Relating Information and Proof

TL;DR

This paper explores a formal connection between information as uncertainty and proof as justification, proposing a probabilistic measure of informativeness based on proof sets, though practical applications remain unclear.

Contribution

It introduces a novel conceptual framework linking information theory and proof theory through probabilistic measures and entropy of proof sets.

Findings

Proposes a formal measure of informativeness of proofs

Defines a probabilistic measure over sets of related proofs

Suggests a new perspective on information in mathematical and linguistic contexts

Abstract

In mathematics information is a number that measures uncertainty (entropy) based on a probabilistic distribution, often of an obscure origin. In real life language information is a datum, a statement, more precisely, a formula. But such a formula should be justified by a proof. I try to formalize this perception of information. The measure of informativeness of a proof is based on the set of proofs related to the formulas under consideration. This set of possible proofs (`a knowledge base') defines a probabilistic measure, and entropic weight is defined using this measure. The paper is mainly conceptual, it is not clear where and how this approach can be applied.