Five theories of reasoning:Inter-connections and applications to mathematics

The last century has seen many disciplines place a greater priority on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumenta- tion have been developed. Perhaps owing to their diverse backgrounds, there are several connections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce's development of abductive reasoning [3], Toulmin's argumentation layout [5], Lakatos's theory of reasoning in mathematics [2], Pollock's notions of counterexample [4] and argumentation schemes constructed by Walton et al [6], and explore some connections between, as well as within, the theories. For instance, we investigate Peirce's abduction to deal with surprising situations in mathematics, represent Pollock's examples in terms of Toulmin’s layout, discuss connections between Toulmin's layout and Walton’s argu- mentation schemes, and suggest new argumentation schemes to cover the sort of reasoning that Lakatos describes, in which arguments may be accepted as faulty, but revised, rather than being accepted or rejected. We also consider how such theories may apply to reasoning in mathematics: in particular, we aim to build on ideas such as Dove's [1], which help to show ways in which the work of Lakatos fits into the informal reasoning community.[1] I. J. Dove. On mathematical proofs and arguments: Johnson and Lakatos. In F. H. Van Eemeren and B. Garssen, editors, Proceedings of the Sixth Conference of the International Society for the Study of Argumentation, volume 1, pages 346–351. Sic Sat, Amsterdam, 2007.[2] I. Lakatos. Proofs and Refutations. CUP, Cambridge, UK, 1976.[3] C. S. Peirce. Collected Papers of Charles Sanders Peirce. Harvard University Press, Cambridge, Mass, 1931–58. Eight Volumes.[4] J. Pollock. Cognitive Carpentry. The MIT press, Cambridge, MA., 1995.[5] S. Toulmin. The uses of argument. CUP, Cambridge, 1958.[6] D. Walton, C. Reed, and F. Macagno. Argumentation Schemes. Cambridge University Press, New York, USA, 2008.