
Re: Euclid's Elements Book 1 Proposition 1  something Euclid missed?
Posted:
Feb 18, 2014 7:40 AM


@KQ: To say that a line consists of points is truly absurd.
A point is an invisible marker that denotes or marks off the distance from a fixed origin. There are many such distances, so that they can't be counted. But does that mean that a line consists of distances? Of course not, a line is the shortest distance between two points.
Can one say that a line is the union of all such distances? I suppose so, but the points are not the distances, they are the reifiable points that denote the length of each distance by means of a rational number.
Sqrt(2) is not a number. If we could measure sqrt(2), then we would be able to reify the invisible marker. We can reify 1.414, but this does not denote sqrt(2). We can reify 3.14159, but this does not denote pi.
In one of its aspects, a point marks off *distance*, and in another, it denotes the *length of that distance*.
For example:
mxxxxxxxxn 0________1
m, n and x are all points, but m and n are also invisible markers denoting 0 and 1. In the next example,
mxxxxxxxxnxxxxxxxxp 0________1________2
we reify points 0, 1 and 2. But we could have drawn the same number line as:
xxxxxxxx xxxxxxxx 0________1________2
It is due to the fact that points have no dimension that we can use them as markers to denote length. We do not introduce holes in the number line, when we insert or remove points, unless of course, we can reify those points.
Does this make more sense to you? :)

