Varieties of Confluence Arguments, Part 1: Practical Applications

This paper is the self-contained first part of a two-part series that examines the wide varieties of ways that mathematicians and philosophers have appealed to confluence phenomena in their work. In this part, we focus on the practical roles such phenomena can play, paying special attention to how they facilitate the communication of mathematical ideas (proofs, definitions, and conjectures) in actual practice. Through surveying the wide array of such arguments, I shall eventually hone in on two subtly distinct facets concerning the uses of confluence arguments (and another that is revealed to be rather trivial upon analysis), which are sometimes conflated in philosophical discussions: one (dubbed Rigor Assurance) guarantees the formal rigor of a proof, while the other (Coding Invariance) ensures the results obtained are not a mere artifact of the coding that is used. I will then attempt to tease these two facets apart with actual examples in print, citing crucial evidence witnessing the distinction. With this setup in place, we will see how certain recent philosophical debates about the practical use of the Church-Turing Thesis may naturally dissolve, as the combatants do not share the same assumptions of the roles confluence arguments play in each case, and thus end up merely talking past each other.