Arithmetical investigations: a study of models of arithmetic and purity of methods

In the mathematical part, we focus on computability-theoretic issues concerning models of first-order Peano arithmetic (PA). In Chapter 2, we investigate the complexity of m-diagrams of models of various completions of PA. We obtain characterizations that extend Solovay's results for open diagrams of models of completions of PA. In Chapter 3, we characterize sequences of Turing degrees that occur as {deg(T cap Sigma_n: ninomega}, where T is a completion of PA. In Chapter 4, we answer three questions asked by J. Knight concerning potential simplifications to Solovay's results. We show that these simplifications cannot be made, by proving some new independence results. In Chapter 5, we extend those independence results, using methods from higher recursion theory. In the philosophical part, we focus on purity constraints in mathematics. A proof is pure, roughly, if it uses methods `close' or `akin' to the statement being proved. We consider three different types of purity, which we call systematic, elementary and cognitive purity. Systematic purity, which we study in Chapters 7 and 8, has roots in Aristotle's views concerning scientific knowledge. Elementary purity, which we study in Chapters 9 through 12, has roots in Pappus' work in geometry, in Descartes' work in both geometry and epistemology, and in Hilbert's foundational work. Cognitive purity, which we discuss in Chapter 13, has roots in Kant's distinctions between philosophical and mathematical cognition, and between different sources of knowledge. We explain in detail what are each of the types of purity, consider what epistemic benefits are conferred by restricting ourselves to pure proofs, and discuss the consequences of apparent violations of these constraints in mathematical practice.