A Constructive Real Projective Plane

TL;DR

This paper explores the constructive approach to real projective plane geometry, combining synthetic and analytic methods to verify classical theorems within a constructivist framework, revealing new insights and open problems.

Contribution

It introduces constructive axioms for projective geometry and verifies classical theorems using a constructive model based on Euclidean space.

Findings

Constructive axioms for projective geometry are established.

Classical theorems are verified within a constructive framework.

The approach reveals hidden constructive content in classical geometry.

Abstract

The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify the consistency of the axioms; it is based on the usual model in three-dimensional Euclidean space, using only constructive properties of the real numbers. The methods of strict constructivism, following principles put forward by Errett Bishop, reveal the hidden constructive content of a portion of classical geometry. A number of open problems remain for future studies.