Abstract
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 (Proofs and Refutations, 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, in which we use work by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods.
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References
Aberdein A. (2005) The uses of argument in mathematics. Argumentation 19(3): 287–301
Aberdein A. (2006) Managing informal mathematical knowledge: Techniques from informal logic. In: Borwein J.M., Farmer W.M. (eds) MKM 2006, LNAI 4108. Springer-Verlag, Berlin, pp 208–221
Alcolea, B. J. (1998). L’argumentació en matemàtiques. In E. Casaban i Moya (Ed.), XIIè Congrés Valencià de Filosofia (pp. 135–147). Valencià.
Burton D. (1985) The history of mathematics. Allyn and Bacon, Boston
Cauchy A.L. (1813) Recherches sur les polyèdres. Journal de l’École Polytechnique 9: 68–86
Cauchy A.L. (1821) Cours d’analyse de l’école Royale Polytechnique. de Bure, Paris
Colton S. (2002) Automated theory formation in pure mathematics. Springer-Verlag, New York
Corfield D. (1997) Assaying Lakatos’s philosophy of mathematics. Studies in History and Philosophy of Science 28(1): 99–121
Crawshay-Williams R. (1957) Methods of criteria of reasoning: An inquiry into the structure of controversy. Routledge and Kegan Paul, London
Crelle A.L. (1826) Lehrbuch der Elemente der Geometrie (Vol. 1, 2). Reimer, Berlin
Elvang-Gøransson, M., Krause, P., & Fox, J. (1993). Dialectical reasoning with inconsistent information. In Proceedings of the 9th Conference on Uncertainty in AI (pp. 114–121). San Mateo, CA: Morgan Kaufmann.
Ernest P. (1997) The legacy of Lakatos: Reconceptualising the philosophy of mathematics. Philosophia Mathematica 5(3): 116–134
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
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: 112–116
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, LNAI 1562 (pp. 242–291). Berlin: Springer-Verlag
Haggith, M. (1996). A meta-level argumentation framework for representing and reasoning about disagreement. Ph.D. thesis, Department of Artificial Intelligence, University of Edinburgh.
Hallett, M., & Majer, U. (Eds.), (2004). David Hilbert’s lectures on the foundations of geometry: 1891–1902. Berlin: Springer-Verlag.
Hardy G H. (1928) Mathematical proof. Mind 38: 11–25
Hilbert, D. (1902). The foundations of geometry (English translation by E. J. Townsend). Open Court
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
Lakatos I. (1976) Proofs and refutations. Cambridge University Press, Cambridge
Larvor B. (1998) Lakatos: An introduction. Routledge, London
Matthiessen L. (1863) Über die Scheinbaren Einschränkungen des Euler’schen Satzes von den Polyedern. Zeitschrift für Mathematik und Physik 8: 1449–1450
Meikle, L., & Fleuriot, J. (2003). Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In D. Basin & B. Wolff (Eds.), Proceedings of the 16th International Conference on Theorem Proving in Higher Order Logics, LNCS 2758 (pp. 319–334). Berlin: Springer-Verlag.
Naess A. (1953) Interpretation and preciseness: A contribution to the theory of communication. Skrifter utgitt ar der norske videnskaps academie, Oslo
Naess, A. (1966). Communication and argument: Elements of applied semantics. London: Allen and Unwin. (Translation of En del elementaere logiske emner. Universitetsforlaget, Oslo, 1947).
Pease, A. (2007). A computational model of Lakatos-style reasoning. Ph.D. thesis, School of Informatics, University of Edinburgh. http://hdl.handle.net/1842/2113.
Pease, A., Colton, S., Smaill, A., & Lee, J. (2004). A model of Lakatos’s philosophy of mathematics. Proceedings of Computing and Philosophy (ECAP).
Pedemonte, B. (2000). Some cognitive aspects of the relationship between argumentation and proof in mathematics. In M. van den Heuvel-Panhuizen (Ed.), 25th Conference of the International Group for the Psychology of Mathematics Education. Utrecht.
Pieri M. (1895) Sui principi che reggiono la geometria di posizione. Atti della Reale Accademia delle scienze di Torino 30: 54–108
Pieri M. (1897–98) I principii della geometria di posizione composti in sistema logico deduttivo. Memorie della Reale Accademia delle Scienze di Torino 48(2): 1–62
Pollock J. (1970) The structure of epistemic justification. In: Rescher N. (eds) Studies in the theory of knowledge. American Philosophical Quarterly Monograph Series 4. Blackwell, Oxford, pp 62–78
Pollock J. (1995) Cognitive carpentry. The MIT press, Cambridge, MA
Polya G. (1945) How to solve it. Princeton University Press, Princeton NJ
Polya G. (1954) Mathematics and plausible reasoning: Vol. 1, Induction and analogy in mathematics. Princeton University Press, Princeton NJ
Popper K.R. (1959) The logic of scientific discovery. Basic Books, New York
Reed C., Rowe G. (2005) Translating Toulmin diagrams: Theory neutrality in argument representation. Argumentation 19(3): 267–286
Roth R.L. (2001) A history of Lagrange’s theorem on groups. Mathematics Magazine 74(1): 99–108
Russell B. (1971) Logic and knowledge: Essays 1901–1950. George Allen and Unwin, London
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: ACM.
Toulmin S. (1958) The uses of argument. Cambridge University Press, Cambridge
Toulmin S., Rieke R., Janik A. (1979) An introduction to reasoning. Macmillan, London
Walton D. (2006) Fundamentals of critical argumentation. Cambridge University Press, Cambridge
Wilder R.L. (1944) The nature of mathematical proof. The American Mathematical Monthly 51(6): 309–323
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Pease, A., Smaill, A., Colton, S. et al. Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics. Found Sci 14, 111–135 (2009). https://doi.org/10.1007/s10699-008-9150-y
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DOI: https://doi.org/10.1007/s10699-008-9150-y