Proofs, reliable processes, and justification in mathematics

Although there exist today a variety of non-deductive reliable processes able to determine the truth of certain mathematical propositions, proof remains the only form of justification accepted in mathematical practice. Some philosophers and mathematicians have contested this commonly accepted epistemic superiority of proof on the ground that mathematicians are fallible: when the deductive method is carried out by a fallible agent, then it comes with its own level of reliability, and so might happen to be equally or even less reliable than existing non-deductive reliable processes—I will refer to this as the reliability argument. The aim of this paper is to examine whether the reliability argument forces us to reconsider the commonly accepted epistemic superiority of the deductive method over non-deductive reliable processes. I will argue that the reliability argument is fundamentally correct, but that there is another epistemic property differentiating the deductive method from non-deductive reliable processes. This property is based on the observation that, although mathematicians are fallible agents, they are also self-correcting agents. This means that when a proof is produced which only contains repairable mistakes, given enough time and energy, a mathematician or a group thereof should be able to converge towards a correct proof through a finite number of verification and correction rounds, thus providing a guarantee that the considered proposition is true, something that non-deductive reliable processes will never be able to produce. From this perspective, the standard of justification adopted in mathematical practice should be read in a diachronic way: the demand is not that any proof that is ever produced be correct—which would amount to require that mathematicians are infallible—but rather that, over time, proofs that contain repairable mistakes be corrected, and proofs that cannot be repaired be rejected.