Representations and the foundations of mathematics

TL;DR

This paper examines how different levels of representation for continuous functions impact the logical strength of foundational theorems, with implications for reverse mathematics and foundational programs.

Contribution

It analyzes the effects of second- versus third-order representations of continuous functions on theorem strength and discusses foundational implications.

Findings

Changing representations alters the logical strength of key theorems.

Third-order functions provide a more robust foundation for continuous functions.

The paper proposes criteria to avoid problematic representations.

Abstract

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.