However, Hilbert's very interesting exploration was more for the benefit of logicians and the foundations of eulcidean geometry. It is not usable for teaching new-comers, or for assisting or describing how people now doing/ using geometry practice geometric reasoning.
To paraphrase my advisor (Gian-Carlo Rota) 'this axiomatic presentation of geometry is to the practice of geometry as medicine is to food'. The axioms point out some areas of concern that should be raised to our attention under certain circumstances, but do not serve the teaching and learning of geometry most of the time.
In my practice of discrete applied geometry, and in my teaching / research on mathematics education, Felix Klein has been a substantially larger influence with his Erlanger Program. It is no accident that one of the two most prestigious prizes in Mathematics Education is named after Klein. The other is named after Hans Freudenthal, who has also written about geometry. There is a nice article by Freudenthal, in JSTOR: Geometry between the devil and the deep sea. I note that to Freudenthal, 'the devil' was the axiomatic approach to geometry! He preferred to be in the deep sea.
Walter Whiteley York University
On 10-Feb-08, at 8:44 PM, Kirby Urner wrote:
"David Hilbert was one of the outstanding mathematicians of the modern era. He proposed 21 geometry axioms--the greatest influence in geometry since Euclid (325 BC)."