UC Irvine

The dissertation presents a collection of interrelated works in the philosophy of mathematics. They are roughly unified by their focus on descriptive set theory, which is investigated through the lens of mathematical practice. Chapters 2 and 3 examine the roles that confluence plays in mathematical practice, such as providing justification for the Church-Turing Thesis. An extensive survey of the technical literature will attest to the ubiquity of justification by confluence, and it will be shown to serve a wide variety of justificational purposes that are largely orthogonal to each other. Chapter 4 presents a series of theorems and proofs that involve increasingly substantial use of metamathematical methods. Reflecting on the question of whether the metamathematical elements can be translated away without loss of insight, it attempts to shed light on our practical taxonomy of proofs by their methodology, as well as on the specific question of whether a proof can be said to make substantial use of metamathematical methods. Chapter 5 traces the pre-history of the theory of Borel equivalence relations, with the specific aim of identifying the early ancestors to this theory prior to its sudden emergence in the 1990s.