Constructive Coordinatization of Desarguesian Planes

TL;DR

This paper develops a constructive approach to Desarguesian geometry, transforming classical nonconstructive axioms into a constructive framework while preserving core results and addressing nonconstructivities with Brouwerian counterexamples.

Contribution

It introduces a constructive coordinatization theory for Desarguesian planes, strengthening axioms and definitions to align with constructive principles and revealing the constructive content of classical results.

Findings

Classical Desarguesian geometry results are established constructively.

Counterexamples highlight nonconstructivities in classical theory.

Constructive definitions are equivalent to traditional ones in classical logic.

Abstract

A classical theory of Desarguesian geometry, originating with D. Hilbert in his 1899 treatise, Grundlagen der Geometrie, leads from axioms to the construction of a division ring from which coordinates may be assigned to points, and equations to lines; this theory is highly nonconstructive. The present paper develops this coordinatization theory constructively, in accordance with the principles introduced by Errett Bishop in his 1967 book, Foundations of Constructive Analysis. The traditional geometric axioms are adopted, together with two supplementary axioms which are constructively stronger versions of portions of the usual axioms. Stronger definitions, with enhanced constructive meaning, are also selected; these are based on a single primitive notion, and are classically equivalent to the traditional definitions. Brouwerian counterexamples are included; these point out specific nonconstructivities in the classical theory, and the consequent need for strengthened definitions and results in a constructive theory. All the major results of the classical theory are established, in their original form, revealing their hidden constructive content.