Natural Reasoning in Mathematical Theorem Proving

The study of mathematical reasoning has been guided by complementary distinctions between deductive inference and individual reasoning, between objective logic and heuristic strategies, and between the context of justification and the context of discovery. This paper develops an alternative approach, examining the reasoning involved in proving mathematical theorems as belonging to a collectivity of mathematicians first and viewing individual theorem provers as taking part in that collective reasoning. Mathematical reasoning, in this way, is 'natural' to the collective practices of proving. An analogy of practice involving tangram puzzles is used to clarify the more mathematical aspects of the paper. The paper concludes by arguing that the congregational character of mathematical reasoning is sustained by individual theorem provers in, as, and through the material detail, material specificity, and material definiteness of mathematical argumentation.