On Tuesday, February 18, 2014 7:40:00 AM UTC-5, John Gabriel wrote: > @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? :-)
You use the term 'reified' simply to say that one point is rational and another is not. There's no need for this extra word, altho people do seem to like creating new words to express the same thing considered from another point of view. I'm all for that if the new point of view brings something new to the table, but if it's just putting old wine in new bottles it seems to confuse things rather than clarify them.
You do grant that a point which is the marker for a rational distance can be removed and leave a hole in the line, but a point which is the marker for an irrational distance does not leave a hole if it is removed because it does not represent a rational distance (leaving aside for the moment the more philosophical question of how can you remove something and NOT leave a hole, if you have already granted its existence).
But consider a line segment AC which contains the intermediate point B, so that AB is length sqrt(2) and BC is length 1. Then B is both from your point of view irrational as a distance measured from A, and rational, as a distance measured from C.
So the point is both reifiable and non-reifiable. This seems to imply that the notion of reifiable as meaning anything more than rational vs irrational, at least as you have so far characterized it, is unusable since it is internally contradictory - in ALL cases where it can be applied.