The metamathematics of ergodic theory

Jeremy Avigad

doi:10.1016/j.apal.2008.09.001

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”.

Highlights

The late nineteenth century inaugurated an era of sweeping changes in mathematics
For example, wrote of his development of the theory of ideals: It is preferable, as in the modern theory of functions, to seek proofs based immediately on fundamental characteristics, rather than on calculation, and to construct the theory in such a way that it is able to predict the results of calculation. . . [21, page 102]. Such attitudes paved the way to the adoption of the infinitary, nonconstructive, set theoretic, algebraic, and structural methods that are characteristic of modern mathematics
Something more is needed to justify the choice of symbolic rules with respect to our understanding of the mathematical enterprise; at the bare minimum, we wish to know that the universal assertions we derive in the system will not be contradicted by our experiences, and the existential predictions will be born out by calculation

Summary

Introduction

The late nineteenth century inaugurated an era of sweeping changes in mathematics. Whereas mathematics had, until that point, been firmly rooted in explicit construction and symbolic calculation, the new developments emphasized a kind of understanding that was often at odds with computational. Something more is needed to justify the choice of symbolic rules with respect to our understanding of the mathematical enterprise; at the bare minimum, we wish to know that the universal assertions we derive in the system will not be contradicted by our experiences, and the existential predictions will be born out by calculation This is exactly what Hilbert’s program was designed to do. The theory of dynamical systems and ergodic theory provide fruitful arenas for such analysis These subjects arose from the study of physical and statistical phenomena, they make full use of modern structural methods that do not directly bear on the original computational concerns. Most of the work I will describe here has been carried out jointly with Philipp Gerhardy, Ksenija Simic, and Henry Towsner

Dynamical systems and ergodic theory

Analysis of the ergodic theorems

The Furstenberg structure theorem and ergodic Ramsey theory

Analysis of the structure theorem

The constructive content of ID1

Conclusion