On Monday, February 17, 2014 9:42:41 AM UTC-5, David Hartley wrote: > In message <email@example.com>, Ken > > Quirici <firstname.lastname@example.org> writes > > >It would be an interesting exercise (at least it seems to me) to > > >imagine this Euclidean axiomatic system INDEPENDENT of ANY pictures. > > >How many 'realities' would this system model? I suspect there are many > > >that have absolutely nothing to do with geometry. That is, take the > > >axioms as simply words that relate to each other in logical ways and > > >see what they can represent. > > > > I was going to suggest this to you, but you've got there on your own. > > Imagine trying to explain Euclid's proofs to a blind man. If it can't be > > done solely from the axioms and postulates, with no diagrams, then it's > > not a valid proof. > > > > It is indeed true that the explicit axioms and postulates in the > > Elements are not sufficient for even the simplest of his proofs. > > > > Most trivially, there's nothing to ensure that there even exist any > > points or lines. > > > > More importantly, as was raised at the start of this thread, without an > > axiom that ensures that certain circles intersect, most of the > > constructions fail. > > > > Consider a system consisting of a single straight line and the points > > thereon. It satisfies all Euclid's explicit postulates (and Pasch's > > axiom) but circles have only two points in their circumferences and > > circles with a common radius do not intersect. There are no equilateral > > triangles. > >
there's problem in that his definitions include (at least in Heath's translation) surface - that which has length and breadth only.
Does this vitiate your statement?
> > So even if you take Euclid's postulates and add the existence of point > > and line, Pasch's axiom and any necessary order axioms, you still don't > > have enough to prove Proposition 1. > > > > Add an axiom that ensures there is a point not on a given line and it > > may be possible, I don't know. > > > > -- > > David Hartley
Thanks. I'm hoping Moise's book on Elementary Geometry from an Advanced Standpoint will provide a sufficiently rigorous approach. Have you read it? If so, is it sufficiently rigorous?