[Paper Review] Constructive Coordinatization of Desarguesian Planes

Mark Mandelkern|arXiv (Cornell University)|Feb 12, 2024

Advanced Numerical Analysis Techniques16 references5 citations

TL;DR

The paper develops Desarguesian plane coordinatization constructively in the Bishop sense, strengthening axioms and using a single primitive notion to derive a division-ring coordinatization.

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.

Motivation & Objective

• Rebuild the classical coordinatization of Desarguesian planes using constructive methods.
• Adopt Bishop's constructive framework and a single primitive notion of distinct points.
• Strengthen definitions and provide finite procedures that replace nonconstructive parts of the classical theory.
• Show how Desargues’s and Pappus’s theorems relate to the symmetry and commutativity properties constructively.

Proposed method

• Define geometry with a single primitive notion of distinct points and a constructive principal relation “point outside a line.”
• Adopt three axiom groups G, L, and K, with L providing constructive handling of nonparallel lines.
• Strengthen concepts of parallelism, dilatations, and translations to obtain constructive maps and extensions.
• Prove Desargues’s Theorem equivalent to the symmetry axioms and Pappus’s Theorem equivalent to commutativity of the coordinatizing division ring.
• Construct coordinatization from homomorphisms of the translation group and establish injectivity of nonzero homomorphisms.
• Demonstrate the real plane R^2 as a constructive model satisfying the axioms and discuss Brouwerian counterexamples that motivate axiom strengthening.

Experimental results

Research questions

• RQ1 How can Desarguesian plane coordinatization be achieved constructively while preserving classical results?
• RQ2 What constructive axiom system suffices to derive a division-ring coordinatization from translations and dilatations?
• RQ3 What is the constructive relationship between Desargues’s Theorem, symmetry axioms, and Pappus’s Theorem?
• RQ4 Can standard models like the real plane R^2 satisfy the strengthened constructive axioms, and what are the Brouwerian counterexamples?
• RQ5 How do Heyting-style approaches compare to the Bishop-style constructive framework in this geometric setting?

Key findings

• The major classical results of Desarguesian geometry are established in their original form with enhanced constructive meaning.
• A constructive framework centers on a strong notion of nonparallel lines and a principal relation of a point outside a line.
• Desargues’s Theorem is shown equivalent to the symmetry axioms, and Pappus’s Theorem to commutativity of the resulting division ring.
• Dilatations and their extensions and inverses are developed constructively, enabling coordinatization via translation group homomorphisms.
• The real plane R^2 is shown to be a Desarguesian plane under the constructive axioms, with Brouwerian counterexamples illustrating nonconstructivities in the classical theory.