A Common Type of Rigorous Proof that Resists Hilbert’s Programme

  • Alan BundyEmail author
  • Mateja Jamnik
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 14)


Following Hilbert, there seems to be a simple and clear definition of mathematical proof: it is a sequence of formulae each of which is either an axiom or follows from earlier formulae by a rule of inference. Automated theorem provers are based on this Hilbertian concept of proof, in which the formulae and rules of inference are represented in a formal logic. These logic-based proofs are typically an order of magnitude longer than the rigorous proofs produced by human mathematicians. There is a consensus, however, that rigorous proofs could, in principle, be unpacked into logical proofs, but this programme is rarely carried out because it would be tedious and uninformative. We argue that, for at least one class of rigorous proofs, which we will call schematic proofs, such a simple unpacking is not available. We will illustrate schematic proofs by analysing Cauchy’s faulty proof of Euler’s Theorem V − E + F = 2, as reported in Lakatos (1976) and giving further examples from Nelsen (1993). We will then give a logic-based account of schematic proofs, distinguishing them from Hilbertian proofs, and showing why they are error prone.



The research reported in this chapter is based on Bundy et al. (2005), Jamnik and Bundy (2005), Bundy (2012). It was supported by EPSRC grants GR/S01771, GR/S31099 and EP/N014758/1. Many thanks to Predrag Janic̆ić and Alison Pease for drawing some of the images and to Andy Fugard for permission to use a diagram drawn by one of the participants in his study. Thanks to Gila Hanna and Andy Fugard for comments on an earlier version.


  1. Baker, S. (1993). Aspects of the constructive omega rule within automated deduction. Unpublished Ph.D. thesis, Edinburgh.Google Scholar
  2. de Bruijn, N. G. (1980). A survey of the project Automath. In J. P. Seldin & J. R. Hindley (Eds.), To H. B. Curry; Essays on combinatory logic, lambda calculus and formalism (pp. 579–606). Academic Press.Google Scholar
  3. Bundy, A. (2012). Reasoning about representations in autonomous systems: What Pólya and Lakatos have to say. In D. McFarland, K. Stenning, & M. McGonigle-Chalmers (Eds.), The complex mind: An interdisciplinary approach, chapter 9 (pp. 167–183). Palgrave Macmillan.Google Scholar
  4. Bundy, A., Jamnik, M., & Fugard, A. (2005). What is a proof? Philosophical Transactions of the Royal Society A, 363(1835), 2377–2392.CrossRefGoogle Scholar
  5. Fugard, A. J. B. (2005). An exploration of the psychology of mathematical intuition. Unpublished M.Sc. thesis, School of Informatics, Edinburgh University.Google Scholar
  6. Gödel, K. (1931). Über formal unentscheidbare sätze der principia mathematica und verwandter systeme i. Monatshefte fÜr Mathematik und Physik, 38, 173–198. English translation in [van Heijenoort, 1967].CrossRefGoogle Scholar
  7. Hilbert, D. (1930). Die Grundlebung der elementahren Zahlenlehre (Vol. 104). Mathematische Annalen.Google Scholar
  8. Jamnik, M., & Bundy, A. (2005). Psychological validity of schematic proofs. In Volume LNCS 2605 of lecture notes in computer science (pp. 321–341). Springer-Verlag GmbH.Google Scholar
  9. Jamnik, M. (2001). Mathematical reasoning with diagrams: From intuition to automation. Stanford, CA: CSLI Press.Google Scholar
  10. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press.Google Scholar
  11. Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. The Mathematical Association of America.Google Scholar
  12. Robinson, A., & Voronkov, A. (Eds.). (2001). Handbook of automated reasoning, 2 volumes. Elsevier.Google Scholar
  13. Shoenfield, J. R. (1959). On a restricted \(\omega \)-rule. Bulletin de l’Académie Polonaise des Sciences: S’erie des sciences mathematiques, astronomiques et physiques, 7, 405–407.Google Scholar
  14. van Heijenoort, J. (1967). From Frege to Gödel: A source book in mathematical logic, 1879–1931. Cambridge, Mass: Harvard University Press.Google Scholar
  15. Zach, R. (2009). Hilbert’s program. Stanford Encyclopedia of Philosophy.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of EdinburghEdinburghUK
  2. 2.University of CambridgeCambridgeUK

Personalised recommendations