Argumentation and mathematics

When Stephen Toulmin published The Uses of Argument (1958), logicians fiercely opposed his views since he challenged the view accepted at the time that all sorts of arguments should be evaluated according to laws of logic. He advocated a dynamic and informal, rather than a formal, notion of validity for arguments. Logic was superseded by what resembled a judicial procedure. His model of the ‘layout of arguments’ (conclusions driven from data supported by warrants and possibly justified by backing, refutation, etc.) is certainly a landmark in Argumentation Theory. Although it seems very general, Toulmin convincingly showed that this layout is governed by distinctive norms in each domain - that they are field-dependent norms according to which arguments are to be judged differently in every domain. Also, Toulmin insisted that his model is not applicable in mathematics and that in pure mathematics, arguments could be judged according to laws of logic. The fact that nowadays mathematical educators use his model in their teaching and that researchers use it to model the object of mathematical activity in classroom discussions suggests that Toulmin’s revolution went farther than he himself originally thought.