The independence of the parallel postulate and the acute angles hypothesis on a two right-angled isosceles quadrilateral

TL;DR

This paper explores the independence of the parallel postulate and the acute angles hypothesis in non-Euclidean geometry, analyzing historical arguments and presenting a model for formal proof using natural language.

Contribution

It introduces a new model for studying geometric hypotheses with natural language, linking philosophical development to formal geometric proofs.

Findings

Arguments for non-Euclidean geometry became more rigorous with formalization.

A model using natural language supports proofs of geometric hypotheses.

The independence of the parallel postulate is contextualized within geometric development.

Abstract

We trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. We use these philosophical views to explain why the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Finally, we provide a model that creates the basis for proof and demonstration using natural language in order to study the acute angles hypothesis on a two right-angled isosceles quadrilateral, with a rectilinear summit side.