Re: What happens to the midpoint when a line is cut in two?
Danny,
Yes, that's the issue.
> Hmm, isn't the reasoning a little back to front here? When you
> say two equal halves, you are ruling out the possibility of a
> single, indivisible midpoint. If the line is composed of points,
> then the cut must, to be consistent with "equal halves", occur
> not at a point, but /between/ two points (if the points cannot
> be divided in such a fashion, the line can't be cut). Seems to
> me like there isn't a paradox, just a contradiction in the model:
> if a line is composed of points, then "midpoint" says there is
> a point in the middle, "equal halves" says there isn't. Take
> your pick...
As pure mathematical theories, considered in isolation, there is
no way to decide which model is better. In that case, you can
indeed "take your pick".
But what I was trying to show is that if you want to support
Euclidean geometry, there are very good reasons for rejecting
the model that assumes space "consists of" a set of points --
namely, you get a contradiction:
1. The axioms of Euclidean geometry imply that line segments
can be bisected into identical halves.
2. But the axioms of point-set topology imply that there is
exactly one midpoint of any segment, and it must go into
one half or the other -- it cannot be split in two.
What Tarksi did with his geometry based on spheres is to avoid
that problem by *not* claiming that space consists of a set of
points. Instead, he assumed the following:
1. Any 3-D volume is the union of one or more spheres,
possibly infinitely many.
2. Any 2-D area is the union of one or more circles.
3. Any 1-D line segment is the union of one or more line
segments.
Then points are defined as the limit of a convergent sequence of
spheres (in 3-D), of circles (in 2-D), or of line segments (in 1-D).
These points are "virtual" or "fictitious" entities, which are not
actually "part" of the line. When you cut a line in two, each end
has a limiting point, but neither point can be said to be part of
the line or part of the original midpoint.
This definition is consistent with the assumptions by Aristotle
and Euclid that points are locations on a line, but not parts that
constitute the line.
It is also significant that this approach solves Zeno's paradox
of motion: At a single instant motion is impossible. If a time
interval were merely a set of instants, there is no way to see
how motion through the interval would be possible when motion
at any instant of the interval is impossible.
But if the parts of any time interval are other time intervals
during which motion is possible then motion during the union
of the intervals would also be possible.
John