Hilbert's Programs and Beyond

by Sieg, Wilfried

• ISBN13: 9780195372229
• ISBN10: 0195372220
• Format: Hardcover
• Copyright: 2013-02-21
• Publisher: Oxford University Press

Summary

Hilbert's Programs & Beyondpresents the foundational work of David Hilbert in a sequence of thematically organized essays. They first trace the roots of Hilbert's work to the radical transformation of mathematics in the 19th century and bring out his pivotal role in creating mathematical logic and proof theory. They then analyze techniques and results of "classical" proof theory as well as their dramatic expansion in modern proof theory. This intellectual experience finally opens horizons for reflection on the nature of mathematics in the 21st century: Sieg articulates his position ofreductive structuralismand explores mathematical capacities via computational models.

Author Biography

Wilfried Sieg is the Patrick Suppes Professor of Philosophy at Carnegie Mellon University. He received his Ph.D. from Stanford University in 1977. From 1977 to 1985, he was Assistant and Associate Professor at Columbia University. In 1985, he joined the Carnegie Mellon faculty as a founding member of the University's Philosophy Department and served as its Head from 1994 to 2005. He is internationally known for mathematical work in proof theory, historical work on modern logic and mathematics, and philosophical essays on the nature of mathematics. Sieg is a Fellow of the American Academy of Arts and Sciences.

Table of Contents

Introduction
In.1. A perspective on Hilbert's Programs
In.2. Milestones
I. Mathematical roots
I.3. Dedekind's analysis of number
I.4. Methods for real arithmetic
I.5. Hilbert's programs: 1917-1922
II. Analyses
Historical
II.1. Finitist proof theory: 1922-1934
II.2. After Königsberg
II.3. In the shadow of incompleteness
II.4. Gödel at Zilsel's
II.5. Hilbert and Bernays: 1939
Systematical
II.6. Foundations for analysis and proof theory
II.7. Reductions of theories for analysis
II.8. Hilbert's program sixty years later
II.9. On reverse mathematics
II.10. Relative consistency and accessible domains
III. Philosophical horizons
III.1. Aspects of mathematical experience
III.2. Beyond Hilbert's reach?
III.3. Searching for proofs