Don Berry

It is an established part of mathematical practice that mathematicians demand deductive proof before accepting a new result as a theorem. However, a wide variety of probabilistic methods of justification are also available. Though such procedures may endorse a false conclusion even if carried out perfectly, their robust structure may mean they are actually more reliable in practice once implementation errors are taken into account. Can mathematicians be rational in continuing to reject these probabilistic methods as a means of establishing a mathematical claim? In this paper, I give reasons in favour of their doing so. Rather than appealing directly to individual epistemological considerations, the discussion offers a normative constraint on what constitutes a good mathematical argument. This I call ‘Univocality’, the requirement that the underlying concepts all have clear defining conditions.

This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes us with a partial rational justification for mathematicians strict insistence on proof.