[snip]
There's a sense in which this is trivially true, since Euclid's geometry
was axiomatized, and Euclidean geometry led to non-Euclidean geometry by
way of concern over the status of the parallel postulate in one or more of
its equivalent forms. However, wasn't there more to it? For example,
don't you suppose Gauss when he mused about some non-Euclidean ideas had in
mind some geometrical considerations about curved surfaces of the sort
exhibited in his work on that subject? Don't you suppose Lobachewski (or
however you want to spell it) also had geometrical considerations in mind?
Bolyai, on the other hand, if I remember correctly, was more of a thinker
along axiomatic rather than intuitive geometrical lines. (I use
"intuitive" to refer to thinking based on geometrical visualization,
including what I'll call "internal imaging", cf. German Anschauung.)
Gordon Fisher gfisher@shentel.net