On Tue, 28 Jan 2014 15:50:51 -0700, Joe Niederberger <email@example.com> wrote:
> Neighbor says: >> If we leave M and take A(M) and (M)B without it, then the two parts are >> not really halves of AB, for the sum of the two halves in which one >> thing is divided is the thing itself, but A(M)+(M)B will be less than >> AB, because AB = A(M)+M+(MB). Now if we take M on either AM or MB, let >> it be taken on AM, then when we separate AB we will have straight-lines >> AM and (M)B, but they will be unequal, inasmuch as A(M)=(M)B ==> >> AM>(M)B. > > You are at least correct in discovering that you can't divide a > mathematical real number line segment into two segments, such that they > are both non-overlapping and "symmetrical" (in the sense that they are > both either open or closed at the dividing point.) > > That seems to bother you. I also get the feeling the very notion of an > open ended interval bothers you. You could read up on it and think about > it for awhile. > > Cheers, > Joe N
It has already been pointed out, but it needs to be said again:
Euclid's system of geometry doesn't live up to his claims.
His "definitions" are seriously flawed, for one thing. For another, not all of Euclid's propositions can be proven from the Postulates, Axioms, "Common Notions", and Definitions that he gives---even though Euclid asserts that he gives such proofs. Moreover, propositions Euclid did not intend, and which seem to contradict what he probably intended, *can* be "proved" from what he gives using logic he probably would've admitted (because he admitted it elsewhere). Any attempt to reconcile Euclid's "straight lines" with modern conceptions of straight lines by means of what he says is therefore doomed to failure. From the get-go.
This fact and the fact that both Euclid's conception and the modern conception of straight lines are mental constructs, and not physical realities---and so also irreconcilable with physical reality, probably lie at the root of Neighbor's difficulties dealing with the explanations that have been given here.
- --Lou Talman Department of Mathematical & Computer Sciences Metropolitan State University of Denver