Archive for History of Exact Sciences

, Volume 72, Issue 2, pp 99–141 | Cite as

The modernity of Dedekind’s anticipations contained in What are numbers and what are they good for?



We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set theory, which undoubtedly point towards the mathematics of twentieth and twenty-first centuries.

Mathematics Subject Classification

01A55 11-03 03B30 03D20 03E30 08A60 18A30 



We would like to thank all who helped us by reading parts of the manuscript and suggesting mathematical improvements: Enric Cosme, Emily Riehl, Roberto Rubio, Sergiu Rudeanu, and Rafael Sivera. Our thanks are further due to our dear friend José García Roca—example of intelligence, goodness, and integrity—for correcting the English and suggesting improvements. And also for his tireless encouragement and invaluable continued moral support. Moreover, we are greatly indebted to the reviewer for his helpful comments and important stylistic suggestions.


  1. Albers, D.J., and G.L. Alexanderson (eds.). 2008. Mathematical people, 2nd ed. Wellesley, MA: A. K. Peters Ltd.MATHGoogle Scholar
  2. Alexandroff, P. 1928. Untersuchungen über Gestalt und Lage abgeschlossneser Mengen. Annals of Mathmatics 2 (30): 101–187.CrossRefMATHGoogle Scholar
  3. Aristotle, 1984. The complete works of Aristotle. (The Revised Oxford Translation), vol. 2, ed. J. Barnes. Princeton, NJ: Princeton University Press.Google Scholar
  4. Baer, R. 1937. Abelian fields and duality of Abelian groups. American Journal of Mathematics 59: 869–888.MathSciNetCrossRefMATHGoogle Scholar
  5. Birkhoff, G. 1933. On the combination of subalgebras. Proceedings of the Cambridge Philosophical Society 29: 441–464.CrossRefMATHGoogle Scholar
  6. Birkhoff, G. 1934. Note on the paper “On the combination of subalgebras.”. Proceedings of the Cambridge Philosophical Society 30: 200.CrossRefMATHGoogle Scholar
  7. Birkhoff, G. 1935. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society 31: 433–454.CrossRefMATHGoogle Scholar
  8. Birkhoff, G. 1987. Selected papers on algebra and topology. Boston: Birkhäuser Boston Inc.MATHGoogle Scholar
  9. Büchi, J.R. 1989. Finite automata, their algebras and grammars. Towards a theory of formal expressions. Edited and with a preface by Dirk Siefkes. New York: Springer-Verlag.Google Scholar
  10. Cantor, G. 1955. Contributions to the founding of the theory of transfinite numbers. Translated, and provided with an introduction and notes, by Philip E. B. Jourdain. New York: Dover Publications, Inc.Google Scholar
  11. Dedekind, R. 1888. Was sind und was sollen die Zahlen? Braunschweig: Friedr. Vieweg & Sohn. (English trans: R. Dedekind). Essays on the theory of numbers. I: Continuity and irrational numbers. II: The nature and meaning of numbers. (Authorized trans: Wooster Woodruff Beman), Dover, New York, 1963. Beman’s translation has been extensively revised by William Ewald in (Ewald, 1996, pp. 787–833).Google Scholar
  12. Dedekind, R. 1930–1932. Gessammelte mathematische Werke. Herausgegeben von Robert Fricke, Emmy Noether, Øystein Ore. Vols. I–III. Braunschweig: Friedr. Vieweg & Sohn.Google Scholar
  13. de Vries, J. 1993. Elements of topological dynamics. Mathematics and its applications, vol. 257. Dordrecht: Kluwer Academic Publishers Group.CrossRefGoogle Scholar
  14. Dieudonné, J. 1989. A History of algebraic and differential topology 1900–1960. Boston, Basel: Birkhäuser.MATHGoogle Scholar
  15. Dugac, P. 1976. Richard Dedekind et les fondements des mathématiques (avec de nombreux textes inédites). Préface de Jean Dieudonné. Collection des Travaux de l’Académie Internationale d’Histoire des Sciences No. 24. L’Histoire des Sciences, Textes et Études. Paris: Librairie Philosophique J. Vrin.Google Scholar
  16. Ewald, W. 1996. From Kant to Hilbert: A source book in the foundations of mathematics. Vol. II. Ed. William Ewald. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press.Google Scholar
  17. Frege, G. 1879. Begriffsschrift, Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Verlag von L. Nebert.Google Scholar
  18. Frege, G. 1966. Grundgesetze der Arithmetik. Band I, II. Hildesheim: Georg Olms Verlagsbuchhandlung.MATHGoogle Scholar
  19. Freudenthal, H. 1937. Entwicklungen von Räumen und ihren Gruppen. Compositio Mathematica 4: 145–234.MathSciNetMATHGoogle Scholar
  20. Gödel, K. 1931. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik 38: 173–198.MathSciNetCrossRefMATHGoogle Scholar
  21. Gray, J. 2000. Linear differential equations and group theory from Riemann to Poincaré, 2nd ed. Boston: Birkhäuser Boston Inc.MATHGoogle Scholar
  22. Grassmann, H. 1861. Lehrbuch der Arithmetik für höhere Lehranstalten. Berlin: Verlag von T. C. F. Enslin (Adolph Enslin).Google Scholar
  23. Grothendieck, A. 1971. Revêtements étales et groupe fondamental (SGA1), éd. par A. Grothendieck, Lecture notes in mathematics, No. 224. Berlin, New York: Springer.Google Scholar
  24. Heck Jr., R.G. 2012. Reading Frege’s Grundgesetze. Oxford: The Clarendon Press, Oxford University Press.MATHGoogle Scholar
  25. Herbrand, J. 1933. Théorie arithmétique des corps de nombres de degré infini II. Mathematische Annalen 108: 699–717.MathSciNetCrossRefMATHGoogle Scholar
  26. Hilbert, D. 1931. Die Grundlegun der elementaren Zahlenlehre. Mathematische Annalen 104: 485–494.MathSciNetCrossRefMATHGoogle Scholar
  27. Hocking, J.G., and G.S. Young. 1988. Topology, 2nd ed. New York: Dover Publications Inc.MATHGoogle Scholar
  28. Krömer, R. 2010. From Cantor to Sheaves: The development of the concepts of direct and inverse limits—A case study on shifts in mathematical methodology in the prehistory of category theory. In Symposium on the philosophy of the logic of sheaves, October 19–21. 2010. Universidad del Valle, Cali, Colombia.Google Scholar
  29. Kurosh, A.G. 1935. Kombinatorischer Aufbau der bikompakten topologischen Räume. Compositio Mathematica 2: 471–476.MathSciNetMATHGoogle Scholar
  30. Lawvere, F.W. 1964. An elementary theory of the category of sets. Proceedings of National Academy of Sciences USA 52: 1506–1511.MathSciNetCrossRefMATHGoogle Scholar
  31. Lipschitz, R. 1986. Briefwechsel mit Cantor, Dedekind, Helmholtz, Kronecker, Weierstrass und anderen. Ed. Winfried Scharlau. Dokumente zur Geschichte der Mathematik, 2. Freiburg: Friedr. Vieweg & Sohn, Braunschweig; Deutsche Mathematiker Vereinigung.Google Scholar
  32. MacLane, S. 1970. Hamiltonian mechanics and geometry. American Mathematical Monthly 77: 570–586.MathSciNetCrossRefGoogle Scholar
  33. MacLane, S., and G. Birkhoff. 1988. Algebra, 3rd ed. New York: Chelsea Publishing Co.MATHGoogle Scholar
  34. Manes, E.G. 1976. Algebraic theories, vol. 26., Graduate texts in mathematics New York, Heidelberg: Springer.MATHGoogle Scholar
  35. Manes, E.G., and M.A. Arbib. 1986. Algebraic approaches to program semantics., Texts and monographs in computer science, AKM series in theoretical computer science New York: Springer.CrossRefMATHGoogle Scholar
  36. Minkowski, H. 1905. Peter Gustav Lejeune Dirichlet und seine Bedeutung für die heutige Mathematik. Jahresber. Deutsch. Math.-Verein. 14: 149–163.MATHGoogle Scholar
  37. Peano, I. 1889. Arithmetices principia, nova methodo exposita. Turin: Fratres Bocca. (English trans: G. Peano). The principles of arithmetic, presented by a new method. (Trans: Jean van Heijenoort in van Heijenoort) (1967, pp. 83–97).Google Scholar
  38. Peirce, C.S. 1881. On the logic of number. American Journal of Mathematics 4: 85–95.MathSciNetCrossRefMATHGoogle Scholar
  39. Pontrjagin, L. 1931. Über den algebraischen Inhalt topologischer Dualitätssätze. Mathematische Annalen 105: 165–205.MathSciNetCrossRefMATHGoogle Scholar
  40. Pontrjagin, L. 1934. Sur les groupes topologiques compacts et le cinquième problème de Hilbert. Comptes rendus de l’Académie des Sciences Paris 198: 238–240.MATHGoogle Scholar
  41. Raussen, M., and Chr Skau. 2004. Interview with Michael Atiyah and Isadore Singer. Newsletter of the European Mathematical Society 53: 24–30.MATHGoogle Scholar
  42. Russell, B. 1903. The principles of mathematics, vol. 1. Cambridge: Cambridge University Press.MATHGoogle Scholar
  43. Sancho Guimerá, J.B. 1959. Teoría de cuerpos algebraicos con ley de composición conmutativa. Revista Matemática Hispano-Americana, \(4.^{a}\) Serie 19: 18–39.Google Scholar
  44. Smith, H.J.S. 1875. On the integration of discontinuous functions. Proceedings of the London Mathematical Society 6: 140–153.MathSciNetMATHGoogle Scholar
  45. van der Waerden, B.L. 1975. On the sources of my book Moderne Algebra. Historia Mathematica 2: 31–40.MathSciNetCrossRefMATHGoogle Scholar
  46. van Heijenoort, J. (ed.). 1967. From Frege to Gödel. A source book in mathematical logic, 1879–1931. Cambridge, MA: Harvard University Press.MATHGoogle Scholar
  47. Weber, H. 1891/92. Leopold Kronecker. Jahresber. Deutsch. Math.-Verein 2: 5–31.Google Scholar
  48. Weil, A. 1975. L’intégration dans les groupes topologiques et ses applications. Hermann: Deuxième édition.MATHGoogle Scholar
  49. Zermelo, E. 1967. A new proof of the possibility of a well-ordering. In From Frege to Gödel. A source book in mathematical logic, 1879–1931, ed. J. van Heijenoort, 183–198. Cambridge, MA: Harvard University Press.Google Scholar
  50. Zermelo, E. 2010. Ernst Zermelo: Collected Works/Gesammelte Werke. Vol. I/Band I. Set Theory, Miscellanea/Mengenlehre, Varia. Ed. Heinz-Dieter Ebbinghaus, Craig G. Fraser and Akihiro Kanamori. Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften [Publications of the Mathematics and Natural Sciences Section of Heidelberg Academy of Sciences], 21. Berlin: Springer.Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Lógica y Filosofía de la CienciaUniversidad de ValenciaValenciaSpain

Personalised recommendations