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Hilbert's Programs & Beyond presents 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 of reductive structuralism and explores mathematical capacities via computational models.
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.
- 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
