Looking from the inside and from the outside
Alessandra Carbone, S. Semmes

TL;DR
This paper explores the relationship between inner and outer descriptions of mathematical objects, using proof-based geometries and topological concepts to reveal new structures and symmetries.
Contribution
It introduces new geometries on groups and fields through proofs, linking proof theory with topological and dynamical structures in mathematics.
Findings
New geometries on finitely generated groups via proofs
Structured rational numbers under dynamical actions like SL(2,Z)
Connections between proof complexity and internal symmetries
Abstract
One often sees a sharp distinction in mathematics between descriptions from the outside and from the inside. Think of defining a set in the plane through an algebraic equation, or dynamically as the closure of the orbit of some point under iterations of a given mapping. In logic one sees this dichotomy in the descriptions of sets of tautologies through semantics and proofs. Logic provides several tools for making outer descriptions of mathematical objects. This paper concerns a slightly complicated mixture of themes related to inner descriptions and formal proofs. We use the notion of feasibility to embed mathematical structures into spaces of logical formulas, from which we can obtain new structures through proofs. We present new geometries on finitely generated groups through proofs, and new structure on the rational numbers (or other fields) which is susceptible to dynamical…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Logic, programming, and type systems · Advanced Topology and Set Theory
