Can Non-classical Logic Treat Mathematics as Exceptional?

The paper criticizes a ‘lazy’ strategy popular amongst contemporary advocates of non-classical logic motivated by non-mathematical phenomena (e.g. vagueness), which is to argue that, since the relevant phenomena do not occur in pure mathematics (e.g. because it is precise), classical logic (with e.g. the law of excluded middle) remains valid as restricted to the special case of pure mathematics, so no revision of classical mathematics is needed. The lazy strategy is shown to be inadequate because it fails for applied mathematics, when theorems of pure mathematics are instantiated with respect to physical situations as characterized in terms for which the phenomena motivating the non-classical logic still arise (e.g. imprecise physical descriptions). Thus, such non-classical logicians are under pressure to reconstruct as much mathematics as they can on the basis of their favoured non-classical logic. In most cases, the results will be too weak for purposes of the natural and social sciences. The chapter also includes a short memoir of the author’s friendship with Alan Weir and their experience of philosophy and sectarian troubles when they both taught in Ireland.