The role of audiences in mathematical proof has largely been neglected, in part due to misconceptions like those in Perelman and Olbrechts-Tyteca which bar mathematical proofs from bearing reflections of audience consideration. In this paper, I argue that mathematical proof is typically argumentation and that a mathematician develops a proof with his universal audience in mind. In so doing, he creates a proof which reflects the standards of reasonableness embodied in his universal audience. Given this framework, we can better understand (...) the introduction of proof methods based on the mathematician’s likely universal audience. I examine a case study from Alexander and Briggs’s work on knot invariants to show that we can fruitfully reconstruct mathematical methods in terms of audiences. (shrink)

In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of argumentative products, but little attention has been paid to the creative process of arguing. Mathematics can be used as a clear example to illustrate some significant theoretical differences between mathematical practice and the products of it, to differentiate the distinct components of the arguments, and to emphasize the need to address the different types of argumentative discourse and argumentative situation in the practice. I (...) consider some issues of recent papers associated with mathematical argumentation in an attempt to contribute to the discussion about the role of arguing in mathematical practice and in the evaluation of the products of this practice. I apply this discussion to learning environments to defend the thesis that argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity. (shrink)

Within three of the most well-known contemporary approaches to argumentation, the notions of solo argumentation and arguing with one’s self are given little attention and are typically argued to be able to be subsumed within the dialectical aspects of the approach being propounded. Challenging these claims, this paper has two main aims. The first is to argue that while dialogical argumentation may be most common, there exists individual dialectical argumentation, which is not so easily subsumed within these theories. Second, in (...) order characterize this type of argumentation the paper also offers distinctions between the interrelated notions of dialectical, dialogical, and quasi-dialogical, reasoning and argumentation, within an individual or between multiple participants, which I hope provide useful precision for the field. (shrink)