Re: What happens to the midpoint when a line is cut in two?
On 8/27/06, John F. Sowa <sowa@bestweb.net> wrote:
> I received an offline question about the following passage
> from pp. 109-110 of my Knowledge Representation book:
A good prompt for me to get it off the shelf and see what sinks in
this time, thanks John ;-)
> > The representation of space as a set of points introduces
> > other puzzles. At the top of Fig. 2.16 is a line segment
> > that is being sliced in two equal halves, which are separated
> > at the bottom [of this fig.]. A question arises about what
> > happens to the midpoint. If the original lines were considered
> > a set of points, it would have just one midpoint. The two
> > halves of the line would not be identical, since only one of
> > them would contain the former midpoint. The other part would
> > contain an uncountable infinite set of points leading up to,
> > but not including the old midpoint.... The paradox of Fig. 2.16
> > did not occur to Aristotle or Euclid who never said that a
> > line consisted of points....
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...
Cheers,
Danny.
--
http://dannyayers.com