This volume brings together a revised and annotated selection of Solomon Feferman's most important writings, covering the relation between logic and mathematics, proof theory, objectivity and intentionality in mathematics, and key issues in the work of Godel, Hilbert and Turing. Contents; I: FOUNDATIONAL PROBLEMS; 1 Deciding the undecidable: Wrestling with Hilbert's problems; 2 Infinity in mathematics: Is Cantor necessary?; II: FOUNDATIONAL WAYS; 3 The logic of mathematical discovery versus the logical structure of mathematics; 4 Foundational ways; 5 Working foundations; III: GÖDEL; 6 Gödel's life and work; 7 Kurt Gödel: Conviction and caution; 8 Introductory note to Gödel's 1933 lecture; IV: PROOF THEORY; 9 What does logic have to tell us about mathematical proofs?; 10 What rests on what? The proof-theoretic analysis of mathematics. 11 Gödel's Dialectica interpretation and its two-way stretchV: COUNTABLY REDUCIBLE MATHEMATICS; 12 Infinity in mathematics: Is Cantor necessary? (Conclusion); 13 Weyl vindicated: Das Kontinuum seventy years later; 14 Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics; Symbols; References; Index; A; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; Q; R; S; T; U; V; W; Z