Comments on ‘Strategic Maneuvering in Mathematical Proofs’

Abstract

In ‘‘Strategic Maneuvering in Mathematical Proofs,’’ Krabbe (2007) aims to show that argumentation theory, and specifically pragma-dialectics, can help us better understand the nature of mathematical proofs. Proofs are not always arguments, by Krabbe’s account, but he holds that ‘‘whenever in a proof the reasoning displays persuasive functions, the proof can be regarded as an argument.’’ In these cases, where proofs do have a persuasive function, their persuasive effectiveness may depend on strategic maneuvering, and this brings proofs within the scope of the recent extensions of pragma-dialectics that integrate rhetorical and dialectical principles. Agreeing with Krabbe’s overall point that proofs are often shaped by rhetorical considerations as well as dialectical ones, I wish to focus my own remarks on what an examination of proofs and proving might contribute to a broader understanding of argumentation. In contemporary argumentation theory, argumentation is assumed to have resolution of disagreement or difference of opinion as its characteristic purpose. In argumentative discourse, we drill down from disagreeing speech acts in search of contradictory source beliefs, and once the source of disagreement is identified and dispatched, argumentation typically returns to the original interactional business. The idea that argumentation is aimed at repair of disagreement is central to my own work (see my own article in this volume). The normal course of interaction is purposeful exchange of speech acts, from which an open-ended set of beliefs and intentions may be reconstructed by each participant. People search and examine these underlying beliefs and intentions when they need to in order to continue with a purposeful exchange. Whether as an expression of Gricean cooperativity (as argued by Jacobs and Jackson 1983) or as output from ‘a simple heuristic that makes us smart’ (as might be inferred from Gigerenzer et al. 1999; see also Jackson 1996),