Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory
Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reductions. This mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frameworks. Against that background, this chapter examines Bernays’s reflections on proof-theoretic reductions of mathematical structures to methodological frames via projections. However, these reflections are focused on narrowly arithmetic features of frames. Drawing on broadened meta-mathematical experience, this chapter proposes a more general characterization of frames that has ontological and epistemological significance. The characterization is given in terms of accessibility: domains of objects are accessible if their elements are inductively generated, and principles for such domains are accessible if they are grounded in our understanding of the generating processes.