Proof, Message and Certificate

  • Andrea Asperti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7362)

Abstract

The recent achievements obtained by means of Interactive Theorem Provers in the automatic verification of complex mathematical results have reopened an old and interesting debate about the essence and purpose of proofs, emphasizing the dichotomy between message and certificate. We claim that it is important to prevent the divorce between these two epistemological functions, discussing the implications for the field of mathematical knowledge management.

Keywords

Formal Proof Mathematical Proof Proof Assistant Chess Player Declarative Language 
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. 1.
    Alama, J., Brink, K., Mamane, L., Urban, J.: Large Formal Wikis: Issues and Solutions. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) Calculemus/MKM 2011. LNCS, vol. 6824, pp. 133–148. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Asperti, A., Armentano, C.: A page in number theory. Journal of Formalized Reasoning 1, 1–23 (2008)MathSciNetMATHGoogle Scholar
  3. 3.
    Asperti, A., Avigad, J.: Zen and the art of formalization. Mathematical Structures in Computer Science 21(4), 679–682 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Asperti, A., Sacerdoti Coen, C.: Some Considerations on the Usability of Interactive Provers. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 147–156. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Asperti, A., Geuvers, H., Natarajan, R.: Social processes, program verification and all that. Mathematical Structures in Computer Science 19(5), 877–896 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Asperti, A., Ricciotti, W.: About the Formalization of Some Results by Chebyshev in Number Theory. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds.) TYPES 2008. LNCS, vol. 5497, pp. 19–31. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Asperti, A., Ricciotti, W.: A Web Interface for Matita. In: Jeuring, J., et al. (eds.) CICM 2012. LNCS (LNAI), vol. 7362, pp. 417–421. Springer, Heidelberg (2012)Google Scholar
  8. 8.
    Asperti, A., Ricciotti, W., Sacerdoti Coen, C., Tassi, E.: The Matita Interactive Theorem Prover. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 64–69. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Asperti, A., Tassi, E.: Superposition as a logical glue. EPTCS 53, 1–15 (2011)CrossRefGoogle Scholar
  10. 10.
    Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Trans. Comput. Log. 9(1) (2007)Google Scholar
  11. 11.
    Barendregt, H.: Towards an interactive mathematical proof language. In: Kamareddine, F. (ed.) Thirty Five Years of Automath, pp. 25–36. Kluwer (2003)Google Scholar
  12. 12.
    Bertot, Y., Gonthier, G., Ould Biha, S., Pasca, I.: Canonical Big Operators. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 86–101. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Bourbaki, N.: The architecture of mathematics. Monthly 57, 221–232 (1950)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Bourbaki, N.: Theory of Sets. Elements of mathematics. Addison Wesley (1968)Google Scholar
  15. 15.
    Boutin, S.: Using Reflection to Build Efficient and Certified Decision Procedures. In: Ito, T., Abadi, M. (eds.) TACS 1997. LNCS, vol. 1281, pp. 515–529. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  16. 16.
    De Bruijn, N.G.: Memories of the automath project. Invited Lecture at the Mathematics Knowledge Management Symposium, November 25-29. Heriot-Watt University, Edinburgh (2003)Google Scholar
  17. 17.
    De Millo, R.A., Lipton, R.J., Perlis, A.J.: Social processes and proofs of theorems and programs. Commun. ACM 22(5), 271–280 (1979)CrossRefGoogle Scholar
  18. 18.
    Garillot, F., Gonthier, G., Mahboubi, A., Rideau, L.: Packaging Mathematical Structures. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 327–342. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Geuvers, H.: Proof Assistants: history, ideas and future. Sadhana 34(1), 3–25 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Gonthier, G., Mahboubi, A.: An introduction to small scale reflection in coq. Journal of Formalized Reasoning 3(2), 95–152 (2010)MathSciNetMATHGoogle Scholar
  21. 21.
    Gonthier, G.: Formal proof – the four color theorem. Notices of the American Mathematical Society 55, 1382–1394 (2008)MathSciNetMATHGoogle Scholar
  22. 22.
    Hardy, G.H.: Mathematical proof. Mind 38, 1–25 (1928)Google Scholar
  23. 23.
    Hardy, G.H.: A Mathematician’s Apology. Cambridge University Press, London (1940)Google Scholar
  24. 24.
    Harrison, J.: Formal proof – theory and practice. Notices of the American Mathematical Society 55, 1395–1406 (2008)MathSciNetMATHGoogle Scholar
  25. 25.
    Hayes, B.: Gauss’s day of reckoning. American Scientist 4(3), 200–207 (2006)Google Scholar
  26. 26.
    Howard, W.A.: The formulae-as-types notion of construction. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 479–490. Academic Press, Boston (1980)Google Scholar
  27. 27.
    Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer (2006)Google Scholar
  28. 28.
    Jamnik, M.: Mathematical Reasoning with Diagrams: from intuition to automation. CSLI Press, Stanford (2001)MATHGoogle Scholar
  29. 29.
    Kaliszyk, C.: Web interfaces for proof assistants. Electr. Notes Theor. Comput. Sci. 174(2), 49–61 (2007)CrossRefGoogle Scholar
  30. 30.
    Kerber, M.: Proofs, Proofs, Proofs, and Proofs. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 345–354. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  31. 31.
    Knuth, D.E.: Literate Programming. Center for the Study of Language and Information (1992)Google Scholar
  32. 32.
    Lakatos, I.: Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press (1976)Google Scholar
  33. 33.
    Lamport, L.: Letter to the editor. Communications of the ACM 22, 624 (1979)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Lee, J.K.: Philosophical perspectives on proof in mathematics education. Philosophy of Mathematics Education Journal 16 (2002)Google Scholar
  35. 35.
    MacKenzie, D.: What in the name of Euclid is going on here? Science 207(5714), 1402–1403 (2005)MathSciNetGoogle Scholar
  36. 36.
    Mackenzie, D.: Mechanizing Proof. MIT Press (2001)Google Scholar
  37. 37.
    Nelsen, R.B.: Proofs without Words: Exercises in Visual Thinking. The Mathematical Association of America (1997)Google Scholar
  38. 38.
    Coen, C.S., Tassi, E., Zacchiroli, S.: Tinycals: step by step tacticals. In: Proceedings of User Interface for Theorem Provers 2006. Electronic Notes in Theoretical Computer Science, vol. 174, pp. 125–142. Elsevier Science (2006)Google Scholar
  39. 39.
    Tankink, C., Geuvers, H., McKinna, J., Wiedijk, F.: Proviola: A Tool for Proof Re-animation. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 440–454. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  40. 40.
    Urban, J., Alama, J., Rudnicki, P., Geuvers, H.: A Wiki for Mizar: Motivation, Considerations, and Initial Prototype. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 455–469. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  41. 41.
    Wiedijk, F.: Formal proof sketches. In: Fokkink, W., van de Pol, J. (eds.) 7th Dutch Proof Tools Day, Program + Proceedings. CWI, Amsterdam (2003)Google Scholar
  42. 42.
    Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andrea Asperti
    • 1
  1. 1.Department of Computer ScienceUniversity of BolognaItaly

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