  Informal logic is a method of argument analysis which is complementary to
that of formal logic, providing for the pragmatic treatment of features of
argumentation which cannot be reduced to logical form. The central claim of
this paper is that a more nuanced understanding of mathematical proof and
discovery may be achieved by paying attention to the aspects of mathematical
argumentation which can be captured by informal, rather than formal, logic. Two
accounts of argumentation are considered: the pioneering work of Stephen
Toulmin [The uses of argument, Cambridge University Press, 1958] and the more
recent studies of Douglas Walton, [e.g. The new dialectic: Conversational
contexts of argument, University of Toronto Press, 1998]. The focus of both of
these approaches has largely been restricted to natural language argumentation.
However, Walton's method in particular provides a fruitful analysis of
mathematical proof. He offers a contextual account of argumentational
strategies, distinguishing a variety of different types of dialogue in which
arguments may occur. This analysis represents many different fallacious or
otherwise illicit arguments as the deployment of strategies which are sometimes
admissible in contexts in which they are inadmissible. I argue that
mathematical proofs are deployed in a greater variety of types of dialogue than
has commonly been assumed. I proceed to show that many of the important
philosophical and pedagogical problems of mathematical proof arise from a
failure to make explicit the type of dialogue in which the proof is introduced.

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