Maarten McKubre-Jordens: Mathematics in Contradiction

Advances in paraconsistent logics have begun attracting the attention of the mathematics community. Motivations for the development of these logics are wide-ranging: expressiveness of language; a more principled  approach to implication; robustness of formal theories in the face of  (local) contradiction; founding naive intuitions; and more.
In this accessible survey talk, we outline what paraconsistency is, why  one might use it to do mathematics, discuss some of the triumphs of and challenges to doing mathematics paraconsistently, and present some recent results in both foundations and applications.
Applying such logics within mathematics gives insight into the nature of proof, teases apart some subtleties that are not often recognized, and gives new responses to old problems. It will turn out that despite the relative weakness of these logics, long chains of mathematical reasoning can be carried out. Moreover, in handling contradictions more carefully, paraconsistent proofs often bring out subtle differences between proofs that are often overlooked - not all proofs are created equal, even when truth (or falsity!) is preserved.