Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics

Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 30)


We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (Lakatos, 1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, which uses work by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods.


argumentation theory computational model Euler’s conjecture Lakatos philosophy of mathematics theory refinement. 



A prior version of this paper was published in Foundations of Science (2009), 14(1–2):111–135. We would like to thank our three reviewers who all gave comprehensive, constructive and thought-provoking reviews. One of the editors of this collection, Andrew Aberdein, made particularly helpful comments. This work was partly funded by EPSRC grant EP/F035594/1.


  1. Aberdein, A. (2005). The uses of argument in mathematics. Argumentation, 19, 287–301.CrossRefGoogle Scholar
  2. Aberdein, A. (2006). Managing informal mathematical knowledge: Techniques from informal logic. In J. M. Borwein & W. M. Farmer (Eds.), MKM 2006, LNAI 4108 (pp. 208–221). Berlin: Springer.Google Scholar
  3. Alcolea Banegas, J. (1998). L’argumentació en matemàtiques. In E. Casaban i Moya (Ed.), XIIè Congrés Valencià de Filosofia (pp. 135–147). Valencià. [Trans. by A. Aberdein & I.J. Dove (Eds.), The Argument of Mathematics (pp. 47–60). Dordrecht: Springer].Google Scholar
  4. Burton, D. (1985). The history of mathematics. Boston, MA: Allyn and Bacon.Google Scholar
  5. Cauchy, A. L. (1813). Recherches sur les polyèdres. Journal de l’École Polytechnique, 9, 68–86.Google Scholar
  6. Cauchy, A. L. (1821). Cours d’Analyse de l’École Royale Polytechnique. Paris: de Bure.Google Scholar
  7. Colton, S. (2002). Automated theory formation in pure mathematics. Berlin: Springer.CrossRefGoogle Scholar
  8. Corfield, D. (1997). Assaying Lakatos’s philosophy of mathematics. Studies in History and Philosophy of Science, 28(1), 99–121.CrossRefGoogle Scholar
  9. Crawshay-Williams, R. (1957). Methods of criteria of reasoning: An inquiry into the structure of controversy. London: Routledge and Kegan Paul.Google Scholar
  10. Crelle, A. L. (1826–1827). Lehrbuch der Elemente der Geometrie (Vols. 1, 2). Berlin: Reimer.Google Scholar
  11. Elvang-Goransson, M., Krause, P., & Fox, J. (1993). Dialectical reasoning with inconsistent information. In Proceedings of the ninth conference annual conference on uncertainty in artificial intelligence (UAI-93) (pp. 114–121). San Francisco, CA: Morgan Kaufmann.Google Scholar
  12. Ernest, P. (1997). The legacy of Lakatos: Reconceptualising the philosophy of mathematics. Philosophia Mathematica, 5(3), 116–134.CrossRefGoogle Scholar
  13. Feferman, S. (1978). The logic of mathematical discovery vs. the logical structure of mathematics. In P. D. Asquith & I. Hacking (Eds.), Proceedings of the 1978 biennial meeting of the Philosophy of Science Association (Vol. 2, pp. 309–327). East Lansing, MI: Philosophy of Science Association.Google Scholar
  14. Fourier, J. (1808). Mémoire sur la propagation de la chaleur dans les corps solides (extrait). Nouveau Bulletin des Sciences, par la Société Philomathique de Paris, 1, 112–16.Google Scholar
  15. Goguen, J. (1999). An introduction to algebraic semiotics, with application to user interface design. In C. L. Nehaniv (Ed.), Computation for metaphors, analogy, and agents: Vol. 1562, Lecture notes in artificial intelligence (pp. 242–291). Berlin: Springer.Google Scholar
  16. Haggith, M. (1996). A meta-level argumentation framework for representing and reasoning about disagreement. PhD thesis, Department of Artificial Intelligence, University of Edinburgh, Edinburgh, UK.Google Scholar
  17. Hardy, G. H. (1928). Mathematical proof. Mind, 38, 11–25.Google Scholar
  18. Hilbert, D. (1901). The foundations of geometry (1st ed.) (E. J. Townsend, Trans.). Chicago, IL: Open Court.Google Scholar
  19. Hilbert, D. (2004). David Hilbert’s lectures on the foundations of geometry, 1891–1902: Vol. 1 of David Hilbert’s lectures on the foundations of mathematics and physics, 1891–1933. M. Hallett & U. Majer (Eds.). Berlin: Springer.Google Scholar
  20. de Jonquières, E. (1890). Note sur un Point Fondamental de la Théorie des Polyèdres. Comptes Rendus des Séances de l’Académie des Sciences, 110, 110–115.Google Scholar
  21. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery (edited by J. Worrall & E. Zahar). Cambridge: Cambridge University Press.Google Scholar
  22. Larvor, B. (1998). Lakatos: An introduction. London: Routledge.Google Scholar
  23. Matthiessen, L. (1863). Über die Scheinbaren Einschränkungen des Euler’schen Satzes von den Polyedern. Zeitschrift für Mathematik und Physik, 8, 1449–1450.Google Scholar
  24. Meikle, L., & Fleuriot, J. (2003). Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In Theorem proving in higher order logics (Vol. 2758, pp. 319–334). Berlin: Springer.Google Scholar
  25. Naess, A. (1953). Interpretation and preciseness: A contribution to the theory of communication. Oslo, Norway: Skrifter utgitt ar der norske videnskaps academie.Google Scholar
  26. Naess, A. (1966 [1947]). Communication and argument: Elements of applied semantics. London: Allen and Unwin. (Translation of En del Elementaere Logiske Emner. Oslo: Universitetsforlaget, 1947).Google Scholar
  27. Pease, A. (2007). A computational model of Lakatos-style reasoning. PhD thesis, School of Informatics, University of Edinburgh, Edinburgh, UK. Online at
  28. Pease, A., & Aberdein, A. (2011). Five theories of reasoning: Interconnections and applications to mathematics. Logic and Logical Philosophy, 20(1–2), 7–57.Google Scholar
  29. Pease, A., Colton, S., Smaill, A., & Lee, J. (2004). A model of Lakatos’s philosophy of mathematics. Computing, philosophy and cognition: Proceedings of the European Computing and Philosophy Conference (ECAP 2004). London: College Publications.Google Scholar
  30. Pedemonte, B. (2001). Some cognitive aspects of the relationship between argumentation and proof in mathematics. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the international group for the Psychology of Mathematics Education PME-25, Utrecht, The Netherlands (Vol. 4, pp. 3–40). Nottingham: PME Proceedings.Google Scholar
  31. Pieri, M. (1895). Sui principi che reggiono la geometria di posizione. Atti della Reale Accademia delle scienze di Torino, 30:54–108.Google Scholar
  32. Pieri, M. (1897–1898). I principii della geometria di posizione composti in sistema logico deduttivo. Memorie della Reale Accademia delle Scienze di Torino 2, 48, 1–62.Google Scholar
  33. Pollock, J. (1970). The structure of epistemic justification. American Philosophical Quarterly, Monograph Series, 4, 62–78.Google Scholar
  34. Pollock, J. (1995). Cognitive carpentry. Cambridge, MA: MIT Press.Google Scholar
  35. Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.Google Scholar
  36. Pólya, G. (1954). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. I). Princeton, NJ: Princeton University Press.Google Scholar
  37. Popper, K. R. (1959). The logic of scientific discovery. New York: Basic Books.Google Scholar
  38. Reed, C., & Rowe, G. (2005). Translating Toulmin diagrams: Theory neutrality in argument representation. Argumentation, 19(3), 267–286.CrossRefGoogle Scholar
  39. Roth, R. L. (2001). A history of Lagrange’s theorem on groups. Mathematics Magazine, 74(1), 99–108.CrossRefGoogle Scholar
  40. Russell, B. (1971). Logic and knowledge: Essays 1901–1950. London: George Allen and Unwin.Google Scholar
  41. Sartor, G. (1993). A simple computational model for nonmonotonic and adversarial legal reasoning. In Proceedings of the fourth international conference on artificial intelligence and law, Amsterdam, The Netherlands (pp. 19–201). New York: ACM.Google Scholar
  42. Toulmin, S. (1958). The uses of argument. Cambridge: Cambridge University Press.Google Scholar
  43. Toulmin, S., Rieke, R., & Janik, A. (1979). An introduction to reasoning. London: Macmillan.Google Scholar
  44. Walton, D. (2006). Fundamentals of critical argumentation. Cambridge: Cambridge University Press.Google Scholar
  45. Wilder, R. L. (1944). The nature of mathematical proof. The American Mathematical Monthly, 51(6), 309–323.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Alison Pease
    • 1
    • 2
  • Alan Smaill
    • 3
  • Simon Colton
    • 1
  • John Lee
    • 3
  1. 1.Department of ComputingImperial College LondonLondonUK
  2. 2.School of Electronic Engineering and Computer ScienceQueen Mary, University of LondonLondonUK
  3. 3.School of InformaticsUniversity of Edinburgh, Informatics ForumEdinburghUK

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