Full Idea
There are two approaches to axiomatising geometry. The 'metric' approach uses a function which maps a pair of points into the real numbers. The 'synthetic' approach is that of Euclid and Hilbert, which does without real numbers and functions.
Gist of Idea
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space
Source
Hartry Field (Science without Numbers [1980], 5)
Book Reference
Field,Hartry: 'Science without Number' [Blackwell 1980], p.42
Related Idea
Idea 18218 Hilbert explains geometry, by non-numerical facts about space [Field,H]