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The metamathematics of ergodic theory (2009)
| Venue: | THE ANNALS OF PURE AND APPLIED LOGIC |
| Citations: | 13 - 1 self |
BibTeX
@INPROCEEDINGS{Avigad09themetamathematics,
author = {Jeremy Avigad},
title = {The metamathematics of ergodic theory},
booktitle = {THE ANNALS OF PURE AND APPLIED LOGIC},
year = {2009},
publisher = {}
}
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Abstract
The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how proof-theoretic methods can be used to locate their “constructive content.”
Keyphrases
ergodic theory recent application recent contribution seventeenth century dynamic metamathematical tradition explicit term constructive character rich opportunity computational concern metamathematical study furstenberg structure theorem number theory force proof-theoretic method nineteenth century statistical mechanic contemporary mathematics central goal pointwise ergodic theorem infinitary method constructive content syntactic modeling structural method
