Abstract
In contrast to the widespread use of computer algebra systems in mathematics automated theorem provers have largely met with indifference. There are signs that this is at last beginning to change. We argue that it is inevitable that automated provers will be adopted as a practical tool for the working mathematician. Mathematical applications of automated provers raises profound challenges for their developers.
Similar content being viewed by others
References
Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. I: Discharging. Ill. J. Math. 21, 429–490 (1977)
Aschbacher, M.: Highly complex proofs and the implications of such proofs. In: The Nature of Mathematical Proof. Phil. Trans. R. Soc. A., vol. 363(1835), pp. 2331–2461 (2005)
Autexier, S., Hutter, D.: Formal software development in MAYA. In: Mechanizing Mathematical Reasoning. LNCS, vol. 2605, pp. 407–432. Springer, Berlin/Heidelberg (2005)
Boyer, R., et al.: The QED manifesto. In Bundy, A. (ed.) Automated Deduction, CADE 12: 12th International Conference on Automated Deduction. LNCS, vol. 814, pp. 238–251. Springer (1994)
Bundy, A., Atiyah, M., Macintyre, A., Mackenzie, D.: The nature of mathematical proof. Phil. Trans. R. Soc. A. 363(1835), 2331–2461 (2005)
Bundy, A., Basin, D., Hutter, D., Ireland, A.: Rippling: Meta-level Guidance for Mathematical Reasoning, Cambridge Tracts in Theoretical Computer Science, vol. 56. Cambridge University Press (2005)
Church,A.:Anote on the Entscheidungsproblem. J. Symb. Log. 1(1), 40–41; 1(3), 101–102 (1936)
Church, A.: An unsolvable problem of elementary number theory. Am. J. Math. 58, 345–363 (1936)
De Millo, R.A., Lipton, R.J., Perlis, A.J.: Social processes and proofs of theorems and programs. Commun. ACM 22(5), 214–225 (1979)
Denney, E., Power, J., Tourlas, K.: Hiproofs: a hierarchical notion of proof tree. ENTCS 155 341–359 (2006)
Fleuriot, J.D., Paulson, L.C.: Proving Newton’s Propositio Kepleriana using geometry and nonstandard analysis in Isabelle. In: Automated Deduction in Geometry 1998. Lecture Notes in Artificial Intelligence, vol. 1669, pp. 47–66 (1999)
Gonthier, G.: Formal proof—the four-color theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)
Gordon, M.J., Milner, A.J., Wadsworth, C.P.: Edinburgh LCF—A mechanised logic of computation. Lecture Notes in Computer Science, vol. 78. Springer (1979)
Gorenstein, D. (1982). Finite simple groups: An introduction to their classification. Plenum Press (New York).
Gowers, W.T.: Rough structure and classification. GAFA: Geom. funct. anal. 1–39 (Special volume) (2000)
Grechuk, B.: Isabelle primer for mathematicians. Available from http://dream.inf.ed.ac.uk/projects/isabelle/ (2010)
Hales, T.C.: The Flyspeck project fact sheet. http://code.google.com/p/flyspeck/wiki/FlyspeckFactSheet (2005)
Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162(3), 1065–1185 (2005)
Hales, T.C.: Formal proof. Not. Am. Math. Soc. 55(11), 1370–80 (2008)
Mackenzie, D.: Mechanizing Proof. MIT Press (2001)
Martin, U.: Computers, reasoning and mathematical practice. In: Berger, U., Schwichtenberg, H. (eds.) Computational Logic, Proceedings of the NATO Advanced Study Institute on Computational Logic, Marktoberdorf, Germany, 1997. NATO ASI Series, vol. 165, pp. 301–346. Springer (1999)
McCasland, R.L., Bundy, A., Smith, P.F.: Ascertaining mathematical theorems. ENTCS 151, 21–38 (2006)
Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science, pp. 77–90. Academic Press (1990)
Rudnicki, P.: An overview of the Mizar project. In: 1992 Workshop on Types for Proofs and Programs, Bastad. Chalmers University of Technology. See http://mizar.org for up-to-date information on Mizar and the Journal of Formalized Mathematics (1992)
Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. In: Proceedings of the London Mathematical Society (2), vol. 42, pp. 230–265 (1936)
Wenzel, M.: Isabelle/Isar—a generic framework for human-readable proof documents. In: Matuszewski, R., Zalewska, A. (eds.) From Insight to Proof—Festschrift in Honour of Andrzej Trybulec. Studies in Logic, Grammar, and Rhetoric, vol. 10(23). University of Bialystok (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research reported in this paper was supported by EPSRC grants EP/F033559/1 and EP/H023119/1. I am indebted to Bogdan Grechuk, Lucas Dixon, Ursula Martin and an anonymous referee for valuable feedback on earlier drafts, to Yuhui Lin for help with the figures, and to Michael Chan for help with the submission.
Rights and permissions
About this article
Cite this article
Bundy, A. Automated theorem provers: a practical tool for the working mathematician?. Ann Math Artif Intell 61, 3–14 (2011). https://doi.org/10.1007/s10472-011-9248-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-011-9248-8