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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.
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