On Monday, February 17, 2014 4:42:41 PM UTC+2, David Hartley wrote: > In message <firstname.lastname@example.org>, Ken
> 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.
Well, it's simply not true that "If it can't be done solely from the axioms and postulates, with no diagrams".
Your statement is immediately disproved by the fact that Euclid arrived at these concepts before he ever used any visualisation. This did not occur to you? :-) Do you think Euclid started with the visualisations first and then developed the theory?! Absurd! For all intents and purposes, Euclid developed the theory just as if he were BLIND. Did you get this? :-)
> 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.
Is it just me or does that statement make no sense whatsoever?
> 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.
Your wrong opinion.
> Add an axiom that ensures there is a point not on a given line and it > may be possible, I don't know.
You don't seem to understand axioms very well at all.