Rationality in Mathematical Proofs

Mathematical proofs are not sequences of arbitrary deductive steps-each deductive step is, to some extent, rational. This paper aims to identify and characterize the particular form of rationality at play in mathematical proofs. The approach adopted consists in viewing mathematical proofs as reports of proof activities-that is, sequences of deductive inferences-and in characterizing the rationality of the former in terms of that of the latter. It is argued that proof activities are governed by specific norms of rational planning agency, and that a deductive step in a mathematical proof qualifies as rational whenever the corresponding deductive inference in the associated proof activity figures in a plan that has been constructed rationally. It is then shown that mathematical proofs whose associated proof activities violate these norms are likely to be judged as defective by mathematical agents, thereby providing evidence that these norms are indeed present in mathematical practice. We conclude that, if mathematical proofs are not mere sequences of deductive steps, if they possess a rational structure, it is because they are the product of rational planning agents.