Review of Basic Proof Theory ( second edition )

Beweistheorie, or “Proof Theory,” was the phrase that David Hilbert used to describe the program by which he hoped to secure the foundations of mathematics. Set forth in the early 1920’s, his plan was to represent mathematical reasoning by formal deductive systems, and show, using safe, “finitary,” methods, that such reasoning could never lead to contradiction. This particular goal was shown by Gödel to be infeasible. But the more general goal of using formal methods to explore various aspects of mathematical provability, including the relationship between classical and constructive methods in mathematics and the strengths and limitations of various axiomatic frameworks, has proved to be remarkably robust. Today, these goals represent the traditional, metamathematical branch of proof theory. Since Hilbert’s time, the subject has expanded in two important respects. First, it has moved well beyond the study of specifically mathematical reasoning. Proof theorists now consider a wide range of deductive systems, designed to model diverse aspects of logical inference; for example, systems of modal logic can be used to model reasoning about possible states of affairs, knowledge, or time, and linear logic provides a means of reasoning about computational resources. The second change is that now more attention is paid to specific features of the deductive systems themselves. For the Hilbert school, deductive calculi were used primarily as a means of exploring the deductive consequences of various axiomatic theories; from this point of view, the particular choice of a calculus is largely a matter of convenience. In much of modern proof theory, however, deductive systems have become objects of study in their own right. For example, in the branch of proof theory known as proof complexity, one aims to determine the efficiency of various systems with respect to various measures proof length, just as the field of computational complexity explores issues of computational efficiency with respect to various measures of complexity. In the field of structural proof theory, the focus is on the particular syntactic representations, axioms, and rules. A central theme is the exploration of transformations, or “reductions,” that preserve the validity of a proof. One