Title：Justification by confluence

Speaker：Jason Chen (Deparment of Logic and Philosophy of Science, UC Irvine)

Time: 2024/04/22 15:00-17:00

Location: Room B112, Lee Shau Kee Humanities Buildings No.2（李兆基人文学苑2号楼）, Peking University

Abstract：

　　The Church-Turing Thesis states that any function intuitively computable by an effective method can be computed by a Turing machine. The received wisdom is that the thesis expresses the conviction that our formalization of effectivity by means of Turing machines is correct. Aside from Turing’s insightful analysis of the meaning of “effective method”, the thesis is crucially justified by a certain phenomenon: all attempts to formalize the notion of effective methods have led to the same class of functions.

　　I propose to understand this last phenomenon in the more general context of justification by confluence. Roughly put, confluence refers to situations whereby a number of distinct attempts (to define, formalize, solve, etc.) have all produced results that are in some sense equivalent or conversely that a particular object or concept admits numerous seemingly distant equivalent definitions. In particular, this paper will focus on the roles confluence phenomena play in terms of evidence and justification.

　　Via an extensive survey of the technical literature, I will show that this kind of justification turns out to be quite  across a diverse range of mathematical subfields, pervasive in actual mathematical practice. However, I shall argue that, despite an apparently singular application in the case of the Church-Turing Thesis, appeals to confluence actually serve a wide variety of justificational purposes that are largely orthogonal to each other. Inspired by Maddy’s poignant classification of what “foundational” means in mathematical/philosophical discourse (in her 2019 paper “What Do We Want a Foundation to Do?”), I shall present a taxonomy of what these purposes are and show how they all arise, in some fuzzy and conflated fashion, in the case of computability. I shall also attempt to tease apart these roles in the surveyed examples, showing that they are in fact orthogonal to each other. At appropriate conjunctures throughout, I will also point to disagreements, debates, or open-ended philosophical questions in the literature, and apply this framework to show that they may be fruitfully resolved.