 
Summary: "Clarifying the nature of the infinite":
the development of metamathematics
and proof theory
Jeremy Avigad and Erich H. Reck
December 11, 2001
Abstract
We discuss the development of metamathematics in the Hilbert school,
and Hilbert's prooftheoretic program in particular. We place this pro
gram in a broader historical and philosophical context, especially with
respect to nineteenth century developments in mathematics and logic.
Finally, we show how these considerations help frame our understanding
of metamathematics and proof theory today.
Contents
1 Introduction 2
2 Nineteenth century developments 4
2.1 Developments in mathematics . . . . . . . . . . . . . . . . . . . . 5
2.2 Changing views of mathematics . . . . . . . . . . . . . . . . . . . 7
2.3 Developments in logic . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Hilbert's proof theory 15
3.1 Hilbert as mathematician . . . . . . . . . . . . . . . . . . . . . . 15
