Introduction: General Remarks on Proof Interpretations

This chapter discusses applied aspects of Gödel's functional (‘Dialectica’) interpretation, which was originally designed for foundational purposes. The reorientation of proof theory toward applications to concrete proofs in different areas of mathematics, which was begun in the 1950s by G. Kreisel's pioneering work on the ‘unwinding of proofs,’ also led to a reassessment of possible uses of functional interpretations. Since the 1990s, this has resulted in a systematic development of specially designed versions of functional interpretation and their use in numerical analysis, functional analysis, metric fixed point theory, and geodesic geometry. Whereas [67] presents a comprehensive survey of the new results that were obtained in these areas in the course of this investigation, this chapter focuses on the underlying logical aspects of these developments. We start, however, with a general discussion of so-called proof interpretations (and their role in Gödel's work), of which functional interpretation is a particularly interesting instance, and explain the original motivation behind the latter.

Proof interpretations play an important role in Gödel's work and seem to be first used systematically by him.

Let T1 and T2 be theories in languages ℒ(T1) and ℒ(T2).