 
Summary: Proof Theory
Jeremy Avigad
June 8, 2011
1 Introduction
At the turn of the nineteenth century, mathematics exhibited a style of argu
mentation that was more explicitly computational than is common today. Over
the course of the century, the introduction of abstract algebraic methods helped
unify developments in analysis, number theory, geometry, and the theory of
equations; and work by mathematicians like Dedekind, Cantor, and Hilbert to
wards the end of the century introduced settheoretic language and infinitary
methods that served to downplay or suppress computational content. This shift
in emphasis away from calculation gave rise to concerns as to whether such
methods were meaningful, or appropriate to mathematics. The discovery of
paradoxes stemming from overly naive use of settheoretic language and meth
ods led to even more pressing concerns as to whether the modern methods were
even consistent. This led to heated debates in the early twentieth century and
what is sometimes called the "crisis of foundations."
In lectures presented in 1922, David Hilbert launched his Beweistheorie, or
Proof Theory, which aimed to justify the use of modern methods and settle the
problem of foundations once and for all. This, Hilbert argued, could be achieved
