Logica Universalis

, Volume 6, Issue 3–4, pp 459–475 | Cite as

Which Mathematical Logic is the Logic of Mathematics?

  • Jaakko Hintikka


The main tool of the arithmetization and logization of analysis in the history of nineteenth century mathematics was an informal logic of quantifiers in the guise of the “epsilon–delta” technique. Mathematicians slowly worked out the problems encountered in using it, but logicians from Frege on did not understand it let alone formalize it, and instead used an unnecessarily poor logic of quantifiers, viz. the traditional, first-order logic. This logic does not e.g. allow the definition and study of mathematicians’ uniformity concepts important in analysis. Mathematicians’ stronger logic was rediscovered around 1990 as the form of independence-friendly logic which hence is not a new logic nor a further development of ordinary first-order logic but a richer version of it.

Mathematics Subject Classification (2010)

03 01 


Quantifier independence (in logic) first-order logic IF logic epsilon–delta technique 


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Bibliographies of the original literature are found in [3, 11], and [12].

  1. 1.
    Alexander A.: Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics. Harvard University Press, Cambridge (2010)zbMATHGoogle Scholar
  2. 2.
    Bottazzini, U.: The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer, Heidelberg/New York (1986)Google Scholar
  3. 3.
    Bourbaki, N.: Théorie des ensembles. Hermann, Paris (1938)Google Scholar
  4. 4.
    Brabenee, R.Z.: Resources for the Study of Real Analysis. The Mathematical Association of America, Washington, D.C. (2004)Google Scholar
  5. 5.
    Cauchy, A.-L.: Cours d’analyse de l’Ecole Royale Polytechnique. De Bure frères, Paris (1821)Google Scholar
  6. 6.
    Forsyth A.R.: Theory of Functions of a Complex Variable. Cambridge University Press, Cambridge (1893)zbMATHGoogle Scholar
  7. 7.
    Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. L. Nebert, Halle (1879)Google Scholar
  8. 8.
    Frege, G.: Review of H. Cohen, Das Prinzip der Infinitesimal-Methode und seine Geschichte. Zeitschrift für Philosophie und Philosophishe Kritik. 87, 324–329 (1885)Google Scholar
  9. 9.
    Frege, G.: Grundgesetze der Arithmetik. Begriffsschriftlich abegeleitet, vol. 1. H. Pohle, Jena (1893)Google Scholar
  10. 10.
    Gleason, A. M.: Fundamentals of Abstract Analysis. Jones and Bartlett, Boston (1991)Google Scholar
  11. 11.
    Grabiner J.: The Origins of Cauchy’s Rigorous Calculus. MIT Press, Cambridge (1981)zbMATHGoogle Scholar
  12. 12.
    Grattan-Guinness, I.: The Development of the Foundations of Mathematical Analysis from Euler to Riemann. MIT Press, Cambridge, MA (1970)Google Scholar
  13. 13.
    Hilbert, D.: Grundlagen der Geometrie. Teubner, Leipzig (1899; many later editions)Google Scholar
  14. 14.
    Hilbert D.: Axiomatisches Denken. Mathematische Annalen. 78, 405–415 (1918)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hilbert, D., Bernays, P.: Grundlagen der Mathematik, vols. 1, 2., Springer, Heidelberg/Berlin (1936–1939)Google Scholar
  16. 16.
    Hintikka, J.: What the bald man can tell us. In: Biletzky, A. (ed.) Hues of Philosophy: Essays in Memory of Ruth Manor. College Publications, London (2011)Google Scholar
  17. 17.
    Hintikka, J.: On the significance of incompleteness results (forthcoming)Google Scholar
  18. 18.
    Hintikka, J.: Kant’s theory of mathematics: What theory? What mathematics? (forthcoming)Google Scholar
  19. 19.
    Hintikka, J.: IF logic, definitions and the Vicious Circle principle. J. Philos. Logic (forthcoming, probably 2012)Google Scholar
  20. 20.
    Hintikka, J., Sandu, G.: Game theoretical semantics. In: van Benthem, J., ter Meulen, A. (eds.) Handbook of Logic and Language, pp. 361–410. Elsevier, Amsterdam (1996)Google Scholar
  21. 21.
    Moore, G.H.: Zermelo’s Axiom of Choice: Its Origins, Development and Influence. Springer, Heidelberg/New York (1982)Google Scholar
  22. 22.
    Peirce C.S.: A review of Forsyth, Harkness and Picard. Nation. 58, 197–199 (1894)Google Scholar
  23. 23.
    Peirce, C.S.: In: Harstshorne, C., Weiss, P., Burk, A. (eds.) Collected Papers, vols. 1–8. Harvard University Press, Cambridge, Mass. (1931–1958) (Referred to as CP.)Google Scholar
  24. 24.
    Pietarinen, A.-V.: Signs of Logic: Peircean Themes in the Philosophy of Logic, Springer, Dordrecht (2006)Google Scholar
  25. 25.
    Seidel, P.L.: Note über eine Eigenschaft der Reihen, welche discontinuerliche Funktionen darstellen, in Ostwald‘s Klassiker, vol. 116, pp. 35–45 (1900; original 1948)Google Scholar
  26. 26.
    Tulenheimo, T.: Independence-friendly logic, In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Summer 2009 Edition). URL: (2009)
  27. 27.
    Weierstrass K.T.W.: Abhandlungen aus der Functionenlehre. Springer, Berlin (1886)zbMATHGoogle Scholar
  28. 28.
    Weierstrass, K.T.W.: Mathematische Werke, vols. 1–7. Mayer & Müller, Berlin (1894–1915) (reprinted: G. Olms, Hildesheim 1967)Google Scholar
  29. 29.
    Whitaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1952)Google Scholar

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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of PhilosophyBoston UniversityBostonUSA

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