Abstract
In The Uses of Argument, Stephen Toulmin (1958) introduced a model of argumentation, in which what may be called the ‘layout of arguments’ is represented. This model has become a classic in argumentation theory and has been used in the analysis, evaluation and construction of arguments. Toulmin’s main thesis is that, in principle, one can make a claim of rationality for any type of argument, and that the criterion of validity depends on the nature of the problem in question. He rejects the idea of universal norms for evaluation of argumentation and that formal logic provides these norms. There is an essential difference between the norms which are relevant to the evaluation of everyday argumentation or in diverse disciplines, on the one hand, and the criterion of formal validity used by formal logic, on the other. In this article, I will begin by reviewing Toulmin’s model of argumentation, focusing on what is described as the elements of argument; secondly, I will attempt to apply this model to argumentation in mathematics, and finally, I will pursue the consequences that connect it to recent ideas on the similarity between mathematics and science in the philosophy of mathematics.
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References
Alcolea Banegas, J. (1997). Demostración como communicación. In A. Estany & D. Quesada (Eds.), Actas del II Congreso de la Sociedad de Lógica, Metodología y Filosofia de la Ciencia en España (pp. 73–77). Barcelona: Servicio de Publicaciones de la Universidad Autónoma Barcelona.
Appel, K., & Haken, W. (1977). The solution of the four-color-map problem. Scientific American, 237(4), 108–121.
Gödel, K. (1995). Unpublished philosophical essays. Basel: Birkhäuser.
Hersh, R. (1997). Prove—Once more and again. Philosophia Mathematica, 5, 153–165.
Moore, G. H. (1982). Zermelo’s axiom of choice. Berlin: Springer.
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30, 161–171.
Toulmin, S. (1958). The uses of argument. Cambridge: Cambridge University Press.
Toulmin, S., Rieke, R., & Janik, A. (1979). An introduction to reasoning. London: Macmillan.
Zermelo, E. (1967 [1904]). Proof that every set can be well-ordered. In J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931 (pp. 139–141). Cambridge, MA: Harvard University Press.
Zermelo, E. (1967 [1908a]). Investigations in the foundations of set theory I. In J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931 (pp. 199–215). Cambridge, MA: Harvard University Press.
Zermelo, E. (1967 [1908b]). A new proof for the possibility of a well-ordering. In J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931 (pp. 183–198). Cambridge, MA: Harvard University Press.
Acknowledgements
Originally published in Catalan as ‘L’argumentació en matemàtiques’, in E. Casaban i Moya, editor, XIIè Congrés Valencià de Filosofia, València, 1998, 135–147. Translated into English by Miguel Gimenez and Andrew Aberdein.
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Banegas, J.A. (2013). Argumentation in Mathematics. In: Aberdein, A., Dove, I. (eds) The Argument of Mathematics. Logic, Epistemology, and the Unity of Science, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6534-4_4
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