5. Bishop, Errett. 1975. The crisis in contemporary mathematics. Historia Mathematica 2(4): 507–517.CrossRefGoogle Scholar
  6. Black, Max. 1946. Critical thinking. New York: Prentice-Hall.Google Scholar
  7. Born, Max. 1966. Symbol and reality. Dialectica 20(2): 143–157. Archives de l’institut international des sciences théoriques, vol. 14: objectivité et réalité dans les différentes sciences (colloque de l’académie internationale de philosophie des sciences).Google Scholar
  8. Bourbaki, Nicolas. 1950. Architecture of mathematics. The American Mathematical Monthly 57(4): 221–232.CrossRefGoogle Scholar
  9. Corry, Leo. 2004. Modern algebra and the rise of mathematical structures, 2nd revised ed. Basel: Birkhäuser.CrossRefGoogle Scholar
  10. Dedekind, Richard. 1888. Was sind und was sollen die Zahlen? Braunschweig: Vieweg.Google Scholar
  11. Feferman, Solomon, Harvey M. Friedman, Penelope Maddy, and John R. Steel. 2000. Does mathematics need new axioms? Bulletin of Symbolic Logic 6(4): 401–446.CrossRefGoogle Scholar
  12. Fairtlough, Matthew V. H., and Stanley S. Wainer. 1998. Hierarchies of provably recursive functions. In Handbook of proof theory, ed. S. Buss, 149–207. Amsterdam: North-Holland.CrossRefGoogle Scholar
  13. Hardy, G. H. 1921. Srinivasa Ramanujan. Proceedings of the London Mathematical Society s2–19(1): xl–lviii.CrossRefGoogle Scholar
  14. Hermes, Hans, and Werner Markwald. 1958. Grundlagen der Mathematik. In Grundzüge der Mathematik, vol. I, ed. H. Behnke, K. Fladt, and W. Süss, 1–89. Göttingen: Vandenhoek & Ruprecht.Google Scholar
  15. Kahle, Reinhard. 2017. Mathematical truth revisited: Mathematics as a toolbox. In Varieties of scientific realism, ed. Evandro Agazzi, 395–406. Cham: Springer.CrossRefGoogle Scholar
  16. Kahle, Reinhard, and Wilfried Keller. 2015. Syntax versus semantics. In 4th International Conference on Tools for Teaching Logic, ed. M.A. Huertas, J. Marcos, M. Manzano, S. Pinchinat, and F. Schwarzentruber, 75–84. University of Rennes 1.Google Scholar
  17. Kohlenbach, Ulrich. 2008. Applied proof theory. Berlin: Springer.Google Scholar
  18. Kreisel, Georg. 1967. Informal rigour and completeness proofs. In Problems in the philosophy of mathematics. Studies in logic and the foundations of mathematics, vol. 47, ed. I. Lakatos, 138–186. Amsterdam: North-Holland.Google Scholar
  19. Peano, Giuseppe. 1889. Arithmetices Principia Novo Methodo Exposita. Augustae Taurinorum: Bocca.Google Scholar
  20. Rautenberg, Wolfgang. 2006. A concise introduction to mathematical logic, 2nd ed. New York: Springer.Google Scholar
  21. Reich, Karin. 2006. Große Forschung, Große Lehre: Emil Artin. In Zum Gedenken an Emil Artin (1898–1962). Hamburger Universitätsreden. Neue Folge, vol 9, ed. Der Präsident der Universität Hamburg, 17–41. Hamburg:Hamburg University Press.Google Scholar
  22. Shoenfield, Joseph R. 2000. Mathematical logic. Reading, MA: Addison-Wesley, 1967. Reprinted by ASL, AK Peters.Google Scholar
  23. Thiel, Christian. 2006. Kreativität in der mathematischen Grundlagenforschung. In Kreativität, ed. G. Abel, 360–375. Hamburg: Mainer. Kolloquienbeträge vom XX. Deutschen Kongreß für Philosophie, 26.–30. September 2005 an der Technischen Universität Berlin.Google Scholar
  24. Zweistein (alias Thomas von Randow). 1963. Logelei. Die Zeit, Ausgabe 31, 2.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CMA & DM, FCT, Universidade Nova de LisboaCaparicaPortugal

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