Proofs, Proofs, Proofs, and Proofs

  • Manfred Kerber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6167)


In logic there is a clear concept of what constitutes a proof and what not. A proof is essentially defined as a finite sequence of formulae which are either axioms or derived by proof rules from formulae earlier in the sequence. Sociologically, however, it is more difficult to say what should constitute a proof and what not. In this paper we will look at different forms of proofs and try to clarify the concept of proof in the wider meaning of the term. This has implications on how proofs should be represented formally.


Theorem Prove Deduction System Mathematical Proof Plausible Reasoning Natural Deduction 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abrams, P.S.: An APL machine. SLAC-114 UC-32 (MISC). Stanford University, Stanford, California (1970)Google Scholar
  2. 2.
    Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth through Proof. Academic Press, Orlando (1986)MATHGoogle Scholar
  3. 3.
    Ayer, A.J.: Language, Truth and Logic, 2nd edn., 1951 edn. Victor Gollancz Ltd. London (1936)Google Scholar
  4. 4.
    Bourbaki, N.: Théorie des ensembles. In: Éléments de mathématique, Fascicule 1, Hermann, Paris, France (1954)Google Scholar
  5. 5.
    de Bruijn, N.G.: A survey of the project Automath. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry - Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 579–606. Academic Press, London (1980)Google Scholar
  6. 6.
    Davis, M.: The early history of automated deduction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, pp. 5–14. Elsevier Science, Amsterdam (2001)Google Scholar
  7. 7.
    Frege, G.: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle (1879)Google Scholar
  8. 8.
    Hales, T.: The Flyspek Project (2010),
  9. 9.
    Hardy, G.: A Mathematician’s Apology. Cambridge University Press, London (1940)Google Scholar
  10. 10.
    van Heijenoort, J. (ed.): From Frege to Gödel – A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press, Cambridge (1967)MATHGoogle Scholar
  11. 11.
    Jamnik, M.: Mathematical Reasoning with Diagrams: From Intuitions to Automation. CSLI Press, Stanford (2001)Google Scholar
  12. 12.
    Kline, M.: Mathematics – The Loss of Certainty. Oxford University Press, New York (1980)Google Scholar
  13. 13.
    Lakatos, I.: Proofs and Refutations. Cambridge University Press, Cambridge (1976)MATHGoogle Scholar
  14. 14.
    Maxwell, E.A.: Fallacies in Mathematics. Cambridge University Press, Cambridge (1959)MATHCrossRefGoogle Scholar
  15. 15.
    McCune, W.: Solution of the Robbins problems. Journal of Automated Reasoning 19(3), 263–276 (1997), MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Nederpelt, R., Geuvers, H., de Vrijer, R. (eds.): Selected Papers on Automath. Studies in Logic and the Foundations of Mathematics, vol. 133. North-Holland, Amsterdam (1994)MATHGoogle Scholar
  17. 17.
    Nederpelt, R., Kamareddine, F.: An abstract syntax for a formal language of mathematics. In: The Fourth International Tbilisi Symposium on Language, Logic and Computation (2001),
  18. 18.
    Pólya, G.: Mathematics and Plausible Reasoning. Princeton University Press, Princeton (1954); Two volumes, Vol. 1: Induction and Analogy in Mathematics, Vol. 2: Patterns of Plausible InferenceGoogle Scholar
  19. 19.
    Prawitz, D.: Natural Deduction – A Proof Theoretical Study. Almqvist & Wiksell, Stockholm (1965)MATHGoogle Scholar
  20. 20.
    Rips, L.J.: The Psychology of Proof – Deductive Reasoning in Human Thinking. The MIT Press, Cambridge (1994)MATHGoogle Scholar
  21. 21.
    Robinson, J.A.: A machine oriented logic based on the resolution principle. Journal of the ACM 12, 23–41 (1965)MATHCrossRefGoogle Scholar
  22. 22.
    Trybulec, A.: The Mizar logic information language. In: Studies in Logic, Grammar and Rhetoric, Białystok, Poland, vol. 1 (1980)Google Scholar
  23. 23.
    Whitehead, A.N., Russell, B.: Principia Mathematica, vol. I. Cambridge University Press, Cambridge (1910)MATHGoogle Scholar
  24. 24.
    Wittgenstein, L.: Bemerkungen über die Grundlagen der Mathematik. In: Suhrkamp-Taschenbuch Wissenschaft, 3rd edn., Frankfurt, Germany, vol. 506 (1989)Google Scholar
  25. 25.
    Claus Zinn. Understanding Informal Mathematical Discourse. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Manfred Kerber
    • 1
  1. 1.Computer ScienceUniversity of BirminghamBirminghamEngland

Personalised recommendations