Advertisement

Synthese

, Volume 153, Issue 1, pp 105–159 | Cite as

Mathematical Method and Proof

  • Jeremy Avigad
Article

Abstract

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.

Keywords

Deductive System Mathematical Proof Mathematical Understanding Proof Assistant Automate 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aigner, M., Ziegler, G. M. 2001Proofs from The Book2Springer-VerlagBerlinGoogle Scholar
  2. Avigad, J. 2000‘Interpreting Classical Theories in Constructive Ones’Journal of Symbolic Logic6517851812CrossRefGoogle Scholar
  3. Baaz, M. 1999‘Note on the Generalization of Calculations’Theoretical Computer Science224311CrossRefGoogle Scholar
  4. Beeson, M. J. 1985Foundations of Constructive MathematicsSpringer-VerlagBerlinGoogle Scholar
  5. Berger, U., Schwichtenberg, H., Seisenberger, M. 2001‘The Warshall Algorithm and Dickson’s Lemma: Two Examples of Realistic Program Extraction’Journal of Automated Reasoning26205221CrossRefGoogle Scholar
  6. Birkhoff, G., Mac Lane, S. 1965A Survey of Modern Algebra3The Macmillan CoNew YorkGoogle Scholar
  7. Bridges, D., Reeves, S. 1999‘Constructive Mathematics in Theory and Programming Practice’Philosophia Mathematica765104Google Scholar
  8. Bundy, A.: 1988, ‘The Use of Explicit Plans to Guide Inductive Proofs’, in E. Lusk and R. Overbeek (eds.), 9th International Conference on Automated Deduction, Vol. 310 of Lecture Notes in Computer Science. Berlin, pp. 111–120.Google Scholar
  9. Buss, S. R. (ed.): 1998, The Handbook of Proof Theory. North-Holland, Amsterdam.Google Scholar
  10. Cauchy, A.-L.: 1821, Cours d’analyse de l’École Royale Polytechnique. Première partie: Analyse algébrique. Paris:. Reprinted in Cauchy’s Ouvres completes, Gauthier-Villars, Paris, 1882–1919, deuxième série, vol. 3.Google Scholar
  11. Colton, S., A. Bundy, and T. Walsh: 1999, ‘Automatic Concept Formation in Pure Mathematics’, in T. L. Dean (ed.), Automatic Concept Formation in Pure Mathematics. Proceedings of the 16th International Joint Conference on Artificial Intelligence. San Francisco, pp. 786–793.Google Scholar
  12. Conway, J. H.: 1997, The Sensual (Quadratic) Form, Vol. 26 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC.Google Scholar
  13. Corfield, D. 2003Towards a Philosophy of Real MathematicsCambridge University PressCambridgeGoogle Scholar
  14. Corry, L.: 1996, Modern Algebra and the Rise of Mathematical Structures, Vol. 17 of Science Networks. Historical Studies, Birkhäuser Verlag, Basel.Google Scholar
  15. Coxeter, H. S. M. 1969Introduction to Geometry2WileyNew YorkGoogle Scholar
  16. Dedekind, R.: 1877, Sur la théorie des nombres entiers algébrique. Paris: Gauthier-Villars. Also Bulletin des sciences mathématiques et astronomiques (1), 11 (1876) 278–288, (2), 1 (1877) 17–41, 69–92, 144–164, 207–248; parts also in Dedekind’s Werke, vol. 3, 263–296. Translated as Theory of Algebraic Integers with an editorial introduction by John Stillwell, Cambridge University Press, Cambridge, 1996.Google Scholar
  17. Dickson, L. E.: 1966, History of the Theory of Numbers. Vol. 1: Divisibility and Primality. Vol. 2: Diophantine Analysis. Vol. 3: Quadratic and higher forms. Chelsea Publishing Co, New York. Originally published by the Carnegie Institute of Washington, Washington, D.C., 1919–1923.Google Scholar
  18. Ebbinghaus, H.-D., H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert: 1990, Numbers, Vol. 123 of Graduate Texts in Mathematics, Springer-Verlag, New York. With an introduction by K. Lamotke, Translated from the second German edition by H. L. S. Orde, Translation edited and with a preface by J. H. Ewing.Google Scholar
  19. Edwards, H. M. 1980‘The Genesis of Ideal Theory’Archive for History of Exact Sciences23321378CrossRefGoogle Scholar
  20. Edwards, H. M. 1992‘Mathematical Ideas, Ideals, and Ideology’Math. Intelligencer14619CrossRefGoogle Scholar
  21. Edwards, H. M.: 1996, Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory, Vol. 50 of Graduate Texts in Mathematics. Springer-Verlag New York. Corrected reprint of the 1977 original.Google Scholar
  22. Euler, L.: (1732/3) 1738, ‘Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus’. Comm. Ac. Petrop. 6, 103–107. Reprinted in Volume 2 of Euler (1911–1956), pp. 1–5.Google Scholar
  23. Euler, L.: (1747/48) 1750, ‘Theoremata circa divisores numerorum’. N. Comm. Ac. Petrop. 1, 20–48. Reprinted in Volume 2 of Euler (1911–1956), pp. 62–85.Google Scholar
  24. Euler, L.: 1770, Vollständige Anleitung zur Algebra. St. Petersberg: Kays. Akademie der Wissenschaften. Reproduced in Volume 1 of Euler (1911–1956).Google Scholar
  25. Euler, L.: 1911–1956, Opera Omnia. Series Prima: Opera Mathematica. Geneva: Societas Scientiarum Naturalium Helveticae. 29 volumes.Google Scholar
  26. Ferreirós, J.: 1999, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, Vol. 23 of Science Networks, Historical Studies, Birkhäuser Verlag, Basel.Google Scholar
  27. Gauss, C. F.: 1801, Disquisitiones Arithmeticae, G. Fleischer, Leipzig.Google Scholar
  28. Goldman, J. R. 1998The Queen of Mathematics: A Historically Motivated Guide to Number TheoryA K Peters LtdWellesley, MAGoogle Scholar
  29. Gray, J.: 1992, ‘The Nineteenth-century Revolution in Mathematical Ontology’, in Revolutions in Mathematics, Oxford Univ. Press, Oxford Sci. Publ. New York. pp. 226–248.Google Scholar
  30. Hafner, J. and P. Mancosu: 2005, ‘The Varieties of Mathematical Explanation’. K. Jørgensen et al., (Eds.) Visualization, Explanation, and Reasoning Styles in Mathematics, Kluwer.Google Scholar
  31. Hardy, G. H. and E. M. Wright: 1979, An Introduction to the Theory of Numbers, 5th edn., Oxford.Google Scholar
  32. Hermite, C.: 1848, ‘Théorème relatif aux nombres entiers’. Journal de Mathématique pures et appliquées 13, 15. Reprinted in Hermite’s Ouevres, Gauthier-Villars, Paris, 1905–1917, p. 264.Google Scholar
  33. Knobloch, E.: 1994, ‘From Gauss to Weierstrass: Determinant Theory and its Historical Evaluations’. in The Intersection of History and Mathematics, Vol. 15 of Sci. Networks Hist. Stud. Birkhäuser, Basel. pp. 51–66.Google Scholar
  34. Kohlenbach, U. 2005‘Some Logical Metatheorems with Applications in Functional Analysis’Transactions of the American Mathematical Society35789128CrossRefGoogle Scholar
  35. Laugwitz, D.: 1999, Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics, Birkhäuser Boston Inc, Boston, MA. Translated from the 1996 German original by Abe Shenitzer with the editorial assistance of the author, Hardy Grant, and Sarah Shenitzer.Google Scholar
  36. Lemmermeyer, F.: 2000, Reciprocity Laws: From Euler to Eisenstein, Springer Monographs in Mathematics. Springer-Verlag, Berlin.Google Scholar
  37. Lenat, D. B.: 1976, ‘AM : An Artificial Intelligence Approach to Discovery in Mathematics as Heuristic Search’. Ph.D. thesis, Stanford.Google Scholar
  38. Mancosu, P. 2000‘On Mathematical Explanation’Grosholz, E.Breger, H. eds. The Growth of Mathematical KnowledgeKluwer Academic PublishersThe Netherlands103119Google Scholar
  39. Mancosu, P. 2001‘Mathematical Explanation: Problems and Prospects’Topoi2097117CrossRefGoogle Scholar
  40. Nipkow, T.: 2003, ‘Structured Proofs in Isar/HOL’, in H. Geuvers and F. Wiedijk (eds.), Types for Proofs and Programs (TYPES 2002), Vol. 2646 of Lecture Notes in Computer Science. Berlin, pp. 259–278. Available under “documentation” at http://www.cl.cam.ac.uk/Research/HVG/Isabelle/index.html
  41. Nipkow, T., L. C. Paulson, and M. Wenzel: 2002, Isabelle/HOL. A Proof Assistant for Higher-order Logic, Vol. 2283 of Lecture Notes in Computer Science. Springer-Verlag, Berlin.Google Scholar
  42. Niven, I., Zuckerman, H. S., Montgomery, H. L. 1991An Introduction to the Theory of Numbers5WileyNew YorkGoogle Scholar
  43. Rudnicki, P., Trybulec, A. 1999Equivalents of Well-foundedness’Journal of Automated Reasoning23197234CrossRefGoogle Scholar
  44. Scharlau, W. and H. Opolka: 1985, From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development, Undergraduate Texts in Mathematics, Springer-Verlag, New York. Translated from the German by Walter K. Bühler and Gary Cornell.Google Scholar
  45. Schoenfeld, A. H. 1985Mathematical Problem SolvingAcademic PressOrlando, FloridaGoogle Scholar
  46. Smith, H. J. S., ‘Report on the Theory of Numbers’. Originally published as a report to the British Association in six parts between 1859 and 1865. Reprinted by Chelsea both as a separate volume and in Smith (1965).Google Scholar
  47. Smith, H. J. S.: 1855, ‘De compositione numerorum primorum formae 4λ  +  1 ex duobus quadratis’, Journal für die reine und angewandte Mathematik (Crelle’s Journal) L., 91–92. Reprinted in Smith (1965), pp. 33–34.Google Scholar
  48. Smith, H. J. S.: 1965, Collected Mathematical Papers, Chelsea Publishing Company, Bronx. Edited by J.W.L. Glaisher. Originally published by Clarendon Press, Oxford, 1894.Google Scholar
  49. Stein, H.: 1988, ‘Logos, Logic, and Logistiké’, in W. Aspray and P. Kitcher (eds.), History and Philosophy of Modern Mathematics. University of Minnesota, pp. 238–259.Google Scholar
  50. Steiner, M. 1978‘Mathematical Explanation’Philosophical Studies34133151CrossRefGoogle Scholar
  51. Tappenden, J.: 1995, ‘Extending Knowledge and ‘Fruitful Concepts’: Fregean Themes in the Philosophy of Mathematics’. Noûs.Google Scholar
  52. Troelstra, A. S. and D. van Dalen: 1988, Constructivism in Mathematics: An Introduction, vols. 1 and 2 North-Holland, Amsterdam.Google Scholar
  53. Weil, A. 1984Number theory: An Approach Through History, from Hammurapi to LegendreBirkhäuser Boston IncBoston, MAGoogle Scholar
  54. Wenzel, M.: 1999, ‘Isar – A Generic Interpretative Approach to Readable Formal Proof Documents.’, in Y. Bertot, G. Dowek, A. Hirschowitz, C. Paulin, and L. Thèry (eds.), Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs’99, Nice, France, September, 1999, Proceedings, Vol. 1690 of Lecture Notes in Computer Science. Berlin, pp. 167–184.Google Scholar
  55. Wenzel, M.: 2002, ‘Isabelle/Isar – A Versatile Environment for Human-Readable Formal Proof Documents’. Ph.D. thesis, Institut für Informatik, Technische Universität München.Google Scholar
  56. Wussing, H.: 1984, The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, MIT Press, Cambridge, MA. Translated from the German by Abe Shenitzer and Hardy Grant.Google Scholar
  57. Zagier, D.: 1990, ‘A One-Sentence Proof that Every Prime p≡1 (mod 4) is a Sum of Two Squares’, American Mathematical Monthly 97(2), 144.Google Scholar
  58. ‘The Isabelle theorem proving environment’, Developed by Larry Paulson at Cambridge University and Tobias Nipkow at TU Munich. http://www.cl.cam.ac.uk/Research/HVG/Isabelle/index.html.

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of PhilosophyCarnegie Mellon UniversityPittsburghUS

Personalised recommendations