Practical reasoning and the witnessably rigorous proof

Abstract

This paper introduces an anthropological approach to the foundations of mathematics. Traditionally, the philosophy of mathematics has focused on the nature and origins of mathematical truth. Mathematicians, however, treat mathematical arguments as determining mathematical truth: if an argument is found to describe a witnessably rigorous proof of a theorem, that theorem is considered—until the need for further examination arises—to be true. The anthropological question is how mathematicians, as a practical matter and as a matter of mathematical practice, make such determinations. This paper looks first at the ways that the logic of mathematical argumentation comes to be realized and substantiated by provers as their own immediate, situated accomplishment. The type of reasoning involved is quite different from deductive logic; once seen, it seems to be endemic to and pervasive throughout the work of human theorem proving. A number of other features of proving are also considered, including the production of notational coherence, the foregrounding of proof-specific proof-relevant detail, and the structuring of mathematical argumentation. Through this material, the paper shows the feasibility and promise of a real-world anthropology of disciplinary mathematical practice.