Meaning of absolute geometry | Babel Free

Definitions

1. The single (up to logical equivalence) geometry whose axiomatic system is equivalent to that of Euclidean geometry without the parallel postulate or any alternative.
2. Any geometry whose axiomatic system extends that of absolute geometry (in the singular sense) and neither assumes nor contradicts the parallel postulate.

Examples

“1993 [Princeton University Press⟳], Donald M. Davis, The Nature and Power of Mathematics, 2004, Dover, page 85, Recall⟳ that absolute geometry is the set⟳ of statements that can be deduced from Euclid's first four postulates. Then existence of parallel lines is certainly a theorem of absolute geometry, while the question⟳ being addressed by most of the mathematicians discussed in this section is whether uniqueness of parallels is also a theorem of absolute geometry.”

“1970, J. F. Rigby, Axioms for Absolute Geometry, III, Canadian Journal of Mathematics, Vol. XXII, No. 1, Canadian Mathematical Congress, University of Toronto Press⟳, page 185, A discussion of one-dimensional absolute geometries, with examples, will be given in a separate⟳ paper.”

CEFR level