An Elementary System of Axioms for Euclidean Geometry based on Symmetry Principles

TL;DR

This paper introduces an elementary axiomatic system for Euclidean geometry based on symmetry principles like homogeneity, isotropy, and scale invariance, linking classical and modern geometric frameworks.

Contribution

It develops an axiomatic system grounded in symmetry principles that simplifies to Weyl's axioms, offering new insights into the philosophy and pedagogy of geometry.

Findings

The axiomatic system is based on symmetry principles.

It simplifies to Weyl's system of axioms.

Supports the view of Euclidean geometry as a priori.

Abstract

In this article, I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides Weyl's system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (i) it supports the thesis that Euclidean geometry is a priori, (ii) it supports the thesis that in modern mathematics the Weyl's system of axioms is dominant to Euclid's system because it reflects the a priori underlying symmetries, (iii) it gives a new and promising approach to learn geometry which, through the Weyl's system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.