Korean Journal of Logic (논리연구)

Korean Association for Logic (한국논리학회)
권호 표지

Frege's influence on the modern practice of doing mathematics

현대수학의 정형화에 대한 프레게의 영향

  • Lee, Gyesik (Department of Computer Science and Engineering, Hankyong National University)
  • 이계식 (한경대학교 컴퓨터공학과)
  • Received : 2016.11.11
  • Accepted : 2017.02.11
  • Published : 2017.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

We discuss Frege's influence on the modern practice of doing mathematical proofs. We start with explaining Frege's notion of variables. We also talk of the variable binding issue and show how successfully his idea on this point has been applied in the field of doing mathematics based on a computer software.

컴퓨터를 이용한 수학적 증명의 정형화는 현대수학의 중요한 연구도구로 활용되고 있다. 본 논문에서는 정형증명에 대한 프레게의 영향을 살펴본다. 이를 위해 자유변항과 구속변항을 정형증명에서 다룰 때 발생하는 문제를 설명한 후, 프레게의 Begriffsschrift에서 언급된 아이디어를 이용하여 변항을 정형적으로 다룰 수 있는 해결책을 소개한다.

Keywords

References

  1. Avigad, J. and Harrison, J. (2014), "Formally verified mathematics", Communications of the ACM, 57(4), pp. 66-75. https://doi.org/10.1145/2591012
  2. Aydemir, B. E., Charguéraud, A., Pierce, B. C., Pollack, R., and Weirich, S. (2008), "Enginerring formal metatheory", ACM SIGPLAN Notices, 43(1), pp. 3-15. https://doi.org/10.1145/1328897.1328443
  3. Barendregt, H. (1981), The Lambda Calculus - Its Syntax and Semantics, North-Holland.
  4. Bertot , Y. and Casteran, P. (2004), Interactive Theorem Proving and Program Development - Coq'Art: The Calculus of Inductive Constructions, Springer.
  5. Church, A. (1940), "A formulation of the simple theory of types", Jounal of Symbolic Logic, 5(2), pp. 56-68. https://doi.org/10.2307/2266170
  6. Coquand, T. (1991), "An algorithm for testing conversion in Type Theory", in Huet and Plotkin (eds.), Logical Frameworks, Cambridge University Press, pp. 255-279.
  7. Curry, H.B. and Feys, R. (1958), Combinatory Logic Volume 1, North Holland.
  8. Ebert, P. and Rossberg (2013), M. Gottlog Frege: Basic Laws of Arithmetic, Oxford University Press.
  9. Frege, G. (1879), Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Verlag von Nebert.
  10. Frege, G. (1893, 1903), Grundgesetze der Arithmetik Volume 1 and 2, Verlag Hermann Pohle.
  11. Gabbay, M. and Pitts, A. (2002), "A new approach to abstract syntax with variable binding", Formal aspects of computing, 13(3-5), pp. 341-363. https://doi.org/10.1007/s001650200016
  12. Gacek, A. (2008), "The Abella interactive theorem prover (system description)", Lecture Notes in Computer Science, 5195, pp. 154-161.
  13. Gentzen, G. (1934), "Untersuchungen über das logische Schliesen. I", Mathematische Zeitschrift, 39(2), pp. 176-210.
  14. Geuvers, H. (2009), "Proof assistants: History, ideas and future", Sadhana Journal, 34(1), pp. 3-25. https://doi.org/10.1007/s12046-009-0001-5
  15. Godel, K. (1958), "Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes", Dialectica, 12, pp. 280-287. https://doi.org/10.1111/j.1746-8361.1958.tb01464.x
  16. Hales, T. et al. (2015), "A formal proof of the Kepler conjecture", arXiv.org, https://arxiv.org/abs/1501.02155.
  17. Harrison, J. (2009), "HOL Light: An overview", Lecture Notes in Computer Science, 5674, pp. 60-66.
  18. Hilbert, D. (1899), Grundlagen der Geometrie, Teubner Verlag.
  19. Huet, G. and Plotkin, G. (1991), Logical Frameworks, Cambridge University Press.
  20. McKinna, J. and Pollack, R. (1993), "Pure type systems formalized", Lecture Notes in Computer Science, 664, pp. 69-111.
  21. McKinna, J. and Pollack, R. (1999), "Some lambda calculus and type theory formalized", Journal of Automated Reasoning, 23(3-4), pp. 373-409. https://doi.org/10.1023/A:1006294005493
  22. Momigliano, A., Martin, A. J., and Felty, A. P. (2008), "Two-level hybred: A system for reasoning using higher-order abstract syntax", Electronic Notes in Theoretical Computer Science, 196, pp. 85-93. https://doi.org/10.1016/j.entcs.2007.09.019
  23. Nipkow, T., Paulson, L.C., and Wenzel, M. (2002), Isabelle/HOL: A proof assistant for higher-order logic, Springer.
  24. Peano, J. (1899), Arithmetices principia, nova methodo exposita, Fratres Bocca.
  25. Pfenning, F. and Schurmann, C. (1999), "System description: Twelf - A meta-logical framework for deductive systems", Lecture Notes in Computer Science, 1632, pp. 202-206.
  26. Prawitz, D. (1965), Natural Deduction, Almqvist & Wiksell.
  27. Sato, M. and Pollack, R. (2010), "External and internal syntax of the lambda-calculus", Journal of symbolic computation, 45(5), pp. 598-616. https://doi.org/10.1016/j.jsc.2010.01.010
  28. Urban, C. (2008), "Nominal Techniques in Isabelle/HOL", Journal of Automated Reasoning, 40(4), pp. 327-356. https://doi.org/10.1007/s10817-008-9097-2
  29. van Heijennoort, J. (1967), From Frege to Godel, Harvard University Press.
  30. Whitehead, A. N. and Russell, B. (1910, 1912, 1913), Principia mathematica Vol. 1 - 3, Cambridge University Press.
  31. Zermelo, E. (1908), "Untersuchungen uber die Grundlagen der Mengenlehre. I", Mathematische Annalen, 65, pp. 261-281. https://doi.org/10.1007/BF01449999